Efficient hydroclimatic data accessing with HyRiver for Python

This tutorial highlights the HyRiver software stack for Python, which is a very powerful tool for acquiring large sets of data from various web services.

I have uploaded a Jupyter-Notebook version of this post here if you would like to execute it yourself.

HyRiver Introduction

The HyRiver software suite was developed by Taher Chegini who, in their own words, describes HyRiver as:

“… a software stack consisting of seven Python libraries that are designed to aid in hydroclimate analysis through web services.”

This description does not do justice to the capability of this software. Through my research I have spent significant amounts of time wrangling various datasets – making sure that dates align, or accounting for spatial misalignment of available data. The HyRiver suite streamlines this process, and makes acquisition of different data from various sources much more efficient.

Here, I am going walk through a demonstration of how to easily access large amounts of data (streamflow, geophysical, and meteorological) for a basin of interest.

Before going through the code, I will highlight the three libraries from the HyRiver stack which I have found most useful: PyGeoHydro, PyNHD, and PyDaymet.

PyGeohydro

PyGeoHydro allows for interaction with eight different online datasets, including:

In this tutorial, I will only be interacting with the USGS NWIS, which provides daily streamflow data.

PyNHD

The PyNHD library is designed to interact with the National Hydrography Dataset (NHD)and the Hydro Network-Linked Data Index (NLDI).

NHDPlus (National Hydrography Dataset)

The NHD defines a high-resolutioon network of stream linkages, each with a unique idenfier (ComID).

The NLDI aids in the discovery of indexed information along some NHD-specified geometry (ComIDs). The NLDI essentially tranverses the linkages specified by the NHD geometry and generates data either local or basin-aggregated data relative to a specific linkage (ComID).

As will be seen later in the tutorial, the NLDI is able to retrieve at least 126 different types of data for a given basin…

PyDaymet

The PyDaymet GirHub repository summarizes the package as:

“[providing] access to climate data from Daymet V4 database using NetCDF Subset Service (NCSS). Both single pixel (using get_bycoords function) and gridded data (using get_bygeom) are supported which are returned as pandas.DataFrame and xarray.Dataset, respectively.”

Tutorial outline:

1. Installation
2. Retrieving USGS Water Data
3. Retrieving Geophysical (NLDI) Data
4. Retrieving Daymet Data

The HyRiver repository contains various examples demonstrating the use of the various libraries. I would definitely recommend digging in deeper to these, and other HyRiver documentation if this post piques your interest.

Step 0: Installation

In this tutorial, I only only interact with the PyNHD, PyGeoHydro, and PyDaymet libraries, so I do not need to install all of the HyRiver suite.

If you operate through pip, you can install these libraries using:

pip install pynhd pygeohydro pydaymet

If you use Anaconda package manager, you can install these packages using:

conda install -c conda-forge pynhd pygeohydro pydaymet

For more information on installation, refer to the HyRiver GitHub repository and related documentation.

Now, onto the fun part!

Step 1: Retreiving USGS Water Data

I am beginning here because streamflow data is typically the first point of interest for most hydrologic engineers or modelers.

Personally, I have gone through the process of trying to download data manually from the USGS NWIS website… My appreciation for the USGS prevents me from saying anything too negative, but let’s just say it was not a pleasant experience.

Pygeohydro allows for direct requests from the USGS National Water Information System (NWIS), which provides daily streamflow data from all USGS gages. The data is conveniently output as a Pandas DataFrame.

With this functionality alone, the PyGeoHydro library is worth learning.

1.1 Initialize PyGeoHydro NWIS tool

# Import common libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

# Import the PyGeohydro libaray tools
import pygeohydro as gh
from pygeohydro import NWIS, plot

# Use the national water info system (NWIS)
nwis = NWIS()


1.2 Requesting USGS Streamflow Data

The get_streamflow() function does exactly as the name entails and will retrieve daily streamflow timeseries, however USGS gage station IDs must be provided. If you are only interested in a single location, then you can enter 8-digit gage ID number along with a specified date range to generate the data:

get_streamflow('########', dates = ('Y-M-D', 'Y-M-D'))

However, I am want to explore larger sets of data over an entire region. Thus, I am going to use PyGeoHydro's get_info() function to identify all gages within some region of interest.

First, I specify a region via (latitude, longitude) bounds, then I send a query which retrieves meta-data information on all the gages in the specified region. In this case, I am exploring the data available near Ithaca, NY.

# Query specifications
region = (-76.7, 42.3, -76, 42.6) # Ithaca, NY

# Send a query for all gage info in the region
query = {"bBox": ",".join(f"{b:.06f}" for b in region),
"hasDataTypeCd": "dv",
"outputDataTypeCd": "dv"}

info_box = nwis.get_info(query)

print(f'PyGeoHydro found {len(set(info_box.site_no))} unique gages in this region.')

# [Out]: PyGeoHydro found #N unique gages in this region.


Although, this info_box identify many gages in the region which have very old or very brief data records. Knowing this, I want to filter out data which does not have a suitable record length.

For the sake of this tutorial, I am considering data between January 1st, 2020 and December 31st, 2020.

# Specify date range of interest
dates = ("2020-01-01", "2020-12-31")

# Filter stations to have only those with proper dates
stations = info_box[(info_box.begin_date <= dates[0]) & (info_box.end_date >= dates[1])].site_no.tolist()

# Remove duplicates by converting to a set
stations = set(stations)


Now, I am ready to use the gage IDs contained in stations to request the streamflow data!

# Retrieve the flow data
flow_data = nwis.get_streamflow(stations, dates, mmd=False)

# Remove gages with nans
flow_data = flow_data.dropna(axis = 1, how = 'any')


After removing duplicates and gages with nans, I have data from five unique gages in this region.

Additionally, PyGeoHydro has a convenient plotting feature to help quickly visualize the streamflow data.

from pygeohydro import plot

# Plot flow data summary
plot.signatures(flow_data)


There is a lot more to be explored in the PyGeoHydro library, but I will leave that up to the curious reader.

Step 2: Retrieving Geophysical (NLDI) Data

So, you’ve got some streamflow data but you don’t know anything about the physical watershed…

This is where the PyNHD library comes in. Using this library, I can identify entire upstream network from a gage, then extract the NLDI data associated with the watershed linkages.

# Import the PyNHD library
import pynhd as pynhd
from pynhd import NHD
from pynhd import NLDI, WaterData


First, we can take a look at all possible local basin characteristic data that are available:

# Get list of local data types (AKA characteristics, or attributes)
possible_attributes = NLDI().get_validchars("local").index.to_list()


There are 126 characteristics available from the NLDI! These characteristics range from elevation, to reservoir capacity, to bedrock depth. Many if these are not of immediate interest to me, so I will specify a subset of select_attributes to retrieve (basin area, max elevation, and stream slope).

I then loop through all of my USGS stations for which I have data in flow_data, identifying the upstream basin linkages using NLDI().navigate_byid(). Once the basin is identified, I extract the ComID numbers for each linkage and use that number to retrieve the NLDI data of interest. I then store the data in nldi_data. This process is done by the following:

# Specify characteristics of interest
select_attributes = ['CAT_BASIN_AREA', 'CAT_ELEV_MAX', 'CAT_STREAM_SLOPE']

# Initialize a storage matrix
nldi_data = np.zeros((len(flow_data.columns), len(select_attributes)))

# Loop through all gages, and request NLDI data near each gage
for i, st in enumerate(flow_data.columns):

# Navigate up all flowlines from gage
flowlines = NLDI().navigate_byid(fsource = 'nwissite',
fid = f'{st}',
source = 'flowlines',
distance = 10)

# Get the nearest comid
station_comid = flowlines.nhdplus_comid.to_list()[0]

# Source NLDI local data
nldi_data[i,:] = NLDI().getcharacteristic_byid(station_comid, "local", char_ids = select_attributes)


So far, I have timeseries streamflow data for five locations in the Ithaca, NY area, along with the basin area, max basin elevation, and stream slope for each stream. If I can access hydro-climate data, maybe I could begin studying the relationships between streamflow and physical basin features after some rain event.

Step 3: Meteorological data

The PyDaymet library allows for direct requests of meteorological data across an entire basin.

The available data includes:

• Minimum and maximum temperature (tmin, tmax)
• Precipitation (prcp)
• Vapor pressure (vp)
• Snow-Water Equivalent (swe)
• Shortwave radiation (srad)

All data are reported daily at a 1km x 1km resolution. Additionally, the PyDaymet library has the ability to estimate potential evapotranspiration, using various approximation methods.

Here, I choose to only request precipitation (prcp) and max temperature (tmax).

NOTE:
So far, the Daymet data retrieval process has been the slowest aspect of my HyRiver workflow. Due to the high-resolution, and potential for large basins, this may be computationally over-intensive if you try to request data for many gages with long time ranges.

# Import the  PyDayment library
import pydaymet as daymet

## Specify which data to request
met_vars = ["prcp", "tmax"]
met_data_names = np.array(['mean_prcp','sd_prcp','mean_tmax','sd_tmax'])

## Initialize storage
daymet_data = np.zeros((len(flow_data.columns), len(met_data_names)))


Similar to the NLDI() process, I loop through each gage (flow_data.columns) and (1) identify the up-gage basin, (2) source the Daymet data within the basin, (3) aggregate and store the data in daymet_data.

## Loop through stations from above
for i, st in enumerate(flow_data.columns):

# Get the up-station basin geometry
geometry = NLDI().get_basins(st).geometry[0]

# Source Daymet data within basin
basin_met_data = daymet.get_bygeom(geometry, dates, variables= met_vars)

## Pull values, aggregate, and store
# Mean and std dev precipitation
daymet_data[i, 0] = np.nan_to_num(basin_met_data.prcp.values).mean()
daymet_data[i, 1] = np.nan_to_num(basin_met_data.prcp.values).std()

# Mean and std dev of max temperature
daymet_data[i, 2] = np.nan_to_num(basin_met_data.tmax.values).mean()
daymet_data[i, 3] = np.nan_to_num(basin_met_data.tmax.values).std()

daymet_data.shape

# [Out]: (5, 4)


Without having used a web-browsers, I have been able to get access to a set of physical basin characteristics, various climate data, and observed streamflow relevant to my region of interest!

Now this data can be exported to a CSV, and used on any other project.

Conclusion

I hope this introduction to HyRiver has encouraged you to go bigger with your hydroclimate data ambitions.

If you are curious to learn more, I’d recommend you see the HyRiver Examples which have various in-depth Jupyter Notebook tutorials.

Citations

Chegini, Taher, et al. “HyRiver: Hydroclimate Data Retriever.” Journal of Open Source Software, vol. 6, no. 66, 27 Oct. 2021, p. 3175, 10.21105/joss.03175. Accessed 15 June 2022.

A Hidden-Markov Modeling Based Approach to Creating Synthetic Streamflow Scenarios

As Dave mentioned in his prior post, our recently published eBook, Addressing Uncertainty in Multisector Dynamics Research, provides several interactive tutorials for hands on training in model diagnostics and uncertainty characterization. This blog post is a preview of an upcoming extension of these trainings featuring an interactive tutorial on creating synthetic streamflow scenarios using a Hidden-Markov Modeling (HMM)- based approach specifically for the Upper Colorado River Basin. This training heavily utilizes Julie Quinn’s prior tutorial on fitting an HMM-based streamflow generator and is inspired by her Earth’s Future paper on understanding if exploratory modeling can truly be scenario-neutral.

In this post, we will primarily be focusing on decadal hydrologic drought in the region as a metric of interest to explore in the historic period, the stationary-synthetic, and non-stationary synthetic ensembles. We will also explore how to place CMIP5 climate projections in the context of our synthetically-created ensembles.  All code for this demo is available in a Jupyter Notebook on Github that follows the progression of this blog step by step (though some content may be subject to change before going into the eBook formally). The code is written in Python.

Background

In the Western United States, and particularly the Colorado River Basin, recent studies have used tree-ring reconstructions to suggest that the megadrought that has been occurring in the Southwest over the past 22 years is the regions worst drought since about 800 AD (Williams et al., 2022). The study’s lead author, UCLA climatologist Park Williams, suggested that had the sequence of wet-dry years occurred as observed but without the human-caused drying trend, the 2000s would have likely still been dry, but not on the same level as the worst of the last millennium’s megadroughts.

The recent trend of warming and reduced soil moisture in the SW is becoming extremely problematic from a water systems planning and management perspective for the Colorado River Basin. It has becoming rapidly clear that the river is completely over-allocated and won’t be able to sustain flow requirements as dictated by the Colorado Compact. Thus, there has been an increasing focus in understanding how susceptible water systems in this region are to plausible future streamflow scenarios, particularly those that may contain persistent droughts. In this tutorial, we’ll discuss how to create these scenarios using a Hidden Markov Model (HMM)- based synthetic generator.

Observed Data

First let’s take a look at the observed data from 1909-2013 for the outlet gauge of the Upper Colorado River that is located at the CO-UT stateline. Below we create a plot of the annual streamflow. Let’s also add an 11-year rolling mean which will give us a sense of long-term averaged behavior.

The Colorado Compact prescribing flows between the Upper and Lower Colorado Basins was negotiated using data prior to 1922, which we now know was one of the consistently wetter periods in the record. It’s clear today that since the 1980s, the Southwest has been experiencing imminent aridification and that this observed record alone isn’t an accurate representation of what future climate might look like in this region.

Let’s get a little more specific and formally quantify decadal drought risk in the observed period. We use a metric proposed in Ault et al. (2014). The authors define a decadal drought as a streamflow value that is more than a half a standard deviation below the 11-year running mean of the available record.

By this metric, the Upper Colorado Basin region has experienced two decadal droughts in the past 100 years.

Synthetic Stationary Generator to Understand Natural Variability

It is important to remember that the historical record of the region is only one instance of streamflow, which ultimately comes from an inherently stochastic and chaotic system (the atmosphere). We require a tool that will allow us to see multiple plausible realizations of that same variability. The tool that we use to develop synthetic flows for the region is a Gaussian Hidden Markov Model. If a system follows a Markov process, it switches between a number of “hidden states” dictated by a transition matrix. Each state has its own Gaussian probability distribution (defined by a mean and standard deviation) and one can draw from this distribution to create synthetic flows that fit the properties of the historical distribution. HMMs are an attractive choice for this region because they can simulate long persistence, particularly long droughts, which is an important consideration for water systems vulnerabilities. The figure below shows an example of a 2-state Gaussian HMM that we will fit for the region.

At this point, I will direct readers to either Julie’s blog or my own Jupyter notebook, which have all the details and code required to fit the model and investigate the parameters. Now let’s create a sample 105-year synthetic realization and see what the drought dynamics look like in the synthetic scenario that we created.

Wow, it looks like we’ve created something like a megadrought type situation by just sampling within the bounds of the historical distribution! You can keep sampling from the model to create more 105-year traces and note how the location and number of decadal droughts changes. Maybe you will experience two like in the the historical period, or fewer or more. Re-sampling from the model will demonstrate how different 105-year traces can look just within the range of natural variability and that what we’ve observed historically is only one trace. It’s also important to remember that when droughts occur can also define the ultimate effect of the drought (i.e. is it at a time when there is a large population growth or a time when humans can adapt by conserving or building more infrastructure?). A hydrologic drought need not manifest into an agricultural drought of the same magnitude if stored surface water is available, for example.

Non-Stationary Synthetic Generator to Impose Climate Changes

Now, we want to be able to create flows under non-stationary conditions to get a better understanding of what flows can look like under climate changes. In order to create flows under non-stationary conditions, we can toggle the parameters of the HMM model in order to create systematic changes to the model that can represent a changing climate. The HMM has 6 parameters that define it. In the historical distribution, we fit a baseline value for these parameters. In this non-stationary generator, we define a range to sample these parameters from.

Now refer to the Jupyter notebook for the code for the following steps. We first sample from these parameter ranges 1000 times using the Latin Hypercube Sample functionality within SALib. We then swap out the baseline parameters with the newly sampled parameters and sample from the model. Below is an example trace from the new non-stationary model in which we are generating more instances of decadal drought.

Placing Non-Stationary Flows in the Context of CMIP5 Projections

By creating a non-stationary generator, we can broaden the drought conditions that we are creating which that can be very useful to understand how our water systems model performs under potentially extreme scenarios. However, it’s useful to place our synthetically generated flows in the context of physically-driven CMIP5 projections to get a better understanding of how the two approaches compare. This example makes use of 97 CMIP5 projections used in the Colorado River Water Availability Study (CWCB, 2012). In each of these projections, monthly precipitation factor changes and temperature delta changes were computed between mean projected 2035–2065 climate statistics and mean historical climate statistics from 1950–2013. These 97 different combinations of 12 monthly precipitation multipliers and 12 monthly temperature delta shifts were applied to historical precipitation and temperature time series from 1950–2013. The resulting climate time series were run through a Variable Infiltration Capacity (VIC) model of the UCRB, resulting in 97 time series of projected future streamflows at the Colorado‐Utah state line.

We can fit an HMM to each trace of projected streamflow and retain the fitted parameters. Then we take the ratio between these parameters and the baseline HMM parameters in order to ultimately have a set of multipliers associated with each CMIP5 projection. This allows us to place the CMIP5 projections into the same space that we are sampling for our synthetic realizations.

In order to visualize these two ensembles in the same space, we plot a response surface that allows us to map how combinations of HMM parameters tend to lead to increased decadal drought occurrence. In order to get a continuous surface, we’ll fit a non-linear regression that ultimately maps the parameter values to decadal drought occurrence over a set of grid points. This response surface is shown by the color gradient below.

We choose two parameters, the dry-dry transition probability and the dry mean, that would most likely influence the decadal drought occurrence and create the response surface (note that a more formal sensitivity analysis can be used to identify the top most influential parameters as well). From the response surface, you can see that we’re more likely to see more instances of decadal droughts when we (1) increase the dry-dry transition probability, which inherently will increase persistence of the dry state, and (2) when we make the dry state log mean drier. We also place those multipliers from the CMIP5 projections into the space as white dots. Note that these CMIP5 projections tend to span the extent of the dry mean sample space, but are less representative of the dry transition probability sample space, particularly the increase in the dry-dry transition probability which leads to more persistent droughts. This suggests that the types of hydrological droughts represented in the projections tend to only be wetter to slightly drier than our baseline.

Both methods of producing these scenarios are valid, though if your goal is to produce a variety of ensembles that are characterized by many different drought characteristics, you will likely find that a generator approach will serve this purpose better.

Concluding Remarks

If you have any questions about the notebook or feedback as you go through the tutorial, please post below in the comments section or email me. We love getting as much feedback as possible before publication in the eBook.

References

Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., & Meko, D. M. (2014). Assessing the risk of persistent drought using climate model simulations and paleoclimate data. Journal of Climate27(20), 7529-7549.

CWCB (2012).Colorado River Water Availability Study Phase I Report. Colorado Water Conservation Board

Williams, A. P., Cook, B. I., & Smerdon, J. E. (2022). Rapid intensification of the emerging southwestern North American megadrought in 2020–2021. Nature Climate Change12(3), 232-234.

Time-evolving scenario discovery for infrastructure pathways

Our recently published eBook, Addressing Uncertainty in Multisector Dynamics Research, provides several interactive tutorials for hands on training in model diagnostics and uncertainty characterization. This blog post is a preview of an upcoming extension of these trainings featuring an interactive tutorial on time-evolving scenario discovery for the development of adaptive infrastructure pathways for water supply planning. This post builds off the prior tutorial on gradient-boosted trees for scenario discovery.

I’ll first introduce a styled water supply test case featuring two water utilities seeking to develop a cooperative infrastructure investment and management policy over a 45-year planning horizon. I’ll then demonstrate how the utilities can explore evolving vulnerability across the planning period. All code for this demo is available on Github. The code is written in Python, but the workflow is model agnostic and can be paired with simulation models in any language.

Background

The Bedford-Greene metropolitan area (Figure 1) is a stylized water resources test case containing two urban water utilities seeking to develop an infrastructure and investment and management strategy to confront growing demands and changing climate. The utilities have agreed to jointly finance and construct a new water treatment plant on Lake Classon, a large regional resource. Both utilities have also identified a set of individual infrastructure options to construct if necessary.

The utilities have formulated a cooperative and adaptive regional water supply management strategy that uses a risk-of-failure (ROF) metric to trigger both short-term drought mitigation actions (water use restrictions and treated transfers between utilities) and long-term infrastructure investment decisions (Figure 2a). ROFs represent a dynamic measure of the utilities’ evolving capacity-to-demand ratios. Both utilities have specified a set of ROF thresholds to trigger drought mitigation actions and plan to actively monitor their short-term ROF on a weekly basis. When a utility’s ROF crosses a specified threhold, the utility will implement drought mitigation actions in the following week. The utilities will also monitor long-term ROF on an annual basis, and trigger infrastructure investment if long-term risk crosses a threshold dictated by the policy. The utilities have also specified a construction order for available infrastructure options.

The utilities performed a Monte Carlo simulation to evaluate how this policy responds to a wide array of future states of the world (SOWs), each representing a different sample of uncertainties including demand growth rates, changes to streamflows, and financial variables.

The ROF-based policies respond to each SOW by generating a unique infrastructure pathway – a sequence of infrastructure investment decisions over time. Infrastructure pathways across 2,000 SOWs are shown in Figure 2b. Three clusters summarizing infrastructure pathways are plotted as green lines which represent the median week that options are triggered. The frequency that each option is triggered across all SOWs is plotted as the shading behind the lines. Bedford relies on the Joint Water Treatment facility and short-term measures (water use restrictions and transfers) to maintain supply reliability. Greene constructs the Fulton Creek reservoir in addition to the Joint Treatment plant and does not strongly rely on short-term measures to combat drought.

The utilities are now interested in evaluating the robustness of their proposed policy, characterizing how uncertainties generate vulnerability and understanding how this vulnerability may evolve over time.

Time-evolving robustness

To measure the robustness of the infrastructure investment and management policy, the two utilities employ a satisficing metric, which measures the fraction of SOWs where the policy is able to meet a set of performance criteria. The utilities have specified five performance criteria that measure the policy’s ability to maintain both supply reliability and financial stability. Performance criteria are shown in Table 1.

Figure 3 shows the evolution of robustness over time for the two utilities. While the cooperative policy is very robust after the first ten years of the planning horizon, it degrades sharply for both utilities over time. Bedford meets the performance criteria in nearly 100% of sampled SOWs after the first 10 years of the planning horizon, but its robustness is reduced to around 30% by the end of the 45-year planning period. Greene has a robustness of over 90% after the first 10 years and degrades to roughly 60% after 45 years. These degradations suggest that the cooperative infrastructure investment and management policy is insufficient to successfully maintain long-term performance in challenging future scenarios. But what is really going on here? The robustness metric aggregates performance across the five criteria shown in Table 1, giving us a general picture of evolving performance, but leaving questions about the nature of the utilities’ vulnerability.

Figure 4 provides some insight into how utility vulnerability evolves over time. Figure 4 shows the fraction of failure SOWs that can be attributed to each performance criterion when performance is measured in the near term (next 10 years), mid-term (next 22 years), and long-term (next 45 years). Figure 4 reveals that the vulnerability of the two utilities evolves in very different ways over the planning period. Early in the planning period, all of Bedford’s failures can be attributed to supply reliability. As the planning horizon progresses, Bedford’s failures diversify into failures in restriction frequency and worst-case drought management cost, indicating that the utility is generally unable to manage future drought. Bedford likely needs more infrastructure investment than is specified by the policy to maintain supply reliability.

In contrast to Bedford’s performance, Greene begins with vulnerability to supply reliability, but its vulnerability shifts over time to become dominated by failures in peak financial cost and stranded assets – measures of the utility’s financial stability. This shift indicates that while the infrastructure investments specified by the cooperative policy mitigate supply failures by the end of the planning horizon, these investments drive the utility into financial failure in many future scenarios.

Factor mapping and factor ranking

To understand how and why the vulnerability evolves over time, we perform factor mapping. Figure 5 below, shows the uncertainty space projected onto the two most influential factors for Bedford, across three planning horizons. Each point represents a sampled SOW, red points represent SOWs that resulted in failure, while white points represent SOWs that resulted in success. The color in the background shows the predicted regions of success and failure from the boosted trees classification.

Figure 4 indicates that Bedford’s vulnerability is primarily driven by rapid and sustained demand growth and this vulnerability increases over time. When evaluated using a 22-year planning horizon, the utility only appears vulnerable to extreme values of near-term demand growth, combined with low values of restriction effectiveness. This indicates that the utility is relying on restrictions to mitigate supply failures, and is vulnerable when they do not work as anticipated. When evaluated over the 45-year planning horizon, Bedford’s failure is driven by rapid and sustained demand growth. If near-term demand grows faster than anticipated (scaling factor > 1.0 on the horizontal axis), the utility will likely fail to meet its performance criteria. If near-term demand is lower than anticipated, the utility may still fail to meet performance criteria if under conditions of high mid-term demand growth. These results provide further evidence that the infrastructure investment and management policy is insufficient to meet Bedford’s long-term water supply needs.

Greene’s vulnerability (Figure 6) evolves very differently from Bedford’s. Greene is vulnerable to high-demand scenarios in the near term, indicating that its current infrastructure is insufficient to meet rapidly growing demands. Greene can avoid this failure under scenarios where the construction permitting time multiplier is the lowest, indicating that new infrastructure investment can meet the utility’s near-term supply needs. When evaluated across a 22-year planning horizon, the utility fails when near-term demand is high and restriction effectiveness is low, a similar failure mode to Bedford. However, the 22-year planning horizon reveals a second failure mode – low demand growth. This failure mode is responsible for the stranded assets failures shown in Figure 3. This failure mode increases when evaluated across the 45-year planning horizon, and is largely driven by low-demand futures when the utility does not generate the revenue to cover debt service payments needed to fund infrastructure investment.

The factor maps in Figures 5 and 6 only show the two most influential factors determined by gradient boosted trees, however, the utilities are vulnerable to other sampled uncertainties. Figure 7 shows the factor importance as determined by gradient boosted trees for both utilities across the three planning horizons. While near-term demand growth is important for both utilities under all three planning horizons, the importance of other factors evolves over time. For example, restriction effectiveness plays an important role for Greene under the 22-year planning horizon but disappears under the 45-year planning horizon. In contrast, the bond interest rate is important for predicting success over the 45-year planning horizon, but does not appear important over the 10- or 22-year planning horizons. These findings highlight how assumptions about the planning period can have a large impact on modeling outcomes.

Fisheries Training Part 1 – Harvest Optimization and MOEA Diagnostics

Welcome to the second post in the Fisheries Training Series, in which we are studying decision making under deep uncertainty within the context of a complex harvested predator-prey fishery. The accompanying GitHub repository, containing all of the source code used throughout this series, is available here. The full, in-depth Jupyter Notebook version of this post is available in the repository as well.

This post builds off of the initial post, Fisheries Training 0: Exploring Predator-Prey Dynamics, and presents the following:

1. A brief re-cap of the harvested predator-prey model
2. Formulation of the harvesting policy and an overview of radial basis functions (RBFs)
3. Formulation of the policy objectives
4. A simulation model for the harvested system
5. Optimization of the harvesting policy using the PyBorg MOEA
• Installation of Platypus and PyBorg*
• Optimization problem formulation
• Basic MOEA diagnostics

Note
*The PyBorg MOEA used in this demonstration is derived from the Borg MOEA and may only be used with permission from its creators. Fortunately, it is freely available for academic and non-commercial use. Visit BorgMOEA.org to request access.

Now, onto the tutorial!

Harvested predator-prey model

In the previous post, we introduced a modified form of the Lotka-Volterra system of ordinary differential equations (ODEs) defining predator-prey population dynamics.

This modified version includes a non-linear predator population growth dynamic original proposed by Arditi and Akçakaya (1990), and includes a harvesting parameter, $z$. This system of equations is defined in Hadjimichael et al. (2020) as:

$\frac{dx}{dt} = bx\Big(1 - \frac{x}{K}\Big) - \frac{\alpha xy}{y^m + \alpha hx} - zx$

$\frac{dy}{dt} = \frac{c\alpha xy}{y^m + \alpha hx} - dy$

Where $x$ is the prey population being harvested and $y$ is the predator population. Please refer to Post 0 of this series for the rest of the parameter descriptions, and for insights into the non-linear dynamics that result from these ODEs. It also demonstrates how the system alternates between ‘basins’ of stability and population collapse.

Harvesting policy

In this post, we instead focus on the generation of harvesting policies which can be operated safely in the system without causing population collapse. Rather than assigning a deterministic (specific, pre-defined) harvest effort level for every time period, we instead design an adaptive policy which is a function of the current state of the system:

$z_t = f(\cdot)$

The problem then becomes the optimization of the control rule, $f(\cdot)$, rather than specific parameter values, $z = [z_1, z_2, ..., z_t]$. The process of optimizing the parameters of a state-aware control rule is known as Direct Policy Search (DPS; Quinn et al, 2017).

Previous work done by Quinn et al. (2017) showed that an adaptive policy, generated using DPS, was able to navigate deeply uncertain ecological tipping points more reliably than intertemporal policies which prescribed specific efforts at each timestep.

The core of the DPS method are radial basis functions (RBFs), which are flexible, parametric function formulations that map the current state of the system to policy action. A previous study by Giuliani et al (2015) demonstrated that RBFs are highly effective in generating Pareto-approximate sets of solutions, and that they perform well when applied to horizons different from the optimized simulation horizon.

There are various RBF approaches available, such as the cubic RBF used by Quinn et al. (2017). Here, we use the Gaussian RBF introduced by Hadjimichael et al. (2020), where the harvest effort during the next timestep, $z_{t+1}$, is mapped to the current prey population levels, $x_t$ by the function:

$z_{t+1} = \sum_{i=1}^n w_i \Big[exp\Big[-\Big(\frac{x_t-c_i}{b_i}\Big)^2\Big]\Big]$

In this formulation $c_i, r_i,$ and $w_i$ are the center, radius, and weights of each RBF $i$ respectively. Additionally, $n$ is the number of RBFs used in the function; in this study we use $n = 2$ RBFs. With two RBFs, there are a total of 6 parameters. Increasing the number of RBFs allows for more flexible function forms to be achieved. However, two RBFs have been shown to be sufficient for this problem.

The sum of the weights must be equal to one, such that:

$\sum_{i=1}^n w_i= 1$

The function harvest_streategy() is contained within the fish_game_functions.py script, which can be accessed here in the repository.

A simplified rendition of the harvest_strategy() function, evaluate_RBF(), is shown below and uses the RBF parameter values (i.e., $[c_1, b_1, w_1, c_2, b_2, w_2]$), and the current prey population, to calculate the next year’s harvesting effort.

import numpy as np
import matplotlib.pyplot as plt

def evaluate_RBF(x, RBF_params, nRBFs):
"""
Parameters:
-----------
x : float
The current state of the system.
RBF_params : list [3xnRBFs]
The RBF parameters in the order of [c, r, w,...,c, r, w].
nRBFs : int
The number of RBFs used in the mapping function.

Returns:
--------
z : float
The policy action.
"""

c = RBF_params[0::3]
r = RBF_params[1::3]
w = RBF_params[2::3]

# Normalize the weights
w_norm = []
if np.sum(w) != 0:
for w_i in w:
w_norm.append(w_i / np.sum(w))
else:
w_norm = (1/nRBFs)*np.ones(len(w))

z = 0.0

for i in range(nRBFs):

# Avoid division by zero
if r[i] != 0:
z = z + w[i] * np.exp(-((x - c[i])/r[i])**2)
else:
z = z + w[i] * np.exp(-((x - c[i])/(10**-6))**2)

# Impose limits on harvest effort
if z < 0:
z = 0
elif z > 1:
z = 1

return z


To better understand the nature of the harvesting policy, it is helpful to visualize the policy function, $z = f(\cdot)$.

For some arbitrary selection of RBF parameters:

$[c_1, b_1, w_1, c_2, b_2, w_2] = [0.2, 1.1, 0.41, 0.34,0.7, 0.59]$

The following function will plot the harvesting strategy:

def plot_RBF_policy(x_range, x_label, y_range, y_label, RBF_params, nRBFs):
"""
Parameters:
-----------
RBF_params : list [3xnRBFs]
The RBF parameters in the order of [c, r, w,...,c, r, w].
nRBFs : int
The number of RBFs used in the mapping function.

Returns:
--------
None.
"""
# Step size
n = 100
x_min = x_range[0]
x_max = x_range[1]
y_min = y_range[0]
y_max = y_range[1]

# Generate data
x_vals = np.linspace(x_min, x_max, n)
y_vals = np.zeros(n)

for i in range(n):
y = evaluate_RBF(x_vals[i], RBF_params, nRBFs)

# Check that assigned actions are within range
if y < y_min:
y = y_min
elif y > y_max:
y = y_max

y_vals[i] = y

# Plot
fig, ax = plt.subplots(figsize = (5,5), dpi = 100)
ax.plot(x_vals, y_vals, label = 'Policy', color = 'green')
ax.set_xlabel(x_label)
ax.set_ylabel(y_label)
ax.set_title('RBF Policy')
plt.show()
return


Let’s take a look at the policy that results from the random RBF parameters listed above. Setting my problem-specific inputs, and running the function:

# Set the RBF parameters
nRBFs = 2
RBF_params = [0.2, 1.1, 0.41, 0.34,0.7, 0.59]

# Specify plot ranges
x_range = [0, 1]
x_label = 'Population ($x$)'
y_range = [0,1]
y_label = 'Harvest Effort ($z$)'

# Plot the policy curve
plot_RBF_policy(x_range, x_label, y_range, y_label, RBF_params, nRBFs)


This policy does not make much intuitive sense… why should harvesting efforts be decreased when the fish population is large? Well, that’s because we chose these RBF parameter values randomly.

To demonstrate the flexibility of the RBF functions and the variety of policy functions that can result from them, I generated a few (n = 7) policies using a random sample of parameter values. The parameter values were sampled from a uniform distribution over each parameters range: $c_i, b_i, w_i \in [0,1]$. Below is a plot of the resulting random policy functions:

Finding the sets of RBF parameter values that result in Pareto-optimal harvesting policies is the next step in this process!

Harvest strategy objectives

We take a multi-objective approach to the generation of a harvesting strategy. Given that the populations are vulnerable to collapse, it is important to consider ecological objectives in the problem formulation.

Here, we consider five objectives, described below.

Objective 1: Net present value

The net present value (NPV) is an economic objective corresponding to the amount of fish harvested.

During the simulation-optimization process (later in this post), we simulate a single policy $N$ times, and take the average objective score over the range of simulations. This method helps to account for variability in expected outcomes due to natural stochasticity. Here, we use $N = 100$ realizations of stochasticity.

With that in mind, the NPV ($O_1$) is calculated as:

$O_1 = \frac{1}{N} \sum_{i=1}^N\Big( \sum_{t=0}^T \frac{z_{t+1,i}x_{t,i}}{(1+\delta)^t}\Big)$

where $\delta$ is the discount rate which converts future benefits to present economic value, here $\delta = 0.05$.

Objective 2: Prey population deficit

The second objective aims to minimize the average prey population deficit, relative to the prey population carrying capacity, $K$:

$O_2 = \frac{1}{N} \sum_{i=1}^N\Big( \frac{1}{T} \sum_{t=1}^T \frac{K - x_{t,i}}{K}\Big)$

Objective 3: Longest duration of consecutive low harvest

In order to maintain steady harvesting levels, we minimize the longest duration of consecutive low harvests. Here, a subjective definition of low harvest is imposed. In a practical decision making process, this threshold may be solicited from the relevant stakeholders.

Objective 3 is defined as:

$O_3 = \frac{1}{N} \sum_{i=1}^N(max_T(\phi_{t,i}))$

where

And the low harvest limit is: $limit = 5\%$.

Objective 4: Worst harvest instance

In addition to avoiding long periods of continuously low harvest, the stakeholders have a desire to limit financial risks associated with very low harvests. Here, we minimize the worst 1% of harvest.

The fourth objective is defined as:

$O_4 = \frac{1}{N} \sum_{i=1}^N(percentile_T(z_{t+1,i}x_{t,i}, 1))$

Objective 5: Harvest variance

Lastly, policies which result in low harvest variance are more easily implemented, and can limit corresponding variance in fish populations.

The last objective minimizes the harvest variance, with the objective score defined as:

$O_5 = \frac{1}{N} \sum_{i=1}^N(Var_T(z_{t+1,i}x_{t,i}))$

Constraint: Avoid collapse of predator population

During the optimization process, we are able to include constraints on the harvesting policies.

Since population collapse is a stable equilibrium point, from which the population will not regrow, it is imperative to consider policies which prevent collapse.

With this in mind, the policy must not result in any population collapse across the $N$ realizations of environmental stochasticity. Mathematically, this is enforced by:

$\frac{1}{N} \sum_{i=1}^N(\Psi_{t,i})) = 0$

where

Problem formulation

Given the objectives described above, the optimization problem is:

$Minimize \ F(z_x) = (-O_1, O_2, O_3, -O_4, O_5)$

Simulation model of the harvested system

Here, we provide an overview of the fish_game_5_objs() model which combines many of the preceding topics. The goal for this model is to take a set of RBF parameters, which define the harvesting strategy, simulate the policy for some length of time, and then return the objective scores resulting from the policy.

Later, this model will allow for the optimization of the harvesting policy RBF parameters through a Multi-Objective Evolutionary Algorithm (MOEA). The MOEA will evaluate many thousands of policies (RBF parameter combinations) and attempt to find, through evolution, those RBF parameters which yield best objective performance.

A brief summary of the model process is described here, but the curious learner is encouraged to take a deeper look at the code and dissect the process.

The model can be understood as having three major sections:

1. Initialization of storage vectors, stochastic variables, and assumed ODE parameters.
2. Simulation of policy and fishery populations over time period T.
3. Calculation of objective scores.
def fish_game_5_objs(vars):
"""
Defines the full, 5-objective fish game problem to be solved

Parameters
----------
vars : list of floats
Contains the C, R, W values

Returns
-------
objs, cnstr
"""

# Get chosen strategy
strategy = 'Previous_Prey'

# Define variables for RBFs
nIn = 1 # no. of inputs (depending on selected strategy)
nOut = 1 # no. of outputs (depending on selected strategy)
nRBF = 2 # no. of RBFs to use

nObjs = 5
nCnstr = 1 # no. of constraints in output

tSteps = 100 # no. of timesteps to run the fish game on
N = 100 # Number of realizations of environmental stochasticity

# Define assumed system parameters
a = 0.005
b = 0.5
c = 0.5
d = 0.1
h = 0.1
K = 2000
m = 0.7
sigmaX = 0.004
sigmaY = 0.004

# Initialize storage arrays for populations and harvest
x = np.zeros(tSteps+1) # Prey population
y = np.zeros(tSteps+1) # Predator population
z = np.zeros(tSteps+1) # Harvest effort

# Create array to store harvest for all realizations
harvest = np.zeros([N,tSteps+1])
# Create array to store effort for all realizations
effort = np.zeros([N,tSteps+1])
# Create array to store prey for all realizations
prey = np.zeros([N,tSteps+1])
# Create array to store predator for all realizations
predator = np.zeros([N,tSteps+1])

# Create array to store metrics per realization
NPV = np.zeros(N)
cons_low_harv = np.zeros(N)
harv_1st_pc = np.zeros(N)
variance = np.zeros(N)

# Create arrays to store objectives and constraints
objs = [0.0]*nObjs
cnstr = [0.0]*nCnstr

# Create array with environmental stochasticity for prey
epsilon_prey = np.random.normal(0.0, sigmaX, N)

# Create array with environmental stochasticity for predator
epsilon_predator = np.random.normal(0.0, sigmaY, N)

# Go through N possible realizations
for i in range(N):

# Initialize populations and values
x[0] = prey[i,0] = K
y[0] = predator[i,0] = 250
z[0] = effort[i,0] = harvest_strategy([x[0]], vars, [[0, K]], [[0, 1]], nIn, nOut, nRBF)
NPVharvest = harvest[i,0] = effort[i,0]*x[0]

# Go through all timesteps for prey, predator, and harvest
for t in range(tSteps):

# Solve discretized form of ODE at subsequent time step
if x[t] > 0 and y[t] > 0:
x[t+1] = (x[t] + b*x[t]*(1-x[t]/K) - (a*x[t]*y[t])/(np.power(y[t],m)+a*h*x[t]) - z[t]*x[t])* np.exp(epsilon_prey[i]) # Prey growth equation
y[t+1] = (y[t] + c*a*x[t]*y[t]/(np.power(y[t],m)+a*h*x[t]) - d*y[t]) *np.exp(epsilon_predator[i]) # Predator growth equation

# Solve for harvesting effort at next timestep
if t <= tSteps-1:
if strategy == 'Previous_Prey':
input_ranges = [[0, K]] # Prey pop. range to use for normalization
output_ranges = [[0, 1]] # Range to de-normalize harvest to
z[t+1] = harvest_strategy([x[t]], vars, input_ranges, output_ranges, nIn, nOut, nRBF)

# Store values in arrays
prey[i,t+1] = x[t+1]
predator[i,t+1] = y[t+1]
effort[i,t+1] = z[t+1]
harvest[i,t+1] = z[t+1]*x[t+1]
NPVharvest = NPVharvest + harvest[i,t+1]*(1+0.05)**(-(t+1))

# Solve for objectives and constraint
NPV[i] = NPVharvest
low_hrv = [harvest[i,j]<prey[i,j]/20 for j in range(len(harvest[i,:]))] # Returns a list of True values when there's harvest below 5%
count = [ sum( 1 for _ in group ) for key, group in itertools.groupby( low_hrv ) if key ] # Counts groups of True values in a row
if count: # Checks if theres at least one count (if not, np.max won't work on empty list)
cons_low_harv[i] = np.max(count)  # Finds the largest number of consecutive low harvests
else:
cons_low_harv[i] = 0
harv_1st_pc[i] = np.percentile(harvest[i,:],1)
variance[i] = np.var(harvest[i,:])

# Average objectives across N realizations
objs[0] = -np.mean(NPV) # Mean NPV for all realizations
objs[1] = np.mean((K-prey)/K) # Mean prey deficit
objs[2] = np.mean(cons_low_harv) # Mean worst case of consecutive low harvest across realizations
objs[3] = -np.mean(harv_1st_pc) # Mean 1st percentile of all harvests
objs[4] = np.mean(variance) # Mean variance of harvest

cnstr[0] = np.mean((predator < 1).sum(axis=1)) # Mean number of predator extinction days per realization

# output should be all the objectives, and constraint
return objs, cnstr


The next section shows how to optimize the harvest policy defined by vars, using the PyBorg MOEA.

A (Very) Brief Overview of PyBorg

PyBorg is the secondary implementation of the Borg MOEA written entirely in Python by David Hadka and Andrew Dircks. It is made possible using functions from the Platypus optimization library, which is a Python evolutionary computing framework.

As PyBorg is Borg’s Python wrapper and thus derived from the original Borg MOEA, it can only be used with permission from its creators. To obtain permission for download, please visit BorgMOEA and complete the web form. You should receive an email with the link to the BitBucket repository shortly.

Installation

The repository you have access to should be named ‘Serial Borg MOEA’ and contain a number of folders, including one called Python/. Within the Python/ folder, you will be able to locate a folder called pyborg. Once you have identified the folder, please follow these next steps carefully:

1. Check your current Python version. Python 3.5 or later is required to enable PyBorg implementation.
2. Download the pyborg folder and place it in the folder where this Jupyter Notebook all other Part 1 training material is located.
3. Install the platypus library. This can be in done via your command line by running one of two options:

If you are using pip:
pip install platypus-opt


If you are using conda:

conda config --add channels conda-forge
conda install platypus-opt

1. Make sure the following training startup files are located within the same folder as this Jupyter Notebook:
a) fish_game_functions.py: Contains all function definitions to setup the problem, run the optimization, plot the hypervolume, and conduct random seed analysis.
b) Part 1 - Harvest Optimization and MOEA Diagnostics.ipynb: This is the current notebook and where the Fisheries fame will be demonstrated.

We are now ready to proceed!

Optimization of the Fisheries Game

Import all libraries

All functions required for this post can be found in the fish_game_functions.py file. This code is adapted from Antonia Hadjimichael’s original post on exploring the Fisheries Game dynamics using PyBorg.

# import all required libraries
from platypus import Problem, Real, Hypervolume, Generator
from pyborg import BorgMOEA
from fish_game_functions import *
from platypus import Problem, Real, Hypervolume, Generator
from pyborg import BorgMOEA
import time
import random


Formulating the problem

Define number of decision variables, constraints, and specify problem formulation:

# Set the number of decision variables, constraints and performance objectives
nVars = 6   # Define number of decision variables
nObjs = 5   # Define number of objectives
nCnstr = 1      # Define number of decision constraints

# Define the upper and lower bounds of the performance objectives
objs_lower_bounds = [-6000, 0, 0, -250, 0]
objs_upper_bounds = [0, 1, 100, 0, 32000]


Initialize the problem for optimization

We call the fisheries_game_problem_setup.py function to set up the optimization problem. This function returns a PyBorg object called algorithm in this exercise that will be optimized in the next step.

def fisheries_game_problem_setup(nVars, nObjs, nCnstr, pop_size=100):
"""
Sets up and runs the fisheries game for a given population size

Parameters
----------
nVars : int
Number of decision variables.
nObjs : int
Number of performance objectives.
nCnstr : int
Number of constraints.
pop_size : int, optional
Initial population size of the randomly-generated set of solutions.
The default is 100.

Returns
-------
algorithm : pyBorg object
The algorthm to optimize with a unique initial population size.

"""
# Set up the problem
problem = Problem(nVars, nObjs, nCnstr)
nVars = 6   # Define number of decision variables
nObjs = 5   # Define number of objective -- USER DEFINED
nCnstr = 1      # Define number of decision constraints

problem = Problem(nVars, nObjs, nCnstr)

# set bounds for each decision variable
problem.types[0] = Real(0.0, 1.0)
problem.types[1] = Real(0.0, 1.0)
problem.types[2] = Real(0.0, 1.0)
problem.types[3] = Real(0.0, 1.0)
problem.types[4] = Real(0.0, 1.0)
problem.types[5] = Real(0.0, 1.0)

# all values should be nonzero
problem.constraints[:] = "==0"

# set problem function
if nObjs == 5:
problem.function = fish_game_5_objs
else:
problem.function = fish_game_3_objs

algorithm = BorgMOEA(problem, epsilons=0.001, population_size=pop_size)
return algorithm

# initialize the optimization
algorithm = fisheries_game_problem_setup(nVars, nObjs, nCnstr)


Define parameters for optimization

Before optimizing, we have to define our desired population size and number of function evaluations (NFEs). The NFEs correspond to the number of evolutions of the set of solutions. For complex, many-objective problems, it may be necessary for a large NFE.

Here, we start with a small limit on NFE, to test the speed of the optimization. Limiting the optimization to 100 NFE is going to produce relatively poor performing solutions, however it is a good starting point for our diagnostic tests.

init_nfe = 100
init_pop_size = 100


Begin the optimization

In addition to running the optimization, we also time the optimization to get a general estimate on the time the full hypervolume analysis will require.

# begin timing the Borg run
borg_start_time = time.time()

algorithm = fisheries_game_problem_setup(nVars, nObjs, nCnstr, pop_size=int(init_pop_size))
algorithm.run(int(init_nfe))

# end timing and print optimization time
borg_end_time = time.time()

borg_total_time = borg_end_time - borg_start_time

print(f"borg_total_time={borg_total_time}s")

Output: borg_total_time=33.62936472892761s

NOTICE:
Running the PyBrog MOEA 100 times took ~34 seconds (on the machine which this was written on…). Keep this in mind, that increasing the NFE will require correspondingly more time. If you increase the number too much, your machine may take a long time to compute the final Pareto-front.

Here, we plot a 3-dimensional plot showing the tradeoff between a select number of objectives. If you have selected the 5-objective problem formulation, you should select the three objectives you would like to analyze the tradeoff surface for. Please select the (abbreviated) objective names from the following list:

Objective 1: Mean NPV
Objective 2: Mean prey deficit
Objective 3: Mean WCLH
Objective 4: Mean 1% harvest
Objective 5: Mean harvest variance

# Plot objective tradeoff surface
fig_objs = plt.figure(figsize=(8,8))

# Select the objectives to plot from the list provided in the description above
obj1 = 'Mean NPV'
obj2 = 'Mean prey deficit'
obj3 = 'Mean 1% harvest'

plot_3d_tradeoff(algorithm, ax_objs, nObjs, obj1, obj2, obj3)


The objectives scores arn’t very good, but that is because the number of function evaluations is so low. In order to get a better set of solutions, we need to run the MOEA for many function evaluations.

The next section demonstrates the change in objective performance with respect to the number of function evaluations.

MOEA Diagnostics

A good MOEA is assessed by it’s ability to quickly converge to a set of solutions (the Pareto-approximate set) that is also diverse. This means that the final set of solutions is close to the true set, as well as covers a large volume of the multi-dimensional problem space. There are three quantitative metrics via which convergence and diversity are evaluated:

1. Generational distance approximates the average distance between the true Pareto front and the Pareto-approximate reference set that your MOEA identifies. It is the easiest metric to meet.
2. Epsilon indicator is a harder metric than generational distance to me et. A high-performing MOEA will have a low epsilon indicator value when the distance of its worst-performing approximate solution from the true Pareto set is small.
3. Hypervolume measures the ‘volume’ that a Pareto front covers across all dimensions of a problem. It is the hardest metric to meet and the most computationally intensive.

Both the generational distance and epsilon indicator metrics require a reference set, which is the known, true Pareto front. Conversely, the hypervolume does not have such a requirement. Given that the Fisheries Game is a complex, multi-dimensional, many-stakeholder problem with no known solution, the hypervolume metric is thus the most suitable to evaluate the ability of PyBorg to quickly converge to a diverse Pareto-approximate set of solutions.

More detailed descriptions of each metric are provided in this handy blog post by Joe Kasprzyk.

Hypervolume

The hypervolume is a measure of the multi-dimensional volume dominated by the approximated Pareto front. As the Pareto front advances toward the “ideal” solution, this value approaches 1.

The efficiency of an MOEA in optimizing a solution can be considered by measuring the hypervolume with respect to the number of function evaluations. This allows the user to understand how quickly the MOEA is converging to a good set of solutions, and how many function evaluations are needed to achieve a good set of solutions.

Defining hypervolume parameters

First, we define the maximum number of function evaluations (maxevals) and the NFE step size (frequency) for which we would like to evaluate the problem hypervolume over. Try modifying these values to see how the plot changes.

Mind that the value of maxevals should always be more than that of your initial NFE, and that the value of frequency should be less than that of the initial NFE. Both values should be integer values.

Also be mindful that increasing the maxevals > 1000 is going to result in long runtimes.

maxevals = 500
frequency = 100


Plotting the hypervolume

Using these parameters, we then plot the hypervolume graph, showing the change in hypervolume value over the NFEs.

fig_hvol = plt.figure(figsize=(10,7))

plot_hvol(algorithm, maxevals, frequency, objs_lower_bounds, objs_upper_bounds, ax_hvol)

plt.title('PyBorg Runtime (Hypervolume)')
plt.xlabel('Number of Function Evaluations')
plt.ylabel('Hypervolume')
plt.show()


Perform random seed analysis

Next, we perform random seed analysis (RSA).

Generally, RSA is performed to track an algorithm’s performance during search. In addition, it is also done to determine if an algorithm has discovered an acceptable approximation of the true Pareto set. More details on RSA can be found here in a blog post by Dave Gold.

For the Fisheries Game, we conduct RSA to determine if PyBorg’s performance is sensitive to the size of its initial population. We do this using the folllowing steps:

1. Run an ensemble of searches, each starting with a randomly sampled set of initial conditions (aka “random seeds”)
2. Combine search results across all random seeds to generate a “reference set” that contains only the best non-dominated solutions across the ensemble
3. Repeat steps 1 and 2 for an initial population size of 200, 400, etc.
pop_size_list = [100, 200, 400, 800, 1000]

fig_rand_seed = plt.figure(figsize=(10,7))

for p in range(len(pop_size_list)):
fisheries_game_problem_setup(nVars, nObjs, nCnstr, pop_size_list[p])
algorithm = fisheries_game_problem_setup(nVars, nObjs, nCnstr, pop_size=int(init_pop_size))
algorithm.run(int(init_nfe))

plot_hvol(algorithm, maxevals, frequency, objs_lower_bounds, objs_upper_bounds,
ax_rand_seed, pop_size_list[p])

plt.title('PyBorg Random Seed Analysis')
plt.xlabel('Number of Function Evaluations')
plt.ylabel('Hypervolume')
plt.legend()
plt.show()


Notice that the runs performed with different initial population sizes tend to converge toward a similar hypervolume value after 500 NFEs.

This reveals that the PyBorg MOEA is not very sensitive to the specific initial parameters; it is adaptable enough to succeed under different configurations.

Conclusion

A classic decision-making idiom says ‘defining the problem is the problem’. Hopefully, this post has revealed that to be true; we have shown that changes to the harvesting strategy functions, simulation model, or objective scores can result in changes to the resulting outcomes.

And if you’ve made it this far, congratulations! Take a minute to think back on the progression of this post: we revisited the harvested predator-prey model, formulated the harvesting policy using RBFs, and formulated the policy objectives and its associated simulation model. Next, we optimized the harvesting policy using the PyBorg MOEA and performed basic MOEA diagnostics using hypervolume as our measure, and executed random seed analysis.

If you’ve progressed through this tutorial using the Jupyter Notebook, we encourage you to re-visit the source code involved in this process. The next advisable step is to re-produce this problem from scratch, as this is the best way to develop a detailed understanding of the process.

Next time, we will explore the outcomes of this optimization, by considering the tradeoffs present across the Pareto set of solutions.

Constructing interactive Ipywidgets: demonstration using the HYMOD model

Last week, Lillian and I published the first post in a series of training post studying the “Fisheries Game, which is a decision making problem within a complex, non-linear, and uncertain ecological context.

In preparing for that post, I learned about the Ipywidgets python library for widget construction. It stood out to me as a tool for highlighting the influence of parametric uncertainty on model performance. More broadly, I think it has great as an educational or data-narrative device.

This blogpost is designed to highlight this potential, and provide a basic introduction to the library. A tutorial demonstration of how an interactive widget is constructed is provided, this time using the HYMOD rainfall-runoff model.

This post is intended to be viewed through a Jupyter Notebook for interaction, which can be accessed through a Binder at this link!

The Binder was built with an internal environment specification, so it should not be necessary to install any packages on your local machine! Because of this, it may take a minute to load the page.

Alternatively, you can pull the source code and run the Jupyter Notebook from your local machine. All of the source code is available in a GitHub repository: Ipywidget_Demo_Interactive_HYMOD.

If using your local machine, you will first need to install the Ipywidget library:

pip install ipywidgets

Let’s begin!

HYMOD Introduction

HYMOD is a conceptual rainfall-runoff model. Given some observed precipitation and evaporation, a parameterized HYMOD model simulates the resulting down-basin runoff.

This post does not focus on specific properties or performance of the HYMOD model, but rather uses the model as a demonstration of the utility of the Ipywidget library.

I chose to use the HYMOD model for this, because the HYMOD model is commonly taught in introductory hydrologic modeling courses. This demonstration shows how an Ipywidget can be used in an educational context. The resulting widget can allow students to interact in real-time with the model behavior, by adjusting parameter values and visualizing the changes in the resulted streamflow.

If you are interested in the technical details of implementing the HYMOD model, you can dig into the source code, available (and throughly commented/descriptive) in the repository for this post: Ipywidget_Demo_Interactive_HYMOD.

HYMOD represents surface flow as a series of several quick-flow reservoirs. Groundwater flow is represented as a single slow-flow reservoir. The reservoirs have constant flow rates, with the quick-flow reservoir rate, Kq, being greater than the slow-flow reservoir rate, Ks.

HYMOD Parameters:

Like any hydrologic model, the performance of HYMOD will be dependent upon the specified parameter values. There are several parameters that can be adjusted:

• Cmax: Max soil moisture storage (mm) [10-2000]
• B: Distribution of soil stores [0.0 – 7.0]
• Alpha: Division between quick/slow routing [0.0 – 1.0]
• Kq: Quick flow reservoir rate constant (day^-1) [0.15 – 1.0]
• Ks: Slow flow reservoir rate constant. (day^-1) [0.0 – 0.15]
• N: The number of quick-flow reservoirs.

Interactive widget demonstration

I’ve constructed an Ipywidets object which allows a user to visualize the impact of the HYMOD model parameters on the resulting simulation timeseries. The user also has the option to select from three different error metrics, which display in the plot, and toggle the observed timeseries plot on and off.

Later in this post, I will give detail on how the widget was created.

Before provided the detail, I want to show the widget in action so that you know the expectation for the final product.

The gif below shows the widget in-use:

Ipywidgets Introduction

The Ipywdiget library allows for highly customized widgets, like the one above. As with any new tool, I’d recommend you check out the documentation here.

Below, I walk through the process of generating the widget shown above.

Lets begin!

Import the library

# Import the library
import ipywidgets as widgets


Basic widget components

Consider an Ipywidget as being an arrangement of modular components.

The tutorial walks through the construction of five key widget components:

1. Variable slider
2. Drop-down selectors
3. Toggle buttons
4. Label objects
5. Interactive outputs (used to connect the plot to the other three components)

In the last section, I show how all of these components can be arranged together to construct the unified widget.

Sliders

Sliders are one of the most common ipywidet tools. They allow for manual manipulation of a variable value. The slider is an object that can be passed to the interactive widget (more on this further down).

For my HYMOD widget, I would like to be able to manipulate each of the model parameters listed above. I begin by constructing a slider object for each of the variables.

Here is an example, for the C_max variable:

# Construct the slider
Cmax_slider = widgets.FloatSlider(value = 500, min = 10, max = 2000, step = 1.0, description = "C_max",
disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')

# Display the slider
display(Cmax_slider)


Notice that each slider recieves a specified minmax, and step corresponding to the possible values. For the HYMOD demo, I am using the parameter ranges specified in Herman, J.D., P.M. Reed, and T. Wagener (2013), Time-varying sensitivity analysis clarifies the effects of watershed model formulation on model behavior.

I will construct the sliders for the remaining parameters below. Notice that I don’t assign the description parameter in any of these sliders… this is intentional. Later in this tutorial I will show how to arrange the sliders with Label() objects for a cleaner widget design.

# Construct remaining sliders
Cmax_slider = widgets.FloatSlider(value = 100, min = 10, max = 2000, step = 1.0, disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')
B_slider = widgets.FloatSlider(value = 2.0, min = 0.0, max = 7.0, step = 0.1, disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')
Alpha_slider = widgets.FloatSlider(value = 0.30, min = 0.00, max = 1.00, step = 0.01, disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')
Kq_slider = widgets.FloatSlider(value = 0.33, min = 0.15, max = 1.00, step = 0.01, disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')
Ks_slider = widgets.FloatSlider(value = 0.07, min = 0.00, max = 0.15, step = 0.01, disabled = False, continuous_update = False, readout = True, readout_format = '0.2f')
N_slider = widgets.IntSlider(value = 3, min = 2, max = 7, disabled = False, continuous_update = False, readout = True)

# Place all sliders in a list
list_of_sliders = [Kq_slider, Ks_slider, Cmax_slider, B_slider, Alpha_slider, N_slider]


The Dropdown() allows the user to select from a set of discrete variable options. Here, I want to give the user options on which error metric to use when comparing simulated and observed timeseries.

I provide three options:

1. RMSE: Root mean square error
2. NSE: Nash Sutcliffe efficiency
3. ROCE: Runoff coefficient error

See the calculate_error_by_type inside the HYMOD_components.py script to see how these are calculated.

To provide this functionality, I define the Dropdown() object, as below, with a list of options and the initial value:

# Construct the drop-down to select from different error metrics
drop_down = widgets.Dropdown(options=['RMSE','NSE','ROCE'], description='',
value = 'RMSE', disabled = False)

# Display the drop-down
display(drop_down)


ToggleButton

The ToggleButton() allows for a bool variable to be toggled between True and False. For my streamflow plot function, I have an option plot_observed = False which determines if the observed streamflow timeseries is shown in the figure.

# Construct the button to toggle observed data On/Off
plot_button = widgets.ToggleButton(value = False, description = 'Toggle', disabled=False, button_style='', tooltip='Description')

# Display the button
display(plot_button)


Labels

As mentioned above, I choose to not include the description argument within the slider, drop-down, or toggle objects. This is because it is common for these labels to get cut-off when displaying the widget object.

For example, take a look at this slider below, with a long description argument:

# Make a slider with a long label
long_title_slider = widgets.FloatSlider(value = 2.0, min = 0.0, max = 7.0, step = 0.1, description = 'This slider has a long label!', readout = True)

# Display: Notice how the label is cut-off!
display(long_title_slider)


The ipywidgets.Label() function provides a way of avoiding this while allowing for long descriptions. Using Label() will ultimately provide you with a lot more control over your widget layout (last section of the tutorial).

The Label() function generates a separate object. Below, I create a unique Label() object for each HYMOD parameter.

# Import the Label() function
from ipywidgets import Label

# Make a list of label strings
param_labs = ['Kq : Quick flow reservoir rate constant (1/day)',
'Ks : Slow flow reservoir rate constant (1/day)',
'C_max : Maximum soil moisture storage (mm)',
'B : Distribution of soil stores',
'Alpha : Division between quick/slow routing',
'N : Number of quick-flow reservoirs']

# Make a list of Label() objects
list_of_labels = [Label(i) for i in param_labs]

# Display the first label, for example.
list_of_labels[0]


Interactive_output

Now that we have constructed interactive

The interactive_output function takes two inputs, the function to interact with, and a dictionary of variable assignments:

interactive_output( function, {‘variable_name’ : variable_widget, …} )

I have created a custome function plot_HYMOD_results which:

1. Loads 1-year of precipitation and evaporation data for the Leaf River catchment.
2. Runs the HYMOD simulation using the provided parameter values.
3. Calculates the error of the simulated vs. observed data.
4. Plots the timeseries of runoff.

The source code for this function can be found in the GitHub repository for this post, or specifically here.

The function receives parameter values for each of the HYMOD parameters discussed above, a bool indicator if observed data should be plotted, and a specified error metric.

plot_HYMOD_results(C_max, B, Alpha, Ks, Kq, N_reservoirs, plot_observed = False, error_type = ‘RMSE’):

I have already generated widget components corresponding to each of these variables! (If you are on the Jupyter Notebook version of this post, make sure to have Run every cell before this, or else the following code wont work.

I can now use the interactive_output function to link the widget components generated earlier with the function inputs:

# Import the interactive_output function
from ipywidgets import interactive_output

# Import my custom plotting function
from HYMOD_plots import plot_HYMOD_results

result_comparison_plot = interactive_output(plot_HYMOD_results, {'C_max' : Cmax_slider, 'B': B_slider, 'Alpha':Alpha_slider,
'Ks':Ks_slider, 'Kq':Kq_slider,'N_reservoirs':N_slider,
'plot_observed' : plot_button,'error_type': drop_down})

# Show the output
result_comparison_plot


Displaying the interactive_output reveals only the plot, but does not include any of the widget functionality…

Despite this, the plot is still linked to the widget components generated earlier. If you don’t believe me (and are reading the Jupyter Notebook version of this post), scroll up and click the ToggleButton a few cells up, then come back and look at the plot again.

Using the interactive_output() function, rather than other variations of the interact() functions available, allows for cleaner widgets to be produced, because now the arrangment of the widget components can be entirely customizable.

Keep reading for more detail on this!

Arranging widget components

Rather than using widget features one-at-a-time, Ipywidgets allow for several widgets to be arranged in a unified layout. Think of everything that has been generated previously as being a cell within the a gridded widget; the best part is that each cell is linked with one another.

Once the individual widget features (e.g., sliders, buttons, drop-downs, and output plots) are defined, they can be grouped using the VBox() (vertical box) and HBox() (horizontal box) functions.

I’ve constructed a visual representation of my intended widget layout, shown below. The dashed orange boxes show those components grouped by the HBox() function, and the blue boxes show those grouped by the VBox() function.

Before getting started, import some of the basic layout functions:

# Import the various
from ipywidgets import HBox, VBox, Layout


Before constructing the entire widget, it is good to get familiar with the basic HBox() and VBox() functionality.

Remember the list of sliders and list of labels that we created earlier?

# Stack the list of label objects vertically:
VBox(list_of_labels)

# Try the same thing with the sliders (remove comment #):
#VBox(list_of_sliders)


In the final widget, I want the column of labels to be located on the left of the column of sliders. HBox() allows for these two columns to be arrange next to one another:

# Putting the columns side-by-side
HBox([VBox(list_of_labels), VBox(list_of_sliders)])


Generating the final widget

Using the basic HBox() and VBox() functions shown above, I arrange all of the widget components I’ve defined previously. I first define each row of the widget using HBox(), and finally stack the rows using VBox().

The script below will complete the arrangement, and call the final widget!

# Define secifications for the widgets: center and wrap
box_layout = widgets.Layout(display='flex', flex_flow = 'row', align_items ='center', justify_content = 'center')

# Create the rows of the widets
title_row = Label('Select parameter values for the HYMOD model:')
slider_row = HBox([VBox(list_of_labels), VBox(list_of_sliders)], layout = box_layout)
error_menu_row = HBox([Label('Choose error metric:'), drop_down], layout = box_layout)
observed_toggle_row = HBox([Label('Click to show observed flow'), plot_button], layout = box_layout)
plot_row = HBox([result_comparison_plot], layout = box_layout)

# Combine label and slider box (row_one) with plot for the final widget
HYMOD_widget = VBox([title_row, slider_row, plot_row, error_menu_row, observed_toggle_row])

# Call the widget and have fun!
HYMOD_widget


Concluding remarks

I hope that you are able to find some fun/interesting/educational use for the Ipywidget skills learned in this post.

Viewing your Scientific Landscape with VOSviewer

In this post, we’re taking a look at a cool tool for visualizing bibliometric networks called VOSviewer. In our group, we have been thinking more about our scientific landscape (i.e. what fields are we reaching with our work and who are we collaborating with most). VOSviewer provides a great way to interactively engage with this network in maps like this:

(The web app cannot always accommodate high traffic, so if the link appears broken, please check back in a couple hours!)

In this blog post, I’m going to highlight this tool by looking at the reach of the Borg Multi-Objective Evolutionary Algorithm (Borg MOEA) developed by Dave Hadka and Patrick Reed and first published in 2013. I also really like that these conceptual networks can be embedded into a website. Slight caveat: if you use WordPress, you must have a Business account with plugins enabled to embed these maps directly into your website. You’ll see that in order to view the interactive maps, I’ll have links to VosViewer’s online application, but unfortunately you will not be able to interact with the maps directly in this post.

The first step to creating these maps is to get our bibliographic network information. I’m using the Dimensions database which has a free web app here. It should be noted that Vosviewer also supports exports from Scopus, Web of Science, Lens, and PubMed. Once I get to the Dimensions web app, I type in “Borg MOEA” to find the original paper and download the citations associated with that paper.

You can choose the Borg paper as shown above and then scroll down and click on “show all” next to “Publication Citations”.

Then we export this by clicking the save/export button at the top of the page.

Make sure to choose “bibliographic mapping” so that the data are compatible with VosViewer. Dimensions will send an email with the link to download a zipped file which contains a .csv file with all the meta data needed to create the network.

Step 2: Create the map in VOSviewer

The user interface for the desktop version of VosViewer is fairly intuitive but also quite powerful. First we choose “Create” on the left  and then specify that we want to create a map based on bibliographic data.

Next, we specify that we want to read our data from a database file. Note that VosViewer also supports RIS files and has an API that you can directly query with. On the right, we choose the Dimensions tab and upload the metadata .csv file.

Now comes the fun part! We can create maps based on various components of the metadata that we downloaded. For example, let’s do the co-authorship analysis. This will show the strength of co-authorship among all the authors that have cited the Borg paper.

Click “Next>” and then fill out some narrowing thresholds. I decrease the number of documents that the author must have from 5 to 1 and then choose to show only connected authors to create a clean map. The result is this:

Here the colors represent the different clusters of authors and the size represents the number of documents that have been published by that author that have cited Borg. If we switch to the “Overlay Visualization” tab, we get more information on the years that the authors have published most together.

Step 3: Clean-Up and Customization

One thing you might have noticed is that there are duplicates of some names. For example, Patrick M. Reed and Patrick Reed. We want to merge these publications so that we only have one bubble to represent Pat. For this, we need to create thesaurus file. In the folder that contains the VOSviewer application, navigate to the “data” folder and look for “thesaurus_authors.txt”. Here we can specify which names to merge.

Now, I can recreate my map, but this time, I supply a thesaurus file when creating the map.

Below, we have our final merged map.

We can create another map that instead shows the journals that the citations come from. Now instead we create a map and choose “Citation” and “Sources”.

So we are seeing strong usage of Borg in journals pertaining to water resources, evolutionary computation, and some miscellaneous IEEE journals.

Note that you can also create maps associated with countries from where the authors’ institutions are located.

We see that the US is the strongest hub, followed by China, Italy, and the UK. We can finally show the institutions that the authors are affiliated with. Here we see lots of publications from Cornell, Penn State, and Politecnico di Milano.

Step 4: Creating a shared link or embedding in a website

Screenshots are not an ideal way to view these maps, so let’s see how we can go about sharing them as interactive applications that others can use. We can go to the right hand side of the application and choose “share” as a google drive link. After going through authentication, you now have a shareable link as follows:

If your website supports it, you can also embed these applications directly into a webpage using the following html code. You would just need to swap out the source location, src, with your own links. When swapping out, be sure to include &simple_ui=true after your link as shown below.

<iframe
allowfullscreen="true"
width="100%"
height="75%"
style="border: 1px solid #ddd; max-width: 1200px; min-height: 500px"
>
</iframe>

This is just scraping the surface of what VOSviewer is capable of, so definitely take time to explore the website, the examples, and the manual. If you make any cool maps, please link them in the comments below!

Introduction to Drought Metrics

In this blog post, we’re going to discuss meteorological and hydrological drought metrics. Droughts are an important but difficult hazard to characterize. Whereas hurricanes or tornadoes occur suddenly and persist for a short and clearly defined interval, a drought is usually thought of as a “creeping disaster”, in which it is often difficult to pinpoint exactly when a drought begins and when it ends. Droughts can last for decades (termed a megadrought) where dryness is chronic but still can be interspersed with brief wet periods.

Droughts are expected to become more prominent globally in the upcoming decades due to the warming of the climate and even now, many locations in the Western US are experiencing droughts that exceed or rival some of the driest periods in the last millennium (California and the Southwest US). Thus, substantial literature is focused on investigating drought through observed gauge data, reconstructions using paleodata, and projections of precipitation and streamflow from climate models to better understand what future droughts may look like. However, many studies differ in their characterization of drought risk. Different GCMs can create contrasting representations of drought risk which often is caused by differences in the underlying representations of future warming and precipitation. Further, it is important to consider that a drought does not have a universal definition. It manifests due to the complex interactions across the atmosphere, land and hydrology, and the human system [1]. To account for this complexity, different definitions and metrics for droughts have been developed to account for single or combinations of these sectors (meteorological, hydrological, agricultural, etc.). Though a single definition has not been agreed upon, it has been widely suggested that using any one measure of drought can be limiting. For example, Swann (2018) suggests that ignoring plant response in any metric will overestimate drought. Furthermore, Hao et al. (2013) explains that a meteorological (precipitation) drought may not lead to an agricultural (soil moisture) drought in tropical areas where there is high soil moisture content. In this blog post, we’re going to define and discuss some of these metrics that are used to represent droughts to aid in deciding what metrics are appropriate for your case study.

SPI and SSI

The Standardized Precipitation Index (SPI) and Standardized Streamflow Index (SSI) are the most common means of characterizing a meteorological or hydrologic drought respectively. These metrics are highly popular because they only require a monthly time series (ideally of length > 30 years) of either precipitation or streamflow. These metrics quantify how much a specific month’s precipitation or streamflow deviates from the long-term monthly average. The steps to calculate SPI are as follows (SSI is the same):

1. Determine a moving window or averaging period. This is usually a value from the following: 3, 6, 12, or 24 months. Any moving window from 1-6 months is better for capturing meteorological and soil moisture conditions whereas 6-24 month averaging windows will better represent droughts affecting streamflow, reservoir storage, and groundwater. If calculating a 6-month SPI,  the way to factor in the windows is as follows: the aggregate precipitation attributed to month j is accumulated over month j-5 to j [4]. Repeat this procedure for the whole time series.
2. Next an appropriate probability density function is fit to the accumulated precipitation. Generally a gamma or Pearson Type III distribution works well for precipitation data, but be sure to check for goodness of fit.
3. The associated cumulative probability distribution from Step 2 is then estimated and subsequently transformed to a normal distribution. Now each data point, j, can be worked through this framework to ultimately calculate a precipitation deviation for a normally distributed probability density with a mean of zero and standard deviation of 1. Because the SPI is normalized, it means that you can use this metric to characterize both dry and wet periods.

The range that the SPI values can take are summarized below [5]:

From the SPI, you can define various drought metrics of interest. We’ll use an SPI of <= -1 to define the consequential drought threshold.

Percentage of Time in Drought: How many months in the time series have an SPI less than -1 divided by the total months in the time period in question?

Drought Duration: What is the number of consecutive months with an SPI below -1?

Drought Intensity or Peak: What is the minimum value of a drought’s period’s SPI?

Drought Recovery: How many months does it take to get from the peak of the drought to an SPI above -1?

There are lots of packages that exist to calculate SPI and SSI and it’s not difficult to code up these metrics from scratch either. Today, I’m going to demonstrate one of the easiest toolboxes to use for MATLAB, the Standardized Drought Analysis Toolbox (SDAT). We’ll use the provided example time series, precip.txt, which is 500 months long. I specify that I want to utilize a window of 6 months (line 50). Running the script creates the SPI index and plots it for you.

Let’s investigate these results further. I’ll change the window to 12 months and plot the results side by side. In the plotting window, I isolate a piece of the time series that is of length 25 months. The instances of drought are denoted by the black squares and the peak of each drought is denoted by the circle. We can then calculate some of those metrics defined above.

Note how different the SPI metric and perception of drought is depending on your chosen window. The window will greatly influence your perception of the number of droughts experienced and their intensity which is worth addressing. As much as SPI and SSI are easy metrics to calculate, that simplicity comes with limitations. Because these metrics only use the precipitation or streamflow time series, there is no consideration of evapotranspiration which means that any information about how much warming plays a role in these drought conditions is neglected, which can be a large piece of the puzzle. These metrics tend to be very sensitive to the length of the record as well [6].

SPEI

Given the limitations of SPI, the Standardized Precipitation Evapotranspiration Index (SPEI) was proposed by Vicente-Serrano et al. (2010) that is based on both temperature and precipitation data. Mathematically, the SPEI is calculated similarly to the SPI, thought instead of using precipitation only, the aggregated value to generate the index on is now precipitation minus PET. PET can be calculated using a more physically-based equation such as the Penman-Montieth equation or the simpler Thornthwaite equation (which only uses air temperature to derive PET). After the aggregate data is created, one can simply outlined in the SPI section. Vicente-Serrano et al. (2010) compare SPI and SPEI in Indore, India and demonstrate that only SPEI can identify an increase in drought severity associated with higher water demand as a result of evapotranspiration. The key limitation of SPEI is mostly based on the fact that it requires a complete temperature and precipitation dataset which may limit its use due to insufficient data being available.

MSDI

Another metric to be aware of is the Multivariate Standardized Drought Index (MSDI) from Hao et al. (2013). This metric probabilistically combines SPI and SSI for a more comprehensive drought characterization that covers both meteorological and agricultural drought conditions. The SPI and SSI metrics are determined as usual and then a copula is fit to derive a joint probability distribution across the two indices. The MSDI index is ultimately found by pushing the joint probabilities into an inverse normal distribution, thus placing it in the same space as the original SPI and SSI. The authors demonstrate that (1) the difference between SPI and SSI decreases as drought duration increases and (2) the MDSI can capture the evolution of both meteorological and hydrological droughts and particularly show that the onset of droughts is dominated by the SPI index and drought persistence is dominated more by the SSI index.

If you are interested more in capturing drought at longer time scales, such as decadal to multi-decadal, a similar style of metric is proposed in Ault et al. (2014). The authors define a decadal drought, such as the 1930s dustbowl or 1950s drought in the Southwest as a precipitation value that is more than ½ sigma below the 11-year running mean. A multi-decadal drought is characterized as a precipitation value that is more than ½ sigma below a 35-year running mean. This definition is somewhat arbitrary and more of a worse-case planning scenario because it corresponds to drought events that are worse than what has been reconstructed over the past millennium.

I hope this post is useful to you! If you use other drought metrics that are not listed here or have certain preferences for metrics based on experiences, please feel free to comment below!

References

[1] Touma, D., Ashfaq, M., Nayak, M. A., Kao, S. C., & Diffenbaugh, N. S. (2015). A multi-model and multi-index evaluation of drought characteristics in the 21st century. Journal of Hydrology526, 196-207.

[2] Swann, A. L. (2018). Plants and drought in a changing climate. Current Climate Change Reports4(2), 192-201.

[3] Hao, Z., & AghaKouchak, A. (2013). Multivariate standardized drought index: a parametric multi-index model. Advances in Water Resources57, 12-18.

[4] Guenang, G. M., & Kamga, F. M. (2014). Computation of the standardized precipitation index (SPI) and its use to assess drought occurrences in Cameroon over recent decades. Journal of Applied Meteorology and Climatology53(10), 2310-2324.

[5] McKee, T. B., Doesken, N. J., & Kleist, J. (1993, January). The relationship of drought frequency and duration to time scales. In Proceedings of the 8th Conference on Applied Climatology (Vol. 17, No. 22, pp. 179-183).

[7] Vicente-Serrano S.M., Santiago Beguería, Juan I. López-Moreno, (2010) A Multi-scalar drought index sensitive to global warming: The Standardized Precipitation Evapotranspiration Index – SPEI. Journal of Climate 23: 1696-1718.

[8] Ault, T. R., Cole, J. E., Overpeck, J. T., Pederson, G. T., & Meko, D. M. (2014). Assessing the risk of persistent drought using climate model simulations and paleoclimate data. Journal of Climate27(20), 7529-7549.

Continuous Deployment with GitHub Actions (or, What Gives Life to a Living eBook?)

Last month we introduced our free eBook Addressing Uncertainty in MultiSector Dynamics Research. Did you know that, to date, we have published 74 revisions? Without an automated release and deployment process, this would have been very tedious! But, with the help of GitHub Actions, every small edit, minor clarification, or major improvement can go live within moments of merging the code. Read on to learn how the eBook team leverages GitHub Actions for CI/CD (Continuous Integration / Continuous Delivery).

GitHub Workflow

A reliable CI/CD strategy depends on a robust code review process⁠—continuously delivering bugs and typos will not impress anyone! There are many acceptable workflows; the best one to use will depend on team composition and codebase. In our case, a feature branching workflow suffices:

The feature branching workflow consists of a main code branch representing published assets. New features or bug fixes require authors to create a new branch from main, implement their changes, and then submit a pull request back into main. A pull request is a formal code review process that gives other authors a chance to provide feedback on the proposed changes. Once consensus has been reached, the feature branch is merged into the main branch, kicking off the CI/CD process.

Automation

While the code review process is designed to catch content and conceptual errors, subtle process and system based errors can often slip through the cracks. Thus the first step in the CI/CD process should be running a suite of automated tests that span a range of systems, behaviors, and any known pain points. The depth and breadth of these tests should be sufficient to ensure an adequate degree of publication readiness without being overly burdensome to maintain. The test suite can grow over time!

For the eBook, our tests simply ensure that the Python dependencies install correctly on Linux, Mac, and Windows containers, that the supporting code can be imported on these systems without error, and that the HTML and PDF versions of the publication generate successfully. If any tests fail, the publication process is cancelled and the authors are notified with details of the failure.

GitHub Actions

GitHub Actions are available to any project hosted in GitHub—totally free for public repositories, and free to a limited extent for private repositories. An Action can be defined as a unit of work performed using a virtual machine in response to an event. In GitHub, Actions are defined using YAML files places into the .github/workflows directory of a repository. YAML (YAML Ain’t Markup Language) is a concise, human-readable syntax for conveying semi-structured data. The minimal content of a GitHub Action includes a name, one or more event triggers, and one or more jobs. A job consists of a name, one or more virtual machine targets, and one or more steps. For example, the eBook test Action looks like this:

.github/workflows/01_test.yml
name: Test
on:
push:
branches: [ main ]
jobs:
test:
runs-on: ${{ matrix.os }} strategy: matrix: os: [ubuntu-latest, macos-latest, windows-latest] steps: - uses: actions/checkout@v2 - name: Set up Python uses: actions/setup-python@master with: python-version: 3.9 - name: Install dependencies run: | python -m pip install --upgrade pip pip install -r requirements.txt - name: Run tests run: | pip install pytest pytest  There is a lot going on here, so let’s take it step by step! name: Test on: push: branches: [ main ] This snippet gives our Action a name, and specifies that it should trigger on updates to the main branch of the repository. jobs: test: runs-on:${{ matrix.os }}
strategy:
matrix:
os: [ ubuntu-latest, macos-latest, windows-latest ]

This snippet defines a job within our “Test” Action named “test”, and then uses special syntax to declare that the job should be run on three different virtual machines: the latest Ubuntu Linux, macOS, and Windows containers. Running the tests on multiple operating systems helps catch bugs with system-specific dependencies.

      steps:
- uses: actions/checkout@v2
- name: Set up Python
uses: actions/setup-python@master
with:
python-version: 3.9
- name: Install dependencies
run: |
python -m pip install --upgrade pip
pip install -r requirements.txt
- name: Run tests
run: |
pip install pytest
pytest

This snippet outlines the actual units of work within the job; each “-” separates a unique task. The uses syntax is special in that it allows one to leverage tasks written by others hosted in the GitHub Actions Marketplace. The actions/checkout@v2 task clones the repository onto the virtual machine, and the actions/setup-python@master task installs and configures the specified Python version. The final two steps use the run directive to invoke custom code, in this case installing Python dependencies and running the Python test suites.

Deployment

Once the tests successfully pass, it’s time to publish! Since the eBook is essentially a web app, GitHub Pages is the perfect deployment platform. Pages hosts the content of a branch as a website, and is free for public repositories.

If you followed along with the previous eBook post, you learned about the Python, Sphinx, and Restructured Text workflow for compiling the eBook content into a polished product. Let’s create a GitHub Action to compile the eBook and deploy it to GitHub Pages! Here’s the full YAML file:

.github/workflows/02_deploy.yml
name: Deploy
on:
push:
branches: [ main ]
jobs:
build:
runs-on: ubuntu-latest
steps:
- uses: actions/checkout@v2
- uses: actions/setup-python@v2
with:
python-version: '3.9'
- name: Install latex dependencies
run: sudo apt-get update -y && sudo apt-get install -y texlive latexmk texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended ghostscript
- name: Update pip and install python dependencies
working-directory: 'docs/'
run: |
python -m pip install --upgrade pip
pip install -r requirements.txt
- name: Build html and pdf ebook
working-directory: 'docs/'
run: |
make html latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
make latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
make latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
continue-on-error: true
- name: Get current datetime in ISO format
id: date
run: echo "::set-output name=date::$(date -u +'%Y-%m-%d')" - name: Create GitHub release id: gh_release uses: softprops/action-gh-release@v1 with: files: docs/build/latex/addressinguncertaintyinmultisectordynamicsresearch.pdf tag_name:${{ steps.date.outputs.date }}v${{ github.run_number }} - name: Commit the compiled files run: | cd docs/build/html git init git add -A git config --local user.email "action@github.com" git config --local user.name "GitHub Action" git commit -m 'deploy' -a || true - name: Push changes to gh-pages uses: ad-m/github-push-action@master with: branch: gh-pages directory: docs/build/html force: true github_token:${{ secrets.GITHUB_TOKEN }}


A lot to unpack here! Let’s take it step by step. As before, we start by naming the Action, triggering it on updates to the main branch, declaring that it should run only on an ubuntu-latest virtual machine, checking out out the code, and setting up Python. Then we get into the new job steps:

      - name: Install latex dependencies
run: sudo apt-get update -y && sudo apt-get install -y texlive latexmk texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended ghostscript

This step installs all the operating system dependencies needed to support the Latex syntax and compilation to PDF. There was some trial and error involved in getting this right, but once correct it should be pretty stable.

      - name: Build html and pdf ebook
working-directory: 'docs/'
run: |
make html latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
make latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
make latexpdf --keep-going LATEXMKOPTS="-interaction=nonstopmode" || true
continue-on-error: true

This step runs the Sphinx makefile to compile the HTML and PDF versions of the eBook. The verbosity and repetitiveness of these commands works around some unusual oddities of the Latex and PDF compilation. --keep-going LATEXMKOPTS="-interaction=nonstopmode" prevents the command from waiting for user input. || true and the repeated make latexpdf lines allow the PDF engine to fully resolve all the references in the restructured text files; otherwise the PDF file would be incomplete and garbled (this one stumped us for awhile!).

      - name: Get current datetime in ISO format
id: date
run: echo "::set-output name=date::$(date -u +'%Y-%m-%d')" - name: Create GitHub release id: gh_release uses: softprops/action-gh-release@v1 with: files: docs/build/latex/addressinguncertaintyinmultisectordynamicsresearch.pdf tag_name:${{ steps.date.outputs.date }}v${{ github.run_number }} To make it easier to chronologically place our eBook releases, we wanted to include a date stamp in our version tags. The first step above assigns the date to a variable. The second step above creates and tags an official GitHub release (using the date and an auto-incrementing run number), and includes the PDF version of the eBook as an asset attached to the release.  - name: Commit the compiled files run: | cd docs/build/html git init git add -A git config --local user.email "action@github.com" git config --local user.name "GitHub Action" git commit -m 'deploy' -a || true - name: Push changes to gh-pages uses: ad-m/github-push-action@master with: branch: gh-pages directory: docs/build/html force: true github_token:${{ secrets.GITHUB_TOKEN }}

These two steps cause the GitHub Actions user to commit the compiled HTML files and force push them to the gh-pages branch of our repository, using a secret token. This is a common “hack” to enable publishing only the desired web assets and not the entire repository. Never force push to other shared code branches!

Action Status

Check the status of the CI/CD pipeline using the Actions tab of the GitHub repository. Successful Actions show a check mark, in progress Actions show a spinner, and failed Actions show an X. Clicking into a particular Action will show more details, log messages, and error traces if relevant.

Slack Integration

To take our workflow to the next level (and to avoid the need to read even more email 😅 ), we added the GitHub app to our eBook Slack channel. This adds a bot that can subscribe to GitHub repositories and report on activity; for instance: new issues, new pull requests, and new releases. We can then discuss and iterate inline in Slack, without having to jump to other apps or sites.

To add the GitHub bot to a channel, right click on the channel name, select “Open channel details”, and navigate to the “Integrations” tab. From here, you can choose “Add apps” and search for GitHub. Once added, type a bot command such as /github subscribe [repository name] to start receiving notifications in the channel. The bot can also be used to open and close issues!

Conclusion

Using the GitHub Actions workflow to automate the testing and publication of our eBook enabled our team to focus more on the content and quality of release rather than stumbling over the intricacies of the publication process. CI/CD has been a powerful tool in software communities, but can greatly benefit researchers and academics as well. We hope our learnings presented above will speed you along in your own workflows!

MORDM VIII: Characterizing the effects of deep uncertainty

In the previous post, we defined robustness using the satisficing metric where (1) reliability should be at least 98%, (2) restriction frequency should be not more than 10% and (3) worst-case cost of drought mitigation action should not be more than 10% of annual net volumetric income. To calculate the robustness of these set of actions (portfolios) against the effects of challenging states of the world (SOWs) on the initial set of actions, we once again re-simulated them to discover how they fail.

In this penultimate post, we will perform simple sensitivity analysis across the average performance of all sixty-nine portfolios of actions to understand which uncertainties control the performance of each utility (Raleigh, Durham and Cary) and the regions across all uncertain SOWs.

Calculating average performance across 100 DU SOWs

First, we create a new folder to hold the output of the next few post-processing steps. Navigate to the WaterPaths/ folder and create a folder called post_processing. Now, let’s calculate the average performance of each of the sixty-nine initial portfolios across the 100 DU SOWs that we previously simulated them over. The code for this can be found in the post_processing_code folder under gather_objs.py file and should look like this:

# -*- coding: utf-8 -*-
"""
Created on Mon April 26 2022 11:12

@author: Lillian Bei Jia Lau
Organizes output objectives by mean across RDMs and across solutions

"""
import numpy as np

obj_names = ['REL_C', 'RF_C', 'INF_NPC_C', 'PFC_C', 'WCC_C', \
'REL_D', 'RF_D', 'INF_NPC_D', 'PFC_D', 'WCC_D', \
'REL_R', 'RF_R', 'INF_NPC_R', 'PFC_R', 'WCC_R', \
'REL_reg', 'RF_reg', 'INF_NPC_reg', 'PFC_reg', 'WCC_reg']

'''
Performs regional minimax
'''
def minimax(N_SOLNS, objs):
for i in range(N_SOLNS):
for j in range(5):
if j == 0:
objs[i,15] = np.min([objs[i,0],objs[i,5], objs[i,10]])
else:
objs[i, (j+15)] = np.max([objs[i,j],objs[i,j+5], objs[i,j+10]])
return objs

'''
Calculates the mean performance acorss all SOWs
'''
def mean_performance_across_rdms(objs_by_rdm_dir, N_RDMS, N_SOLNS):
objs_matrix = np.zeros((N_SOLNS,20,N_RDMS), dtype='float')
objs_means = np.zeros((N_SOLNS,20), dtype='float')

for i in range(N_RDMS):
filepathname = objs_by_rdm_dir + str(i) + '_sols0_to_' + str(N_SOLNS) + '.csv'
objs_matrix[:,:15,i] = objs_file

objs_file_wRegional = minimax(N_SOLNS, objs_matrix[:,:,i])

objs_matrix[:,:,i] = objs_file_wRegional

array_has_nan = np.isnan(np.mean(objs_matrix[:,3,i]))
if(array_has_nan == True):
print('NaN found at RDM ', str(i))

for n in range(N_SOLNS):
for n_objs in range(20):
objs_means[n,n_objs] = np.mean(objs_matrix[n,n_objs,:])

return objs_means

'''
Calculates the mean performance acorss all SOWs
'''
def mean_performance_across_solns(objs_by_rdm_dir, N_RDMS, N_SOLNS):
objs_matrix = np.zeros((N_SOLNS,20,N_RDMS), dtype='float')
objs_means = np.zeros((N_RDMS,20), dtype='float')

for i in range(N_RDMS):
filepathname = objs_by_rdm_dir + str(i) + '_sols0_to_' + str(N_SOLNS) + '.csv'
objs_matrix[:,:15,i] = objs_file
objs_file_wRegional = minimax(N_SOLNS, objs_matrix[:,:,i])

objs_matrix[:,:,i] = objs_file_wRegional

array_has_nan = np.isnan(np.mean(objs_matrix[:,3,i]))
if(array_has_nan == True):
print('NaN found at RDM ', str(i))

for n in range(N_RDMS):
for n_objs in range(20):
objs_means[n,n_objs] = np.mean(objs_matrix[:,n_objs,n])

return objs_means

# change number of solutions available depending on the number of solutions
# that you identified
N_SOLNS = 69
N_RDMS = 100

# change the filepaths
objs_by_rdm_dir = '/yourFilePath/WaterPaths/output/Objectives_RDM'
objs_og_dir = '/yourFilePath/WaterPaths/'

fileoutpath = '/yourFilePath/WaterPaths/post_processing/'

fileoutpath_byRDMs = fileoutpath + 'meanObjs_acrossRDM.csv'
fileoutpath_bySoln = fileoutpath + 'meanObjs_acrossSoln.csv'

# should have shape (N_SOLNS, 20)
objs_byRDM = mean_performance_across_rdms(objs_by_rdm_dir, N_RDMS, N_SOLNS)
# should have shape (N_RDMS, 20)
objs_bySoln = mean_performance_across_solns(objs_by_rdm_dir, N_RDMS, N_SOLNS)

np.savetxt(fileoutpath_byRDMs, objs_byRDM, delimiter=",")
np.savetxt(fileoutpath_bySoln, objs_bySoln, delimiter=",")



This will output two .csv files: meanObjs_acrossRDM.csv contains the average performance of each of the sixty-nine objectives evaluated over 100 DU SOWs, while meanObjs_acrossSoln.csv contains the average performance of all solutions within one SOW. Take some time to understand this difference, as this will be important when performing sensitivity analysis and scenario discovery.

Calculate the robustness of each portfolio to deep uncertainty

Now, let’s identify how each of these solutions perform under a set of more challenging SOWs. Within post_processing_code/, identify the file called calc_robustness_across_rdms.py. It should look like this:

# -*- coding: utf-8 -*-
"""
Created on Mon April 26 2022 11:12

@author: Lillian Bei Jia Lau

Calculates the fraction of RDMs over which each perturbed version of the solution meets all four satisficing criteria
"""
import numpy as np
import pandas as pd

obj_names = ['REL_C', 'RF_C', 'INF_NPC_C', 'PFC_C', 'WCC_C', \
'REL_D', 'RF_D', 'INF_NPC_D', 'PFC_D', 'WCC_D', \
'REL_R', 'RF_R', 'INF_NPC_R', 'PFC_R', 'WCC_R', \
'REL_reg', 'RF_reg', 'INF_NPC_reg', 'PFC_reg', 'WCC_reg']

utilities = ['Cary', 'Durham', 'Raleigh', 'Regional']

'''
Performs regional minimax
'''
def minimax(N_SOLNS, objs):
for i in range(N_SOLNS):
for j in range(5):
if j == 0:
objs[i,15] = np.min([objs[i,0],objs[i,5], objs[i,10]])
else:
objs[i, (j+15)] = np.max([objs[i,j],objs[i,j+5], objs[i,j+10]])
return objs

'''
For each rdm, identify if the perturbed solution version x satisfies the satisficing criteria
'''
def satisficing(df_objs):
for i in range(4):
df_objs['satisficing_C'] = (df_objs['REL_C'] >= 0.98).astype(int) *\
(df_objs['WCC_C'] <= 0.10).astype(int) *\
(df_objs['RF_C'] <= 0.10).astype(int)

df_objs['satisficing_D'] = (df_objs['REL_D'] >= 0.98).astype(int) *\
(df_objs['WCC_D'] <= 0.10).astype(int) *\
(df_objs['RF_D'] <= 0.10).astype(int)

df_objs['satisficing_R'] = (df_objs['REL_R'] >= 0.98).astype(int) *\
(df_objs['WCC_R'] <= 0.10).astype(int) *\
(df_objs['RF_R'] <= 0.10).astype(int)

df_objs['satisficing_reg'] = np.max(df_objs.iloc[:, 20:23])
return df_objs

def calc_robustness(objs_by_rdm_dir, N_RDMS, N_SOLNS):

# matrix structure: (N_SOLNS, N_OBJS, N_RDMS)
objs_matrix = np.zeros((N_SOLNS,20,N_RDMS), dtype='float')

satisficing_matrix = np.zeros((N_SOLNS,4,N_RDMS), dtype='float')
solution_robustness = np.zeros((N_SOLNS,4), dtype='float')

for i in range(N_RDMS):
# get one perturbed instance's behavior over all RDMs
filepathname = objs_by_rdm_dir + str(i) + '_sols0_to_' + str(N_SOLNS) + '.csv'

objs_matrix[:,:15,i] = objs_file

objs_file_wRegional = minimax(N_SOLNS, objs_matrix[:,:,i])

objs_matrix[:,:,i] = objs_file_wRegional

# NaN check
array_has_nan = np.isnan(np.mean(objs_matrix[:,3,i]))
if(array_has_nan == True):
print('NaN found at RDM ', str(i))

# for the perturbed instances
for r in range(N_RDMS):

df_curr_rdm = pd.DataFrame(objs_matrix[:, :, r], columns = obj_names)

df_curr_rdm = satisficing(df_curr_rdm)
satisficing_matrix[:N_SOLNS,:,r] = df_curr_rdm.iloc[:,20:24]

for n in range(N_SOLNS):
solution_robustness[n,0] = np.sum(satisficing_matrix[n,0,:])/N_RDMS
solution_robustness[n,1] = np.sum(satisficing_matrix[n,1,:])/N_RDMS
solution_robustness[n,2] = np.sum(satisficing_matrix[n,2,:])/N_RDMS

solution_robustness[:,3] = np.min(solution_robustness[:,:3], axis=1)

return solution_robustness

'''
Change number of solutions available depending on the number of solutions
that you identified and the number of SOWs that you are evaluating them over.
'''
N_RDMS = 100
N_SOLNS = 69

objs_by_rdm_dir = '/scratch/lbl59/blog/WaterPaths/output/Objectives_RDM'

fileoutpath_robustness = '/scratch/lbl59/blog/WaterPaths/post_processing/' + \
'robustness_' + str(N_RDMS) + '_SOWs.csv'

robustness = calc_robustness(objs_by_rdm_dir, N_RDMS, N_SOLNS)

np.savetxt(fileoutpath_robustness, robustness, delimiter=",")



When you run this script from the terminal, you should have a .csv file called ‘robustness_100_SOWs.csv‘ appear in your post_processing/ folder. Now, let’s get onto some sensitivity analysis!

Delta moment-independent sensitivity analysis

The Delta moment-independent (DMI) method (Borgonovo, 2007) is sensitivity analysis method that compares the entire probability distribution of both input and output parameters to estimate the sensitivity of the output to a specific input parameter. It is one of many global sensitivity analysis methods, which in itself is one of two main categories of sensitivity analysis that enables the assessment of the degree to which uncertainty in model inputs map to the degree of uncertainty in model output. For purposes of this test case, the DMI is preferable as it does not rely on any one statistical moment (variance, mean and skew) to describe the dependence of model output to its input parameters. It is also time-sensitive, reflecting the current state of knowledge within the system, which philosophically pairs well with our use of the ROF triggers. More information on alternative global sensitivity methods can be found here.

Within the context of our test case, we will be using the DMI method to identify uncertainties in our decision variables that most strongly influence our performance over the 100 DU SOWs. To perform DMI sensitivity analysis, first navigate to the post_processing/ folder. Within the folder, create two sub-folders called delta_output_DV/ and delta_output_DUF/. This is where all your DMI output will be stored. Next, locate the delta_sensitivity.py file within the post_processing_code/ folder. The code should look similar to the script provided below:

import sys
from SALib.analyze import delta
import numpy as np
import pandas as pd

'''
Finds the upper and lower bounds of input parameters
'''
def find_bounds(input_file):
bounds = np.zeros((input_file.shape[1],2), dtype=float)
for i in range(input_file.shape[1]):
bounds[i,0] = min(input_file[:,i])
bounds[i,1] = max(input_file[:,i])

return bounds
'''
Performs delta moment-independent sensitivity analysis
Source: https://github.com/SALib/SALib/tree/main/examples/delta
'''
def delta_sensitivity(dec_vars, measured_outcomes, names, mo_names, bounds, rdm, mode):
X = dec_vars
Y = measured_outcomes

problem = {
'num_vars': int(dec_vars.shape[1]),
'names': names,
'bounds': bounds
}

for i in range(measured_outcomes.shape[1]):
mo_label = mo_names[i]
if i == 2 and mode == 'objs':
break
else:
filename = '../post_processing/delta_output_' + rdm + '/S1_' + mo_label + '.csv'
S1 = delta.analyze(problem, X, Y[mo_label].values, num_resamples=10, conf_level=0.95, print_to_console=False)
numpy_S1 = np.array(S1["S1"])
fileout = pd.DataFrame([names, numpy_S1], index = None, columns = None)
fileout.to_csv(filename, sep=",")

'''
0 - Name all file headers and compSol to be analyzed
'''
obj_names = ['REL_C', 'RF_C', 'INF_NPC_C', 'PFC_C', 'WCC_C', \
'REL_D', 'RF_D', 'INF_NPC_D', 'PFC_D', 'WCC_D', \
'REL_R', 'RF_R', 'INF_NPC_R', 'PFC_R', 'WCC_R', \
'REL_reg', 'RF_reg', 'INF_NPC_reg', 'PFC_reg', 'WCC_reg']

dv_names = ['RT_C', 'RT_D', 'RT_R', 'TT_D', 'TT_R', 'LMA_C', 'LMA_D', 'LMA_R',\
'RC_C', 'RC_D', 'RC_R', 'IT_C', 'IT_D', 'IT_R', 'IP_C', 'IP_D', \
'IP_R', 'INF_C', 'INF_D', 'INF_R']

rdm_headers_dmp = ['Cary restr. eff', 'Durham restr. eff', 'Raleigh restr. eff']
rdm_headers_utilities = ['Demand growth\nmultiplier', 'Bond term\nmultiplier', \
'Bond interest\nrate multiplier', 'Infrastructure interest\nrate multiplier']
rdm_headers_ws = ['Streamflow amp', 'SCR PT', 'SCR CT', 'NRR PT', 'NRR CT', 'CR Low PT', 'CR Low CT',\
'CR High PT', 'CR High CT', 'WR1 PT', 'WR1 CT', 'WR2 PT', 'WR2 CT',\
'DR PT', 'DR CT', 'FR PT', 'FR CT']

duf_names = ['Cary restr. eff', 'Durham restr. eff', 'Raleigh restr. eff', 'Demand growth\nmultiplier',\
'Bond term\nmultiplier', 'Bond interest\nrate multiplier', 'Infrastructure interest\nrate multiplier',\
'Streamflow amp\nmultiplier', 'SCR PT\nmultiplier', 'SCR CT\nmultiplier', 'NRR PT\nmultiplier',\
'NRR CT\nmultiplier', 'CR Low PT\nmultiplier', 'CR Low CT\nmultiplier', 'CR High PT\nmultiplier',\
'CR High CT\nmultiplier', 'WR1 PT\nmultiplier', 'WR1 CT\nmultiplier', 'WR2 PT\nmultiplier',\
'WR2 CT\nmultiplier', 'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier',\
'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier']

utilities = ['Cary', 'Durham', 'Raleigh', 'Regional']

N_RDMS = 100
N_SOLNS = 69

'''
1 - Load DU factor files and DV files
'''
# change to your own filepath
rdm_factors_directory = '/yourFilePath/WaterPaths/TestFiles/'
rdm_dmp_filename = rdm_factors_directory + 'rdm_dmp_test_problem_reeval.csv'
rdm_utilities_filename = rdm_factors_directory + 'rdm_utilities_test_problem_reeval.csv'
rdm_watersources_filename = rdm_factors_directory + 'rdm_water_sources_test_problem_reeval.csv'

dufs = pd.concat([rdm_dmp, rdm_utilities, rdm_ws], axis=1, ignore_index=True)
duf_numpy = dufs.to_numpy()

# change to your own filepath
dv_directory = '/yourFilePath/WaterPaths/'

'''
2 - Get bounds for DU factors and DVs
'''
duf_bounds = find_bounds(duf_numpy)
dv_bounds = find_bounds(dvs)

'''
3 - Load robustness file and objectives file
'''
# change to your own filepath
main_dir = '/yourFilePath/WaterPaths/post_processing/'

robustness_filename = main_dir + 'robustness_100_SOWs.csv'
robustness_df = pd.DataFrame(robustness_arr, columns=utilities)

objs_mean_rdm_filename = main_dir + 'meanObjs_acrossRDM.csv'
objs_mean_rdm_df = pd.DataFrame(objs_mean_rdm_arr, columns=obj_names)

objs_mean_soln_filename = main_dir + 'meanObjs_acrossSoln.csv'
objs_mean_soln_df = pd.DataFrame(objs_mean_soln_arr, columns=obj_names)

# to change  depending on whether DV or DUF is being analyzed
dec_vars = dvs
measured_outcomes = objs_mean_rdm_df
names = dv_names
mo_names = obj_names
bounds = dv_bounds
rdm = 'DV'
mode = 'objs'
###

delta_sensitivity(dec_vars, measured_outcomes, names, mo_names, bounds, rdm, mode)


The code above identifies the sensitivity of the average values of all sixty-nine performance objective sets over all 100 deeply-uncertain SOWs to the decision variables. This is why we use the meanObjs_acrossRDM.csv file – this file contains sixty-nine mean values of the performance objectives, where each set of performance objectives inversely maps to their corresponding portfolio of actions.

To identify the sensitivity of performance objectives to the DU factors, change lines 121 to 127 to the following:

# to change  depending on whether DV or DUF is being analyzed
dec_vars = duf_numpy[:100,:]
measured_outcomes = objs_mean_soln_df
names = duf_names
mo_names = obj_names
bounds = duf_bounds[:100,:]
rdm = 'DUF'
mode = 'objs'
###


The code above identifies the sensitivity of the average values of all twenty performance objectives over each of the sixty-nine different portfolios to the set of deeply uncertain hydroclimatic and demand scenarios. This is why we use the meanObjs_acrossSoln.csv file – this file contains one hundred mean values of the performance objectives, where each set of performance objectives inversely maps to their corresponding DU SOW.

Great job so far! Now let’s visualize the sensitivity of our output to our input parameters using heatmaps. Before being able to visualize each utility’s performance sensitivity, we must first organize the sensitivity indices of the decision variables into a file containing the indices over all objectives for each utility. The gather_delta.py script does this. Simply change the value of mode on line 11 to ‘DUF‘ to gather the indices for the DU factors.

"""
Created on Tue April 26 2022 16:12

@author: Lillian Bei Jia Lau

Gathers the delta sensitivity indices into files per utility
"""
import numpy as np
import pandas as pd

mode = 'DV'
main_dir = '/yourFilePath/WaterPaths/post_processing/delta_output_' + mode + '/'
utilities = ['_C', '_D', '_R', '_reg']
objs = ['REL', 'RF', 'INF_NPC', 'PFC', 'WCC']
utilities_full = ['Cary', 'Durham', 'Raleigh', 'Regional']

dv_names = ['RT_C', 'RT_D', 'RT_R', 'TT_D', 'TT_R', 'LMA_C', 'LMA_D', 'LMA_R',\
'RC_C', 'RC_D', 'RC_R', 'IT_C', 'IT_D', 'IT_R', 'IP_C', 'IP_D', \
'IP_R', 'INF_C', 'INF_D', 'INF_R']

duf_names = ['Cary restr. eff', 'Durham restr. eff', 'Raleigh restr. eff', 'Demand growth\nmultiplier',\
'Bond term\nmultiplier', 'Bond interest\nrate multiplier', 'Infrastructure interest\nrate multiplier',\
'Streamflow amp\nmultiplier', 'SCR PT\nmultiplier', 'SCR CT\nmultiplier', 'NRR PT\nmultiplier',\
'NRR CT\nmultiplier', 'CR Low PT\nmultiplier', 'CR Low CT\nmultiplier', 'CR High PT\nmultiplier',\
'CR High CT\nmultiplier', 'WR1 PT\nmultiplier', 'WR1 CT\nmultiplier', 'WR2 PT\nmultiplier',\
'WR2 CT\nmultiplier', 'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier',\
'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier']

s1_dv_cary = np.zeros((len(objs), len(dv_names)), dtype=float)
s1_dv_durham = np.zeros((len(objs), len(dv_names)), dtype=float)
s1_dv_raleigh = np.zeros((len(objs), len(dv_names)), dtype=float)
s1_dv_regional = np.zeros((len(objs), len(dv_names)), dtype=float)

s1_dv_dict = {
'_C': s1_dv_cary,
'_D': s1_dv_durham,
'_R': s1_dv_raleigh,
'_reg': s1_dv_regional
}

s1_duf_cary = np.zeros((len(objs), len(duf_names)), dtype=float)
s1_duf_durham = np.zeros((len(objs), len(duf_names)), dtype=float)
s1_duf_raleigh = np.zeros((len(objs), len(duf_names)), dtype=float)
s1_duf_regional = np.zeros((len(objs), len(duf_names)), dtype=float)

s1_duf_dict = {
'_C': s1_duf_cary,
'_D': s1_duf_durham,
'_R': s1_duf_raleigh,
'_reg': s1_duf_regional
}

for i in range(len(utilities)):
s1_util = []
hdrs = []
if mode == 'DV':
s1_util = s1_dv_dict[utilities[i]]
hdrs = dv_names
elif mode == 'DUF':
s1_util = s1_duf_dict[utilities[i]]
hdrs = duf_names

for j in range(len(objs)):
curr_file = main_dir + 'S1_' + objs[j] + utilities[i] + '.csv'

s1_util_df = pd.DataFrame(s1_util, columns=hdrs)
out_filepath = main_dir + utilities_full[i] + '.csv'

s1_util_df.to_csv(out_filepath, sep=',', index=False)



Now, let’s plot our heatmaps! The code to do so can be found in sensitivity_heatmap.py, and should look similar to the code provided below:

import numpy as np
import pandas as pd
import seaborn as sns
import matplotlib.pyplot as plt
from mpl_toolkits.axes_grid1 import AxesGrid

sns.set_theme()

# change depending on compromise solution and whether its sensitivity to DUF or DVs
mode = 'DUF'
rot = 90    # if DV use 0; if DUF use 45
main_dir = '/YourFilePath/WaterPaths/post_processing/delta_output_' + mode + '/'
c_filepath = main_dir + 'Cary.csv'
d_filepath = main_dir + 'Durham.csv'
r_filepath = main_dir + 'Raleigh.csv'
reg_filepath = main_dir + 'Regional.csv'

savefig_dir = '/YourFilePath/WaterPaths/post_processing/'
savefig_name = savefig_dir + 'dmi_heatmap_' + mode + '.svg'

grid_kws = {"height_ratios": (0.20, 0.20, 0.20, 0.20, .02), "hspace": 0.5}
f, (ax1, ax2, ax3, ax4, cbar_ax) = plt.subplots(5, figsize=(15, 20), gridspec_kw=grid_kws)
plt.subplots_adjust(top = 0.95, bottom = 0.05,
hspace = 0, wspace = 0.05)

y_objs=['REL', 'RF', 'INPC', 'PFC', 'WCC']

x_dvs=['$RT_{C}$', '$RT_{D}$', '$RT_{R}$', '$TT_{D}$', '$TT_{R}$', '$LM_{C}$', '$LM_{D}$', '$LM_{R}$',\
'$RC_{C}$', '$RC_{D}$', '$RC_{R}$', '$IT_{C}$', '$IT_{D}$', '$IT_{R}$', '$IP_{C}$', \
'$IP_{D}$', '$IP_{R}$','$INF_{C}$', '$INF_{D}$', '$INF_{R}$']
x_dufs = ['Cary\nrestr. eff', 'Durham\nrestr. eff', 'Raleigh\nrestr. eff', 'Dem. growth\nmultiplier',\
'Bond term\nmultiplier', 'Bond interest\nrate multiplier', 'Infra. interest\nrate multiplier',\
'Streamflow amp\nmultiplier', 'SCR PT\nmultiplier', 'SCR CT\nmultiplier', 'NRR PT\nmultiplier',\
'NRR CT\nmultiplier', 'CR Low PT\nmultiplier', 'CR Low CT\nmultiplier', 'CR High PT\nmultiplier',\
'CR High CT\nmultiplier', 'WR1 PT\nmultiplier', 'WR1 CT\nmultiplier', 'WR2 PT\nmultiplier',\
'WR2 CT\nmultiplier', 'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier',\
'DR PT\nmultiplier', 'DR CT\nmultiplier', 'FR PT\nmultiplier', 'FR CT\nmultiplier']

x_labs = []
if mode == 'DV':
x_labs = x_dvs
elif mode == 'DUF':
x_labs = x_dufs

plt.rc('xtick', labelsize=1)
plt.rc('ytick', labelsize=3)
plt.rc('axes', labelsize=5)
plt.rc('axes', titlesize=14)

ax1.set_title("Cary")
sns.heatmap(cary, linewidths=.05, cmap="YlOrBr", xticklabels=[],
yticklabels=y_objs, ax=ax1, cbar=False)
ax1.set_yticklabels(y_objs, rotation=0)

ax2.set_title("Durham")
sns.heatmap(durham, linewidths=.05, cmap="YlOrBr", xticklabels=[],
yticklabels=y_objs, ax=ax2, cbar=False)
ax2.set_yticklabels(y_objs, rotation=0)

ax3.set_title("Raleigh")
sns.heatmap(raleigh, linewidths=.05, cmap="YlOrBr", xticklabels=[],
yticklabels=y_objs, ax=ax3, cbar=False)
ax3.set_yticklabels(y_objs, rotation=0)

ax4.set_title("Regional")
ax4 = sns.heatmap(regional, linewidths=.05, cmap="YlOrBr", xticklabels=x_labs,
yticklabels=y_objs, ax=ax4, cbar=True, cbar_ax=cbar_ax,
cbar_kws={'orientation': 'horizontal'})     # change depending on whether analyzing DUF or DV
ax4.set_xticklabels(x_labs, rotation=rot, fontsize=10)
ax4.set_yticklabels(y_objs, rotation=0)

plt.savefig(savefig_name)



Running this for the sensitivity to decision variables and DU factors will generate the following images:

In the figure above, the color of each box represents the sensitivity of a performance objective (y-axis) to a specific decision variable (x-axis). It is interesting to note that the restriction triggers (RT) of all utilities strongly influence each of their individual and regional reliability and restriction frequency. This indicates the potential for regional conflict, as possible errors is operating one utility’s restriction trigger may adversely affect other utilities’ reliabilities and ability to maintain full control over their own use of water-use restrictions. Furthermore, Raleigh’s performance is sensitive to more decision variables than its remaining two counterparts, with it’s worst-case cost (WCC) being affected most by Cary’s infrastructure investments. This observation highlights the importance of careful cooperation between a region’s member utilities to ensure that all partners play their part in maintaining both their own and their counterparts’ performance.

In this next figure, we observe that uncertainty in demand growth is the only DU factor that significantly drives changes in individual and regional performance. This finding can thus help utilities to focus on demand management programs, or formulate operations and management policies that enable them to more quickly adapt to changes in consumer and industrial demand growth.

Overall, in this post, we have performed a simple sensitivity analysis to identify uncertainties in the decision variables and DU factors that control regional and individual performance. All the code for processing the output data can be found in this GitHub repository here. In the next post, we will end the MORDM blogpost series by performing scenario discovery to map regions of success and failure as defined by our robustness metrics.

References

Borgonovo, E. (2007). A new uncertainty importance measure. Reliability Engineering &Amp; System Safety, 92(6), 771-784. doi: 10.1016/j.ress.2006.04.015

Herman, J. D., Reed, P. M., Zeff, H. B., & Characklis, G. W. (2015). How should robustness be defined for water systems planning under change? Journal of Water Resources Planning and Management, 141(10), 04015012. doi:10.1061/(asce)wr.1943-5452.0000509

Reed, P.M., Hadjimichael, A., Malek, K., Karimi, T., Vernon, C.R., Srikrishnan, V., Gupta, R.S., Gold, D.F., Lee, B., Keller, K., Thurber, T.B, & Rice, J.S. (2022). Addressing Uncertainty in Multisector Dynamics Research [Book]. Zenodo. https://doi.org/10.5281/zenodo.6110623

A step-by-step tutorial for scenario discovery with gradient boosted trees

Our recently published eBook, Addressing Uncertainty in Multisector Dynamics Research, provides several interactive tutorials for hands on training in model diagnostics and uncertainty characterization. The purpose of this post is to expand upon these trainings by providing a tutorial demonstrating gradient boosted trees for scenario discovery. I’ll first provide some brief background on scenario discovery and gradient boosted trees, then demonstrate a Python implementation on a water supply planning problem. All code here is written in Python, but the workflow is model agnostic, and can be paired with simulation models in any language. I’ve included my code within the text below, but all code and data for this post can also be found in this git repository.

In water resources planning and management, decision makers are often faced with uncertainty about how their system will change in the future. Traditionally, planners have confronted this uncertainty by developing prespecified narrative scenarios, which reduce the multitude of possible future conditions into a small subset of important future states of the world (a prominent example is the ‘scenario matrix framework’ used to evaluate climate change (O’Neill et al., 2014)). While this approach provides intuitive appeal, it may increase system vulnerability if future conditions do not evolve as decision makers expect (for a detailed critique of scenario based planning see Reed et al., 2022). This vulnerability is especially apparent for systems facing deep uncertainty, where decision makers do not know or cannot agree upon the probability density functions of key system inputs (Kwakkel et al., 2016).

Scenario discovery (Groves and Lempert, 2007) is an exploratory modeling centered approach that seeks to discover consequential future scenarios using computational experiments rather than relying on prespecified information. To perform scenario discovery, decision makers first identify a set of relevant uncertainties and their plausible ranges. Next, an ensemble of these uncertainties is developed by sampling across parameter ranges. Candidate policies are then evaluated across this ensemble and machine learning or data mining algorithms are used to examine which combinations of uncertainties generate vulnerability in the system. These combinations can then be used to develop narrative scenarios to inform implementation and monitoring efforts or new policy development.

A core element of the scenario discovery process is the algorithm used to classify future states of the world. Popular algorithms include the PRIM, CART and logistic regression. Recently, gradient boosted trees have been applied as an alternative classificiation algorithm. Gradient boosted trees have advantages over other scenario discovery algorithms because they can easily capture nonlinear and non-differentiable boundaries in the uncertainty space (which are particularly prevalent in water supply planning problems that have discrete capacity expansion options), are highly resistant to overfitting and provide a clear means of ranking the importance of uncertain factors (Trindade et al., 2020). For a comprehensive overview of gradient boosted trees, see Bernardo’s post here.

Test case: the Sedento Valley

To demonstrate gradient boosted trees for scenario discovery we’ll use the Sedento Valley water supply planning test case (Trindade et al., 2020). In the Sedento Valley, three water utilities seek to discover cooperative water supply managment and infrastructure investment portfolios to meet several conflicting objectives in a system facing deep uncertainty. In this post, we’ll investigate how these deep uncertainties (which include demand growth, the efficacy of water use restrictions, financial variables and parameters governing infrastructure permitting and construction time) impact a utility’s ability to maintain three performance criteria: keeping reliability > 98%, restriction frequency < 20% and worst case cost less than 10% of annual revenue. For simplicity, we’ll focus on one regional water utility named Watertown.

Step 1: create a sample of deeply uncertain states of the world

To start the scenario discovery process, we generate an ensemble of deep uncertainties that represent future states of the world (SOWs). Here, we’ll use Latin Hypercube Sampling with an implementation I found in the Surrogate Modeling Toolbox.

import numpy as np
from smt.sampling_methods import LHS

'''
This script will generate 1000 Latin Hypercube Samples (LHS)
of deeply uncertain system parameters for the Sedento Valley
'''

# create an array storing the ranges of deeply uncertain parameters
DU_factor_limits = np.array([
[0.9, 1.1], # Watertown restriction efficacy
[0.9, 1.1], # Dryville restriction efficacy
[0.9, 1.1], # Fallsland restriction efficacy
[0.5, 2.0], # Demand growth rate multiplier
[1.0, 1.2], # Bond term
[0.6, 1.0], # Bond interest rate
[0.6, 1.4], # Discount rate
[0.75, 1.5], # New River Reservoir permitting time
[1.0, 1.2], # New River Reservoir construction time
[0.75, 1.5], # College Rock Reservoir (low) permitting time
[1.0, 1.2], # College Rock Reservoir (low) construction time
[0.75, 1.5], # College Rock Reservoir (high) permitting time
[1.0, 1.2], # College Rock Reseroir (high) construction time
[0.75, 1.5], # Water Reuse permitting time
[1.0, 1.2], # Water Reuse construction time
[0.8, 1.2], # Inflow amplitude
[0.2, 0.5], # Inflow frequency
[-1.57, 1.57]]) # Inflow phase

# Use the smt package to set up the LHS sampling
sampling = LHS(xlimits=DU_factor_limits)

# We will create 1000 samples
num = 1000

# Create the actual sample
x = sampling(num)

# save to a csv file
np.savetxt('DU_factors.csv', x, delimiter=',')


Step 2: Evaluate performance across SOWs

Next, we’ll evaluate the performance of our policy across the LHS sample of DU factors. For the Sedento Valley test case, we use WaterPaths, an open-source simulation system for integrated water supply portfolio management and infrastructure investment planning (for more see Trindade et al., 2020). This step is not included in the git repository as it requires high-performance computing for this system, but results can be found in the “Model_output.csv” file. For simulation details, see Gold et al., 2022.

Step 3: Convert model output into a boolean array for classification

To perform classification, we need to convert the results of our simulations to a binary array classifying each SOW as a “success” or “failure” based on whether the policy met the performance criteria under the SOW. First, we define a small function to determine if an SOW meets a set of criteria, then we apply this function to our results. We also load the DU factor LHS sample.

# First, define a function to check whether performance criteria are met
def check_criteria(objectives, crit_objs, crit_vals):
"""
Determines if an objective meets a given set of criteria for a set of SOWs

inputs:
objectives: np array of all objectives across a set of SOWs
crit_objs: the column index of the objective in question
crit_vals: an array containing [min, max] of the values

returns:
meets_criteria: an numpy array containing the SOWs that meet both min and max criteria

"""

# check max and min criteria for each objective
meet_low = objectives[:, crit_objs] >= crit_vals[0]
meet_high = objectives[:, crit_objs] <= crit_vals[1]

# check if max and min criteria are met at the same time
meets_criteria = np.hstack((meet_low, meet_high)).all(axis=1)

return meets_criteria

##### Load data and pre-process #####

# load objectives and create input array of boolean values for SD input
REL = check_criteria(Reeval_objectives, [0], [.979, 1])
RF = check_criteria(Reeval_objectives, [1], [0, 0.10])
WCC = check_criteria(Reeval_objectives, [2], [0, 0.10])
SD_input = np.vstack((REL, RF, WCC)).SD_input(axis=0)

DU_names = ['Watertown Rest. Eff.', 'Dryville Rest. Eff.', 'FSD_inputsland Rest. Eff.',
'Demand Growth Rate', 'Bond Term', 'Bond Interest',
'Discount Rate', 'NRR Perm', 'NRR Const', 'CRR L Perm',
'CRR L Const.',	'CRR H Perm.', 'CRR H Const.', 'WR1 Perm.',
'WR1 Const.', 'Inflows A', 'Inflows m','Inflows p']


Step 4: Fit a boosted trees classifier

After we’ve formatted the data, we’re ready to perform boosted trees classification. There are several packages for boosted trees in Python, here we’ll use the implementation from scikit-learn. We’ll use an ensemble of 200 trees with depth 3 and a learning rate of 0.1. These parameters need to be tuned for the individual problem, I found this nice post that goes into detail on parameter tuning.

##### Boosted Tree Classification #####

# create a gradient boosted classifier object
learning_rate=0.1,
max_depth=3)

# fit the classifier
gbc.fit(DU_factors, SD_input)


Step 5: Examine which DU factors have the most impact on performance criteria

Now we’re ready to examine the results of our classification. First, we’ll examine how important each DU factor is to the classification results generated by boosted trees. To rank the imporance of each DU factor, we examine the percentage decrease in impurity of the ensemble of trees that is associated with each factor. In plain english, this is a measure of how helpful each DU factor is to correctly classifying SOWs. This infromation is generated during the fit of the classifier above and is easily accessible as an attribute of our scikit-learn classifier.

For our example, one deep uncertainty, demand growth rate, clearly stands out as the most influential, as shown in the figure below. A second, the restriction efficacy for Watertown (the utility we’re focusing on), also stands out as a higher level of importance. All other DU factors have little impact on the classification in this case.

##### Factor Ranking #####

# Extract the feature importances
feature_importances = deepcopy(gbc.feature_importances_)

# rank the feature importances and plot
importances_sorted_idx = np.argsort(feature_importances)
sorted_names = [DU_names[i] for i in importances_sorted_idx]

fig = plt.figure(figsize=(8,8))
ax = fig.gca()
ax.barh(np.arange(len(feature_importances)), feature_importances[importances_sorted_idx])
ax.set_yticks(np.arange(len(feature_importances)))
ax.set_yticklabels(sorted_names)
ax.set_xlim([0,1])
ax.set_xlabel('Feature Importance')
plt.tight_layout()


Step 6: Create factor maps

Finally, we visualize the results of our classification through factor mapping. In the plot below, we show the uncertainty space projected onto the two most influential factors, demand growth rate and restriciton efficacy. Each point represents a sampled SOW, red points represent SOWs that resulted in failure, while white points represent SOWs that resulted in success. The color in the background shows the predicted regions of success and failure from the boosted trees classification.

Here we observe that high levels of demand growth are the primary source of vulnerability for the water utility. When restriction efficacy is lower than our estimate (multiplier < 1), the utility faces failures at demand growth levels of about 1.7 times the estimated values. When restriction effectiveness is at or above estimates, the acceptable scaling of demand growth raises to about 1.8.

Taken as a whole, these results provide valueable insights for decision makers. From our original 18 deep uncertainties, we have discovered that two are critical for the success of our water supply management policy. Further, we have defined thresholds within the uncertainty space that define scenarios that lead to failure. We can use this information to inform monitoring efforts for the water supply policy, or to inform a new problem formulation that tailors actions to mitigate these vulnerabilities.

##### Factor Mapping #####

# Select the top two factors discovered above
selected_factors = DU_factors[:, [3,0]]

# Fit a classifier using only these two factors
learning_rate=0.1,
max_depth=3)
gbc_2_factors.fit(selected_factors, SD_input)

# plot prediction contours
x_data = selected_factors[:,0]
y_data = selected_factors[:,1]

x_min, x_max = (x_data.min(), x_data.max())
y_min, y_max = (y_data.min(), y_data.max())

# create a grid to makes predictions on
xx, yy = np.meshgrid(np.arange(x_min, x_max * 1.001, (x_max - x_min) / 100),
np.arange(y_min, y_max * 1.001, (y_max - y_min) / 100))

dummy_points = list(zip(xx.ravel(), yy.ravel()))

z = gbc_2_factors.predict_proba(dummy_points)[:, 1]
z[z < 0] = 0.
z = z.reshape(xx.shape)

# plot the factor map
fig = plt.figure(figsize=(10,8))
ax = fig.gca()
ax.contourf(xx, yy, z, [0, 0.5, 1.], cmap='RdBu',
alpha=.6, vmin=0.0, vmax=1)
ax.scatter(selected_factors[:,0], selected_factors[:,1],\
c=SD_input, cmap='Reds_r', edgecolor='grey',
alpha=.6, s= 100, linewidth=.5)
ax.set_xlim([.5, 2])
ax.set_ylim([.9,1.1])
ax.set_xlabel('Demand Growth Multiplier')
ax.set_ylabel('Restriction Eff. Multiplier')


References

Gold, D. F., Reed, P. M., Gorelick, D. E., & Characklis, G. W. (2022). Power and Pathways: Exploring Robustness, Cooperative Stability, and Power Relationships in Regional Infrastructure Investment and Water Supply Management Portfolio Pathways. Earth’s Future, 10(2), e2021EF002472.

Groves, D. G., & Lempert, R. J. (2007). A new analytic method for finding policy-relevant scenarios. Global Environmental Change, 17(1), 73-85.

Kwakkel, J. H., Walker, W. E., & Haasnoot, M. (2016). Coping with the wickedness of public policy problems: approaches for decision making under deep uncertainty. Journal of Water Resources Planning and Management, 142(3), 01816001.

O’Neill, B. C., Kriegler, E., Riahi, K., Ebi, K. L., Hallegatte, S., Carter, T. R., … & van Vuuren, D. P. (2014). A new scenario framework for climate change research: the concept of shared socioeconomic pathways. Climatic change, 122(3), 387-400.

Reed, P.M., Hadjimichael, A., Malek, K., Karimi, T., Vernon, C.R., Srikrishnan, V., Gupta, R.S., Gold, D.F., Lee, B., Keller, K., Thurber, T.B., & Rice, J.S. (2022). Addressing Uncertainty in Multisector Dynamics Research [Book]. Zenodo. https://doi.org/10.5281/zenodo.6110623

Trindade, B. C., Gold, D. F., Reed, P. M., Zeff, H. B., & Characklis, G. W. (2020). Water pathways: An open source stochastic simulation system for integrated water supply portfolio management and infrastructure investment planning. Environmental Modelling & Software, 132, 104772.