# Preparing Data for a Time Series Analysis

This semester, I had the opportunity to take Dr. Scott Steinschneider’s new class, Hydrologic Engineering in a Changing Climate, that is offered here at Cornell. In this class, we covered time series analysis, extreme value modeling, and trend tests. I chose to do a final project which focused on using a time series approach to forecast electricity demand in California. As I worked through my project, it became apparent to me that data are rarely in the form where a time series model can be applied directly. Consequently, multiple transformations are usually necessary before a model can be fit to the data set. In this post, I outline some of the inherent characteristics of data that might warrant a transformation as well as the steps that can be taken to address these problems.

Given a data set that you would like to fit any Box-Jenkins model to, you should ask yourself the following two questions?

1. Is normality a reasonable assumption for the residuals?
2. Are the data stationary?

### Normality

Normality can be checked before fitting the model because if the original data are not normal, then there is a good chance that the residuals won’t be as well. If you fit a histogram to the data and it looks like Figure 1, you probably need to apply some form of normalizing transformation.

###### Figure 1: Histogram of Monthly Flow

Two traditional transformations that you can try are a log transform or a Box-Cox transform, shown in the following two equations, where xt is the original data point.

Log Transform:

Box-Cox Transform:

Sometimes a log transformation can be too drastic and skew the data the opposite way. The Box-Cox transform is effectively a less intense transformation that one can try if the log transform is not suitable. Note that when λ=0, the Box-Cox transform reduces to a simple log transform.

The powerTransform function in the R package, car, can be used to find a lambda that will maximize normality.

### Stationarity

For a data set to exhibit stationarity, the following three principles must be true for us to be confident that our model will represent our data well:

For some lag term, s,

1. E[xt]=E[xt+s] (The mean of the data set does not change with time)
2. Var[xt]=Var[xt+s] (The variance of the data set does not change with time)
3. Cov[xt,xt+s]= γ (The covariance between data points is some constant value,γ)

Outlined below are some of the characteristics of a data set that can cause a violation of one or more of these principles.

Seasonality

Seasonality in data can exist if a time series pattern repeats over a fixed and known period. Figure 2 shows monthly inflow into the Schoharie Creek Reservoir. Periodicity is apparent, but it isn’t until we look at the autocorrelation function (ACF) of the data, shown in Figure 3, that we see that there is a clear repetition occurring every 12 months.

###### Figure 3: ACF of Monthly Inflow

One effective way to get rid of this monthly seasonality is to use the following de-seasonalizing equation:

The seasonality is removed from each data point by subtracting the corresponding monthly mean (xmt) and dividing by the month’s standard deviation ( smt). This equation can also be used to account for daily or yearly seasonality as well.

Differencing is another way to address seasonality in data. A seasonal difference is the difference between an observation and the corresponding observation from the previous year.

Where m=12 for monthly data, m=4 for quarterly data, and so on 1.

Trend

A trend, shown in the first panel of Figure 4, is a clear violation of the first requirement for stationarity. There are a couple options that one can implement to deal with trends: differencing and model fitting.

###### Figure 4: De-trending process1

From the above figures, it is clear that differencing can be used to account for seasonality but can also be used to dampen a trend. A first difference is performed by subtracting the value of the current observation from the one in the time step before. It can be applied as follows:

If the transformed data is plotted and still has a trend, a second difference can further be applied.

It is important to note the distinction between seasonal and first differences. Seasonal differencing is the difference from one year to the next, while first differencing is the difference between one observation and the next. Seasonal and trend differencing can both be applied, but sometimes, if seasonal differencing is performed first, it will remove the need for further differencing1.

In Figure 4, note how a log transform, seasonal differencing, and second differencing is necessary to ultimately remove the trend.

###### Figure 5: Modeling Fitting with Ordinary Least Squares2

If a monotonic trend is observed, such as the one in Figure 5, a model fitting can be performed. In this example, a linear model is fit to the trend by choosing coefficients that minimize the sum of squares. This model is then subtracted from the original data to give residuals. The goal is for the resulting residuals to be stationary. Note that a polynomial model can also be fit to the trend if appropriate2 .

Heteroscedasticity

Heteroscedasticity describes the phenomenon when the data do not exhibit a constant variance. This is a violation of the second principle. Heteroscedasticity tends to appear in financial time series (i.e. prices of stocks and bonds) which can be very volatile, but it appears less so in hydrological data3. I did not have to address heteroscedasticity in the electricity load data for my project, and some statisticians suggest that one doesn’t have to deal with it unless it is very severe as weak heteroscedasticity tends be taken care of with normalization and de-seasonalization.

One way to check for heteroscedasticity in a time series is with the McLeod-Li test for conditional heteroscedasticity. If heteroscedasticity is present, consider using an ARCH/GARCH model, if an AR or ARMA model can be fit to the data, respectively, or a hybrid ARCH-ARIMA model if the latter models are not appropriate.

### Choosing a Time Series Model

Once the necessary transformations have been performed, you are ready to fit a time series model to your data. R has a some useful packages for this: forecast and stats. Some helpful functions in these packages include:

auto.arima (forecast) – This function tells you what model is the best fit for your data, the coefficients for the lag terms, and variance of errors (along with other useful information).

arima.sim (stats) – This function allows you to simulate a set of data from your time series model.

predict (stats) – This function will provide a prediction for n time steps into the future based on the chosen time series model. Keep in mind it is best when used to predict just the next few time step.

Finally, remember that back-transformations must be performed on all simulations or predictions to get them into back into the original space.

*For a really helpful explanation of different time series notation, check this previous post.

### References

*All information or figures not specifically cited came from class notes and homework from Dr. Scott Steinschneider’s class

(1) Stationarity and Differencing: https://www.otexts.org/fpp/8/1

(2) Removal of Trend and Seasonality, UC Berkeley: https://www.stat.berkeley.edu/~gido/Removal%20of%20Trend%20and%20Seasonality.pdf

(3) Heteroscedasticity: http://www.math.canterbury.ac.nz/~m.reale/econ324/Topic2.pdf

# Directed search with the Exploratory Modeling workbench

This is the third blog in a series showcasing the functionality of the Exploratory Modeling workbench. Exploratory modeling entails investigating the way in which uncertainty and/or policy levers map to outcomes. To investigate these mappings, we can either use sampling based strategies (open exploration) or optimization based strategies (directed search) In the first blog, I gave a general overview of the workbench and showed briefly how both investigation strategies can be done. In the second blog, I demonstrated the use of the workbench for open exploration in substantial more detail. In this third blog, I will demonstrate in more detail how to use the workbench for directed search. Like in the previous two blog post, I will use the DPS version of the lake problem.

For optimization, the workbench relies on platypus. You can easily install the latest version of platypus from github using pip

pip install git+https://github.com/Project-Platypus/Platypus.git


By default, the workbench will use epsilon NSGA2, but all the other algorithms available within platypus can be used as well.

Within the workbench, optimization can be used in three ways:
* Search over decision levers for a reference scenario
* Robust search: search over decision levers for a set of scenarios
* worst case discovery: search over uncertainties for a reference policy

The search over decision levers or over uncertainties relies on the specification of the direction for each outcome of interest defined on the model. It is only possible to use ScalarOutcome objects for optimization.

## Search over levers

Directed search is most often used to search over the decision levers in order to find good candidate strategies. This is for example the first step in the Many Objective Robust Decision Making process. This is straightforward to do with the workbench using the optimize method.

from ema_workbench import MultiprocessingEvaluator, ema_logging

ema_logging.log_to_stderr(ema_logging.INFO)

with MultiprocessingEvaluator(model) as evaluator:
results = evaluator.optimize(nfe=10000, searchover='levers',
epsilons=[0.1,]*len(model.outcomes),
population_size=50)


the result from optimize is a DataFrame with the decision variables and outcomes of interest. The latest version of the workbench comes with a pure python implementation of parallel coordinates plot built on top of matplotlib. It has been designed with the matplotlib and seaborn api in mind. We can use this to quickly visualize the optimization results.

from ema_workbench.analysis import parcoords

paraxes = parcoords.ParallelAxes(parcoords.get_limits(results), rot=0)
paraxes.plot(results, color=sns.color_palette()[0])
paraxes.invert_axis('max_P')
plt.show()


Note how we can flip an axis using the invert_axis method. This eases interpretation of the figure because the ideal solution in this case would be a straight line for the four outcomes of interest at the top of the figure.

### Specifying constraints

In the previous example, we showed the most basic way for using the workbench to perform many-objective optimization. However, the workbench also offers support for constraints and tracking convergence. Constrains are an attribute of the optimization problem, rather than an attribute of the model as in Rhodium. Thus, we can pass a list of constraints to the optimize method. A constraint can be applied to the model input parameters (both uncertainties and levers), and/or outcomes. A constraint is essentially a function that should return the distance from the feasibility threshold. The distance should be 0 if the constraint is met.

As a quick demonstration, let’s add a constraint on the maximum pollution. This constraint applies to the max_P outcome. The example below specifies that the maximum pollution should be below 1.

from ema_workbench import MultiprocessingEvaluator, ema_logging, Constraint

ema_logging.log_to_stderr(ema_logging.INFO)

constraints = [Constraint("max pollution", outcome_names="max_P",
function=lambda x:max(0, x-1))]

with MultiprocessingEvaluator(model) as evaluator:
results = evaluator.optimize(nfe=1000, searchover='levers',
epsilons=[0.1,]*len(model.outcomes),
population_size=25, constraints=constraints)


### tracking convergence

To track convergence, we need to specify which metric(s) we want to use and pass these to the optimize method. At present the workbench comes with 3 options: Hyper volume, Epsilon progress, and a class that will write the archive at each iteration to a separate text file enabling later processing. If convergence metrics are specified, optimize will return both the results as well as the convergence information.

from ema_workbench import MultiprocessingEvaluator, ema_logging
from ema_workbench.em_framework.optimization import (HyperVolume,
EpsilonProgress, )
from ema_workbench.em_framework.outcomes import Constraint

ema_logging.log_to_stderr(ema_logging.INFO)

# because of the constraint on pollution, we can specify the
# maximum easily
convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[1,1,1,1]),
EpsilonProgress()]
constraints = [Constraint("max pollution", outcome_names="max_P",
function=lambda x:max(0, x-1))]

with MultiprocessingEvaluator(model) as evaluator:
results_ref1, convergence1 = evaluator.optimize(nfe=25000, searchover='levers',
epsilons=[0.05,]*len(model.outcomes),
convergence=convergence_metrics,
constraints=constraints,
population_size=100)


We can visualize the results using parcoords as before, while the convergence information is in a DataFrame making it also easy to plot.

fig, (ax1, ax2) = plt.subplots(ncols=2, sharex=True)
ax1.plot(convergence1.epsilon_progress)
ax1.set_xlabel('nr. of generations')
ax1.set_ylabel('$\epsilon$ progress')
ax2.plot(convergence1.hypervolume)
ax2.set_ylabel('hypervolume')
sns.despine()
plt.show()


### Changing the reference scenario

Up till now, we have performed the optimization for an unspecified reference scenario. Since the lake model function takes default values for each of the deeply uncertain factors, these values have been implicitly assumed. It is however possible to explicitly pass a reference scenario that should be used instead. In this way, it is easy to apply the extended MORDM approach suggested by Watson and Kasprzyk (2017).

To see the effects of changing the reference scenario on the values for the decision levers found through the optimization, as well as ensuring a fair comparison with the previous results, we use the same convergence metrics and constraints from the previous optimization. Note that the constraints are in essence only a function, and don’t retain optimization specific state, we can simply reuse them. The convergence metrics, in contrast retain state and we thus need to re-instantiate them.

from ema_workbench import Scenario

reference = Scenario('reference', **dict(b=.43, q=3,mean=0.02,
stdev=0.004, delta=.94))
convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[1,1,1,1]),
EpsilonProgress()]

with MultiprocessingEvaluator(model) as evaluator:
results_ref2, convergence2 = evaluator.optimize(nfe=25000, searchover='levers',
epsilons=[0.05,]*len(model.outcomes),
convergence=convergence_metrics,
constraints=constraints,
population_size=100, reference=reference)



### comparing results for different reference scenarios

To demonstrate the parcoords plotting functionality in some more detail, let’s combine the results from the optimizations for the two different reference scenarios and visualize them in the same plot. To do this, we need to first figure out the limits across both optimizations. Moreover, to get a better sense of which part of the decision space is being used, let’s set the limits for the decision levers on the basis of their specified ranges instead of inferring the limits from the optimization results.

columns = [lever.name for lever in model.levers]
columns += [outcome.name for outcome in model.outcomes]
limits = {lever.name: (lever.lower_bound, lever.upper_bound) for lever in
model.levers}
limits = dict(**limits, **{outcome.name:(0,1) for outcome in model.outcomes})
limits = pd.DataFrame.from_dict(limits)
# we resort the limits in the order produced by the optimization
limits = limits[columns]

paraxes = parcoords.ParallelAxes(limits, rot=0)
paraxes.plot(results_ref1, color=sns.color_palette()[0], label='ref1')
paraxes.plot(results_ref2, color=sns.color_palette()[1], label='ref2')
paraxes.legend()
paraxes.invert_axis('max_P')
plt.show()


## Robust Search

The workbench also comes with support for many objective robust optimization. In this case, each candidate solution is evaluated over a set of scenarios, and the robustness of the performance over this set is calculated. This requires specifying 2 new pieces of information:
* the robustness metrics
* the scenarios over which to evaluate the candidate solutions

The robustness metrics are simply a collection of ScalarOutcome objects. For each one, we have to specify which model outcome(s) it uses, as well as the actual robustness function. For demonstrative purposes, let’s assume we are use a robustness function using descriptive statistics: we want to maximize the 10th percentile performance for reliability, inertia, and utility, while minimizing the 90th percentile performance for max_P.

We can specify our scenarios in various ways. The simplest would be to pass the number of scenarios to the robust_optimize method. In this case for each generation a new set of scenarios is used. This can create noise and instability in the optimization. A better option is to explicitly generate the scenarios first, and pass these to the method. In this way, the same set of scenarios is used for each generation.

If we want to specify a constraint, this can easily be done. Note however, that in case of robust optimization, the constrains will apply to the robustness metrics instead of the model outcomes. They can of course still apply to the decision variables as well.

import functools
from ema_workbench import Constraint, MultiprocessingEvaluator
from ema_workbench import Constraint, ema_logging
from ema_workbench.em_framework.optimization import (HyperVolume,
EpsilonProgress)
from ema_workbench.em_framework.samplers import sample_uncertainties

ema_logging.log_to_stderr(ema_logging.INFO)

percentile10 = functools.partial(np.percentile, q=10)
percentile90 = functools.partial(np.percentile, q=90)

MAXIMIZE = ScalarOutcome.MAXIMIZE
MINIMIZE = ScalarOutcome.MINIMIZE
robustnes_functions = [ScalarOutcome('90th percentile max_p', kind=MINIMIZE,
variable_name='max_P', function=percentile90),
ScalarOutcome('10th percentile reliability', kind=MAXIMIZE,
variable_name='reliability', function=percentile10),
ScalarOutcome('10th percentile inertia', kind=MAXIMIZE,
variable_name='inertia', function=percentile10),
ScalarOutcome('10th percentile utility', kind=MAXIMIZE,
variable_name='utility', function=percentile10)]

def constraint(x):
return max(0, percentile90(x)-10)

constraints = [Constraint("max pollution",
outcome_names=['90th percentile max_p'],
function=constraint)]
convergence_metrics = [HyperVolume(minimum=[0,0,0,0], maximum=[10,1,1,1]),
EpsilonProgress()]
n_scenarios = 10
scenarios = sample_uncertainties(model, n_scenarios)

nfe = 10000

with MultiprocessingEvaluator(model) as evaluator:
robust_results, convergence = evaluator.robust_optimize(robustnes_functions,
scenarios, nfe=nfe, constraints=constraints,
epsilons=[0.05,]*len(robustnes_functions),
convergence=convergence_metrics,)

fig, (ax1, ax2) = plt.subplots(ncols=2)
ax1.plot(convergence.epsilon_progress.values)
ax1.set_xlabel('nr. of generations')
ax1.set_ylabel('$\epsilon$ progress')
ax2.plot(convergence.hypervolume)
ax2.set_ylabel('hypervolume')
sns.despine()
plt.show()


paraxes = parcoords.ParallelAxes(parcoords.get_limits(robust_results), rot=45)
paraxes.plot(robust_results)
paraxes.invert_axis('90th percentile max_p')
plt.show()


## Search over uncertainties: worst case discovery

Up till now, we have focused on optimizing over the decision levers. The workbench however can also be used for worst case discovery (Halim et al, 2016). In essence, the only change is to specify that we want to search over uncertainties instead of over levers. Constraints and convergence works just as in the previous examples.

Reusing the foregoing, however, we should change the direction of optimization of the outcomes. We are no longer interested in finding the best possible outcomes, but instead we want to find the worst possible outcomes.

# change outcomes so direction is undesirable
minimize = ScalarOutcome.MINIMIZE
maximize = ScalarOutcome.MAXIMIZE

for outcome in model.outcomes:
if outcome.kind == minimize:
outcome.kind = maximize
else:
outcome.kind = minimize


We can reuse the reference keyword argument to perform worst case discovery for one of the policies found before. So, below we select solution number 9 from the pareto approximate set. We can turn this into a dict and instantiate a Policy objecti.

from ema_workbench import Policy

policy = Policy('9', **{k:v for k, v in results_ref1.loc[9].items()
if k in model.levers})

with MultiprocessingEvaluator(model) as evaluator:
results = evaluator.optimize(nfe=1000, searchover='uncertainties',
epsilons=[0.1,]*len(model.outcomes),
reference=policy)


Visualizing the results is straightforward using parcoords.

paraxes = parcoords.ParallelAxes(parcoords.get_limits(results), rot=0)
paraxes.plot(results)
paraxes.invert_axis('max_P')
plt.show()


## Closing remarks

This blog showcased the functionality of the workbench for applying search based approaches to exploratory modelling. We specifically looked at the use of many-objective optimization for searching over the levers or uncertainties, as well as the use of many-objective robust optimization. This completes the overview of the functionality available in the workbench. In the next blog, I will put it all together to show how the workbench can be used to perform Many Objective Robust Decision Making.

# Nondimensionalization of differential equations – an example using the Lotka-Volterra system of equations

I decided to write about nondimensionalization today since it’s something I only came across recently and found very exciting. It’s apparently a trivial process for a lot of people, but it wasn’t something I was taught how to do during my education so I thought I’d put a short guide out on the interwebs for other people. Nondimensionalization (also referred to as rescaling) refers to the process of transforming an equation to a dimensionless form by rescaling its variables. There are a couple benefits to this:

• All variables and parameters in the new system are unitless, i.e. scales and units not an issue in the new system and the dynamics of systems operating on different time and/or spatial scales can be compared;
• The number of model parameters is reduced to a smaller set of fundamental parameters that govern the dynamics of the system; which also means
• The new model is simpler and easier to analyze, and
• The computational time becomes shorter.

I will now present an example using a predator-prey system of equations. In my last blogpost, I used the Lotka-Volterra system of equations for describing predator-prey interactions. Towards the end of that post I talked about the logistic Lotka-Volterra system, which is in the following form:Where x is prey abundance, y is predator abundance, b is the prey growth rate, d is the predator death rate, c is the rate with which consumed prey is converted to predator abundance, a is the rate with which prey is killed by a predator per unit of time, and K is the carrying capacity of the prey given its environmental conditions.

The first step is to define the original model variables as products of new dimensionless variables (e.g. x*) and scaling parameters (e.g. X), carrying the same units as the original variable.The rescaled models are then substituted in the original model:
Carrying out all cancellations and obvious simplifications:Our task now is to define the rescaling parameters X, Y, and T to simplify our model – remember they have to have the same units as our original parameters.

 Variable/parameter Unit x mass 1 y mass 1 t time b 1/time d 1/time a 1/(mass∙time) 2 c mass/mass 3 K mass

There’s no single correct way of going about doing this, but using the units for guidance and trying to be smart we can simplify the structure of our model. For example, setting X=K will remove that term from the prey equation (notice that this way X has the same unit as our original x variable).

The choice of Y is not very obvious so let’s look at T first. We could go with both T=1/b or T=1/d. Unit-wise they both work but one would serve to eliminate a parameter from the first equation and the other from the second. The decision here depends on what dynamics we’re most interested in, so for the purposes of demonstration here, let’s go with T=1/b.

We’re now left with defining Y, which only appears in the second term of the first equation. Looking at that term, the obvious substitution is Y=b/a, resulting in this set of equations:

Our system of equations is still not dimensionless, as we still have the model parameters to worry about. We can now define aggregate parameters using the original parameters in such a way that they will not carry any units and they will further simplify our model.

By setting p1=caK/b and p2=d/b we can transform our system to:a system of equations with no units and just two parameters.

1 Prey and predator abundance don’t have to necessarily be measured using mass units, it could be volume, density or something else. The units for parameters a, c, K would change equivalently and the rescaling still holds.

2 This is the death rate per encounter with predator per time t.

3 This is the converted predator (mass) per prey (mass) consumed.

# Open exploration with the Exploratory Modelling Workbench

In this blog, I will continue to showcase the functionality of the exploratory modelling workbench. In the previous blog, I have given a general introduction to the workbench, and showed how the Direct Policy Search example that comes with Rhodium can be adapted for use with the workbench. In this blog post, I will showcase how the workbench can be used for open exploration.

## first a short background

In exploratory modeling, we are interested in understanding how regions in the uncertainty space and/or the decision space map to the whole outcome space, or partitions thereof. There are two general approaches for investigating this mapping. The first one is through systematic sampling of the uncertainty or decision space. This is sometimes also known as open exploration. The second one is to search through the space in a directed manner using some type of optimization approach. This is sometimes also known as directed search.

The workbench support both open exploration and directed search. Both can be applied to investigate the mapping of the uncertainty space and/or the decision space to the outcome space. In most applications, search is used for finding promising mappings from the decision space to the outcome space, while exploration is used to stress test these mappings under a whole range of possible resolutions to the various uncertainties. This need not be the case however. Optimization can be used to discover the worst possible scenario, while sampling can be used to get insight into the sensitivity of outcomes to the various decision levers.

## open exploration

To showcase the open exploration functionality, let’s start with a basic example using the DPS lake problem also used in the previous blog post. We are going to simultaneously sample over uncertainties and decision levers. We are going to generate 1000 scenarios and 5 policies, and see how they jointly affect the outcomes. A scenario is understood as a point in the uncertainty space, while a policy is a point in the decision space. The combination of a scenario and a policy is called experiment. The uncertainty space is spanned by uncertainties, while the decision space is spanned by levers. Both uncertainties and levers are instances of RealParameter (a continuous range), IntegerParameter (a range of integers), or CategoricalParameter (an unorder set of things). By default, the workbench will use Latin Hypercube sampling for generating both the scenarios and the policies. Each policy will be always evaluated over all scenarios (i.e. a full factorial over scenarios and policies).

from ema_workbench import (RealParameter, ScalarOutcome, Constant,
ReplicatorModel)
model = ReplicatorModel('lakeproblem', function=lake_model)
model.replications = 150

#specify uncertainties
model.uncertainties = [RealParameter('b', 0.1, 0.45),
RealParameter('q', 2.0, 4.5),
RealParameter('mean', 0.01, 0.05),
RealParameter('stdev', 0.001, 0.005),
RealParameter('delta', 0.93, 0.99)]

# set levers
model.levers = [RealParameter(&quot;c1&quot;, -2, 2),
RealParameter(&quot;c2&quot;, -2, 2),
RealParameter(&quot;r1&quot;, 0, 2),
RealParameter(&quot;r2&quot;, 0, 2),
RealParameter(&quot;w1&quot;, 0, 1)]

def process_p(values):
values = np.asarray(values)
values = np.mean(values, axis=0)
return np.max(values)

#specify outcomes
model.outcomes = [ScalarOutcome('max_P', kind=ScalarOutcome.MINIMIZE,
function=process_p),
ScalarOutcome('utility', kind=ScalarOutcome.MAXIMIZE,
function=np.mean),
ScalarOutcome('inertia', kind=ScalarOutcome.MINIMIZE,
function=np.mean),
ScalarOutcome('reliability', kind=ScalarOutcome.MAXIMIZE,
function=np.mean)]

# override some of the defaults of the model
model.constants = [Constant('alpha', 0.41),
Constant('steps', 100)]



Next, we can perform experiments with this model.

from ema_workbench import (MultiprocessingEvaluator, ema_logging,
perform_experiments)
ema_logging.log_to_stderr(ema_logging.INFO)

with MultiprocessingEvaluator(model) as evaluator:
results = evaluator.perform_experiments(scenarios=1000, policies=5)



### Visual analysis

Having generated these results, the next step is to analyze them and see what we can learn from the results. The workbench comes with a variety of techniques for this analysis. A simple first step is to make a few quick visualizations of the results. The workbench has convenience functions for this, but it also possible to create your own visualizations using the scientific Python stack.

from ema_workbench.analysis import pairs_plotting
fig, axes = pairs_plotting.pairs_scatter(results, group_by='policy',
legend=False)
plt.show()



Writing your own visualizations requires a more in-depth understanding of how the results from the workbench are structured. perform_experiments returns a tuple. The first item is a numpy structured array where each row is a single experiment. The second item contains the outcomes, structured in a dict with the name of the outcome as key and a numpy array as value. Experiments and outcomes are aligned based on index.

import seaborn as sns

experiments, outcomes = results

df = pd.DataFrame.from_dict(outcomes)
df = df.assign(policy=experiments['policy'])

# rename the policies using numbers
df['policy'] = df['policy'].map({p:i for i, p in
enumerate(set(experiments['policy']))})

# use seaborn to plot the dataframe
grid = sns.pairplot(df, hue='policy', vars=outcomes.keys())
ax = plt.gca()
plt.show()



Often, it is convenient to separate the process of performing the experiments from the analysis. To make this possible, the workbench offers convenience functions for storing results to disc and loading them from disc. The workbench will store the results in a tarbal with .csv files and separate metadata files. This is a convenient format that has proven sufficient over the years.

from ema_workbench import save_results

save_results(results, '1000 scenarios 5 policies.tar.gz')

results = load_results('1000 scenarios 5 policies.tar.gz')



In addition to visual analysis, the workbench comes with a variety of techniques to perform a more in-depth analysis of the results. In addition, other analyses can simply be performed by utilizing the scientific python stack. The workbench comes with

• Scenario Discovery, a model driven approach to scenario development.
• Dimensional stacking, a quick visual approach drawing on feature scoring to enable scenario discovery. This approach has received limited attention in the literature (Suzuki et al., 2015). The implementation in the workbench replaces the rule mining approach with a feature scoring approach.
• Feature Scoring, a poor man’s alternative to global sensitivity analysis
• Regional sensitivity analysis

### Scenario Discovery

A detailed discussion on scenario discovery can be found in an earlier blogpost. For completeness, I provide a code snippet here. Compared to the previous blog post, there is one small change. The library mpld3 is currently not being maintained and broken on Python 3.5 and higher. To still utilize the interactive exploration of the trade offs within the notebook, use the interactive back-end as shown below.

from ema_workbench.analysis import prim

experiments, outcomes = results

x = experiments
y = outcomes['max_P'] &lt;0.8

prim_alg = prim.Prim(x, y, threshold=0.8)
box1 = prim_alg.find_box()


%matplotlib notebook

plt.show()



%matplotlib inline
# we go back to default not interactive

box1.inspect(43)
box1.inspect(43, style='graph')
plt.show()



### dimensional stacking

Dimensional stacking was suggested as a more visual approach to scenario discovery. It involves two steps: identifying the most important uncertainties that affect system behavior, and creating a pivot table using the most influential uncertainties. Creating the pivot table involves binning the uncertainties. More details can be found in Suzuki et al. (2015) or by looking through the code in the workbench. Compared to the original paper, I use feature scoring for determining the most influential uncertainties. The code is set up in a modular way so other approaches to global sensitivity analysis can easily be used as well if so desired.

from ema_workbench.analysis import dimensional_stacking

x = experiments
y = outcomes['max_P'] &lt;0.8

dimensional_stacking.create_pivot_plot(x,y, 2, nbins=3)
plt.show()



We can see from this visual that if B is low, while Q is high, we have a high concentration of cases where pollution stays below 0.8. The mean and delta have some limited additional influence. By playing around with an alternative number of bins, or different number of layers, patterns can be coarsened or refined.

### regional sensitivity analysis

A third approach for supporting scenario discovery is to perform a regional sensitivity analysis. The workbench implements a visual approach based on plotting the empirical CDF given a classification vector. Please look at section 3.4 in Pianosi et al (2016) for more details.

from ema_workbench.analysis import regional_sa
from numpy.lib import recfunctions as rf

x = rf.drop_fields(experiments, 'model', asrecarray=True)
y = outcomes['max_P'] &lt; 0.8

regional_sa.plot_cdfs(x,y)
plt.show()



### feature scoring

Feature scoring is a family of techniques often used in machine learning to identify the most relevant features to include in a model. This is similar to one of the use cases for global sensitivity analysis, namely factor prioritisation. In some of the work ongoing in Delft, we are comparing feature scoring with Sobol and Morris and the results are quite positive. The main advantage of feature scoring techniques is that they impose virtually no constraints on the experimental design, while they can handle real valued, integer valued, and categorical valued parameters. The workbench supports multiple techniques, the most useful of which generally is extra trees (Geurts et al. 2006).

For this example, we run feature scoring for each outcome of interest. We can also run it for a specific outcome if desired. Similarly, we can choose if we want to run in regression mode or classification mode. The later is applicable if the outcome is a categorical variable and the results should be interpreted similar to regional sensitivity analysis results. For more details, see the documentation.

from ema_workbench.analysis import feature_scoring

x = experiments
y = outcomes

fs = feature_scoring.get_feature_scores_all(x, y)
sns.heatmap(fs, cmap='viridis', annot=True)
plt.show()



From the results, we see that max_P is primarily influenced by b, while utility is driven by delta, for inertia and reliability the situation is a little bit less clear cut.

### linear regression

In addition to the prepackaged analyses that come with the workbench, it is also easy to rig up something quickly using the ever expanding scientific Python stack. Below is a quick example of performing a basic regression analysis on the results.

experiments, outcomes = results

for key, value in outcomes.items():
params = model.uncertainties #+ model.levers[:]

fig, axes = plt.subplots(ncols=len(params), sharey=True)

y = value

for i, param in enumerate(params):
ax = axes[i]
ax.set_xlabel(param.name)

pearson = sp.stats.pearsonr(experiments[param.name], y)

ax.annotate(&quot;r: {:6.3f}&quot;.format(pearson[0]), xy=(0.15, 0.85),
xycoords='axes fraction',fontsize=13)

x = experiments[param.name]
sns.regplot(x, y, ax=ax, ci=None, color='k',
scatter_kws={'alpha':0.2, 's':8, 'color':'gray'})

ax.set_xlim(param.lower_bound, param.upper_bound)

axes[0].set_ylabel(key)

plt.show()



The workbench can also be used for more advanced sampling techniques. To achieve this, it relies on SALib. On the workbench side, the only change is to specify the sampler we want to use. Next, we can use SALib directly to perform the analysis. To help with this, the workbench provides a convenience function for generating the problem dict which SALib provides. The example below focusses on performing SOBOL on the uncertainties, but we could do the exact same thing with the levers instead. The only changes required would be to set lever_sampling instead of uncertainty_sampling, and get the SALib problem dict based on the levers.

from SALib.analyze import sobol
from ema_workbench.em_framework.salib_samplers import get_SALib_problem

with MultiprocessingEvaluator(model) as evaluator:
sa_results = evaluator.perform_experiments(scenarios=1000,
uncertainty_sampling='sobol')

experiments, outcomes = sa_results
problem = get_SALib_problem(model.uncertainties)

Si = sobol.analyze(problem, outcomes['max_P'],
calc_second_order=True, print_to_console=False)

Si_filter = {k:Si[k] for k in ['ST','ST_conf','S1','S1_conf']}
Si_df = pd.DataFrame(Si_filter, index=problem['names'])


# Animations (2/2)

In the second part of this two-part post we’ll learn to use different tools and techniques to visualize and save animations (first part here). All the code discussed here is available on a GitHub repository, here.

This part focuses on the moviepy Python library, and all the neat things one can do with it. There actually are some nice tutorials for when we have a continuous function t -> f(t) to work with (see here). Instead, we are often working with data structures that are indexed on time in a discrete way.

Moviepy could be used from any data source dependent on time, including netCDF data such as the one manipulated by VisIt in the first part of this post. But in this second part, we are instead going to focus on how to draw time -dependent trajectories to make sense of nonlinear dynamical systems, then animate them in GIF. I will use the well-known shallow lake problem, and go through a first example with detailed explanation of the code. Then I’ll finish with a second example showing trajectories.

## Part I: using state trajectories to understand the concept of stable equilibria

The shallow lake problem is a classic problem in the management of coupled human and natural system. Some human (e.g. agriculture) produce phosphorus that eventually end up in water bodies such as lakes. Too much phosphorus in lake causes a processus called eutrophication which usually destroys lakes’ diverse ecosystems (no more fish) and lower water quality. A major problem with that is that eutrophication is difficult or even sometimes impossible to reverse: lowering phosphorus inputs to what they were pre-eutrophication simply won’t work. Simple nonlinear dynamics, first proposed by Carpenter et al. in 1999 (see here) describe the relationship between phosphorus inputs (L) and concentration (P). The first part of the code (uploaded to GitHub as movie1.py) reads:

 import attractors
import numpy as np
from moviepy.video.io.bindings import mplfig_to_npimage
from moviepy.video.VideoClip import DataVideoClip
import matplotlib.pyplot as plt
import matplotlib.lines as mlines

# Lake parameters
b = 0.65
q = 4

# One step dynamic (P increment rate)
# arguments are current state x and lake parameters b,q and input l
def Dynamics(x, b, q, l):
dp = (x ** q) / (1 + x ** q) - b * x + l
return dp 

Where the first 6 lines contain the usual library imports. Note that I am importing an auxiliary Python function “attractors” to enable me to plot the attractors (see attractors.pyon the GitHub repository). The function “Dynamics” correspond to the evolution of P given L and lake parameters b and q, also given in this bit of code. Then we introduce the time parameters:

# Time parameters
dt = 0.01 # time step
T = 40 # final horizon
nt = int(T/dt+1E-6) # number of time steps

To illustrate that lake phosphorus dynamics depend not only on the phosphorus inputs L but also on initial phosphorus levels, we are going to plot P trajectories for different constant values of L, and three cases regarding the initial P. We first introduce these initial phosphorus levels, and the different input levels, then declare the arrays in which we’ll store the different trajectories

# Initial phosphorus levels
pmin = 0
pmed = 1
pmax = 2.5

# Inputs levels
l = np.arange(0.001,0.401,0.005)

# Store trajectories
low_p = np.zeros([len(l),nt+1]) # Correspond to pmin
med_p = np.zeros([len(l),nt+1]) # Correspond to pmed
high_p = np.zeros([len(l),nt+1]) # Correspond to pmax 

Once that is done, we can use the attractor import to plot the equilibria of the lake problem. This is a bit of code that is the GitHub repository associated to this post, but that I am not going to comment on further here.

After that we can generate the trajectories for P with constant L, and store them to the appropriate arrays:

# Generating the data: trajectories
def trajectory(b,q,p0,l,dt,T):
# Declare outputs
time = np.arange(0,T+dt,dt)
traj = np.zeros(len(time))
# Initialize traj
traj[0] = p0
# Fill traj with values
for i in range(1,len(traj)):
traj[i] = traj[i-1] + dt * Dynamics(traj[i-1],b,q,l)
return traj
# Get them!
for i in range(len(l)):
low_p[i,:] = trajectory(b,q,pmin,l[i],dt,T)
med_p[i, :] = trajectory(b, q, pmed, l[i], dt, T)
high_p[i,:] = trajectory(b,q,pmax,l[i],dt,T)


Now we are getting to the interesting part of making the plots for the animation. We need to declare a figure that all the frames in our animation will use (we don’t want the axes to wobble around). For that we use matplotlib / pyplot libraries:

# Draw animated figure
fig, ax = plt.subplots(1)
ax.set_xlabel('Phosphorus inputs L')
ax.set_ylabel('Phosphorus concentration P')
ax.set_xlim(0,l[-1])
ax.set_ylim(0,pmax)
line_low, = ax.plot(l,low_p[:,0],'.', label='State, P(0)=0')
line_med, = ax.plot(l,med_p[:,0],'.', label='State, P(0)=1')
line_high, = ax.plot(l,high_p[:, 0], '.', label='State, P(0)=2.5')


Once that is done, the last things we need to do before calling the core moviepy functions are to 1) define the parameters that manage time, and 2) have a function that makes frames for the instant that is being called.

For 1), we need to be careful because we are juggling with different notions of time, a) time in the dynamics, b) the index of each instant in the dynamics (i.e., in the data, the arrays where we stored the trajectories), and c) time in the animation. We may also want to have a pause at the beginning or at the end of the GIF, rather than watch with tired eyes as the animation is ruthlessly starting again before we realized what the hell happened. So here is how I declared all of this:

# Parameters of the animation
initial_delay = 0.5 # in seconds, delay where image is fixed before the animation
final_delay = 0.5 # in seconds, time interval where image is fixed at end of animation
time_interval = 0.25 # interval of time between two snapshots in the dynamics (time unit or non-dimensional)
fps = 20 # number of frames per second on the GIF
# Translation in the data structure
data_interval = int(time_interval/dt) # interval between two snapshots in the data structure
t_initial = -initial_delay*fps*data_interval
t_final = final_delay*fps*data_interval
time = np.arange(t_initial,low_p.shape[1]+t_final,data_interval) # time in the data structure

Now for 2), the function to make the frames resets the parts of the plot that change for different time indexes(“t” below is the index in the data). If we don’t do that, the plot will keep the previous plotted elements, and will grow messier at the animation goes on.

# Making frames
def make_frame(t):
t = int(t)
if t<0:
return make_frame(0)
elif t>nt:
return make_frame(nt)
else:
line_low.set_ydata(low_p[:,t])
line_med.set_ydata(med_p[:,t])
line_high.set_ydata(high_p[:, t])
ax.set_title(' Lake attractors, and dynamics at t=' + str(int(t*dt)), loc='left', x=0.2)
if t > 0.25*nt:
alpha = (t-0.25*nt) / (1.5*nt)
lakeAttBase(eqList, 0.001, alpha=alpha)
plt.legend(handles=[stable, unstable], loc=2)
return mplfig_to_npimage(fig) 

In the above mplfig_to_npimage(fig) is a moviepy function that turns a figure into a frame of our GIF. Now we just have to call the function to do frames using the data, and to turn it into a GIF:

# Animating
animation = DataVideoClip(time,make_frame,fps=fps)
animation.write_gif("lake_attractors.gif",fps=fps)

Where the moviepy function DataVideoClip takes as arguments the sequences of indexes defined by the vector “time” defined in the parameters of the animation, the “make_frame” routine we defined, and the number of frame per second we want to output. The last lines integrates each frame to the GIF that is plotted below:

Each point on the plot represent a different world (different constant input level, different initial phosphorus concentration), and the animation shows how these states converge towards an stable equilibriun point. The nonlinear lake dynamics make the initial concentration important towards knowing if the final concentration is low (lower set of stable equilibria), or if the lake is in a eutrophic state (upper set of stable equilibria).

## Part II: plotting trajectories in the 2-D plane

Many trajectories can be plotted at the same time to understand the behavior of attractors, and visualize system dynamics for fixed human-controlled parameters (here, the phosphorus inputs L). Alternatively, if one changes the policy, trajectories evolve depending on both L and P. This redefines how trajectories are defined.

I did a similar bit of code to show how one could plot trajectories in the 2D plane. It is also uploaded on the GitHub repository (under movie2.py), and is similar in its structure to the code above. The definition of the trajectories and where to store them change. We define trajectories where inputs are lowered at a constant rate, with a minimum input of 0.08. For three different initial states, that gives us the following animation that illustrates how the system’s nonlinearity leads to very different trajectories even though the starting positions are close and the management policy, identical:

This could easily be extended to trajectories in higher dimensional planes, with and without sets of equilibria to guide our eyes.

# Animations (1/2)

In the first part of this two-part post we’ll learn to use different tools and techniques to visualize and save animations (second part here). This first part focuses on VisIt, an open-source visualization software developed by the Lawrence Livermore Laboratory – and many others (…that’s an advantage of being open-source). There are a lot of cool things about VisIt, and you can read about that here or here, and probably in future posts. But for today, we’ll just learn how to use VisIt to play and save animations. VisIt supports over a hundred file formats, and this post focuses on reading spatially distributed and time-varying data stored in the netCDF format (more about netCDF here).

Let us assume we have VisIt installed and the VisIt path has been added to our bash profile. Then we just need to cd into the directory where our data are stored, and type visit in the command line to launch the program. VisIt comes with two windows, one for manipulating files and the data stored in them, and another where plots are drawn (it is possible to draw more plots in additional windows, but in this tutorial we’ll stick to one window). Both windows are visible in the screenshot below.

It may be difficult to see, but the left-hand window is the space for managing files and plots. The active “Plot” is also displayed. Here the active file is “air.2m.2000.nc” where nc is th extension for netCDF files. The plot is a “Pseudocolor” of that data, the equivalent of “contourf” in Matlab or Python – Matplotlib. It is plotted in the right-hand window. It is the worldwide distribution of daily air temperature (in Kelvin) with a 1 degree resolution for January 1, 2000 on the left pane (Data from NCEP). Note the toolbar above the plot: it contains “play / stop” buttons that can allow us to play the 365 other days of year 2000 (the cursor is on “stop” in the screenshot). We are going to save the animation of the whole year with the VisIt movie wizard.

But first, let us change the specifications of our plot. In the left-hand window, we go to PlotAtts – > Pseudocolor (recall that is the plot type that we have). We can change the colorbar, set minima and maxima so that the colorbar does not change for each day of the year when we save the movie, and even make the scale skewed so we see more easily the differences in temperatures in non-polar regions where people live (on the image above the temperature in the middle of the colorbar is 254.9 K, which corresponds to -18.1C or roughly -1F). The screenshot below shows our preferences. We can click “Apply” for the changes to take effect, and exit the window.

Now we can save the movie. We go to File – > Save Movie and follow the VisIt Movie Wizard. You will find it very intuitive and easy. We save a “New Simple Movie” in DVIX (other formarts are available), with 20 frames (i.e. 20 days) per second. The resulting movie is produced within 2 minutes, but since WordPress does not support DIVX unless I am willing to pay for premium access, I had to convert the result as a GIF. To be honest, this is a huge letdown as the quality suffers compared to a video (unless I want to upload a 1GB monster). But the point of this post is that VisIt is really easy to use, not that you should convert its results into GIF.

In this animation it is easy to see the outline of land masses as they get either warmer or cooler than the surrounding oceans! Videos would look neater: you can try them for yourself! (at home, wherever you want).

It is actually possible to command VisIt using Python scripts, but I haven’t mastered that yet so it will be a tale for another post.

# An Introduction to Copulas

Modeling multivariate probability distributions can be difficult when the marginal probability density functions of the component random variables are different. Copulas are a useful tool to model dependence between random variables with any marginal distributions. This post will introduce the idea of a copula, run through the basic math that underlies its composition and discuss some common copulas in use. Through researching for this post I found several comprehensive coding examples in Matlab, Python and R, so instead of creating my own I’m focusing on the a theoretical introduction to copulas in this post and will link to the coding tutorials at the end.

### What is a copula?

The word copula is derived from the Latin noun for a link or a tie (as is the English word “couple”) and its purpose is to describe the dependence structure between two variables. Sklar’s theorem states that “Any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the two variables” [1].

To understand how a copula can describe the dependence function between random variables its helpful to first review some simple statistics.

The above statements can be summarized as saying that the values of the CDF of any marginal distribution are uniformly distributed on the interval [0,1] ie. if you make a random draw from any distribution, you have the same probability of drawing the largest value (U=1) of that distribution as the smallest possible value (U=0) or the median value (U=.5).

So what does this have to do with copulas? Great question, a copula is actually a joint distribution of the CDFs of the random variables it is modeling. Put formally:

A k dimensional copula is a function $c:[0,1]^k \rightarrow [0,1]$ and is a CDF with uniform marginals [1].

So now that we’ve defined what a copula is, lets take a look at the form of some commonly used ones.

### The Gaussian Copula

The Gaussian takes the form:

Where:

Φ_R is the joint standard normal CDF with:

ρ_(i,j) is the correlation between random variables X_i and X_j.

Φ^-1 is the inverse standard normal CDF.

It’s important to note that the Pearson Correlation coefficient is a bad choice for ρ row here, a rank based correlation such as Spearman’s ρ or Kendall’s τ are better options since they are scale invariant and do not require linearity.

### Issues with Tail Dependency

The Gaussian copula is a helpful tool and relatively easy to fit, even for relatively large numbers of RVs with different marginal distributions. Gaussian copulas however, do not do a good job capturing tail dependence and can cause one to underestimate risk of simultaneously being in the tails of each distribution. The failure of the Gaussian copula to capture tail dependence has been blamed for contributing the the 2008 financial crisis after it was widely used by investment firms on Wall Street (this is actually a really interesting story, for more details check out this article from the financial times) .

Tail dependency can be quantified by the coefficients of upper and lower tail dependence (λ_u and λ_l) defined as:

### The Student t Copula

Like the student t distribution, the student t copula has a similar shape to the Gaussian copula, but with fatter tails, thus it can do a slightly better job capturing tail dependence.

Where:

t_ν,Σ is the joint student t CDF, Σ is covariance matrix (again don’t use Pearson correlation coefficient), ν is the degrees of freedom and t^-1_ν is the inverse student t CDF.

### Archimedean Copulas

Archimedian copulas are a family of copulas with the following form:

ψ(u|θ) is called the generator function and θ is the parameters for the copula.

3 common Archimedean Copulas are:

• Gumbel: which is good at modeling upper tail dependence
• Clayton: which is good at modeling lower tail dependence
• Frank: has lighter tails and more density in the middle

It’s important to note that these copulas are usually employed for bivariate cases, for more than two variables, the Gaussian or Student t copulas are usually used.

A comparison of the shape of the copulas above can be found in Figure 1.

Image source: wikipedia.org

### Coding Copulas

There are numerous packages for modeling copulas in Matlab, Python and R.

In Matlab, the Statistics and Machine learning Toolbox has some helpful functions. You can find some well narrated examples of copulas here. There’s also the Multivariate Copula Analysis Toolbox from UC Irvine.

In Python, the copulalib package can be used to model the Clayton, Frank and Gumbel copulas. The statsmodels  package also has copulas built in. I found this post on the copulalib package, it has an attached Jupyter notebook with nice coding examples and figures. Here’s a post on statsmodels copula implementation, along with example Jupyter notebook.

Finally, here’s an example of coding copulas in R using the copulas library.

### References and Acknowledgments

1. Rüschendorf, L. (2013). Mathematical risk analysis dependence, risk bounds, optimal allocations and portfolios. Berlin: Springer.

I’d like to note that the majority of the content in the post came from Scott Steinschneider’s excellent course, BEE 6940: Multivariate Analysis, at Cornell.