# Evaluating and visualizing sampling quality

State sampling is a necessary step for any computational experiment, and the way sampling is carried out will influence the experiment’s results. This is the case for instance, for sensitivity analysis (i.e., the analysis of model output sensitivity to values of the input variables). The popular method of Sobol’ (Sobol’, 2001) relies on tailor-made sampling techniques that have been perfected through time (e.g., Joe and Kuo, 2008; Saltelli et al., 2010). Likewise, the method of Morris (Morris, 1991), less computationally demanding than Sobol’s (Herman et al., 2013) and used for screening (i.e., understanding which are the inputs that most influence outputs), relies on specific sampling techniques (Morris, 1991; Campolongo et al., 2007).

But what makes a good sample, and how can we understand the strengths and weaknesses of the sampling techniques (and also of the associated sensitivity techniques we are using) through quick visualization of some associated metrics?

This post aims to answer this question. It will first look at what makes a good sample using some examples from a sampling technique called latin hypercube sampling. Then it will show some handy visualization tools for quickly testing and visualizing a sample.

# What makes a good sample?

Intuitively, the first criterion for a good sample is how well it covers the space from which to sample. The difficulty though, is how we define “how well” it practice, and the implications that has.

Let us take an example. A quick and popular way to generate a sample that covers the space fairly well is latin hypercube sampling (LHS; McKay et al., 1979). This algorithm relies on the following steps for drawing N samples from a hypercube-shaped of dimension p.:

1) Divide each dimension of the space in N equiprobabilistic bins. If we want uniform sampling, each bin will have the same length. Number bins from 1 to N each dimension.

2) Randomly draw points such that you have exactly one in each bin in each dimension.

For instance, for 6 points in 2 dimensions, this is a possible sample (points are selected randomly in each square labelled A to F):

It is easy to see that by definition, LHS has a good space coverage when projected on each individual axis. But space coverage in multiple dimensions all depends on the luck of the draw. Indeed, this is also a perfectly valid LHS configuration:

In the above configuration, it is easy to see that on top of poor space coverage, correlation between the sampled values along both axes is also a huge issue. For instance, if output values are hugely dependent on values of input 1, there will be large variations of the output values as values of input 2 change, regardless of the real impact of input 2 on the output.

Therefore, there are two kinds of issues to look at. One is correlation between sampled values of the input variables. We’ll look at it first because it is pretty straightforward. Then we’ll look at space coverage metrics, which are more numerous, do not look exactly at the same things, and can be sometimes conflicting. In fact, it is illuminating to see that sample quality metrics sometimes trade-off with one another, and several authors have turned to multi-objective optimization to come up with Pareto-optimal sample designs (e.g., Cioppa and Lucas, 2007; De Rainville et al., 2012).

One can look at authors such as Sheikholeslami and Razavi (2017) who summarize similar sets of variables. The goal there is not to write a summary of summaries but rather to give a sense that there is a relationship between which indicators of sampling quality matter, which sampling strategy to use, and what we want to do.

In what follows we note $x_{k,i}$ the kth  sampled value of input variable i, with $1\leq k \leq N$ and $1\leq i \leq p$.

### Correlation

Sample correlation is usually measured through the Pearson statistic. For inputs variables i and j among the p input variables, we note $x_{k,i}$ and $x_{k,j}$ the values of these variables i and j in sample k $(1\leq k \leq N)$ have:

$\rho_{ij} = \frac{\sum_{k=1}^N (x_{k,i}-\bar{x}_i)(x_{k,j}-\bar{x}_j)}{\sqrt{\sum_{k=1}^N (x_{k,i}-\bar{x}_i)^2 \sum_{k=1}^N (x_{k,j}-\bar{x}_j)^2}}$

In the above equation, ${\bar{x}_i}$  and ${\bar{x}_j}$ are the average sampled values of inputs i and j .

Then, the indicator of sample quality looks at the maximal level of correlation across all variables:

$\rho_{\max} = \max_{1\leq i \leq j \leq N} |\rho_{ij}|$

This definition relies on the remark that $\rho_{ij} = \rho_{ji}$.

### Space Coverage

There are different measures of space coverage.

We are best equipped to visualize space coverage via 1D or 2D projections of a sample. In 1D a measure of space coverage is by dividing each dimension in N equiprobable bins, and count the fraction of bins that have at least a point. Since N is the sample size, this measure is maximized when there is exactly one point in each bin — it is a measure that LHS maximizes.

Other measures of space coverage consider all dimension at once. A straightforward measure of space filling is the minimum Euclidean distance between two sampled points X in the generated ensemble:

$D = \min_{1\leq k \leq m \leq N} \left\{ d(\textbf{X}^k, \textbf{X}^m) \right\}$

Other indicators measure discrepancy which is a concept closely related to space coverage. In simple terms, a low discrepancy means that when we look at a subset of a sampled input space, its volume is roughly proportional to the number of points that are in it. In other words, there is no large subset with relatively few sampled points, and there is no small subset with a relatively large density of sampled points. A low discrepancy is desirable and in fact, Sobol’ sequences that form the basis of the Sobol’ sensitivity analysis method, are meant to minimize discrepancy.

# Sample visualization

The figures that follow can be easily reproduced by cloning a little repository SampleVis I put together, and by entering on the command line python readme.py &amp;&gt; output.txt. That Python routine can be used with both latin hypercube and Sobol’ sampling (using the SAlib sampling tool; SAlib is a Python library developed primarily by Jon Herman and Will Usher, and which is extensively discussed in this blog.)

In what follows I give examples using a random draw of latin hypercube sampling with 100 members and 7 sampled variables.

### Correlation

No luck, there is statistically significant pairwise correlation between in three pairs of variables: x1 and x4, x4 and x6, and x5 and x6. Using LHS, it can take some time to be lucky enough until the drawn sample is correlation-free (alternatively, methods to minimize correlations have been extensively researched over the years, though no “silver bullet” really emerges).

This means any inference that works for both variables in any of these pairs may be suspect. The SampleVis toolbox contains also tools to plot whether these correlations are positive or negative.

### Space coverage

The toolbox enables to plot several indicators of space coverage, assuming that the sampled space is the unit hypercube of dimension p (p=7 in this example). It computes discrepancy and minimal distance indicators. Ironically, my random LHS with 7 variables and 100 members has a better discrepancy (here I use an indicator called L2-star discrepancy) than a Sobol’ sequence with as many variables and members. The minimal Euclidean distance as well is better than for Sobol’ (0.330 vs. 0.348). This means that if for our experiment, space coverage is more important than correlation, the drawn LHS is pretty good.

To better grasp how well points cover the whole space, it is interesting to plot the distance of the point that is closest to each point, and to represent that in growing order:

This means that some points are not evenly spaced, and some are more isolated than others. When dealing with a limited number of variables, it can also be interesting to visualize 2D projections of the sample, like this one:

This again goes to show that the sample is pretty-well distributed in space. We can compare with the same diagram for a Sobol’ sampling with 100 members and 7 variables:

It is pretty clear that the deterministic nature of Sobol’ sampling, for so few points, leaves more systematic holes in the sampled space. Of course, this sample is too small for any serious Sobol’ sensitivity analysis, and holes are plugged by a larger sample. But again, this comparison is a visual heuristic that tells a similar story as the global coverage indicator: this LHS draw is pretty good when it comes to coverage.

# References

Campolongo, F., Cariboni, J. & Saltelli, A. (2007). An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software, 22, 1509 – 1518.

Cioppa, T. M. & Lucas, T. W. (2007). Efficient Nearly Orthogonal and Space-Filling Latin Hypercubes. Technometrics, 49, 45-55.

De Rainville, F.-M., Gagné, C., Teytaud, O. & Laurendeau, D. (2012). Evolutionary Optimization of Low-discrepancy Sequences. ACM Trans. Model. Comput. Simul., ACM, 22, 9:1-9:25.

Herman, J. D., Kollat, J. B., Reed, P. M. & Wagener, T. (2013). Technical Note: Method of Morris effectively reduces the computational demands of global sensitivity analysis for distributed watershed models. Hydrology and Earth System Sciences, 17, 2893-2903.

Joe, S. & Kuo, F. (2008). Constructing Sobol Sequences with Better Two-Dimensional Projections. SIAM Journal on Scientific Computing, 30, 2635-2654.

McKay, M.D., Beckman R.J. & Conover, W.J. (1979).A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics, 21(2), 239-245.

Morris, M. D. (1991). Factorial Sampling Plans for Preliminary Computational Experiments. Technometrics, 33, 161-174.

Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M. & Tarantola, S. (2010). Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181, 259 – 270.

Sheikholeslami, R. & Razavi, S. (2017). Progressive Latin Hypercube Sampling: An efficient approach for robust sampling-based analysis of environmental models. Environmental Modelling & Software, 93, 109 – 126.

Sobol’, I. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55, 271 – 280.

# Logistic Regression for Scenario Discovery

As most of you probably know, scenario discovery is an exploratory modeling approach [Bankes, 1993] that involves stress-testing proposed policies over plausible future “states of the world” (SOWs) to discover conditions under which those policies would fail to meet performance goals [Bryant and Lempert, 2010]. The scenario discovery process is therefore an exercise in statistical classification. Two commonly used methods used for the scenario discovery process are the Patient Rule Induction Method (PRIM; Friedman and Fisher [1999]) and Classification and Regression Trees (CART; Breiman et al. [1984]), both of which are included in the OpenMORDM R package and Rhodium Python package.

Another commonly used method in classification that hasn’t been given much attention in the scenario discovery literature is logistic regression. Logistic regression models estimate the probability that an event is classified as a success (1) as opposed to a failure (0) as a function of different covariates. This allows for the definition of “safe operating spaces,” or factor combinations leading to success, based on the probability with which one would like to be able to achieve the specified performance goal(s). We may not know the probability that a particular SOW will occur, but through the logistic regression we can estimate the probability of success in that SOW should it occur. The logistic regression can also identify which factors most influence a policy’s ability to meet those performance goals.

This blog post will illustrate how to build logistic regression models in Python for scenario discovery using the Red River basin as an example. Here we are interested in determining under what streamflow and demand characteristics reservoir operating policies are unable to protect Hanoi from the 100-yr flood. We assume operators want to ensure protection to this event with at least 95% reliability and use logistic regression to estimate under what combination of streamflow and demand characteristics they will be able to do so.

The form of the logistic regression model is given by Equation 1, where pi represents the probability that performance in the ith SOW is classified as a success and Xi represents a vector of covariates (in this case, streamflow and demand characteristics) describing the ith SOW:

1) $\ln\Bigg(\frac{p_i}{1-p_i}\Bigg) = \mathbf{X_i^\intercal}\mathbf{\beta}$.

The coefficients, $\mathbf{\beta}$, on the covariates are estimated using Maximum Likelihood Estimation.

To determine which streamflow and demand characteristics are most important in explaining successes and failures, we can compare the McFadden’s pseudo-R2 values associated with different models that include different covariates. McFadden’s pseudo-R2, $R_{McFadden}^2$, is given by Equation 2:

2) $R_{McFadden}^2 = 1 - \frac{\ln \hat{L}(M_{Full})}{\ln \hat{L}(M_{Intercept})}$

where $\ln \hat{L}(M_{Full})$ is the log-likelihood of the full model and $\ln \hat{L}(M_{Intercept})$ is the log-likelihood of the intercept model, i.e. a model with no covariates beyond the intercept. The intercept model therefore predicts the mean probability of success across all SOWs. $R_{McFadden}^2$ is a measure of improvement of the full model over the intercept model.

A common approach to fitting regression models is to add covariates one-by-one based on which most increase R2 (or in this case, $R_{McFadden}^2$), stopping once the increase of an additional covariate is marginal. The covariate that by itself most increases $R_{McFadden}^2$ is therefore the most important in predicting a policy’s success. To do this in Python, we will use the library statsmodels.

Imagine we have a pandas dataframe, dta that includes n columns of streamflow and demand characteristics describing different SOWs (rows) and a final column of 0s and 1s representing whether or not the policy being evaluated can provide protection to the 100-yr flood in that SOW (0 for no and 1 for yes). Assume the column of 0s and 1s is the last column and it is labeled Success. We can find the value of $R_{McFadden}^2$ for each covariate individually by running the following code:

import pandas as pd
import statsmodels.api as sm
from scipy import stats

# deal with fact that calling result.summary() in statsmodels.api
# calls scipy.stats.chisqprob, which no longer exists
stats.chisqprob = lambda chisq, df: stats.chi2.sf(chisq, df)

def fitLogit(dta, predictors):
# concatenate intercept column of 1s
dta['Intercept'] = np.ones(np.shape(dta)[0])

# get columns of predictors
cols = dta.columns.tolist()[-1:] + predictors

#fit logistic regression
logit = sm.Logit(dta['Success'], dta[cols])
result = logit.fit()

return result

n = len(dta.columns) - 1
for i in range(n):
predictors = dta.columns.tolist()[i:(i+1)]
result = fitLogit(dta, predictors)
print(result.summary())



A sample output for one predictor, Col1 is shown below. This predictor has a pseudo-R2 of 0.1138.

Once the most informative predictor has been determined, additional models can be tested by adding more predictors one-by-one as described above. Suppose that through this process, one finds that the first 3 columns of dta (Col1,Col2 and Col3) are the most informative for predicting success on providing protection to the 100-yr flood, while the subsequent columns provide little additional predictive power. We can use this model to visualize the probability of success as a function of these 3 factors using a contour map. If we want to show this as a 2D projection, the probability of success can only be shown for combinations of 2 of these factors. In this case, we can hold the third factor constant at some value, say its base value. This is illustrated in the code below, which also shows a scatter plot of the SOWs. The dots are shaded light blue if the policy succeeds in providing protection to the 100-yr flood in that world, and dark red if it does not.


import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import seaborn as sns
import pandas as pd
import statsmodels.api as sm

def fitLogit(dta, predictors):
# concatenate intercept column of 1s
dta['Intercept'] = np.ones(np.shape(dta)[0])

# get columns of predictors
cols = dta.columns.tolist()[-1:] + predictors

#fit logistic regression
logit = sm.Logit(dta['Success'], dta[cols])
result = logit.fit()

return result

def plotContourMap(ax, result, constant, dta, contour_cmap, dot_cmap, levels, xgrid, ygrid, \
xvar, yvar, base):

# find probability of success for x=xgrid, y=ygrid
X, Y = np.meshgrid(xgrid, ygrid)
x = X.flatten()
y = Y.flatten()
if constant == 'x3': # 3rd predictor held constant at base value
grid = np.column_stack([np.ones(len(x)),x,y,np.ones(len(x))*base[2]])
elif constant == 'x2': # 2nd predictor held constant at base value
grid = np.column_stack([np.ones(len(x)),x,np.ones(len(x))*base[1],y])
else: # 1st predictor held constant at base value
grid = np.column_stack([np.ones(len(x)),np.ones(len(x))*base[0],x,y])

z = result.predict(grid)
Z = np.reshape(z, np.shape(X))

contourset = ax.contourf(X, Y, Z, levels, cmap=contour_cmap)
ax.scatter(dta[xvar].values, dta[yvar].values, c=dta['Success'].values, edgecolor='none', cmap=dot_cmap)
ax.set_xlim(np.min(X),np.max(X))
ax.set_ylim(np.min(Y),np.max(Y))
ax.set_xlabel(xvar,fontsize=24)
ax.set_ylabel(yvar,fontsize=24)
ax.tick_params(axis='both',labelsize=18)

return contourset

# build logistic regression model with first 3 columns of predictors from dta
predictors = dta.columns.tolist()[0:3]
result = fitLogit(dta, predictors)

# define color map for dots representing SOWs in which the policy
# succeeds (light blue) and fails (dark red)
dot_cmap = mpl.colors.ListedColormap(np.array([[227,26,28],[166,206,227]])/255.0)

# define color map for probability contours
contour_cmap = mpl.cm.get_cmap(‘RdBu’)

# define probability contours
contour_levels = np.arange(0.0, 1.05,0.1)

# define grid of x (1st predictor), y (2nd predictor), and z (3rd predictor) dimensions
# to plot contour map over
xgrid = np.arange(-0.1,1.1,0.01)
ygrid = np.arange(-0.1,1.1,0.01)
zgrid = np.arange(-0.1,1.1,0.01)

# define base values of 3 predictors
base = [0.5, 0.5, 0.5]

fig = plt.figure()
# plot contour map when 3rd predictor ('x3') is held constant
plotContourMap(ax, result, 'x3', dta, contour_cmap, dot_cmap, contour_levels, xgrid, ygrid, \
'Col1', 'Col2', base)
# plot contour map when 2nd predictor ('x2') is held constant
contourset = plotContourMap(ax, result, 'x2', dta, contour_cmap, dot_cmap, contour_levels, xgrid, zgrid, \
'Col1', 'Col3', base)

cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7])
cbar = fig.colorbar(contourset, cax=cbar_ax)
cbar_ax.set_ylabel('Probability of Success',fontsize=20)
yticklabels = cbar.ax.get_yticklabels()
cbar.ax.set_yticklabels(yticklabels,fontsize=18)
fig.set_size_inches([14.5,8])
fig.savefig('Fig1.png')
fig.clf()



This produces the following figure:

We can also use the probability contours discovered above to define “safe operating spaces” as combinations of these 3 factors under which the evaluated policy is able to succeed in providing protection to the 100-yr flood with some reliability, say 95%. The hyperplane of factor combinations defining that 95% probability contour can be determined by setting p to 0.95 in Equation 2. Again, to plot 2-D projections of that hyperplane, the values of the other covariates can be held constant at their base values. The code below illustrates how to do this with a 95% boundary.


# define colormap for classifying boundary between failure and success
class_cmap = mpl.colors.ListedColormap(np.array([[251,154,153],[31,120,180]])/255.0)

# define probability cutoff between failure and success
class_levels = [0.0, 0.95, 1.0]

fig = plt.figure()
# plot contour map when 3rd predictor ('x3') is held constant
plotContourMap(ax, result, 'x3', dta, class_cmap, dot_cmap, class_levels, xgrid, ygrid, \
'Col1', 'Col2', base)

# plot contour map when 2nd predictor ('x2') is held constant
plotContourMap(ax, result, 'x2', dta, class_cmap, dot_cmap, class_levels, xgrid, zgrid, \
'Col1', 'Col3', base)

fig.set_size_inches([14.5,8])
fig.savefig('Fig2.png')
fig.clf()



This produces the following figure, where the light red region is the parameter ranges in which the policy cannot provide protection to the 100-yr flood with 95% reliability, and the dark blue region is the “safe operating space” in which it can.

All code for this example can be found here.

# 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')

from ema_workbench import load_results

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()



## More advanced sampling techniques

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'])


# Python’s template class

Many analyses require the same model to be run many times, but with different inputs.  For instance a Sobol sensitivity analysis requires thousands (or millions) of model runs corresponding to some strategic sampling of the factor space.  Depending on how complicated your model is, facilitating hundreds or thousands of runs may or may not be straightforward.  Some models require a unique configuration file, so performing a Sobol analysis is not as simple as changing a vector of numbers passed to an executable.

A very simple solution suggested by Jon Herman in an earlier post is to use Python string templates.  It is such a handy tool, I thought it deserved its own post.  We’ll use Python’s string module’s Template class.  The Template class has two methods: substitute and safe_substitute.  The difference between substitute and safe_substitute is that substitute will throw an exception if there is some problem filling the template, where as safe_substitute will not.  These two methods work essentially as standard $-based substitutions in Python, but rather than altering a single string, we can alter an entire document, and can then save it with a unique name. Let’s consider a simple example first where we modify a single string: from string import Template s = Template('$who is from $where') d = {} d['who'] = 'Bill' d['where'] = 'Boston' p = s.substitute(d)  Which returns the string Bill is from Boston…lucky Bill. Now we can get a bit fancier with lists of people and places: from string import Template s = Template('$who is from $where') people = ['Bill','Jim','Jack'] places = ['Boston','London','LA'] p = {} cnt = int(0) for person in people: for place in places: d = {} d['who'] = person d['where'] = place p[cnt] = s.substitute(d) cnt = cnt+1  Which returns a p as a dictionary of every combination of people and places: Bill is from Boston Bill is from London Bill is from LA Jim is from Boston Jim is from London Jim is from LA Jack is from Boston Jack is from London Jack is from LA Of course this is a silly example, but this sort of exercise proved really useful for some recent factorial experiments where we wanted to test a model performance for every combination of input factors (specified by filename strings). Getting a bit more complex, let’s consider a long configuration file needed to run your model. For example GCAM, an integrated assessment model I’ve previously discussed, uses a configuration xml file that’s about 100 lines long. We’ll consider a pared down version: <?xml version="1.0" encoding="UTF-8"?> <Configuration> <Files> <Value name="xmlInputFileName">../input/gcam-data-system/xml/modeltime-xml/modeltime.xml</Value> <Value name="BatchFileName">batch_ag.xml</Value> <Value name="policy-target-file">../input/policy/forcing_target_4p5.xml</Value> <Value name="xmldb-location">../output/database_basexdb</Value> </Files> <ScenarioComponents> <Value name = "climate">../input/climate/magicc.xml</Value> </ScenarioComponents> </Configuration>  Now, suppose we want to vary the cost of solar power inside the model over a number of levels, and we want each model run to print to a unique output directory. Our first step is to make a template xml file with a$-place holder where we want to vary the configuration file:

<?xml version="1.0" encoding="UTF-8"?>
<Configuration>
<Files>
<Value name="xmlInputFileName">../input/gcam-data-system/xml/modeltime-xml/modeltime.xml</Value>
<Value name="BatchFileName">batch_ag.xml</Value>
<Value name="policy-target-file">../input/policy/forcing_target_4p5.xml</Value>
<Value name="xmldb-location">../output/database_basexdb_$RN&</Value> </Files> <ScenarioComponents> <Value name = "climate">../input/climate/magicc.xml</Value> <!-- SOLAR -->$SOLAR
</ScenarioComponents>
</Configuration>


We can utilize the template xml file using Python’s template class as follows:

with open(template_name,'r') as T:
SOLAR_1 = ['<Value name="solar">../input/gcam-data-system/xml/energy-xml/solar_low.xml</Value>']
SOLAR_2 = ['']
SOLAR_3 = ['<Value name="solar">../input/gcam-data-system/xml/energy-xml/solar_adv.xml</Value>']
SOLAR = [SOLAR_1,SOLAR_2,SOLAR_3]
for i in range(3):
d = {}
d['SOLAR']=SOLAR[i]
d['RN']=str(i)
S1 = template.safe_substitute(d)
with open('./configuration_' + str(i) + '.xml','w') as f1:
f1.write(S1)


Here we are looping over experimental particles, defined by a unique setting of the solar power level in our experimental design.  For each particle a GCAM, the solar level and run number are substituted in (see S1), and S1 is written to a unique XML file.  If we open configuration_0.xml we get see that the substitution has worked!

<?xml version="1.0" encoding="UTF-8">
<Configuration>
<Files>
<Value name="xmlInputFileName">../input/gcam-data-system/xml/modeltime-xml/modeltime.xml</Value>
<Value name="BatchFileName">batch_ag.xml</Value>
<Value name="policy-target-file">../input/policy/forcing_target_4p5.xml</Value>
<Value name="xmldb-location">../output/database_basexdb_0</Value>
</Files>
<ScenarioComponents>
<Value name = "climate">../input/climate/magicc.xml</Value>
<!-- SOLAR -->
<Value name="solar">../input/gcam-data-system/xml/energy-xml/solar_low.xml</Value>
</ScenarioComponents>
</Configuration>


Of course this is a very simple example, but it has proven incredibly useful in our ongoing work.

That’s all for now!

# SALib v0.7.1: Group Sampling & Nonuniform Distributions

This post discusses the changes to the Python library SALib in version 0.7.1, with some examples of how to use the new capabilities. The two major additions in this version were: group sampling for Sobol’ sensitivity analysis and specifying nonuniform distributions for variables.

## Sobol’ Indices Group Sampling

Previous versions of SALib allowed one to calculate the first-order, total-order, and second-order indices for individual input parameters. These same indices can be defined for groups of input parameters (see Saltelli (2002) for more discussion). The main change is adding an item called ‘groups’  to the problem dictionary, which specifies the group of each parameter. Here is some example code. Notice in the ‘groups’  entry in the problem definition.


from SALib.sample import saltelli
from SALib.analyze import sobol
from SALib.util import read_param_file
import numpy as np

# example function
def sampleModel(x):
y = x[:,0]**1.5 + x[:,1] + 2*x[:,2] + x[:,3] + np.exp(0.3*x[:,4]) + x[:,5] \
+ 2*x[:,1]*x[:,4] + (x[:,0]*x[:,5])**2
return y

# problem definition
prob_gps_code = {
'num_vars':6,
'names': ['P1','P2','P3','P4','P5','P6'],
'bounds':[[0.0, 1.0], [2.0, 3.0], [0.5, 1.0], [0.0, 5.0], [-0.5, 0.5], [0.0, 1.0]],
'groups':['group1','group2','group1','group1','group2','group3']
}
# generating parameter values
param_vals_gps_code = saltelli.sample(prob_gps_code, 10000,calc_second_order=True)

# calculating model output values
Y_gps_code = sampleModel(param_vals_gps_code)

# completing Sobol' sensitivity analysis
Si_gps_code = sobol.analyze(prob_gps_code,Y_gps_code,calc_second_order=True,print_to_console=True)



The output from this code is given below. In this case the first-order indices (S1’s) are the index that is closed for the group. The S1 for group1 ( P1, P3, and P4) would be equivalent to summing:  S1 for P1, P3, and P4; S2 for (P1 & P3), (P1 & P4), and (P3 & P4); and S3 for (P1 & P3 & P4). All of the equations used in calculating the sensitivity indices are the same, but now they are for groups of variables.

Group S1 S1_conf ST ST_conf
group1 0.471121 0.081845 0.472901 0.011769
group2 0.498600 0.078950 0.497005 0.013081
group3 0.030502 0.019188 0.031736 0.001041

Group_1 Group_2 S2 S2_conf
group1 group2 0.000618 0.159951
group1 group3 0.002170 0.161403
group2 group3 -0.003324 0.155224

Note: You can also use the read_param_file() function to define the problem. For the above example the problem file would look like:

P1,0.0,1.0,group1
P2,2.0,3.0,group2
P3,0.5,1.0,group1
P4,0.0,5.0,group1
P5,-.5,.5,group2
P6,0.0,1.0,group3

## Nonuniform Distributions

Often the variables in a sensitivity analysis are assumed to be distributed uniformly over some interval. In the updated version of the SALib it is possible to specify whether the each input parameter is triangular, normal, lognormal, or uniform. Each of these distributions interprets the ‘bounds’ in the problem dictionary separately, as listed below.

• Triangular, “triang” (assumed lower bound of 0)
• first “bound” is width of distribution (scale, must be greater than 0)
• second “bound” is location of peak as a fraction of the scale (must be on [0,1])
• Normal, “norm”
• first “bound” is the mean (location)
• second “bound” is the standard deviation (scale, must be greater than 0)
• Lognormal, “lognorm” (natural logarithms, assumed lower bound of 0)
• first “bound” is the ln-space mean
• second “bound” is the ln-space standard deviation (must be greater than 0)
• Uniform, “unif”
• first “bound” is the lower bound
• second “bound” is the upper bound (must be greater than lower bound)

Triangular and lognormal distributions with a non-zero lower bound can be obtained by adding the lower bound to the generated parameters before sending the input data to be evaluated by the model.

Building on the same example as above, the problem dictionary and related analysis would be completed as follows.

# problem definition
prob_dists_code = {
'num_vars':6,
'names': ['P1','P2','P3','P4','P5','P6'],
'bounds':[[0.0,1.0], [1.0, 0.75], [0.0, 0.2], [0.0, 0.2], [-1.0,1.0], [1.0, 0.25]],
'dists':['unif','triang','norm','lognorm','unif','triang']
}

# generating parameter values
param_vals_dists_code = saltelli.sample(prob_dists_code, 10000,calc_second_order=True)

# calculating model output
Y_dists_code = sampleModel(param_vals_dists_code)

# complete Sobol' sensitivity analysis
Si_dists_code = sobol.analyze(prob_dists_code,Y_dists_code,calc_second_order=True,print_to_console=True)&lt;/pre&gt;


The output from this analysis is given below, which is consistent with the format in previous versions of SALib.

Parameter S1 S1_conf ST ST_conf
P1 0.106313 0.030983 0.110114 0.003531
P2 0.037785 0.027335 0.085197 0.003743
P3 0.128797 0.029834 0.128702 0.003905
P4 0.034284 0.016997 0.034193 0.001141
P5 0.579715 0.071509 0.627896 0.017935
P6 0.062791 0.021743 0.065357 0.002221
Parameter_1 Parameter_2 S2 S2_conf
P1 P2 0.001783 0.060174
P1 P3 0.001892 0.060389
P1 P4 0.001753 0.060155
P1 P5 0.001740 0.062130
P1 P6 0.004774 0.060436
P2 P3 -0.003539 0.051611
P2 P4 -0.003500 0.051186
P2 P5 0.044591 0.054837
P2 P6 -0.003585 0.051388
P3 P4 -0.000562 0.058972
P3 P5 -0.000533 0.059584
P3 P6 -0.000480 0.059923
P4 P5 -0.000364 0.034382
P4 P6 -0.000191 0.034301
P5 P6 -0.001293 0.137576

Note 1: You can also use the read_param_file() function to define the problem. The one catch is when you want to use nonuniform distributions without grouping the variables. In this case the fourth column in the input file (column for ‘groups’) must be the parameter name repeated from the first column. For the above example the problem file would look like:

P1,0.0,1.0,P1,unif
P2,1.0,.75,P2,triang
P3,0.0,.2,P3,norm
P4,0.0,.2,P4,lognorm
P5,-1.0,1.0,P5,unif
P6,1.0,.25,P6,triang

Note 2: If you are uncertain that the distribution transformation yielded the desired results, especially since the ‘bounds’ are interpreted differently by each distribution, you can check by plotting histograms of the data. The histograms of the data used in the example are shown below. (The data was actually saved to a .txt file for reference and then imported to R to plot these histograms, but matplotlib has a function histogram().)

## References

Saltelli, Andrea (2002). Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications 145(2):280-297. doi:10.1016/S0010-4655(02)00280-1

# Extensions of SALib for more complex sensitivity analyses

Over the past few weeks, I’ve had some helpful discussions with users of SALib that I thought would be worth sharing. These questions mostly deal with using the existing library in clever ways for more complicated modeling scenarios, but there is some extra information about library updates at the end.

1. How to sample parameters in log space

All three methods in the library (Sobol, Morris, and extended FAST) currently assume an independent uniform sampling of the parameters to be analyzed. (This is described in the documentation). However, lots of models have parameters that should be sampled in log space. This is especially true of environmental parameters, like hydraulic conductivity for groundwater models. In this case, uniform sampling over several orders of magnitude will introduce bias away from the smaller values.

One approach is instead to uniformly sample the exponent of the parameter. For example, if your parameter value ranges from [0.001, 1000], sample from [-3, 3]. Then transform the value back into real space after you read it into your model (and of course, before you do any calculations!) This way you can still use uniform sampling while ensuring fair representation in your parameter space.

2. How to sample discrete scenarios

In some sensitivity analysis applications, the uncertain factor you’re sampling isn’t a single value, but an entire scenario! This could be, for example, a realization of future streamflow or climate conditions—we would like to compare the sensitivity of some model output to streamflow and climate scenarios, without reducing the latter to a single value.

This can be done in SALib as follows. Say that you have an ensemble of 1,000 possible streamflow scenarios. Sample a uniform parameter on the range [0, 999]. Then, in your model, round it down to the nearest integer, and use it as an array index to access a particular scenario. This is the approach used in the “General Probabilistic Framework” described by Baroni and Tarantola (2014). Discretizing the input factor should not affect the Sobol and FAST methods. It will affect the Morris method, which uses the differences between input factors to determine elementary effects, so use with caution.

This approach was recently used by Matt Perry to analyze the impact of climate change scenarios on forest growth.

3. Dealing with model-specific configuration files

In Matt’s blog post (linked above), he mentioned an important issue: the space-separated columns of parameter samples generated by SALib may not be directly compatible with model input files. Many models, particularly those written in compiled languages, will have external configuration files (plaintext/XML/etc.) to specify their parameters. Currently SALib doesn’t have a solution for this—you’ll have to roll your own script to convert the parameter samples to the format of your model’s configuration file. (Update 11/16/14: here is an example of using Python to replace template variables with parameter samples from SALib).

One idea for how to do this in the future would be to have the user specify a “template file”, which is a configuration file where the parameter values are replaced with tags (for example, “{my_parameter}”. The location of this file could be specified as a command line parameter. Then, while generating parameter samples, SALib could make a copy of the template for each model run, overwriting the tags with parameter values along the way. The downside of this approach is that you would have thousands of input files instead of one. I’m going to hold off on this for now, but feel free to submit a pull request.

4. Confidence intervals for Morris and FAST

Previously, only the Sobol method printed confidence intervals for the sensitivity indices. These are generated by bootstrapping with subsets of the sample matrix. I updated the Morris method with a similar technique, where confidence intervals are bootstrapped by sampling subsets of the distribution of elementary effects.

For FAST (and extended FAST), there does not appear to be a clear way to get confidence intervals by bootstrapping. The original extended FAST paper by Saltelli et al. displayed confidence intervals on sensitivity indices, but these were developed by replicating the experiment, adding a random phase shift to generate alternate sequences of points as given in Section 2.2 of the linked paper. I added this random phase shift to SALib such that a different random seed will produce a different sampling sequence for FAST (previously this was not the case).

However, my attempts to bootstrap the FAST results were unsuccessful. The sequence of model outputs are FFT‘d to develop the sensitivity indices, which means that they cannot be sub-sampled or taken out of order. So for now, FAST does not provide confidence intervals. You can generate your own confidence intervals by replicating the full sensitivity analysis with different random seeds. This is usually very difficult for environmental models, given the computational expense, but not for test functions.

Thanks for reading. Email me at jdh366-at-cornell-dot-edu if you have any questions, or want to share a successful (or unsuccessful) application of SALib!

# Method of Morris (Elementary Effects) using SALib

This post was updated on January 16, 2015 to correct a few errors and update the SALib module structure, and again in 2017.

The Sensitivity Analysis Library (SALib) is an open-source Python library for common sensitivity analysis routines, including the Sobol, Morris, and FAST methods. In 2017 it was published in the Journal of Open Source Software:

Herman, J. and Usher, W. (2017) SALib: An open-source Python library for sensitivity analysis. Journal of Open Source Software, 2(9).

This post describes how to use the command-line interface of the library to run the Method of Morris (also known as the Elementary Effects Method). The Github page gives an example of the Python interface. The default use case for SALib is to perform decoupled sensitivity analysis, i.e. the sampling and analysis steps are performed separately, and the model evaluations that occur in between are left to the user.

Step 0: Get the library

The easiest approach is to pip install SALib, which will pull the latest version from the Python package index. Or, you can download at the above link as a zip/tar file and run python setup.py install. If you’re interested in contributing, you can clone the git repository:

git clone https://github.com/SALib/SALib .


Step 1: Choose sampling bounds for your parameters

Create a simple text file with the form [parameter] [lower bound] [upper bound]. For example, such a file for the “Sobol G” test function might look like this:

x1 0.0 1.0
x2 0.0 1.0
x3 0.0 1.0
x4 0.0 1.0
x5 0.0 1.0
x6 0.0 1.0
x7 0.0 1.0
x8 0.0 1.0


The bounds are used to sample parameter values. The variable names will only appear in the printed output, and they will not affect the method itself. Let’s call this file params.txt.

Step 2: Generate parameter sets

Put your params.txt file in the same directory as the SALib folder. Move to this directory and type the following command:

python -m SALib.sample.morris -n 1000 -p params.txt -o my_samples_file.txt


The -n flag specifies the number of trajectories to generate. The -p flag specifies the parameter bounds file that you created in the first step. Finally, the -o flag tells the program where to output the matrix of parameter samples. A total of N(p+1) samples will be generated; in this case, N = 1000 and p = 8, leading to a total of 9000 model evaluations.

The sampling command also has two options that aren’t required, --num-levels and --grid-jump. By default, these are set to 4 and 2, respectively.

Step 3: Run the parameter sets through your model

The parameter sets are now saved in my_samples_file.txt. Run these parameter sets through your model, and save the output(s) to a text file. The output file should contain one row of output values for each model run. This process is performed outside of SALib, so the details will be language-specific. Be careful to read in your parameter sets in the same order they were sampled.

Step 4: Calculate Morris Indices

You now have the output values for all of your model runs saved to a file. For the sake of example, let’s call that file SGOutput.txt (the output from the Sobol G function). We need to send this information back to SALib to compute the sensitivity indices, using following command:

python -m SALib.analyze.morris -p params.txt -X my_samples_file.txt -Y SGOutput.txt -c 0


The options here are: the parameter file (-p), the file containing calculated outputs (-Y), and the column of the objective values file to read (-c). The columns are assumed to be zero-indexed; if you have calculated multiple objective values, you would continue on to -m 1, etc., repeating the same command as above. By default, this will print sensitivity indices to the command line. You may want to print them to a file using the “>” operator.

Step 5: Interpret your results

Say that you saved the indices from the above command into the file morrisIndices.txt. If you open this file in a text editor, it will look something like this:

Parameter Mu Sigma Mu_Star
x1 0.040674 2.573977 2.077549
x2 -0.113902 1.514879 1.109394
x3 -0.025667 0.620538 0.454424
x4 -0.001532 0.324167 0.229770
x5 -0.001736 0.032333 0.023077
x6 -0.000858 0.032265 0.022693
x7 -0.000976 0.032779 0.022949
x8 -0.000224 0.034278 0.024575


The parameter names will match the ones you specified in params.txt. The mean and variance of each parameter’s elementary effects are given by mu and sigma, respectively. Mu_star is the mean of the absolute values of the elementary effects, following Campolongo et al. (2007). This Mu_star value is the best approximation of “total” sensitivity provided by the Morris method. Note that these indices do not have a direct interpretation as an “attribution of variance”, like we saw in the example results from the Sobol method. Instead, they should be used to understand the ranking of the most sensitive parameters, and to provide an approximate quantification of sensitivity.

For a full description of available methods and options, please consult the readme in the Github repository or on the SALib website. Github users can also post issues or submit pull requests. Thanks for reading!