Note

Go to the end to download the full example code.

# Compute squared SRC indices confidence intervals¶

This example shows how to compute squared SRC indices confidence bounds with bootstrap.
First, we compute squared SRC indices and draw them.
Then we compute bootstrap confidence bounds using the `BootstrapExperiment`

class and draw them.

## Define the model¶

```
import openturns as ot
import openturns.viewer as otv
from openturns.usecases import flood_model
```

Load the flood model.

```
fm = flood_model.FloodModel()
distribution = fm.distribution
g = fm.model.getMarginal(1)
dim = distribution.getDimension()
```

See the distribution

```
distribution
```

See the model

```
g.getOutputDescription()
```

## Estimate the squared SRC indices¶

We produce a pair of input and output sample.

```
ot.RandomGenerator.SetSeed(0)
N = 100
X = distribution.getSample(N)
Y = g(X)
```

Compute squared SRC indices from the generated design.

```
importance_factors = ot.CorrelationAnalysis(X, Y).computeSquaredSRC()
print(importance_factors)
```

```
[0.32175,0.137302,0.15122,0.005747,0.00141732,0.000104678,0.0189879,0.138303]
```

Plot the squared SRC indices.

```
input_names = g.getInputDescription()
graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
importance_factors, input_names, "Importance factors"
)
graph.setYTitle("Squared SRC")
_ = otv.View(graph)
```

## Compute confidence intervals¶

We now compute bootstrap confidence intervals for the importance factors.
This is done with the `BootstrapExperiment`

class.

Create SRC bootstrap sample

```
bootstrap_size = 100
src_boot = ot.Sample(bootstrap_size, dim)
for i in range(bootstrap_size):
selection = ot.BootstrapExperiment.GenerateSelection(N, N)
X_boot = X[selection]
Y_boot = Y[selection]
src_boot[i, :] = ot.CorrelationAnalysis(X_boot, Y_boot).computeSquaredSRC()
```

Compute bootstrap quantiles

```
alpha = 0.05
src_lb = src_boot.computeQuantilePerComponent(alpha / 2.0)
src_ub = src_boot.computeQuantilePerComponent(1.0 - alpha / 2.0)
src_interval = ot.Interval(src_lb, src_ub)
print(src_interval)
```

```
[0.230359, 0.40736]
[0.072598, 0.255676]
[0.0938578, 0.221143]
[0.00165483, 0.0153625]
[5.263e-06, 0.00847213]
[1.78931e-07, 0.00297661]
[0.0095922, 0.0301949]
[0.0873361, 0.225831]
```

```
def draw_importance_factors_with_bounds(
importance_factors, input_names, alpha, importance_bounds
):
"""
Plot importance factors indices with confidence bounds of level 1 - alpha.
Parameters
----------
importance_factors : Point(dimension)
The importance factors.
input_names : list(str)
The names of the input variables.
alpha : float, in [0, 1]
The complementary confidence level.
importance_bounds : Interval(dimension)
The lower and upper bounds of the importance factors
Returns
-------
graph : Graph
The importance factors indices with lower and upper 1-alpha confidence intervals.
"""
dim = importance_factors.getDimension()
lb = importance_bounds.getLowerBound()
ub = importance_bounds.getUpperBound()
palette = ot.Drawable.BuildDefaultPalette(2)
graph = ot.SobolIndicesAlgorithm.DrawCorrelationCoefficients(
importance_factors, input_names, "Importance factors"
)
graph.setColors([palette[0], "black"])
graph.setYTitle("Squared SRC")
# Add confidence bounds
for i in range(dim):
curve = ot.Curve([1 + i, 1 + i], [lb[i], ub[i]])
curve.setLineWidth(2.0)
curve.setColor(palette[1])
graph.add(curve)
return graph
```

sphinx_gallery_thumbnail_number = 2

Plot the SRC indices mean and confidence intervals.

```
src_mean = src_boot.computeMean()
graph = draw_importance_factors_with_bounds(src_mean, input_names, alpha, src_interval)
graph.setTitle(f"Importance factors - CI {(1.0 - alpha) * 100:.2f}%")
_ = otv.View(graph)
```

We see that the variable must be significant, because the lower bound of the confidence interval does not cross the X axis. Furthermore, its bounds are significantly greater than the bounds of the other variables (although perhaps less significantly for the variable). Hence, it must be recognized that is the most important variable in this model, according to the linear regression model.

We see that the variable has an importance factor close to zero, taking into account the confidence bounds (which are very small in this case). Hence, the variable could be replaced by a constant without reducing the variance of the output much.

The variables , and are somewhat in-between these two extreme situations. We cannot state that one of them is of greater importance than the other, because the confidence bounds are of comparable magnitude. Looking only at the importance factors, we may wrongly conclude that has a greater impact than because the estimate of the importance factor for is strictly greater than the estimate for . But taking into account for the variability of the estimators, this conclusion has no foundation, since confidence limits are comparable. In order to distinguish between the impact of these two variables, a larger sample size is needed.

```
otv.View.ShowAll()
```