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# The HSIC sensitivity indices: the Ishigami model¶

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

This example is a brief overview of the HSIC sensitivity indices classes and how to call them. HSIC estimators rely on a reproducing kernel of a Hilbert space. We can use them to compute sensitivity indices. We present the methods on the Ishigami function.

## Definition of the model¶

We load the model from the usecases module.

```
im = ishigami_function.IshigamiModel()
```

We generate an input sample of size 100 (and dimension 3).

```
size = 100
X = im.distributionX.getSample(size)
```

We compute the output by applying the Ishigami model to the input sample.

```
Y = im.model(X)
```

## Setting the covariance models¶

The HSIC algorithms use reproducing kernels defined on Hilbert spaces to estimate independence.
For each input variable we choose a covariance kernel.
Here we choose a `SquaredExponential`

kernel for all input variables.

They are all stored in a list of covariance kernels where is the number of input variables. The remaining one is for the output variable.

```
covarianceModelCollection = []
```

```
for i in range(3):
Xi = X.getMarginal(i)
inputCovariance = ot.SquaredExponential(1)
inputCovariance.setScale(Xi.computeStandardDeviation())
covarianceModelCollection.append(inputCovariance)
```

Likewise we define a covariance kernel associated to the output variable.

```
outputCovariance = ot.SquaredExponential(1)
outputCovariance.setScale(Y.computeStandardDeviation())
covarianceModelCollection.append(outputCovariance)
```

## The Global HSIC estimator¶

In this paragraph, we perform the analysis on the raw data: that is the global HSIC estimator.

### Choosing an estimator¶

After having defined the covariance kernels one has to select an appropriate estimator for the computations.

Two estimators are proposed:

an unbiased estimator through the

`HSICUStat`

classa biased, but asymptotically unbiased, estimator through the

`HSICVStat`

class

Beware that the conditional analysis used later cannot be performed with the unbiased estimator.

```
estimatorType = ot.HSICUStat()
```

We now build the HSIC estimator:

```
globHSIC = ot.HSICEstimatorGlobalSensitivity(
covarianceModelCollection, X, Y, estimatorType
)
```

We get the R2-HSIC indices:

```
R2HSICIndices = globHSIC.getR2HSICIndices()
print("\n Global HSIC analysis")
print("R2-HSIC Indices: ", R2HSICIndices)
```

```
Global HSIC analysis
R2-HSIC Indices: [0.249305,-0.00429972,0.0437032]
```

and the HSIC indices:

```
HSICIndices = globHSIC.getHSICIndices()
print("HSIC Indices: ", HSICIndices)
```

```
HSIC Indices: [0.0204961,-0.000366135,0.00366669]
```

The p-value by permutation.

```
pvperm = globHSIC.getPValuesPermutation()
print("p-value (permutation): ", pvperm)
```

```
p-value (permutation): [0,0.50495,0.00990099]
```

We have an asymptotic estimate of the value for this estimator.

```
pvas = globHSIC.getPValuesAsymptotic()
print("p-value (asymptotic): ", pvas)
```

```
p-value (asymptotic): [4.62161e-12,0.553716,0.0159241]
```

We vizualise the results.

```
graph1 = globHSIC.drawHSICIndices()
view1 = otv.View(graph1)
graph2 = globHSIC.drawPValuesAsymptotic()
view2 = otv.View(graph2)
graph3 = globHSIC.drawR2HSICIndices()
view3 = otv.View(graph3)
graph4 = globHSIC.drawPValuesPermutation()
view4 = otv.View(graph4)
```

## The Target HSIC estimator¶

We now perform the target analysis which consists in using a filter function over the output.

### Defining a filter function¶

We define a filter function on the output variable for the target analysis. In this example we use the function where is the distance to a well-chosen interval.

We first define a critical domain: in our case that is the interval.

```
criticalDomain = ot.Interval(5, float("inf"))
```

We have access to the distance to this domain thanks to the
`DistanceToDomainFunction`

class.

```
dist2criticalDomain = ot.DistanceToDomainFunction(criticalDomain)
```

We define the parameters in our function from the output sample

```
s = 0.1 * Y.computeStandardDeviation()[0]
```

We now define our filter function by composition of the parametrized function and the distance function.

```
f = ot.SymbolicFunction(["x", "s"], ["exp(-x/s)"])
phi = ot.ParametricFunction(f, [1], [s])
filterFunction = ot.ComposedFunction(phi, dist2criticalDomain)
```

We choose an unbiased estimator

```
estimatorType = ot.HSICUStat()
```

and build the HSIC estimator

```
targetHSIC = ot.HSICEstimatorTargetSensitivity(
covarianceModelCollection, X, Y, estimatorType, filterFunction
)
```

We get the R2-HSIC indices:

```
R2HSICIndices = targetHSIC.getR2HSICIndices()
print("\n Target HSIC analysis")
print("R2-HSIC Indices: ", R2HSICIndices)
```

```
Target HSIC analysis
R2-HSIC Indices: [0.260374,0.00207302,-0.00658141]
```

and the HSIC indices:

```
HSICIndices = targetHSIC.getHSICIndices()
print("HSIC Indices: ", HSICIndices)
```

```
HSIC Indices: [0.00108746,8.96771e-06,-2.80515e-05]
```

The p-value by permutation.

```
pvperm = targetHSIC.getPValuesPermutation()
print("p-value (permutation): ", pvperm)
```

```
p-value (permutation): [0,0.237624,0.693069]
```

We have an asymptotic estimate of the value for this estimator.

```
pvas = targetHSIC.getPValuesAsymptotic()
print("p-value (asymptotic): ", pvas)
```

```
p-value (asymptotic): [1.42697e-09,0.316344,0.59066]
```

We vizualise the results.

```
graph5 = targetHSIC.drawHSICIndices()
view5 = otv.View(graph5)
graph6 = targetHSIC.drawPValuesAsymptotic()
view6 = otv.View(graph6)
graph7 = targetHSIC.drawR2HSICIndices()
view7 = otv.View(graph7)
graph8 = targetHSIC.drawPValuesPermutation()
view8 = otv.View(graph8)
```

## The Conditional HSIC estimator¶

In this last section we preprocess the input variables: that is the conditional analysis. To do so, one has to work with a weight function. Here the weight function is the filter function we used previously.

```
weightFunction = filterFunction
```

We have to select a biased -but asymptotically unbiased- estimator

```
estimatorType = ot.HSICVStat()
```

We build the conditional HSIC estimator

```
condHSIC = ot.HSICEstimatorConditionalSensitivity(
covarianceModelCollection, X, Y, weightFunction
)
```

We get the R2-HSIC indices:

```
R2HSICIndices = condHSIC.getR2HSICIndices()
print("\n Conditional HSIC analysis")
print("R2-HSIC Indices: ", R2HSICIndices)
```

```
Conditional HSIC analysis
R2-HSIC Indices: [0.155438,0.017438,0.188395]
```

and the HSIC indices:

```
HSICIndices = condHSIC.getHSICIndices()
print("HSIC Indices: ", HSICIndices)
```

```
HSIC Indices: [0.00492342,0.000779929,0.00858311]
```

For the conditional estimator we only have access to the p-value by permutation:

```
pvperm = condHSIC.getPValuesPermutation()
print("p-value (permutation): ", pvperm)
print("")
```

```
p-value (permutation): [0.029703,0.712871,0]
```

We can vizualise the results thanks to the various drawing methods.

```
graph9 = condHSIC.drawHSICIndices()
view9 = otv.View(graph9)
graph10 = condHSIC.drawR2HSICIndices()
view10 = otv.View(graph10)
graph11 = condHSIC.drawPValuesPermutation()
view11 = otv.View(graph11)
```

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