# Evaluate the mean of a random vector by simulations¶

## Abstract¶

We introduce the `ExpectationSimulationAlgorithm` class which implements an incremental Monte Carlo sampling algorithm to estimate the mean of a random vector.

```import openturns as ot
import openturns.viewer as otv
from matplotlib import pylab as plt
```

We shall use this algorithm for the Ishigami function that we load from the usecases module :

```from openturns.usecases import ishigami_function

im = ishigami_function.IshigamiModel()
```

The Ishigami model and the distribution of the input variables are stored in the im object :

```model = im.model
distribution = im.distributionX
```

We create a random vector that follows the distribution of the input variables.

```inputVector = ot.RandomVector(distribution)
```

The output vector is a `CompositeRandomVector`.

```outputVector = ot.CompositeRandomVector(model, inputVector)
```

The mean of the output vector is

```print("Mean of the output random vector : %.5f" % im.expectation)
```
```Mean of the output random vector : 3.50000
```

We define the algorithm simply by calling it with the output vector :

```algo = ot.ExpectationSimulationAlgorithm(outputVector)
```

We can also set the algorithm parameters :

```algo.setMaximumOuterSampling(80000)
algo.setBlockSize(1)
algo.setCoefficientOfVariationCriterionType("NONE")
```

We are then ready to launch the algorithm and store the result.

```algo.run()
result = algo.getResult()
```

As usual for Monte Carlo estimation we can draw the convergence history.

```graphConvergence = algo.drawExpectationConvergence()
view = otv.View(graphConvergence)
```

The result obtained with the previous algorithm is an instance of the :class:~openturns.ExpectationSimulationResult` class.

The expected value of the mean is given by the getExpectationEstimate method :

```expectation = result.getExpectationEstimate()
print("Estimated mean of the output random vector : %.5f" % expectation[0])
```
```Estimated mean of the output random vector : 3.50261
```

The variance and standard deviation of the estimated mean are respectively given by getVarianceEstimate and getStandardDeviation:

```expectationVariance = result.getVarianceEstimate()
print(
"Variance of the estimated mean of the output random vector : %.5f"
% expectationVariance[0]
)
standardDeviation = result.getStandardDeviation()
print("Standard deviation : %.5f" % standardDeviation[0])
```
```Variance of the estimated mean of the output random vector : 0.00017
Standard deviation : 0.01316
```

This variance and this standard deviation must not to be confused with the variance and the standard deviation of the Ishigami model!

```print("Ishigami variance : %.5f" % im.variance)
print("Ishigami standard deviation : %.5f" % im.variance ** (1 / 2))
```
```Ishigami variance : 13.84459
Ishigami standard deviation : 3.72083
```

The asymptotic confidence distribution of the output random vector mean estimate is

```expectationDistribution = result.getExpectationDistribution()
print(expectationDistribution)
```
```Normal(mu = 3.50261, sigma = 0.0131575)
```

Let us draw it:

```graphExpectationDistribution = expectationDistribution.drawPDF()
graphExpectationDistribution.setTitle(
"Normal asymptotic distribution of the mean estimate"
)
view = otv.View(graphExpectationDistribution)
```

Display all figures

```plt.show()
```