# Draw minimum volume level sets¶

```import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt
ot.Log.Show(ot.Log.NONE)
```

## Draw minimum volume level set in 1D¶

In this paragraph, we compute the minimum volume level set of a univariate distribution.

### With a Normal, minimum volume LevelSet¶

```n = ot.Normal()
```
```graph = n.drawPDF()
view = viewer.View(graph)
``` We want to compute the minimum volume LevelSet which contains alpha`=90% of the distribution. The `threshold is the value of the PDF corresponding the alpha-probability: the points contained in the LevelSet have a PDF value lower or equal to this threshold.

```alpha = 0.9
levelSet, threshold = n.computeMinimumVolumeLevelSetWithThreshold(alpha)
threshold
```

Out:

```0.10313564037537128
```

The LevelSet has a contains method. Obviously, the point 0 is in the LevelSet.

```levelSet.contains([0.])
```

Out:

```True
```
```def computeSampleInLevelSet(distribution, levelSet, sampleSize = 1000):
"""
Generate a sample from given distribution.
Extract the sub-sample which is contained in the levelSet.
"""
sample = distribution.getSample(sampleSize)
dim = distribution.getDimension()
# Get the list of points in the LevelSet.
inLevelSet = []
for x in sample:
if levelSet.contains(x):
inLevelSet.append(x)
# Extract the sub-sample of the points in the LevelSet
numberOfPointsInLevelSet = len(inLevelSet)
inLevelSetSample = ot.Sample(numberOfPointsInLevelSet,dim)
for i in range(numberOfPointsInLevelSet):
inLevelSetSample[i] = inLevelSet[i]
return inLevelSetSample
```
```def from1Dto2Dsample(oldSample):
"""
Create a 2D sample from a 1D sample with zero ordinate (for the graph).
"""
size = oldSample.getSize()
newSample = ot.Sample(size,2)
for i in range(size):
newSample[i,0] = oldSample[i,0]
return newSample
```
```def drawLevelSet1D(distribution, levelSet, alpha, threshold, sampleSize = 100):
'''
Draw a 1D sample included in a given levelSet.
The sample is generated from the distribution.
'''
inLevelSample = computeSampleInLevelSet(distribution,levelSet,sampleSize)
cloudSample = from1Dto2Dsample(inLevelSample)
graph = distribution.drawPDF()
mycloud = ot.Cloud(cloudSample)
graph.setTitle("%.2f%% of the distribution, sample size = %d, " % (100*alpha, sampleSize))
return graph
```
```graph = drawLevelSet1D(n, levelSet, alpha, threshold)
view = viewer.View(graph)
``` ### With a Normal, minimum volume Interval¶

```interval = n.computeMinimumVolumeInterval(alpha)
interval
```

[-1.64485, 1.64485]

```def drawPDFAndInterval1D(distribution, interval, alpha):
'''
Draw the PDF of the distribution and the lower and upper bounds of an interval.
'''
xmin = interval.getLowerBound()
xmax = interval.getUpperBound()
graph = distribution.drawPDF()
yvalue = distribution.computePDF(xmin)
curve = ot.Curve([[xmin,0.],[xmin,yvalue],[xmax,yvalue],[xmax,0.]])
curve.setColor("black")
graph.setTitle("%.2f%% of the distribution, lower bound = %.3f, upper bound = %.3f" % (100*alpha, xmin,xmax))
return graph
```

The computeMinimumVolumeInterval returns an Interval.

```graph = drawPDFAndInterval1D(n, interval, alpha)
view = viewer.View(graph)
``` ### With a Mixture, minimum volume LevelSet¶

```m = ot.Mixture([ot.Normal(-5.,1.),ot.Normal(5.,1.)],[0.2,0.8])
```
```graph = m.drawPDF()
view = viewer.View(graph)
``` ```alpha = 0.9
levelSet, threshold = m.computeMinimumVolumeLevelSetWithThreshold(alpha)
threshold
```

Out:

```0.04667473141153258
```

The interesting point is that a LevelSet may be non-contiguous. In the current mixture example, this is not an interval.

```graph = drawLevelSet1D(m, levelSet, alpha, threshold, 1000)
view = viewer.View(graph)
``` ### With a Mixture, minimum volume Interval¶

```interval = m.computeMinimumVolumeInterval(alpha)
interval
```

[-5.44003, 6.72227]

The computeMinimumVolumeInterval returns an Interval. The bounds of this interval are different from the previous LevelSet.

```graph = drawPDFAndInterval1D(m, interval, alpha)
view = viewer.View(graph)
``` ## Draw minimum volume level set in 2D¶

In this paragraph, we compute the minimum volume level set of a bivariate distribution.

Create a gaussian

```corr = ot.CorrelationMatrix(2)
corr[0, 1] = 0.2
copula = ot.NormalCopula(corr)
x1 = ot.Normal(-1., 1)
x2 = ot.Normal(2, 1)
x_funk = ot.ComposedDistribution([x1, x2], copula)

# Create a second gaussian
x1 = ot.Normal(1.,1)
x2 = ot.Normal(-2,1)
x_punk = ot.ComposedDistribution([x1, x2], copula)

# Mix the distributions
mixture = ot.Mixture([x_funk, x_punk], [0.5,1.])
```
```graph = mixture.drawPDF()
view = viewer.View(graph)
``` For a multivariate distribution (with dimension greater than 1), the computeMinimumVolumeLevelSetWithThreshold uses Monte-Carlo sampling.

```ot.ResourceMap.SetAsUnsignedInteger("Distribution-MinimumVolumeLevelSetSamplingSize",1000)
```

We want to compute the minimum volume LevelSet which contains alpha`=90% of the distribution. The `threshold is the value of the PDF corresponding the alpha-probability: the points contained in the LevelSet have a PDF value lower or equal to this threshold.

```alpha = 0.9
levelSet, threshold = mixture.computeMinimumVolumeLevelSetWithThreshold(alpha)
threshold
```

Out:

```0.0076863340815168865
```
```def drawLevelSetContour2D(distribution, numberOfPointsInXAxis, alpha, threshold, sampleSize= 500):
'''
Compute the minimum volume LevelSet of measure equal to alpha and get the
corresponding density value (named threshold).
Generate a sample of the distribution and draw it.
Draw a contour plot for the distribution, where the PDF is equal to threshold.
'''
sample = distribution.getSample(sampleSize)
X1min = sample[:, 0].getMin()
X1max = sample[:, 0].getMax()
X2min = sample[:, 1].getMin()
X2max = sample[:, 1].getMax()
xx = ot.Box([numberOfPointsInXAxis],
ot.Interval([X1min], [X1max])).generate()
yy = ot.Box([numberOfPointsInXAxis],
ot.Interval([X2min], [X2max])).generate()
xy = ot.Box([numberOfPointsInXAxis, numberOfPointsInXAxis],
ot.Interval([X1min, X2min], [X1max, X2max])).generate()
data = distribution.computePDF(xy)
graph = ot.Graph('', 'X1', 'X2', True, 'topright')
labels = ["%.2f%%" % (100*alpha)]
contour = ot.Contour(xx, yy, data, [threshold], labels)
contour.setColor('black')
graph.setTitle("%.2f%% of the distribution, sample size = %d" % (100*alpha,sampleSize))
cloud = ot.Cloud(sample)
return graph
```

The following plot shows that 90% of the sample is contained in the LevelSet.

```numberOfPointsInXAxis = 50
graph = drawLevelSetContour2D(mixture, numberOfPointsInXAxis, alpha, threshold)
view = viewer.View(graph)
plt.show()
``` ```plt.show()
```

Total running time of the script: ( 0 minutes 0.861 seconds)

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