# Draw minimum volume level set in 1D¶

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

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

## 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:

```0.1031356403794346
```

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()[0]
xmax = interval.getUpperBound()[0]
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:

```0.04667473141178892
```

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)
plt.show()
```

Total running time of the script: ( 0 minutes 0.502 seconds)

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