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Draw a survival function

# sphinx_gallery_thumbnail_number = 9

Introduction

The goal of this example is to show how to draw the survival function of a sample or a distribution, in linear and logarithmic scales.

Let X be a random variable with distribution function F:

F(x) = P(X\leq x)

for any x\in\mathbb{R}. The survival function S is:

S(x) = P(X>x) = 1 - P(X\leq x) = 1 - F(x)

for any x\in\mathbb{R}.

Let us assume that \{x_1,...,x_N\} is a sample from F.

Let \hat{F}_N be the empirical cumulative distribution function:

\hat{F}_N(x) = \frac{1}{N} \sum_{i=1}^N \mathbf{1}_{x_i\leq x}

for any x\in\mathbb{R}. Let \hat{S}_n be the empirical survival function:

\hat{S}_N(x) = \frac{1}{N} \sum_{i=1}^N \mathbf{1}_{x_i>x}

for any x\in\mathbb{R}.

Motivations for the survival function

For many probabilistic models associated with extreme events or lifetime models, the survival function has a simpler expression than the distribution function.

  • More specifically, several models (e.g. Pareto or Weibull) have a simple expression when we consider the logarithm of the survival function. In this situation, the (\log(x),\log(S(x))) plot is often used. For some distributions, this plot is a straight line.

  • When we consider probabilities very close to 1 (e.g. with extreme events), a loss of precision can occur when we consider the 1-F(x) expression with floating point numbers. This loss of significant digits is known as “catastrophic cancellation” in the bibliography and happens when two close floating point numbers are subtracted. This is one of the reasons why we sometimes use directly the survival function instead of the complementary of the distribution.

Define a distribution

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

ot.Log.Show(ot.Log.NONE)

sigma = 1.4
xi = 0.5
u = 0.1
distribution = ot.GeneralizedPareto(sigma, xi, u)

Draw the survival of a distribution

The computeCDF and computeSurvivalFunction computes the CDF F and survival S of a distribution.

p1 = distribution.computeCDF(10.0)
p1

0.9513919027838056

p2 = distribution.computeSurvivalFunction(10.0)
p2

0.048608097216194426

p1 + p2

1.0

The drawCDF and drawSurvivalFunction methods allows one to draw the functions F and S.

graph = distribution.drawCDF()
graph.setTitle("CDF of a distribution")
view = viewer.View(graph)
CDF of a distribution
graph = distribution.drawSurvivalFunction()
graph.setTitle("Survival function of a distribution")
view = viewer.View(graph)
Survival function of a distribution

In order to get finite bounds for the next graphics, we compute the xmin and xmax bounds from the 0.01 and 0.99 quantiles of the distributions.

xmin = distribution.computeQuantile(0.01)[0]
xmin

0.11410588272579382

xmax = distribution.computeQuantile(0.99)[0]
xmax

25.29999999999998

The drawSurvivalFunction methods also has an option to plot the survival with the X axis in logarithmic scale.

npoints = 50
logScaleX = True
graph = distribution.drawSurvivalFunction(xmin, xmax, npoints, logScaleX)
graph.setTitle("Survival function of a distribution where X axis is in log scale")
view = viewer.View(graph)
# graph
Survival function of a distribution where X axis is in log scale

In order to get both axes in logarithmic scale, we use the LOGXY option of the graph.

npoints = 50
logScaleX = True
graph = distribution.drawSurvivalFunction(xmin, xmax, npoints, logScaleX)
graph.setLogScale(ot.GraphImplementation.LOGXY)
graph.setTitle(
    "Survival function of a distribution where X and Y axes are in log scale"
)
view = viewer.View(graph)
# graph
Survival function of a distribution where X and Y axes are in log scale

Compute the survival of a sample

We now generate a sample that we are going to analyze.

sample = distribution.getSample(1000)

sample.getMin(), sample.getMax()

(class=Point name=Unnamed dimension=1 values=[0.100135], class=Point name=Unnamed dimension=1 values=[45.8248])

The computeEmpiricalCDF method of a Sample computes the empirical CDF.

p1 = sample.computeEmpiricalCDF([10])
p1

0.95

Activating the second optional argument allows one to compute the empirical survival function.

p2 = sample.computeEmpiricalCDF([10], True)
p2

0.05

p1 + p2

1.0

Draw the survival of a sample

In order to draw the empirical functions of a Sample, we use the UserDefined class.

  • The drawCDF method plots the CDF.

  • The drawSurvivalFunction method plots the survival function.

userdefined = ot.UserDefined(sample)
graph = userdefined.drawCDF()
graph.setTitle("CDF of a sample")
view = viewer.View(graph)
# graph
CDF of a sample
graph = userdefined.drawSurvivalFunction()
graph.setTitle("Empirical survival function of a sample")
view = viewer.View(graph)
# graph
Empirical survival function of a sample

As previously, the drawSurvivalFunction method of a distribution has an option to set the X axis in logarithmic scale.

xmin = sample.getMin()[0]
xmax = sample.getMax()[0]
pointNumber = sample.getSize()
logScaleX = True
graph = userdefined.drawSurvivalFunction(xmin, xmax, pointNumber, logScaleX)
graph.setTitle("Empirical survival function of a sample; X axis in log-scale")
view = viewer.View(graph)
# graph
Empirical survival function of a sample; X axis in log-scale

We obviously have P(X>X_{max})=0, where X_{max} is the sample maximum. This prevents from using the sample maximum and have a logarithmic Y axis at the same time. This is why in the following example we restrict the interval where we draw the survival function.

xmin = sample.getMin()[0]
xmax = sample.getMax()[0] - 1  # To avoid log(0) because P(X>Xmax)=0
pointNumber = sample.getSize()
logScaleX = True
graph = userdefined.drawSurvivalFunction(xmin, xmax, pointNumber, logScaleX)
graph.setLogScale(ot.GraphImplementation.LOGXY)
graph.setTitle("Empirical survival function of a sample; X and Y axes in log-scale")
view = viewer.View(graph)
# graph
Empirical survival function of a sample; X and Y axes in log-scale

Compare the distribution and the sample with respect to the survival

In the final example, we compare the distribution and sample survival functions in the same graphics.

xmin = sample.getMin()[0]
xmax = sample.getMax()[0] - 1  # To avoid log(0) because P(X>Xmax)=0
npoints = 50
logScaleX = True
graph = userdefined.drawSurvivalFunction(xmin, xmax, pointNumber, logScaleX)
graph.setLogScale(ot.GraphImplementation.LOGXY)
graph.setColors(["blue"])
graph.setLegends(["Sample"])
graphDistribution = distribution.drawSurvivalFunction(xmin, xmax, npoints, logScaleX)
graphDistribution.setLegends(["GPD"])
graph.add(graphDistribution)
graph.setLegendPosition("upper right")
graph.setTitle("GPD against the sample - n=%d" % (sample.getSize()))
view = viewer.View(graph)
# graph
plt.show()
GPD against the sample - n=1000