.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_probabilistic_modeling/distributions/plot_generate_by_inversion.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_probabilistic_modeling_distributions_plot_generate_by_inversion.py: Generate random variates by inverting the CDF ============================================= .. GENERATED FROM PYTHON SOURCE LINES 6-13 Abstract -------- In this example, we show how to generate random variates by inversion of the cumulated distribution function. In simple situations, this method is rarely used in practice because of problems with performance, statistical quality of the generated random variates and numerical accuracy of the generated numbers when we use floating point numbers. However, it is an interesting method to know about because it is a building block for other algorithms and can be used to visualize the distribution of the generated numbers. .. GENERATED FROM PYTHON SOURCE LINES 16-40 The WeibullMin distribution --------------------------- Let :math:`\beta>0` and :math:`\alpha>0` be two real parameters: :math:`\alpha` is a shape parameter and :math:`\beta` is a scale parameter. The cumulated distribution function of the WeibullMin distribution (also referred as `Weibull distribution`) is: .. math:: F(x) = 1 - \exp\left(-\frac{x-\gamma}{\beta}\right)^\alpha, for any :math:`x\geq \gamma`. For the sake of simplicity, we set :math:`\gamma=0` In some situations, this parameters are denoted by :math:`k=\beta` and :math:`\lambda=\alpha`. The inverse of the CDF is: .. math:: F^{-1}(x) = \beta \left(-\log(1-p)\right)^{\frac{1}{\alpha}} for any :math:`x\geq 0`. This is the *quantile* function, because it computes the quantile :math:`F^{-1}(x)` depending on an outcome :math:`x`. .. GENERATED FROM PYTHON SOURCE LINES 42-92 Loss of accuracy when the probability is close to 1 --------------------------------------------------- In practice, if the probability :math:`p` is *very* close to 1, then the complementary probability :math:`1-p` is close to zero. This can lead to a significant loss of accuracy when we evaluate the subtraction :math:`1-p` with floating point numbers because :math:`p` and 1 have lots of common digits. This is called a *loss of accuracy by catastrophic cancellation*, a problem which is common in extreme events. We can use the :math:`\textrm{expm1}` function, defined by the equation: .. math:: \textrm{expm1}(x) = \exp(x)-1, for any :math:`x\in\mathbb{R}`. This is *not* numerically equivalent to computing `exp` and then subtracting 1. Indeed, the `expm1` function is more accurate when its argument `x` is close to zero. The CDF is then: .. math:: F(x) = -\textrm{expm1} \left(\left(-\frac{x}{\beta}\right)^\alpha\right), for any :math:`x\geq 0`. For the quantile function, we can use the :math:`\textrm{log1p}` function, defined by the equation: .. math:: \textrm{log1p}(x) = \log(1+x) for any :math:`x>-1`. Therefore, the quantile function is: .. math:: x = \beta \left(-\textrm{log1p}(-p)\right)^{\frac{1}{\alpha}} for :math:`x\geq 0`. Note that for :math:`\gamma > 0`, the quantile function writes: .. math:: x = \gamma + \beta \left(-\textrm{log1p}(-p)\right)^{\frac{1}{\alpha}} with :math:`x\geq \gamma`. In the following, we will not use these robust equations and this issue will not be taken into account. .. GENERATED FROM PYTHON SOURCE LINES 94-96 Generate by inversion: histogram and density -------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 98-106 .. code-block:: Python import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt import numpy as np ot.Log.Show(ot.Log.NONE) .. GENERATED FROM PYTHON SOURCE LINES 107-111 The following function defines the quantile function of the `WeibullMin` distribution. (Of course, we could use the `computeQuantile` method of the `WeibullMin` class as well. This would create a simpler, but less interesting example: this is a trade off that we accept in order to better understand the algorithm.) .. GENERATED FROM PYTHON SOURCE LINES 114-123 .. code-block:: Python def weibullQ(argument): """ WeibullMin quantile function """ p, alpha, beta = argument quantile = beta * (-np.log1p(-p)) ** (1 / alpha) return [quantile] .. GENERATED FROM PYTHON SOURCE LINES 124-126 .. code-block:: Python quantileFunction = ot.PythonFunction(3, 1, weibullQ) .. GENERATED FROM PYTHON SOURCE LINES 127-128 We define the parameters of the Weibull distribution and create the parametric function. .. GENERATED FROM PYTHON SOURCE LINES 130-133 .. code-block:: Python alpha = 10.0 beta = 1.0 .. GENERATED FROM PYTHON SOURCE LINES 134-137 .. code-block:: Python quantile = ot.ParametricFunction(quantileFunction, [1, 2], [alpha, beta]) quantile .. raw:: html


.. GENERATED FROM PYTHON SOURCE LINES 138-139 In the library, the uniform distribution is by default over the :math:`[-1,1]` interval. To obtain a uniform distribution over :math:`[0,1]`, we need to set the bounds explicitly. .. GENERATED FROM PYTHON SOURCE LINES 141-143 .. code-block:: Python U = ot.Uniform(0.0, 1.0) .. GENERATED FROM PYTHON SOURCE LINES 144-145 Then we generate a sample of size 1000 from the uniform distribution. .. GENERATED FROM PYTHON SOURCE LINES 147-150 .. code-block:: Python n = 1000 uniformSample = U.getSample(n) .. GENERATED FROM PYTHON SOURCE LINES 151-152 To generate the numbers, we evaluate the quantile function on the uniform numbers. .. GENERATED FROM PYTHON SOURCE LINES 154-156 .. code-block:: Python weibullSample = quantile(uniformSample) .. GENERATED FROM PYTHON SOURCE LINES 157-158 In order to compare the results, we use the `WeibullMin` class (using the default value of the location parameter :math:`\gamma=0`). .. GENERATED FROM PYTHON SOURCE LINES 160-162 .. code-block:: Python W = ot.WeibullMin(beta, alpha) .. GENERATED FROM PYTHON SOURCE LINES 163-172 .. code-block:: Python histo = ot.HistogramFactory().build(weibullSample).drawPDF() histo.setTitle("Weibull alpha=%s, beta=%s, n=%d" % (alpha, beta, n)) histo.setLegends(["Sample"]) wpdf = W.drawPDF() wpdf.setColors(["blue"]) wpdf.setLegends(["Weibull"]) histo.add(wpdf) view = viewer.View(histo) .. image-sg:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_generate_by_inversion_001.png :alt: Weibull alpha=10.0, beta=1.0, n=1000 :srcset: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_generate_by_inversion_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 173-174 We see that the empirical histogram of the generated outcomes is close to the exact density of the Weibull distribution. .. GENERATED FROM PYTHON SOURCE LINES 176-178 Visualization of the quantiles ------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 180-181 We now want to understand the details of the algorithm. To do this, we need to compare the distribution of the uniform numbers with the distribution of the generated points. .. GENERATED FROM PYTHON SOURCE LINES 183-186 .. code-block:: Python n = 50 uniformSample = U.getSample(n) .. GENERATED FROM PYTHON SOURCE LINES 187-189 .. code-block:: Python weibullSample = quantile(uniformSample) .. GENERATED FROM PYTHON SOURCE LINES 190-191 We sort the sample by increasing order. .. GENERATED FROM PYTHON SOURCE LINES 193-198 .. code-block:: Python data = ot.Sample(n, 2) data[:, 0] = weibullSample data[:, 1] = uniformSample data.setDescription(["x", "p"]) .. GENERATED FROM PYTHON SOURCE LINES 199-202 .. code-block:: Python sample = ot.Sample(data.sort()) sample[0:5, :] .. raw:: html
xp
00.72122130.0373631
10.74909530.05411842
20.81109280.1159369
30.83494750.1518191
40.84313010.166006


.. GENERATED FROM PYTHON SOURCE LINES 203-206 .. code-block:: Python weibullSample = sample[:, 0] uniformSample = sample[:, 1] .. GENERATED FROM PYTHON SOURCE LINES 207-224 .. code-block:: Python graph = ot.Graph("Weibull alpha=%s, beta=%s, n=%s" % (alpha, beta, n), "x", "U", True) # Add the CDF plot curve = W.drawCDF() curve.setColors(["blue"]) graph.add(curve) # Plot dashed horizontal & vertical lines for i in range(n): curve = ot.Curve( [0.0, weibullSample[i, 0], weibullSample[i, 0]], [uniformSample[i, 0], uniformSample[i, 0], 0.0], ) curve.setColor("red") curve.setLineStyle("dashed") graph.add(curve) view = viewer.View(graph) plt.show() .. image-sg:: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_generate_by_inversion_002.png :alt: Weibull alpha=10.0, beta=1.0, n=50 :srcset: /auto_probabilistic_modeling/distributions/images/sphx_glr_plot_generate_by_inversion_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 225-227 This graphics must be read from the U axis on the left to the blue curve (representing the CDF), and down to the X axis. We see that the horizontal lines on the U axis follow the uniform distribution. On the other hand, the vertical lines (on the X axis) follow the Weibull distribution. .. _sphx_glr_download_auto_probabilistic_modeling_distributions_plot_generate_by_inversion.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_generate_by_inversion.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_generate_by_inversion.py `