Evaluate Probability Function

Updated: March 2, 2017

Fits a specified probability distribution function to a dataset

Category: Statistical Functions

You can use the Evaluate Probability Function module to calculate statistical measures that describe a column’s distribution, such as the Bernoulli, Pareto, or Poisson distributions.

When you evaluate your data against a probability distribution, you are mapping column values against a set of values with known properties. By knowing whether your data corresponds to one of these well-known distributions, you might be able infer other properties of your data. In general, you can get better predictions from a model when you can identify the distribution that fits the data best.

The question of which probability distribution function to use depends on the data and the variables that are being measured. For example, some distributions are designed to describe probabilities of discrete values; others are intended for use only with continuous numerical variables. For some distributions, you must also know in advance an expected mean, degrees of freedom, and so forth. For details, see Supported Probability Distributions.

For all probability distributions, you can compute the cumulative distribution function (cdf), the inverse cumulative distribution function (InverseCdf), or the probability density function (Pdf).

The Probability Function Evaluation module requires a dataset that contains at least one column of numerical values as input. It returns a data table that contains the values computed by the specified probability function. Optionally, you can also return the original values from the analyzed columns.

The options for this module change depending on the type of probability distribution you want to compute, and all other options will be reset if you change the probability distribution method. Therefore, be sure to choose the Distribution option first.

The module requires that you connect a dataset that contains numerical data types, and then select one or more columns to analyze, using a single probability distribution method.

If you want to analyze multiple columns with different methods, use a separate instance of the module for each probability distribution that you apply.

1. Add the Evaluate Probability Function module to your experiment. You can find this module in the Statistical Functions group in the experiment items list in Azure Machine Learning Studio.

2. Connect a dataset containing at least one column of numbers.

3. Use the Distribution option to select the kind of probability distribution that you want to calculate. See Supported Probability Distributions for a list of options and their required arguments.

4. Set parameters as required by the distribution.

5. Choose one of three statistics to create:

   • Cumulative distribution function (cdf)

     Returns the probability for a compound event, defined as the sum of ocurrences when the random variable takes a value smaller than some specific value x. In other words, it answers the question: "How common are samples that are less than or equal to this value?" This function can be used with both continuous and discrete numeric variables.

   • Inverse cumulative distribution function (InverseCdf)

     Returns the value associated with a specific cumulative probability (cdf). In other words, it answers the question: "What is the value of x at which the cdf function returns the cumulative probability y?"

   • Probability density function (pdf)

     Describes the relative likelihood for a random variable to be a specific value. In other words, it answers the question: "How common are samples at exactly this value?"

6. Use the Column Selector to choose the columns over which to compute the selected probability distribution.

   • All the columns you select must have a numerical data type.
   • The range of data in the column must also be valid, given the selected probability function. Otherwise, an error or NaN result may occur.
   • For sparse columns, any values that correspond to background zeros will not be processed.

7. Use the Result mode option to specify how to output the results. You can replace column values with the probability distribution values, append the new values to the dataset, or return only the probability distribution values.

8. Run the experiment, or right-click the Evaluate Probability Function module and click Run selected.

Results

For example, the following table represents partial results, using the Append option, on a single temperature column (from the Forest Fires sample dataset). Note that the headings of the generated columns contain the probability distribution that was used.

You can also chart a simple cumulative distribution and probability density of any numeric column, without using the Evaluate Probability Function module. This chart might be helpful in determining which probability distribution is likely to suit your data.

1. Right-click the dataset or module output, and select Visualize.
2. Select the column of interest, and in the Histogram pane, select cumulative distribution or probability density.
3. A chart of the distribution, like the following, is superimposed on the histogram representing the data.

By using the Evaluate Probability Function module, you can calculate the following distribution types:

Bernoulli

The Bernoulli distribution is a distribution over bits: in other words, a distribution over discrete values with only two possible values. The parameter p specifies the probability that a 1 is generated.

To calculate the Bernoulli distribution, you must set the following options:

• Probability of success
  Type a number (float) between 0.0 and 1.0 that specifies the probability of success. The default is .5.

Beta

The Beta distribution is a continuous univariate distribution.

To calculate the Beta distribution, you must set the following options:

• Shape
  Type a value to change the shape of the distribution.

  A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

  The value must be a number (double). The default is 1.0.

• Scale
  Type a number to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Upper bound
  Type a number (double) that represents the upper bound of the distribution. The default is 1.0.

• Lower bound
  Type a number (double) that represents the lower bound of the distribution. The default is 0.0.

Binomial

This is a discrete univariate distribution.

The binomial distribution is used to model the number of successes in a sample. Replacement is used when sampling. For sampling without replacement, use the Hypergeometric distribution.

To calculate the Binomial distribution, you must set the following options:

• Probability of success
  Type a number (float) between 0.0 and 1.0 that indicates the probability of success. The default is .5.

• Number of trials
  Specify the number of trials.

  Use an integer, with a minimum value of 1. The default is 3.

Cauchy

The Cauchy distribution is a symmetric continuous probability distribution.

To calculate the Cauchy distribution, you must set the following options:

• Location
  Type a number (double) that represents the location of the 0^th element.

  By specifying a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

ChiSquare

This distribution is a sum of the squares of k independent, standard, normal, random variables.

To calculate the ChiSquare distribution, you must set the following options:

• Number of degrees of freedom *
  Type a number (double) to specify the degrees of freedom. The default is 1.0.

ChiSquareRightTailed

To calculate the ChiSquareRightTailed distribution, you must set the following options:

• Number of degrees of freedom *
  Type a number (double) to specify the degrees of freedom. The default is 1.0.

Exponential

The exponential distribution is a distribution over the real numbers parameterized by one non-negative parameter.

To calculate the Exponential distribution, you must set the following options:

• Lambda
  Type a number (double) to use as the lambda parameter. The default is 1.0.

FFisher

The FFisher option generates the probability of the Fisher statistic for a sample, also known as the Fisher F-distribution. This distribution is two-tailed.

To calculate this distribution, you must set the following options:

• Numerator degrees of freedom
  Type a number (double) to specify the degrees of freedom that is used in the numerator. The default is 3.0.

• Denominator degrees of freedom
  Type a number (double) to specify the degrees of freedom that is used in the denominator. The default is 6.0.

FFisherRightTailed

With the FfisherRightTailed option, you can create a right-tailed Fisher distribution. The Fisher distribution is also known as the Fisher F-distribution, Snedecor distribution, or Fisher-Snedecor distribution. This particular form of the distribution is right-tailed.

To calculate the FFisherRightTailed distribution, you must set the following options:

• Numerator degrees of freedom
  Type a number (double) to specify the degrees of freedom that is used in the numerator. The default is 3.0.

• Denominator degrees of freedom
  Type a number (double) to specify the degrees of freedom that is used in the denominator. The default is 6.0.

Gamma

The gamma distribution is a family of continuous probability distributions with two parameters. For example, chi-squared is a special case of the gamma distribution.

To calculate the Gamma distribution, you must set the following options:

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Location
  Type a number (double) that represents the location of the 0^th element.

  By specifying a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

GeneralizedExtremeValues

The GeneralizedExtremeValues option lets you create a distribution developed to handle extreme values. The generalized extreme value (GEV) distribution is actually a group of continuous probability distributions that combines the Gumbel, Fréchet, and Weibull distributions (also known as type I, II, and III extreme value distributions).

For more information about extreme value theory, see this article in Wikipedia: Fisher-Tippet-Gnedenko theorem.

To calculate the GeneralizedExtremeValues distribution, you must set the following options:

• Shape
  Type a value to change the shape of the distribution.

  A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

  The value must be a number (double). The default is 1.0.

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Location
  Type a number (double) that represents the location of the 0^th element.

  By typing a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

Geometric

The Geometric distribution is a distribution over positive integers parameterized by one positive real number. This implementation of the Geometric distribution will never generate zeros.

To calculate the Geometric distribution, you must set the following options:

• Probability of success
  Type a number (float) between 0.0 and 1.0 that indicates the probability of success. The default is .5.

GumbelMax

The Gumbel distribution is one of several extreme value distributions. The GumbelMax option implements the Maximum Extreme Value Type 1 distribution.

To calculate the GumbelMax distribution, you must set the following options:

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Location
  Type a number (double) that represents the location of the 0^th element.

  By typing a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

GumbelMin

The Gumbel distribution is one of several extreme value distributions. The Gumbel distribution is also referred to as the Smallest Extreme Value (SEV) distribution or the Smallest Extreme Value (Type I) distribution. The GumbelMin option implements the Minimum Extreme Value Type 1 distribution.

To calculate the GumbelMin distribution, you must set the following options:

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Location
  Type a number (double) that represents the location of the 0^th element.

  By typing a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

Hypergeometric

This distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement, just as the binomial distribution describes the number of successes for draws with replacement.

To calculate the Hypergeometric distribution, you must set the following options:

• Number of samples
  Type an integer that indicates the number of samples to use. The default is 9.

• Number of success
  Type an integer that defines the value for success. The default is 24.

• Population size
  Specify the population size to use when estimating the hypergeometric distribution.

Laplace

The Laplace distribution is a distribution over the real numbers parameterized by a mean and scale parameter.

To calculate the Laplace distribution, you must set the following options:

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Location
  Type a number (double) that represents the location of the 0^th element.

  By typing a value for the Location parameter, you can shift the probability distribution up or down a numeric scale.

  The default is 0.0.

Logistic

The logistic distribution is similar to the normal distribution, but it has no limit on the left side of the distribution. The logistic distribution is used in logistic regression and neural network models and for modeling life sciences data.

To calculate the Logistic distribution, you must set the following options:

• Scale
  Type a value to use for scaling the distribution.

  By applying a scale value to the distribution, you can shrink or stretch it.

  The default value is 1.0. Values must be positive numbers.

• Mean
  Type a number (double)that indicates the estimated mean value of the distribution. The default is 0.0.

Lognormal

The lognormal distribution is a continuous univariate distribution.

To calculate the Lognormal distribution, you must set the following options:

• Mean
  Type a number (double) that indicates the estimated mean value of the distribution. The default is 0.0.

• Standard deviation
  Type a positive number (double) that indicates the estimated standard deviation of the distribution. The default is 1.0.

NegativeBinomial

The negative binomial distribution is a distribution over the natural numbers with two parameters (r, p). In the special case that r is an integer, you can interpret the distribution as the number of tails before the r^th head when the probability of the head is p.

To calculate the NegativeBinomial distribution, you must set the following options:

• Probability of success
  Type a number (float) between 0.0 and 1.0 that indicates the probability of success. The default is .5.

• Number of success
  Type an integer that specifies the value for success. The default is 24.

Normal

Also known as the Gaussian distribution.

To calculate the Normal distribution, you must set the following options:

• Mean
  Type a number (double) that indicates the estimated mean value of the distribution. The default is 0.0.

• Standard deviation
  Type a positive number (double) that indicates the estimated standard deviation of the distribution. The default is 1.0.

Pareto

The Pareto distribution is a power-law probability distribution that coincides with social, scientific, geophysical, actuarial, and many other types of observable phenomena.

To calculate the Pareto distribution, you must set the following options:

• Shape
  Type a value (optional) to change the shape of the distribution.

A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

The value must be a number (double). The default is 1.0.

• Scale
  Type a value (optional) to change the shape of the distribution.

  A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

  The value must be a number (double). The default is 1.0.

Poisson

For more information about the Poisson distribution, see Poisson Regression. In this implementation, Knuth's method is used to generate Poisson distributed random variables.

To calculate the Poisson distribution, you must set the following options:

• Mean
  Type a number (double) that indicates the estimated mean value of the distribution. The default is 0.0.

Rayleigh

The Rayleigh distribution is a continuous probability distribution. As an example of how it arises, the wind speed will have a Rayleigh distribution if the components of the two-dimensional wind velocity vector are uncorrelated and normally distributed with equal variance.

To calculate the Rayleigh distribution, you must set the following options:

• Lower bound
  Type a number (double) that represents the lower bound of the distribution. The default is 0.0.

StandardNormal

To calculate the StandardNormal distribution, all you need to do is select the columns.

TStudent

The TStudent option implements the univariate Student’s t-distribution.

To calculate the TStudent distribution, you must set the following options:

• Number of degrees of freedom
  Type a number (double) to specify the degrees of freedom. The default is 1.0.

TStudentRightTailed

Implements the univariate Student’s t-distribution by using one right tail.

To calculate the TStudentRightTailed distribution, you must set the following options:

• Number of degrees of freedom
  Type a number (double) to specify the degrees of freedom. The default is 1.0.

TStudentTwoTailed

Implements a two-tailed Student’s t-distribution.

To calculate the TStudentTwoTailed distribution, you must set the following options:

• Number of degrees of freedom
  Type a number (double) to specify the degrees of freedom. The default is 1.0.

Uniform

The uniform distribution is also known as the rectangular distribution.

To calculate the Uniform distribution, you must set the following options:

• Lower bound
  Type a number (double) that represents the lower limit of the distribution. The default is 0.0.

• Upper bound
  Type a number (double) that represents the upper limit of the distribution. The default is 1.0.

Weibull

The Weibull distribution is widely used in reliability engineering. It can use the Shape parameter to model many other distributions.

To calculate the Weibull distribution, you must set the following options:

• Shape
  Type a value (optional) to change the shape of the distribution.

  A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

  The value must be a number (double). The default is 1.0.

• Scale
  Type a value (optional) to change the shape of the distribution.

  A shape parameter is any parameter of a probability distribution that does not define its location or scale. Therefore, when you enter a value for shape, the parameter changes the shape of the distribution rather than moving, stretching, or shrinking it.

  The value must be a number (double). The default is 1.0.

This module supports all distributions that are provided in the open source MATH.NET Numerics library. For more information, see the documentation for the Math.Net.Numerics.Distribution library.

Right-tailed and two-tailed distributions appear as separate distributions, not as parameterized versions of base distributions. The current behavior is to preserve compatibility with Excel.

For a complete list of error messages, see Module Error Codes.

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