Laplace Distribution

The Laplace distribution, also known as the double exponential distribution, is a continuous probability distribution that describes phenomena with heavier tails than the normal distribution. It is characterized by its distinctive peaked center and symmetric exponential decay, making it particularly useful in modeling data with more extreme values than would be expected in a normal distribution.

The Laplace distribution is defined by two parameters: the location parameter $a$ ∈ ℝ and the scale parameter $b$ > 0. The location parameter $a$ represents the center and peak of the distribution, while the scale parameter $b$ determines its spread and tail behavior. For a random variable X, the probability density function (PDF) is:

This distribution is perfectly symmetric around $a$, with exponential tails that decay at a rate determined by $b$.

The cumulative distribution function (CDF) is:

• Signal processing uses the Laplace distribution for modeling noise in communication systems.

• Financial mathematics employs it for modeling asset returns with heavier tails than normal.

• Machine learning applications use it in Bayesian inference and regularization.

• Speech recognition systems use it for modeling audio features.

Notable properties include:

• The distribution is symmetric around its mean.

• The characteristic function is $\varphi (t)=e^{iat}/(1+b^2{t}^2)$.

• The distribution has heavier tails than the normal distribution but lighter than the Cauchy distribution.

• The difference of two independent exponential random variables follows a Laplace distribution.

• The distribution arises as the maximum entropy distribution when the mean absolute deviation from the median is specified.

• It is a special case of the asymmetric Laplace distribution with symmetry parameter equal to 1.

The Laplace distribution is implemented by the LaplaceDistribution class. It has three constructors. The first constructor takes two arguments. The first argument is the location parameter, and corresponds to the mode of the probability density function. The second argument is the scale parameter.

The following constructs the same Laplace distribution with location parameter 6.8 and scale parameter 4.1:

var laplace = new LaplaceDistribution(6.8, 4.1);

Dim laplace = New LaplaceDistribution(6.8, 4.1)

No code example is currently available or this language may not be supported.

let laplace = LaplaceDistribution(6.8, 4.1)

If a variable is assumed to have a Laplace distribution, then the parameter of the distribution can be estimated using the method of matching moments. The second and third constructors perform this calculation. The first parameter is a Vector<T> whose distribution is to be estimated. The optional second parameter is a EstimationMethod value that specifies the method to be used. The default is the method of matching moments.

Note that parameter estimation says nothing about how well the estimated distribution fits the variable's distribution. Use one of the goodness-of-fit tests to verify the appropriateness of the choice of distribution.

The LaplaceDistribution class has two specific properties, LocationParameter and ScaleParameter, which return the location parameter (mode) and scale parameter of the distribution.

LaplaceDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the location and scale parameters of the distribution.

var random = new Pcg32();
double sample = LaplaceDistribution.Sample(random, 6.8, 4.1);

Dim random = New Pcg32()
Dim sample = LaplaceDistribution.Sample(random, 6.8, 4.1)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = LaplaceDistribution.Sample(random, 6.8, 4.1)

The above example uses the Pcg32 to generate uniform random numbers.

For more information on the Laplace distribution, refer to the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 24. Wiley Series in Probability and Statistics.

• Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 26. Wiley Series in Probability and Statistics.

Other Resources