# Laplace distribution

Contents:

## The Laplace distribution

The Laplace distribution, named after the French mathematician Pierre Laplace, is a continuous probability distribution with a strikingly sharp peak. While it shares similarities with the more familiar normal distribution, it boasts a unique unimodal and symmetrical distribution pattern. This distribution is frequently used to model data with a higher peak than normal or to describe phenomena with heavy tails. Interestingly, it emerges as the difference of two independent random variables with identical exponential distributions [1].

## Laplace distribution properties

The Laplace is the result of two exponential distributions — one positive and one negative. Thus, it is sometimes called the double exponential distribution.

While the normal distribution is bell-shaped, the Laplace distribution has a strikingly sharp peak, which translates to a higher probability of occurrences around the central value. This makes it an excellent choice for situations where data has more variability around the median than the mean, such as in image processing, speech recognition, and signal analysis.

In addition, the Laplace has heavy tails, which means it has a higher probability of extreme values. This makes it ideal for modeling volatile data and financial returns, as it can capture sudden and drastic changes in these phenomena.

The probability density function (PDF) is given by

where μ represents a real-valued location parameter and b (>0) represents the scale parameter (also called the diversity). These two values control the shape of the distribution.

The classical univariate Laplace distribution has a location parameter of zero and scale parameter of one. The function simplifies to

f(x) = e-|x| / 2,

Where e-x is the exponential function.

The cumulative distribution function (CDF) is the integral of the PDF:

Where sgn is the sign function.

Other properties include [3]:

Many statistical software packages offer options for evaluating the Laplace distribution, including Maple and SPSS.

In SPSS, RV.LAPLACE(mean, scale) is a numeric function that returns a random value from a Laplace with specified mean and scale parameters [5].

## Uses of the Laplace distribution

The Laplace distribution emerges as the difference of two independent random variables with identical exponential distributions. This makes it an excellent choice for modeling data with additive noise or errors, such as image or signal data that may be distorted due to environmental conditions. The Laplace’s robustness to noise has made it increasingly popular in machine learning applications that require noise-robust regression analysis.

It’s also worth mentioning that the Laplace distribution has a natural connection to the L1 regularization, which is commonly used in statistical learning to improve model interpretation and reduce overfitting. The use of L1 regularization in conjunction with the Laplace distribution has been shown to improve model interpretability while maintaining high prediction accuracy on diverse datasets.

In conclusion, the Laplace distribution is a valuable continuous probability distribution with a unique and advantageous distribution pattern. Its heavy tails make it well-suited for modeling phenomena with sudden and drastic changes while its sharp peak is beneficial for modeling data with a higher peak than normal. The Laplace distribution’s emergence from the difference of exponential distributions makes it an excellent choice for modeling additive noise or error-prone data. It’s exciting to see how the distribution is increasingly being used in diverse settings, such as machine learning, regularization techniques, and statistical physics, to name a few. The Laplace distribution is a great tool to have in one’s statistical toolbox, and it’s worth exploring its properties and applications further.

## References

[1] Leemis, L. (n.d.). Exponential / Laplace. Retrieved January 10, 2018 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/ExponentialLaplace.pdf

[2] CC BY-SA 3.0 File:Laplace distribution pdf.png. MarkSweep via Wikimedia Commons.

[3] Härdle, W. & Simar, L. (2015). Applied Multivariate Statistical Analysis. Springer.

[4] Statistics[Distributions] Laplace, Laplace distribution. Retrieved April 18, 2021 from: https://www.maplesoft.com/support/help/Maple/view.aspx?path=Statistics/Distributions/Laplace

## The asymmetric Laplace distribution

An asymmetric Laplace distribution (ALD), also called a skew Laplace, is a generalization of the Laplace distribution. ALDs are skewed distributions, with a steep peak at the origin and heavier than normal tails. This family of distributions is commonly used for analysis of financial data.

## PDF and CDF

The asymmetric Laplace distribution is, simply speaking, a Laplace distribution with skew (hence its alternate name skew Laplace distribution). It has been introduced in the literature by many authors, many of whom use different notation (and sometimes, different formulas) to describe its properties.

One way to define the probability density function (PDF) for an asymmetric Laplace distribution is [1]

Where

• m = location parameter,
• λ > 0 = scale parameter,
• κ = asymmetry parameter. When κ = 1, the distribution becomes the (symmetric) Laplace distribution.

Yu and Zhang [2], proposed an alternate formula for the PDF which uses alternate notation and form:

Where

• s = scale parameter,
• α = skewness parameter
• I( ) = the indicator function (equals 1, when the condition is satisfied and 0 otherwise).

Note that many other forms do exist.

## Characteristic function

A random variable Y follow an asymmetric Laplace distribution if parameters θ ∈ R, µ ∈ R and σ ≥ 0 exist such that the characteristic function of Y has the form [3]

The characteristic function can be defined as

where κ > 0 is related to µ and σ by

while

Special cases include:

• If θ = µ = σ = 0, then Ψ(t) = 1 for every t ∈ ℝ, then it becomes a degenerate distribution at 0.
• If θ = σ = 0 and µ ≠ 0, it becomes an exponential distribution with mean µ (concentrated on (0, ∞)) for µ > 0 and on (−∞, 0) for µ < 0.
• For µ = 0 and σ ≠ 0, it becomes a symmetric Laplace distribution with mean θ and variance σ2 .

## References

[1] Kozubowski, Tomasz J.; Podgorski, Krzysztof (2000). “A Multivariate and Asymmetric Generalization of Laplace Distribution”. Computational Statistics. 15 (4): 531. doi:10.1007/PL00022717. S2CID 124839639. Retrieved 2015-12-29.

[2] Yu, K., Zhang, J., 2005. A three-parameter asymmetric laplace distribution and its extension. Communications in Statistics – Theory and Methods. 34, 1867–1879. https://doi.org/10.1080/03610920500199018

[3] Jing, H.; Liu, Y.; Zhao, J. Asymmetric Laplace Distribution Models for Financial Data: VaR and CVaR . Symmetry 2022, 14, 807. https://doi.org/10.3390/sym14040807

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