Non-Gaussian time series

TL;DR

• Time series whose observations do not follow a normal (bell-shaped) distribution.
• Can be symmetric or asymmetric, exhibiting shapes such as heavy tails, skewness, or peakedness.
• Requires alternative modeling approaches because it violates Gaussian assumptions used in many standard methods.

Definition

Non-Gaussian time series refers to time series data that does not follow a normal or Gaussian distribution. A Gaussian distribution is characterized by a bell-shaped curve with most data points occurring around the mean and symmetry on either side; non-Gaussian series deviate from this pattern and can exhibit different shapes, including skewed, peaked, or heavy-tailed forms.

Explanation

Non-Gaussian time series can take a variety of shapes that differ from the symmetric bell curve of the normal distribution. They may be:

• Symmetric non-Gaussian: shaped symmetrically around the mean but differing from the normal distribution in other respects.
• Asymmetric non-Gaussian: not symmetric around the mean, showing skewness with a longer tail on one side.

Such series frequently contain data points that are notably farther from the mean than would be expected under a Gaussian model. These deviations affect modeling and inference and often require methods beyond those that assume normality.

Examples

Symmetric non-Gaussian time series: Pareto distribution

The Pareto distribution is presented as a symmetric non-Gaussian time series often used to model income or wealth distributions. It is described as having a long tail on one side, with a small number of data points significantly further from the mean than the majority. This is referred to as a “power law” distribution, where the probability of observing a data point decreases exponentially as the data point becomes further from the mean.

Example scenario from the source: studying the income distribution of a population where most incomes fall within a certain range but a small number are significantly higher or lower; such a pattern would be modeled by a Pareto distribution, because the probability of observing a high income decreases exponentially as the income becomes further from the mean.

Asymmetric non-Gaussian time series: Skew normal distribution

The skew normal distribution is presented as an asymmetric non-Gaussian time series characterized by a skewed shape with a longer tail on one side and a shorter tail on the other. It is used to model data skewed in one direction.

Example scenario from the source: studying the distribution of stock returns in a market where the majority of returns are positive but a small number of negative returns are significantly lower than the mean; this pattern would be modeled by a skew normal distribution, since the probability of observing a negative return decreases as the return becomes further from the mean.

Use cases

• Modeling income and wealth distributions (Pareto example).
• Modeling skewed financial returns such as stock returns (skew normal example).
• Informing policymakers about resource distribution and helping investors assess risks and potential returns.

Notes or pitfalls

• Non-Gaussian time series can be more difficult to model and analyze than Gaussian time series because they do not follow the same patterns and rules as the normal distribution.
• They can be modeled using a variety of statistical techniques mentioned in the source, such as maximum likelihood estimation or Bayesian modeling.

Related terms

• Gaussian distribution (normal distribution)
• Symmetric non-Gaussian
• Asymmetric non-Gaussian
• Pareto distribution
• Power law
• Skew normal distribution
• Maximum likelihood estimation
• Bayesian modeling