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 a symmetrical distribution on either side of the mean. Non-Gaussian time series, on the other hand, can have a variety of shapes, such as skewed or peaked, and can have data points that are significantly further from the mean.

There are two main types of non-Gaussian time series: symmetric and asymmetric. Symmetric non-Gaussian time series have a shape that is symmetrical around the mean, but deviate from the normal distribution in other ways. Asymmetric non-Gaussian time series, on the other hand, have a shape that is not symmetrical around the mean.

Below are two examples of non-Gaussian time series, one symmetric and one asymmetric:

Symmetric non-Gaussian time series: Pareto distribution

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

For example, if we were studying the income distribution of a certain population, we might find that the majority of people have an income that falls within a certain range, with a small number of people having significantly higher or lower incomes. This would be modeled by a Pareto distribution, as 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 an asymmetric non-Gaussian time series that is characterized by a skewed shape, with a longer tail on one side and a shorter tail on the other. This distribution is often used to model data that is skewed in one direction, such as data that is skewed to the left or to the right.

For example, if we were studying the distribution of stock returns in a certain market, we might find that the majority of stock returns are positive, but there is a small number of negative returns that are significantly lower than the mean. This would be modeled by a skew normal distribution, as the probability of observing a negative return decreases as the return becomes further from the mean.

Both of these non-Gaussian time series have important implications for data analysis and modeling. The Pareto distribution, for example, can be used to model the distribution of wealth or income in a population, and can help policymakers understand the distribution of resources within a society. The skew normal distribution, on the other hand, can be used to model data that is skewed in one direction, such as stock returns, and can help investors understand the risks and potential returns of different investments.

It is important to note that non-Gaussian time series can be more difficult to model and analyze than Gaussian time series, as they do not follow the same patterns and rules. However, they can provide valuable insights into real-world data, and can be modeled using a variety of statistical techniques, such as maximum likelihood estimation or Bayesian modeling. In conclusion, non-Gaussian time series are an important tool for understanding and analyzing real-world data, and can provide valuable insights into a wide range of phenomena, from income distribution to stock returns.