### Change-point analysis in nonstationary stochastic models

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White Noise Process : A white noise process is a serially uncorrelated stochastic process with a mean of zero and a constant and finite variance. Note that this implies that every white noise process is a weak stationary process. Very close to the definition of strong stationarity, N -th order stationarity demands the shift-invariance in time of the distribution of any n samples of the stochastic process, for all n up to order N.

### Background

Thus, the same condition is required:. Naturally, stationarity to a certain order N does not imply stationarity of any higher order but the inverse is true.

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An interesting thread in mathoverflow showcases both an example of a 1st order stationary process that is not 2nd order stationary, and an example for a 2nd order stationary process that is not 3rd order stationary. And similarly, having a finite second moment is a sufficient and necessary condition for a 2nd order stationary process to also be a weakly stationary process. The term first-order stationarity is sometimes used to describe a series that has means that never changes with time, but for which any other moment like variance can change.

Cyclostationarity is prominent in signal processing. A stochastic process is trend stationary if an underlying trend function solely of time can be removed, leaving a stationary process. In the presence of a shock a significant and rapid one-off change to the value of the series , trend-stationary processes are mean-reverting; i.

Intuitive extensions exist of all of the above types of stationarity for pairs of stochastic processes. Weak stationarity and N -th order stationarity can be extended in the same way the latter to M - N -th order joint stationarity. A weaker form of weak stationarity, prominent in geostatistical literature see [Myers ] and [Fischer et al. An important class of non-stationary processes are locally stationary LS processes.

Alternatively, [Dahlhaus, ] defines them informally as processes which locally at each time point are close to a stationary process but whose characteristics covariances, parameters, etc. A formal definition can be found in [Vogt, ], and [Dahlhaus, ] provides a rigorous review of the subject. LS processes are of importance because they somewhat bridge the gap between the thoroughly explored sub-class of parametric non-stationary processes see the following section and the uncharted waters of the wider family of non-parametric processes, in that they have received rigorous treatment and a corresponding set of analysis tools akin to those enjoyed by parametric processes.

A great online resource on the topic is the home page of Prof. Guy Nason , who names LS processes as his main research interest. The following typology figure, partial as it may be, can help understand the relations between the different notions of stationarity we just went over:. The definitions of stationarity presented so far have been non-parametric; i. The related concept of a difference stationarity and unit root processes, however, requires a brief introduction to stochastic process modeling.

The topic of stochastic modeling is also relevant insofar as various simple models can be used to create stochastic processes see figure 5. The forecasting of future values is a common task in the study of time series data. To make forecasts, some assumptions need to be made regarding the Data Generating Process DGP , the mechanism generating the data. These assumptions often take the form of an explicit model of the process, and are also often used when modeling stochastic processes for other tasks, such as anomaly detection or causal inference.

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## Change-Point Analysis in Nonstationary Stochastic Models

We will go over the three most common such models. This is a memory-based model, in the sense that each value is correlated with the p preceding values; an AR model with lag p is denoted with AR p. The vector autoregressive VAR model generalizes the univariate case of the AR model to the multivariate case; now each element of the vector x[t] of length k can be modeled as a linear function of all the elements of the past p vectors:.

Like for autoregressive models, a vector generalization, VMA, exists. With a basic understanding of common stochastic process models, we can now discuss the related concept of difference stationary processes and unit roots. This concept relies on the assumption that the stochastic process in question can be written as an autoregressive process of order p, denoted as AR p :.

## Bayesian change-point modeling with segmented ARMA model

We can write the same process as:. The part inside the parenthesis on the left is called the characteristic equation of the process.

We can consider the roots of this equation:. This means that the process can be transformed into a weakly-stationary process by applying a certain type of transformation to it, called differencing. Difference stationary processes have an order of integration , which is the number of times the differencing operator must be applied to it in order to achieve weak stationarity. A process that has to be differenced r times is said to be integrated of order r, denoted by I r.

A common sub-type of difference stationary process are processes integrated of order 1, also called unit root process.

## Dynamics of the quorum sensing switch: stochastic and non-stationary effects

The simplest example for such a process is the following autoregressive model:. Unit root processes, and difference stationary processes generally, are interesting because they are non-stationary processes that can be easily transformed into weakly stationary processes. As a result, while the term is not used interchangeably with non-stationarity, the questions regarding them sometimes are.

I thought it worth mentioning here, as sometime tests and procedures to check whether a process has a unit root a common example is the Dickey-Fuller test are mistakenly thought of as procedures for testing non-stationarity as a latter post in this series touches upon. Another definition of interest is a wider, and less parametric, sub-class of non-stationary processes, which can be referred to as semi-parametric unit root processes. The definition was introduced in [Davidson, ], but a concise overview of it can be found [Breitung, ]. If you are interested in the concept of stationarity, or have stumbled into the topic while working with time series data, then I hope you have found this post a good introduction to the subject.

Some references and useful links are found below. As I have mentioned, a latter post in this series provides a similar overview of methods of detection of non-stationarity , and another will provide the same for transformation of non-stationarity time series data. Also, please feel free to get in touch with me with any comments and thoughts on the post or the topic.

Sign in. Get started. Stationarity in time series analysis. A review of the concept and types of stationarity. Shay Palachy Follow. Why is stationarity important? A formal definition for stochastic processes Before introducing more formal notions for stationarity, a few precursory definitions are required. Stochastic Processes A common approach in the analysis of time series data is to consider the observed time series as part of a realization of a stochastic process.

Definitions of stationarity Having a basic definition of stochastic processes to build on, we can now introduce the concept of stationarity. Strong stationarity Strong stationarity requires the shift-invariance in time of the finite-dimensional distributions of a stochastic process. Parametric notions of non-stationarity The definitions of stationarity presented so far have been non-parametric; i.

Basic concepts in stochastic process modeling The forecasting of future values is a common task in the study of time series data. Difference stationary processes With a basic understanding of common stochastic process models, we can now discuss the related concept of difference stationary processes and unit roots.

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References Academic Literature [Boshnakov, ] G. Linear Algebra Appl. Nonparametric tests for unit roots and cointegration.

Journal of econometrics, 2 , — Costationarity of locally stationary time series. Journal of Time Series Econometrics , 2 2. Locally stationary processes.

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