Simulate an ensemble of paths:

Simulate with given precision:

Simulate a first-order scalar process:

Sample paths for positive and negative values of the parameter:

Compare the serial dependence between consecutive values on scatter plots:

Simulate a weakly stationary process with given initial values:

For a process with a trend, initial values influence the behavior of the whole path:

Simulate a two-dimensional process:

Create a 2D sample path function from the data:

The color of the path is the function of time:

Create a 3D sample path function with time:

The color of the path is the function of time:

Simulate a three-dimensional process:

Create a sample path function from the data:

The color of the path is the function of time:

Estimate process parameters:

Compare the sample covariance functions with that of the estimated process:

Use TimeSeriesModel to automatically find orders:

Compare the sample covariance functions with the best time series model:

Find the maximum likelihood estimator:

Fix the constant and the variance and estimate the remaining parameters:

Plot the log-likelihood function together with the position of the estimated parameters:

Estimate a vector autoregressive process:

Compare covariance functions for each component:

Forecast future values:

Find the forecast for the next 10 steps:

Show the forecast path:

Plot the data and the forecasted values:

Find a forecast for a vector-valued time series process:

Find the forecast for the next 10 steps:

Plot the data and the forecast for each component:

For low order it is possible to find the closed form of the correlation function:

Partial correlation function is zero for lags larger than the process order:

Correlation matrix:

Covariance matrix:

Inverse of the covariance matrix of an ARProcess is symmetric multidiagonal:

Covariance function for a vector-valued process:

Power spectral density:

Vector ARMAProcess:

Check if a time series process is weakly stationary:

For a vector process:

Find conditions for a process to be weakly stationary:

Find conditions for higher order:

Variance is positive only for a weakly stationary process:

Define stationarity conditions:

Variance is positive, assuming weak stationarity:

Autoregressive process is always invertible:

The available methods for estimating an ARProcess:

Compare log-likelihoods:

Method of moments admits the following solvers:

Use a general solver for moments when fixing or repeating parameters:

Maximum conditional likelihood method allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum likelihood method allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Spectral estimator allows specification of windows used for PowerSpectralDensity calculation:

Spectral estimator allows the following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum entropy method:

It is also known as Burg method:

Univariate SliceDistribution:

Multivariate slice distributions:

Slice distribution of a vector-valued time series:

First-order probability density function with zero initial conditions:

Stationary mean and variance:

Compare with the density function of a normal distribution:

Compute the expectation of an expression:

Calculate a probability:

Skewness and kurtosis functions are constant:

Moment:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

Approximate an MA process with an AR process of order 3:

Compare the covariance function for the original and the approximate processes:

Approximate a vector process:

Approximate an ARMA process with an AR process:

Approximate an ARMA with fixed initial values:

Compare sample paths:

Approximate a SARIMA process with an AR process:

Compare sample paths:

TransferFunctionModel representation:

For a vector-valued process:

PoleZeroPlot of a stationary AR process:

A non stationary AR process has at least one pole outside the unit circle:

StateSpaceModel representation:

For a vector-valued process: