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:

Initial values do not influence the process values:

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:

Estimation:

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

Use TimeSeriesModel to automatically find orders:

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

Find 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 moving-average process:

Compare covariance functions for each component:

Forecast future values:

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:

Correlation function exists in closed form:

Closed form of the partial correlation function for the first order:

Covariance matrix:

Covariance matrix of an MAProcess is symmetric multidiagonal:

Correlation matrix:

Covariance function for a vector-valued process:

Power spectral density:

Vector MAProcess:

MAProcess is weakly stationary for any choice of parameters:

For a vector process:

Check if a time series is invertible:

For a vector process:

Find invertible representation for a moving-average process:

The moments are being conserved:

Find invertibility conditions:

Find conditions for higher order:

The available methods for estimating an MAProcess:

Compare log likelihoods:

Method of moments allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum conditional likelihood method allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Maximum likelihood method allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Spectral estimator allows to specify windows used for PowerSpectralDensity calculation:

Spectral estimator allows following solvers:

This method allows for fixed parameters:

Some relations between parameters are also permitted:

Minimum prediction method:

This method allows for fixed parameters:

Single time SliceDistribution:

Multiple time slice distributions:

Slice distribution of a vector-valued time series:

First-order probability density function:

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 are constant:

Moment of order r:

Generating functions:

CentralMoment and its generating function:

FactorialMoment has no closed form for symbolic order:

Cumulant and its generating function:

Approximate an AR process with an MA process of order 5:

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

Approximate a vector process:

Approximate an ARMA process with an MA process:

Compare sample paths:

Approximate a SARIMA process with an MA process:

Compare sample paths:

TransferFunctionModel representation:

For a vector-valued process:

StateSpaceModel representation:

For a vector-valued process: