Filter using an autoregressive model:

Filter noise from a sample path using an ARMA model:

Find filtered data:

Compare the data to the prediction:

Use KalmanFilter with TimeSeriesModel:

Compare filtering with given initial values:

Create a sample from a weakly stationary ARMA:

Filter using ARMA with given initial values:

Compare:

Find a filter for a multivariate model and data:

Compare filters using autoregressive and moving-average processes:

Build an autoregressive filter:

Build a moving-average filter:

Compare data to the filters:

Find residuals for a fitted time series model:

Spread of residuals:

Distribution of residuals:

Consider the following time series data and determine whether it is adequately modeled by a MAProcess:

The correlation function drops off after lag 3. This is evidence of an MAProcess[3]:

Fit an MAProcess[3] model to the data:

Find the model residuals and determine whether they are normally distributed white noise:

The null hypothesis of normality cannot be rejected:

Analyze residuals in TimeSeriesModel:

Calculate residuals from Kalman filtering:

Compare to residuals from TimeSeriesModelFit:

Kalman filter creates a one-step forecast:

Find a filter for the data:

Calculate one-step-ahead predictions from data parts:

Compare the paths:

KalmanFilter requires the process to have all parameters numeric:

The data can be symbolic: