Chapter 6 Common output for fits

This is a brief summary of common outputs for model fits. The output are shown with the following simple fit:

dat <- cumsum(rnorm(100,0,0.5)) + rnorm(100,0,0.5)
fit <- MARSS(dat)

Use the broom package to get the model output in tidy form. The confidence intervals shown are approximate and based on the estimated Hessian matrix. See ?tidy.marssMLE for information on how to change to a different type of confidence interval.

broom::tidy(fit)

   term    estimate  std.error    conf.low    conf.up
1   R.R  0.20232993 0.07256917  0.06009696 0.34456290
2   U.U -0.03710875 0.05988268 -0.15447663 0.08025914
3   Q.Q  0.35208044 0.10659375  0.14316053 0.56100035
4 x0.x0  0.66568193 0.70913982 -0.72420657 2.05557043

The get the estimated states use:

head(stats::tsSmooth(fit, type="xtT"))

  .rownames t .estimate       .se
1      X.Y1 1 0.6287215 0.3194423
2      X.Y1 2 0.6361536 0.3325886
3      X.Y1 3 0.1632734 0.3336694
4      X.Y1 4 0.1711874 0.3337600
5      X.Y1 5 0.3302335 0.3337676
6      X.Y1 6 0.2024342 0.3337682

Note that generic rownames were given since none were specified for dat. You can also get all the Kalman filter and smoother estimates for \(\mathbf{x}\) from MARSSkf(). See ?MARSSkf.

There are two types of fitted values that are used in the state-space literature: the one-step-ahead which uses on the data up to \(t-1\) and the smoothed fitted values which uses all the data. Read up on fitted values for MARSS models at ?fitted.marssMLE.

To get the fitted values, the estimated \(\mathbf{y}\), using all the data:

head(fitted(fit, type="ytT"))

  .rownames t           y   .fitted
1        Y1 1  0.60321037 0.6287215
2        Y1 2  0.91217455 0.6361536
3        Y1 3 -0.11302441 0.1632734
4        Y1 4  0.08433636 0.1711874
5        Y1 5  0.49507490 0.3302335
6        Y1 6  0.36857168 0.2024342

To get the fitted values, the estimated \(\mathbf{y}\), using the data up to time \(t-1\):

head(fitted(fit, type="ytt1"))

  .rownames t           y   .fitted
1        Y1 1  0.60321037 0.6285732
2        Y1 2  0.91217455 0.5753577
3        Y1 3 -0.11302441 0.7752737
4        Y1 4  0.08433636 0.1078043
5        Y1 5  0.49507490 0.0540304
6        Y1 6  0.36857168 0.3301366

Note, this is the default for fitted(fit).

To make some generic plots and diagnostic plots, you can use autoplot() in the ggplot2 package:

ggplot2::autoplot(fit, plot.type = "xtT")

To get residuals, use:

resids <- residuals(fit)
head(format(resids, digits = 2))

  type .rownames  name   t   value .fitted .resids .sigma .std.resids
1 ytt1        Y1 model   1  0.6032  0.6286 -0.0254   0.74     -0.0341
2 ytt1        Y1 model   2  0.9122  0.5754  0.3368   0.83      0.4076
3 ytt1        Y1 model   3 -0.1130  0.7753 -0.8883   0.83     -1.0642
4 ytt1        Y1 model   4  0.0843  0.1078 -0.0235   0.84     -0.0281
5 ytt1        Y1 model   5  0.4951  0.0540  0.4410   0.84      0.5279
6 ytt1        Y1 model   6  0.3686  0.3301  0.0384   0.84      0.0460

There are many types of residuals possible for MARSS models. The default residuals(fit) will return the innovations residuals, which are what are typically used for residuals diagnostics for state-space models. However residuals() and the companion MARSSresiduals() will return all possible residuals for your model. Read about MARSS residuals in ?residuals.marssMLE.

To get your parameter estimates, use:

coef(fit)

Type ?coef.marssMLE to see the different forms that you can get your estimated parameters in. A common form is as a matrix. To get that use coef(fit, type="matrix").

coef(fit, type = "matrix")$Q

          [,1]
[1,] 0.3520804