MARSSresiduals.tt {MARSS}

MARSS Contemporaneous Residuals

Description

Calculates the standardized (or auxiliary) contemporaneous residuals, aka the residuals and their variance conditioned on the data up to time t. Contemporaneous residuals are only for the observations. Not exported. Access this function with MARSSresiduals(object, type="tt").

Usage

MARSSresiduals.tt(object, method = c("SS"), normalize = FALSE, 
    silent = FALSE, fun.kf = c("MARSSkfas", "MARSSkfss"))

Arguments

object

An object of class marssMLE.

method

Algorithm to use. Currently only "SS".

normalize

TRUE/FALSE See details.

silent

If TRUE, don't print inversion warnings.

fun.kf

Can be ignored. This will change the Kalman filter/smoother function from the value in object$fun.kf if desired.

Details

This function returns the conditional expected value (mean) and variance of the model contemporaneous residuals. 'conditional' means in this context, conditioned on the observed data up to time t and a set of parameters.

Model residuals

\mathbf{v}_t is the difference between the data and the predicted data at time t given \mathbf{x}_t:

\mathbf{v}_t = \mathbf{y}_t - \mathbf{Z} \mathbf{x}_t - \mathbf{a} - \mathbf{d}\mathbf{d}_{t}

The observed model residuals \hat{\mathbf{v}}_t are the difference between the observed data and the predicted data at time t using the fitted model. MARSSresiduals.tt fits the model using the data up to time t. So

\hat{\mathbf{v}}_t = \mathbf{y}_t - \mathbf{Z}\mathbf{x}_t^{t} - \mathbf{a} - \mathbf{D}\mathbf{d}_{t}

where \mathbf{x}_t^{t} is the expected value of \mathbf{X}_t conditioned on the data from 1 to t from the Kalman filter. \mathbf{y}_t are your data and missing values will appear as NA. These will be returned in residuals.

var.residuals returned by the function is the conditional variance of the residuals conditioned on the data up to t and the parameter set \Theta. The conditional variance is

\hat{\Sigma}_t = \mathbf{R}+\mathbf{Z} \mathbf{V}_t^{t} \mathbf{Z}^\top

where \mathbf{V}_t^{t} is the variance of \mathbf{X}_t conditioned on the data up to time t. This is returned by MARSSkfss in Vtt.

Standardized residuals

std.residuals are Cholesky standardized residuals. These are the residuals multiplied by the inverse of the lower triangle of the Cholesky decomposition of the variance matrix of the residuals:

\hat{\Sigma}_t^{-1/2} \hat{\mathbf{v}}_t

. These residuals are uncorrelated unlike marginal residuals.

The interpretation of the Cholesky standardized residuals is not straight-forward when the \mathbf{Q} and \mathbf{R} variance-covariance matrices are non-diagonal. The residuals which were generated by a non-diagonal variance-covariance matrices are transformed into orthogonal residuals in \textrm{MVN}(0,\mathbf{I}) space. For example, if v is 2x2 correlated errors with variance-covariance matrix R. The transformed residuals (from this function) for the i-th row of v is a combination of the row 1 effect and the row 1 effect plus the row 2 effect. So in this case, row 2 of the transformed residuals would not be regarded as solely the row 2 residual but rather how different row 2 is from row 1, relative to expected. If the errors are highly correlated, then the Cholesky standardized residuals can look rather non-intuitive.

mar.residuals are the marginal standardized residuals. These are the residuals multiplied by the inverse of the diagonal matrix formed from the square-root of the diagonal of the variance matrix of the residuals:

\textrm{dg}(\hat{\Sigma}_t)^{-1/2} \hat{\mathbf{v}}_t

, where 'dg(A)' is the square matrix formed from the diagonal of A, aka diag(diag(A)). These residuals will be correlated if the variance matrix is non-diagonal.

Normalized residuals

If normalize=FALSE, the unconditional variance of \mathbf{V}_t and \mathbf{W}_t are \mathbf{R} and \mathbf{Q} and the model is assumed to be written as

\mathbf{y}_t = \mathbf{Z} \mathbf{x}_t + \mathbf{a} + \mathbf{v}_t

\mathbf{x}_t = \mathbf{B} \mathbf{x}_{t-1} + \mathbf{u} + \mathbf{w}_t

If normalize=TRUE, the model is assumed to be written

\mathbf{y}_t = \mathbf{Z} \mathbf{x}_t + \mathbf{a} + \mathbf{H}\mathbf{v}_t

\mathbf{x}_t = \mathbf{B} \mathbf{x}_{t-1} + \mathbf{u} + \mathbf{G}\mathbf{w}_t

with the variance of \mathbf{V}_t and \mathbf{W}_t equal to \mathbf{I} (identity).

MARSSresiduals() returns the residuals defined as in the first equations. To get normalized residuals (second equation) as used in Harvey et al. (1998), then use normalize=TRUE. In that case the unconditional variance of residuals will be \mathbf{I} instead of \mathbf{R} and \mathbf{Q}. Note, that the normalized residuals are not the same as the standardized residuals. In former, the unconditional residuals have a variance of \mathbf{I} while in the latter it is the conditional residuals that have a variance of \mathbf{I}.

Value

A list with the following components

model.residuals

The observed contemporaneous model residuals: data minus the model predictions conditioned on the data 1 to t. A n x T matrix. NAs will appear where the data are missing.

state.residuals

All NA. There are no contemporaneous residuals for the states.

residuals

The residuals. model.residuals are in rows 1:n and state.residuals are in rows n+1:n+m.

var.residuals

The joint variance of the residuals conditioned on observed data from 1 to t-. This only has values in the 1:n,1:n upper block for the model residuals.

std.residuals

The Cholesky standardized residuals as a n+m x T matrix. This is residuals multiplied by the inverse of the lower triangle of the Cholesky decomposition of var.residuals. The model standardized residuals associated with the missing data are replaced with NA. Note because the contemporaneous state residuals do not exist, rows n+1:n+m are all NA.

mar.residuals

The marginal standardized residuals as a n+m x T matrix. This is residuals multiplied by the inverse of the diagonal matrix formed by the square-root of the diagonal of var.residuals. The model marginal residuals associated with the missing data are replaced with NA.

bchol.residuals

Because state residuals do not exist, this will be equivalent to the Cholesky standardized residuals, std.residuals.

E.obs.residuals

The expected value of the model residuals conditioned on the observed data 1 to t. Returned as a n x T matrix.

var.obs.residuals

The variance of the model residuals conditioned on the observed data. Returned as a n x n x T matrix. For observed data, this will be 0. See MARSSresiduals.tT() for a discussion of these residuals and where they might be used.

msg

Any warning messages. This will be printed unless Object$control$trace = -1 (suppress all error messages).

Author(s)

Eli Holmes, NOAA, Seattle, USA.

References

Holmes, E. E. 2014. Computation of standardized residuals for (MARSS) models. Technical Report. arXiv:1411.0045.

See Also

MARSSresiduals.tT(), MARSSresiduals.tt1(), fitted.marssMLE(), plot.marssMLE()

Examples

  dat <- t(harborSeal)
  dat <- dat[c(2,11),]
  fit <- MARSS(dat)
  
  # Returns a matrix
  MARSSresiduals(fit, type="tt")$std.residuals
  # Returns a data frame in long form
  residuals(fit, type="tt")