Calculates the standardized (or auxiliary) one-step-ahead residuals, aka the innovations residuals and their variance. Not exported. Access this function with residuals(object, conditioning="t-1").
MARSSresiduals.tt1(object, method=c("SS"), normalize=FALSE, silent=FALSE)An object of class marssMLE.
Algorithm to use. Currently only "SS".
TRUE/FALSE
If TRUE, don't print inversion warnings.
A list with the following components
The model residuals conditioned on the data up to time \(t-1\) and the set of model parameters. Called the innovations. Residuals associated with missing data will appear as NA.
The variance of the model residuals as a n x n x T matrix. The variance exists for all t values including missing data.
The Cholesky standardized model residuals as a n x T matrix. This is residuals multiplied by the inverse of the Cholesky decomposition of var.residuals.
The marginal standardized model residuals as a n 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.
Any warning messages. This will be printed unless Object$control$trace = -1 (suppress all error messages).
This function returns the conditional expected value (mean) and variance of the model one-step-ahead residuals. 'conditional' means in this context, conditioned on the observed data up to time \(t-1\) and a set of parameters.
Model residuals
\(\mathbf{v}_t\) is the difference between the data and the predicted data at time \(t\) given \(\mathbf{x}_t\):
$$ \mathbf{v}_t = \mathbf{y}_t - \mathbf{Z} \mathbf{x}_t - \mathbf{a} $$
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.tt1 fits the model using the data up to time \(t-1\). So
$$ \hat{\mathbf{v}}_t = \mathbf{y}_t - \mathbf{Z}\tilde{\mathbf{x}}_t^{t-1} - \mathbf{a} $$
where \(\tilde{\mathbf{x}}_t^{t-1}\) is the expected value of \(\mathbf{X}_t\) conditioned on the data from 1 to \(t-1\) 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 \(t-1\) and the parameter set \(\Theta\). The conditional variance is
$$ \hat{\Sigma}_t = \mathbf{R}+\mathbf{Z}_t \mathbf{V}_t^{t-1} \mathbf{Z}_t^\top $$
where \(\mathbf{V}_t^{t-1}\) is the variance of \(\mathbf{X}_t\) conditioned on the data up to time \(t-1\). This is returned by MARSSkf in Vtt1.
Standardized residuals
std.residuals are Cholesky standardized residuals. These are the residuals muliplied by the inverse of the Cholesky decomposition of the variance matrix of the residuals:
$$ \hat{\Sigma}_t^{-1/2} \hat{\mathbf{v}}_t$$
These residuals are uncorrelated.
The interpretation of the Cholesky standardized residuals is not straight-forward when the Q and 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 MVN(0,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 muliplied by the inverse of the diagonal matrix formed from the square-root of the diagonal of the variance matrix of the residuals:
$$ \textrm{dg}(\hat{\Sigma}_t)^{-1/2} \hat{v}_t$$, 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 \(V_t\) and \(W_t\) are R and Q and the model is assumed to be written as
$$ y_t = Z x_t + a + v_t$$
$$ x_t = B x_{t-1} + u + w_t$$
If normalize=TRUE, the model is assumed to be written
$$ y_t = Z x_t + a + Hv_t$$
$$ x_t = B x_{t-1} + u + Gw_t$$
with the variance of \(V_t\) and \(W_t\) equal to I (identity).
residuals.marssMLE returns the residuals defined as in the first equations. To get the residuals defined as Harvey et al. (1998) define them (second equations), then use normalize=TRUE. In that case the unconditional variance of residuals will be I instead of R and Q. Note, that the `normalized' residuals are not the same as the `standardized' residuals. In former, the unconditional residuals have a variance of I while in the latter it is the conditional residuals that have a variance of I.
R. H. Shumway and D. S. Stoffer (2006). Section on the calculation of the likelihood of state-space models in Time series analysis and its applications. Springer-Verlag, New York.
Holmes, E. E. 2014. Computation of standardized residuals for (MARSS) models. Technical Report. arXiv:1411.0045.
# NOT RUN {
dat <- t(harborSeal)
dat <- dat[c(2,11),]
MLEobj <- MARSS(dat)
residuals(MLEobj, conditioning="t-1")$std.residuals
# }
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