The main idea is to use multivariable time series (as observations) to predict a state variable (one dimension).
Please find the attachments.
For example time series mm (4 variables and 200 observations) was used to learn V, W of a DLM. I have two question in this regard:
1) I supposed that dimensions of DLM should be as follows based on matrix operation:
FF = R4×1
GG = R1×1
V = R4×1 (because the dimension of FF×Ө and V should be the same)
W = R1×1
M0 = R1×1
C0 = R1×1
Therefore, the ¨V vector.R¨ code was developed. But, an error was displayed:
Error in dlm(FF = matrix(1, N, 1), GG = 1, V = matrix(exp(parm [1:4]), : Incompatible dimensions of matrices
Debug result:
m
Mahdieh,
V is the covariance matrix of the observation equation; if you observe four time series, it must be 4 x 4. This explains the first error.
Regarding the second, V has to be 4 x 4, but *not only that*. It has to be a valid covariance matrix. If you were using a diagonal matrix, the trick of taking exp(parm) would ensure that, for exponentiating anything gives a non-negative value, and a diagonal matrix of non-negative values is non-negative definite.
But you are filling all 16 entries of V, and there is no guarantee of symmetry nor of getting a matrix which is non-negative definite, and thus a proper covariance matrix. This explains the second error.
Best, ft.
A good reference for this is West and Harrison, Bayesian forecasting and dynamic models, Springer Verlag, 1997. It appears you are using West and Harrison notation or a variant of it. Chapter 9.2 discusses the “multiple regression dlm”.
I think that subchapter will clarify a few things. DLMs are defined by {F,G,V,W}. F is a 4x1 vector of independent variables. G = I (4x4 identity matrix), the system evolution matrix. V is the observation variance. W is the covariance matrix for the state parameter vector (regression coefficients), 4x4.
The variable predicted is the “observation variable”. West & Harrison do not call it a “state variable”. Rather, W&H refer to the current regression coefficients as the state vector or system vector Θ (subject to change over time).
Again, assuming W&H notation, the observation equation:
Y = F’ Θ + v , v ~ N( 0, V )
(1x1) = (4 x 1)’ (4 x 1) + (1x1) ; N[(1x1), (1x1)]
The system equation:
Θt = G Θt-1 + w, w ~ N(0, W)
(4x1) = (4x4) (4x1) + (4x1); N[(4x1),(4x4)]
I would recode things yourself, don’t use a library “dlm”. The equations are pretty easy (especially with W&H open next to you).
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