Obtain Inputs of RMFD System for Given Data
solve_inverse_RMFD_R.RdCompute the inputs \(u_t\) of RMFD(p, q) systems of the form
$$y_t = d_0 v_t + d_1 v_{t-1} + \cdots + d_q v_{t-q}$$
where
$$v_t + c_1 v_{t-1} + \cdots + c_p v_{t-p} = u_t$$
for given data \(y_t\) which are contained column-wise in the matrix data_output, see below.
Arguments
- polm_c, polm_d
polm objects. Jointly, they reprsent an RMFD. \(c(z)\) is square, has the identity as zero-lag coefficient. \(d(z)\) might be tall, and its zero-lag coefficient matrix is in general free.
- data_output
\((m, N)\) matrix with the outputs \((y_1,...,y_N\).
- t0
integer, start iteration at t=t0.
Value
The R implementation solve_inverse_RMFD_R returns the matrix u with the computed inputs
$$u_t = d^{+}(z) c(z) y_t$$ in its columns.
The RcppArmadillo implementation solve_rmfd_cpp() returns NULL but overwrites its input argument!
Note that the RcppArmadillo implementation has a different user interface (it is intended for internal use only).
Details
Values \(y_t\), \(u_t\) for \(t\leq 0\) are implicitely set to be zero. However, if we start the iteration with some \(t_0>1\) we can enforce non-zero initial values.
The routines are used internally and hence do not check their arguments.
We require the number of outputs to be positive \(m > 0\), the number of inputs to be non-negative \(n \geq 0\),
the degree of \(c(z)\) to be non-negative \(p \geq 0\), the degree of \(d(z)\) is unrestricted, i.e. \((q+1) \geq 0\),
and for the starting value and the sample size \(1 \leq t_0 \leq N\) holds.
Note also that the RcppArmadillo implementation overwrites the input argument y.
Use this procedure with care!
Note the non standard arguments: The polynomial matrices \(c(z)\) and \(d(z)\) are saved as "wide" matrices. The order of the coefficients in \(c(z)\) is reversed and there is no \(c_0\) coefficient (because it is required to be the identity matrix). The order of the coefficients in \(d(z)\) is as usual, \(d_0\) is available too. The data matrices are organized columnwise (to avoid memory shuffling)!