Outputs of a statespace system
outputs_STSP_cpp.RdThis internal helper function computes the outputs and states for a statespace system of the form $$a_{t+1} = A a_t + B u_t, \; y_t = C a_t + D u_t$$
Arguments
- A
\((s,s)\) matrix.
- B
\((s,n)\) matrix.
- C
\((m,s)\) matrix.
- D
\((m,n)\) matrix.
- u
\((n,N)\) matrix with the inputs/disturbances: \((u_1,u_2,\ldots,u_N)\).
- a
\((s,N+1)\) matrix. This matrix is overwritten with the (computed) states: \((a_1,a_2,\ldots,a_N,a_{N+1})\). On input
a[,1]must hold the initial state \(a_1\).- y
\((m,N)\) matrix. This matrix is overwritten with (computed) outputs: \((y_1,y_2,\ldots,y_N)\).
Note
Use this procedure with care!
The procedure does not check the input arguments.
The procedure overwrites the input arguments
aandu.The data matrices are organized columnwise (to avoid memory shuffling)!
See also
outputs_ARMA_cpp, residuals_ARMA_cpp, cll_theta_ARMA_cpp,
outputs_STSP_cpp, residuals_STSP_cpp, cll_theta_STSP_cpp and
solve_de, solve_inverse_de and ll.
Examples
# generate a random statespace model (3 outputs, 2 inputs and 4 states)
m = 3
n = 2
s = 4
model = test_stspmod(dim = c(m, n), s = s, digits = 2)
# generate random noise sequence (sample size N = 10)
n.obs = 10
u = matrix(rnorm(n.obs*n), nrow = n, ncol = n.obs)
print(u)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.6608268 1.973914 0.2354302 -0.8402714 1.1646030 -0.02291834
#> [2,] 1.2236039 2.072861 -0.4070931 -0.6996284 -0.3120939 0.43262238
#> [,7] [,8] [,9] [,10]
#> [1,] 0.27630991 -0.4755849 0.7693238 0.4517465
#> [2,] -0.06727034 -0.6866931 0.6912570 0.4445293
# generate matrix for the state sequence
a = matrix(0, nrow = s, ncol = n.obs+1)
a[,1] = rnorm(s) # random initial state a[1]
print(a)
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [1,] 0.003780549 0 0 0 0 0 0 0 0 0 0
#> [2,] 0.637364872 0 0 0 0 0 0 0 0 0 0
#> [3,] 0.649775552 0 0 0 0 0 0 0 0 0 0
#> [4,] -0.806978707 0 0 0 0 0 0 0 0 0 0
# generate matrix for the outputs
y = matrix(0, nrow = m, ncol = n.obs)
# call outputs_STSP_cpp()
outputs_STSP_cpp(model$sys$A, model$sys$B, model$sys$C, model$sys$D, u, a, y)
print(u)
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.6608268 1.973914 0.2354302 -0.8402714 1.1646030 -0.02291834
#> [2,] 1.2236039 2.072861 -0.4070931 -0.6996284 -0.3120939 0.43262238
#> [,7] [,8] [,9] [,10]
#> [1,] 0.27630991 -0.4755849 0.7693238 0.4517465
#> [2,] -0.06727034 -0.6866931 0.6912570 0.4445293
print(a) # a is overwritten with the computed states
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.003780549 1.851140674 -1.37841443 -3.9348659 -7.8430227 -11.6527278
#> [2,] 0.637364872 -0.388257448 1.67204528 1.0499707 3.4914483 0.5246119
#> [3,] 0.649775552 3.491592639 4.43244276 6.4804504 9.3259936 21.8985763
#> [4,] -0.806978707 0.002310544 -0.05564513 0.2959016 -0.3714579 -1.8704692
#> [,7] [,8] [,9] [,10] [,11]
#> [1,] -23.941100 -38.84891 -62.36087 -134.81683 -173.80340
#> [2,] 9.665598 11.48839 19.95334 53.30300 41.51877
#> [3,] 30.963632 67.85477 105.83304 191.51137 366.62189
#> [4,] 2.685993 -11.55248 14.00840 -33.17766 17.05150
print(y) # y is overwritten with the computed outputs
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 1.0288442 7.7915364 -1.1305329 -8.218414 -14.405891 -22.23845 -46.25018
#> [2,] 0.9014999 4.2552733 0.6892649 2.650351 2.748217 14.44811 11.99803
#> [3,] -0.2131480 0.9464061 1.7437085 6.425427 8.805214 25.76432 28.39758
#> [,8] [,9] [,10]
#> [1,] -73.43075 -112.62910 -262.24134
#> [2,] 32.64877 52.77545 76.76601
#> [3,] 70.16839 95.04417 192.38061