# Supervised Dimension Reduction for Multivariate Time Series Supervised dimension reduction for multivariate time series data. There are three different algorithms to choose from. TSIR is a time series version of Sliced Inverse Regression (SIR), TSAVE is a time series version of Sliced Average Variance Estimate (TSAVE) and a hybrid of TSIR and TSAVE is TSSH (Time series SIR SAVE Hybrid). For summary of an object of class tssdr, see `summary.tssdr`. ```r tssdr(y, X, ...) ## Default S3 method: tssdr(y, X, algorithm = c("TSIR", "TSAVE", "TSSH"), k = 1:12, H = 10, weight = 0.5, method = c("frjd", "rjd"), eps = 1e-06, maxiter = 1000, ...) ## S3 method for class 'ts' tssdr(y, X, ...) ## S3 method for class 'xts' tssdr(y, X, ...) ## S3 method for class 'zoo' tssdr(y, X, ...) ## S3 method for class 'tssdr' print(x, digits = 3, ...) ## S3 method for class 'tssdr' components(x, ...) ## S3 method for class 'tssdr' plot(x, main = "The response and the directions", ...) ``` ## Arguments - `y`: A numeric vector or a time series object of class `ts`, `xts` or `zoo` (same type as X). Missing values are not allowed. - `X`: A numeric matrix or a multivariate time series object of class `ts`, `xts` or `zoo` (same type as y). Missing values are not allowed. - `algorithm`: Algorithm to be used. The options are `"TSIR"`, `"TSAVE"` and `"TSSH"`. Default is `"TSIR"`. - `k`: A vector of lags. It can be any non-zero positive integer, or a vector consisting of them. Default is `1:12`. - `H`: The number of slices. If `"TSSH"` is used, $H$ is a 2-vector; the first element is used for TSIR part and the second for TSAVE part. Default is $H = 10$. - `weight`: Weight $0 \le a \le 1$ for the hybrid method `TSSH` only. With $a = 1$ it reduces to TSAVE and with $a = 0$ to TSIR. Default is $a = 0.5$. - `method`: The method to use for the joint diagonalization. The options are `"rjd"` and `"frjd"`. Default is `"frjd"`. - `eps`: Convergence tolerance. - `maxiter`: The maximum number of iterations. - `...`: Further arguments to be passed to or from methods. In methods for class 'tssdr' only: - `x`: An object of class tssdr - `digits`: The number of digits when printing an object of class tssdr. Default is 3 - `main`: A title for a plot when printing an object of class tssdr. ## Details Assume that the $p$-variate time series $Z$ with $T$ observations is whitened, i.e. $Z = S^(-1/2)*(X_t - (1/T)*sum_t(X_t))$, where $S$ is a sample covariance matrix of $X$. Divide $y$ into $H$ disjoint intervals (slices) by its empirical quantiles. For each lag $j$, denote $y_j$ for a vector of the last $n - j$ values of the sliced $y$. Also denote $Z_j$ for the first $n - j$ observations of $Z$. Then $Z_jh$ are the disjoint slices of $Z_j$ according to the values of $y_j$. Let $T_jh$ be the number of observations in $Z_jh$. Write $A_jh = 1/T_jh * sum_t((Z_jh)_t),$ and $Aj = (A_j1, \ldots, A_jH)'$. Then for algorithm `TSIR` matrix $$ {\bf \widehat{M}}_{0j} = {\bf \widehat{Cov}}_{A_j}.M_0j = Cov_Aj (a sample covariance matrix of Aj). $$ Denote $Cov_jh$ for a sample covariance matrix of $Z_jh$. Then for algorithm `TSAVE` matrix $$ {\bf \widehat{M}}_{0j} = \frac{1}{H}\sum_{h = 1}^H({\bf I}_p - {\bf \widehat{Cov}_{jh}})^2.M_1j = 1/T_jh * sum_h((I_p - Cov_jh)^2), $$ For `TSSH` then matrix $$ {\bf \widehat{M}}_{2j} = a{\bf \widehat{M}_{1j}} + (1-a){\bf \widehat{M}_{0j}},M_2j = a*M_1j + (1-a)*M_0j, $$ for a chosen $0 \le a \le 1$. Note that the value of $H$ can be different for TSIR and TSAVE parts. The algorithms find an orthogonal matrix $U = (u_1, \ldots, u_p)'$ by maximizing, for $b = 0, 1$ or $2$, $$ \sum_{i \in k} ||diag({\bf U} {\bf \widehat{M}}_{bj} {\bf U}')||^2 = \sum_{i \in 1}^p \sum_{j \in k} ({\bf u}_i' {\bf \widehat{M}}_{bj} {\bf u}_i)^2.sum_j (||diag(U M_bj U')||^2) = sum_i sum_j (u_i' M_bj u_i)^2, $$ The final signal separation matrix is then $W = U S^(-1/2)$. Write $\lambda_ij = c*(u_i' M_aj u_i)^2$, where $c$ is chosen in such way that $sum_i sum_j \lambda_ij = 1,$ Then the $(i, j)$:th element of the matrix $L$ is $\lambda_ij$. To make a choice on which lags and directions to keep, see `summary.tssdr`. Note that when printing a tssdr object, all elements are printed, except the directions S. ## Returns A list of class 'tssdr' containing the following components: - **W**: The estimated signal separation matrix. - **k**: The vector of the used lags. - **S**: The estimated directions as time series object standardized to have mean 0 and unit variances. - **MU**: The mean vector of `X`. - **L**: The Lambda matrix for choosing lags and directions. - **H**: The used number of slices. - **yname**: The name for the response time series $y$. - **Xname**: The name for the predictor time series $X$. - **algorithm**: The used algorithm as a character string. ## References Matilainen, M., Croux, C., Nordhausen, K. and Oja, H. (2017), **Supervised Dimension Reduction for Multivariate Time Series**, Econometrics and Statistics, 4, 57--69. Matilainen, M., Croux, C., Nordhausen, K. and Oja, H. (2019), **Sliced Average Variance Estimation for Multivariate Time Series**. Statistics: A Journal of Theoretical and Applied Statistics, 53, 630--655. Li, K.C. (1991), **Sliced Inverse Regression for Dimension Reduction**, Journal of the American Statistical Association, 86, 316--327. Cook, R. and Weisberg, S. (1991), **Sliced Inverse Regression for Dimension Reduction**, Comment. Journal of the American Statistical Association, 86, 328--332. ## Author(s) Markus Matilainen ## See Also `summary.tssdr`, `dr` ## Examples ```r n <- 10000 A <- matrix(rnorm(9), 3, 3) x1 <- arima.sim(n = n, list(ar = 0.2)) x2 <- arima.sim(n = n, list(ar = 0.8)) x3 <- arima.sim(n = n, list(ar = 0.3, ma = -0.4)) eps2 <- rnorm(n - 1) y <- 2*x1[1:(n - 1)] + eps2 X <- ((cbind(x1, x2, x3))[2:n, ]) %*% t(A) res1 <- tssdr(y, X, algorithm = "TSAVE") res1 summ1 <- summary(res1, type = "alllag", thres = 0.8) summ1 plot(summ1) head(components(summ1)) coef(summ1) # Hybrid of TSIR and TSAVE. For TSIR part H = 10 and for TSAVE part H = 2. tssdr(y, X, algorithm = "TSSH", weight = 0.6, H = c(10, 2)) ```

tssdr function