# Fit a CATCH model and predict categorical response. The `catch` function solves classification problems and selects variables by fitting a covariate-adjusted tensor classification in high-dimensions (CATCH) model. The input training predictors include two parts: tensor data and low dimensional covariates. The tensor data could be matrix as a special case of tensor. In `catch`, tensor data should be stored in a list form. If the dataset contains no covariate, `catch` can also fit a classifier only based on the tensor predictors. If covariates are provided, the method will adjust tensor for covariates and then fit a classifier based on the adjusted tensor along with the covariates. If users specify testing data at the same time, predicted response will be obtained as well. ```r catch(x, z = NULL, y, testx = NULL, testz = NULL, nlambda = 100, lambda.factor = ifelse((nobs - nclass) <= nvars, 0.2, 1E-03), lambda = NULL,dfmax = nobs, pmax = min(dfmax * 2 + 20, nvars), pf = rep(1, nvars), eps = 1e-04, maxit = 1e+05, sml = 1e-06, verbose = FALSE, perturb = NULL) ``` ## Arguments - `x`: Input tensor (or matrix) list of length $N$, where $N$ is the number of observations. Each element of the list is a tensor or matrix. The order of tensor can be any positive integer not less than 2. - `z`: Input covariate matrix of dimension $N*q$, where $qp$. - `lambda`: A sequence of user-specified `lambda` values. `lambda` is the weight of L1 penalty and a smaller `lambda` allows more variables to be nonzero. If `NULL`, then the algorithm will generate a sequence of `nlambda` many potential `lambda`s according to `lambda.factor`. - `dfmax`: The maximum number of selected variables in the model. Default is the number of observations `N`. - `pmax`: The maximum number of potential selected variables during iteration. In middle step, the algorithm can select at most `pmax` variables and then shrink part of them such that the nubmer of final selected variables is less than `dfmax`. Default is $min (dfmax*2+20, p)$. - `pf`: Weight of lasso penalty. Default is a vector of value `1` and length `p`, representing L1 penalty of length $p$. Can be mofidied to use adaptive lasso penalty. - `eps`: Convergence threshold for coordinate descent difference between iterations. Default value is `1e-04`. - `maxit`: Maximum iteration times for all lambda. Default value is `1e+05`. - `sml`: Threshold for ratio of loss function change after each iteration to old loss function value. Default value is `1e-06`. - `verbose`: Indicates whether print out lambda during iteration or not. Default value is `FALSE`. - `perturb`: Perturbation scaler. If it is specified, the value will be added to diagonal of estimated covariance matrix. A small value can be used to accelerate iteration. Default value is `NULL`. ## Details The `catch` function fits a linear discriminant analysis model as follows: $$ \mathbf{Z}|(Y=k)\sim N(\boldsymbol{\phi_k},\boldsymbol{\psi}), $$ $$ \mathbf{X}|(\mathbf{Z}=\mathbf{z}, Y=k)\sim TN(\boldsymbol{\mu}_k+\boldsymbol{\alpha}\bar{\times}_{M+1}\mathbf{z},\boldsymbol{\Sigma}_1,\cdots,\boldsymbol{\Sigma}_M). $$ The categorical response is predicted from the estimated Bayes rule: $$ \widehat{Y}=\arg\max_{k=1,\cdots,K}{a_k+\boldsymbol{\gamma}_k^T\mathbf{Z}+<\boldsymbol{\beta}_k,\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}>}, $$ where $\mathbf{X}$ is the tensor, $\mathbf{Z}$ is the covariates, $a_k$, $\boldsymbol{\gamma}_k$ and $\boldsymbol{\alpha}$ are parameters estimated by CATCH. A detailed explanation can be found in reference. When `Z` is not `NULL`, the function will first adjust tensor on covariates by modeling $$ \mathbf{X}=\boldsymbol{\mu}_k+\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}+\mathbf{E}, $$ where $\mathbf{E}$ is an unobservable tensor normal error independent of $\mathbf{Z}$. Then `catch` fits model on the adjusted training tensor $\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}$ and makes predictions on testing data by using the adjusted tensor list. If `Z` is `NULL`, it reduces to a simple tensor discriminant analysis model. The coefficient of tensor $\boldsymbol{\beta}$, represented by `beta` in package, is estimated by $$ \min_{\boldsymbol{\beta}_2,\ldots,\boldsymbol{\beta}_K}\left[\sum_{k=2}^K\left(\langle\boldsymbol{\beta}_k,[\![\boldsymbol{\beta}_k;\widehat{\boldsymbol{\Sigma}}_{1},\dots,\widehat{\boldsymbol{\Sigma}}_{M}]\!]\rangle-2\langle\boldsymbol{\beta}_k,\widehat{\boldsymbol{\mu}}_{k}-\widehat{\boldsymbol{\mu}}_{1}\rangle\right)+\lambda\sum_{j_{1}\dots j_{M}}\sqrt{\sum_{k=2}^{K}\beta_{k,j_{1}\cdots j_{M}}^2}\right]. $$ When response is multi-class, the group lasso penalty over categories is added to objective function through parameter `lambda`, and it reduces to a lasso penalty in binary problems. The function `catch` will predict categorical response when testing data is provided. If testing data is not provided or if one wishes to perform prediction separately, `catch` can be used to only fit model with a catch object outcome. The object outcome can be combined with the adjusted tensor list from `adjten` to perform prediction by `predict.catch`. ## Returns - **beta**: Output variable coefficients for each `lambda`, which is the estimation of $\boldsymbol{\beta}$ in the Bayes rule. `beta` is a list of length being the number of `lambda`s. Each element of `beta` is a matrix of dimension $nvars*(nclass-1)$. - **df**: The number of nonzero variables for each value in sequence `lambda`. - **dim**: Dimension of coefficient array. - **lambda**: The actual `lambda` sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on `dfmax` and `pmax`. - **obj**: Objective function value for each value in sequence `lambda`. - **x**: The tensor list after adjustment in training data. If covariate is absent, this is the original input tensor list. - **y**: Class label in training data. - **npasses**: Total number of iterations. - **jerr**: Error flag. - **sigma**: Estimated covariance matrix on each mode. `sigma` is a list with the `i`th element being covariance matrix on `i`th mode. - **delta**: Estimated delta matrix $(vec(\widehat{\boldsymbol{\mu}}_2-\widehat{\boldsymbol{\mu}}_1),\cdots,vec(\widehat{\boldsymbol{\mu}}_K-\widehat{\boldsymbol{\mu}}_1))$. - **mu**: Estimated mean array of $\mathbf{X}$. - **prior**: Proportion of samples in each class. - **call**: The call that produces this object. - **pred**: Predicted categorical response for each value in sequence `lambda` when `testx` is provided. ## References Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, **accepted**. ## Author(s) Yuqing Pan, Qing Mai, Xin Zhang ## See Also `cv.catch`, `predict.catch`, `adjten` ## Examples ```r #without prediction n <- 20 p <- 4 k <- 2 nvars <- p*p*p x <- array(list(),n) vec_x <- matrix(rnorm(n*nvars), nrow=n, ncol=nvars) vec_x[1:10,] <- vec_x[1:10,]+2 z <- matrix(rnorm(n*2), nrow=n, ncol=2) z[1:10,] <- z[1:10,]+0.5 y <- c(rep(1,10),rep(2,10)) for (i in 1:n){ x[[i]] <- array(vec_x[i,],dim=c(p,p,p)) } obj <- catch(x,z,y=y) ```

catch function