============ Description: ============ This is the primary function for performing Functional Linear Regression, where the predictor is a function X(t_x) and the response can be either a function Y(t_y) or a scalar. Reference: Yao, F., M\"{u}ller, H.G., Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Annals of Statistics 33, 2873--2903. (Reference [3]) This function includes the following steps: 1) FPCA using the PACE method for X(t_x) and/or Y(t_y) (if the responses are functions/trajectories) 2) Computation of the regression parameter function (regression parameter surface if the responses are functions) 3) Prediction of responses or response functions, given predictor values 4) Computation of the values of functional R-square and Quasi R-square (a simple predictive measure based on the concept of proportion of variance explained by the regression). ======== Usage: ======== [res, xx, yy] = FPCreg(x, t_x, y, t_y, param_X, param_Y, FIT, K_x, K_y, isNewSub, alpha) ======= Input: ======= x : 1*n cell array for predictor function x, where x{i} is the row vector of measurements for the ith subject, i=1,...,n. It may contain data for subjects that are used for prediction; this is controlled by "isNewSub" which is either a vector consisting of 0's and 1's according to whether subject is used for prediction (0) or estimation (1), or is a positive integer nn. In this case, nn is the number of subjects to be used for estimation and n-nn is the number of remaining subjects to be used for prediction, corresponding to the last n-nn data rows. When "isNewSub" is set to [], all n subjects are used for estimation and no prediction will be calculated; see "isNewSub" for details. t_x : 1*n cell array, t_x{i} is the row vector of time points for the ith subject at which corresponding measurements x{i} are taken, i=1,...,n. It contains subjects that are used for prediction. See above for 2 different cases of "isNewSub" and the definition of "isNewSub" for more details. y : i) When no prediction is requested, that is, isNewSub = [] or isNewSub = 0: 1*n cell array for response function y, where y{i} is the response row vector for the ith subject, i = 1,...,n, or a 1*n vector for scalar response y, in which case y(i) is the response value for the ith subject, i = 1,...,n. ii) When prediction is requested, that is isNewSub is either a vector of 0's (subjects used for estimation) and 1's (subjects used for prediction but not estimation) or a positive integer: Cell array of dim 1*nn (functional response case) or a vector of dim 1*nn (scalar response case), in either case, nn is the number of subjects used for estimation. iii) When prediction is requested and y is still 1*n, it will be truncated to 1*nn according to the isNewSub definition below. (see "isNewSub" for more details). t_y : 1*n cell array, where t_y{i} is the row vector of time points for the ith subject at which corresponding measurements y{i} are taken (for each response trajectory), i = 1,...,n, or [] if y is a scalar response. See above and the definition of "isNewSub" for more details. param_X: an object that is returned by setOptions(), that sets the input arguments for FPCA() of the X (predictor) functions (default setting is param_X = []). The default method for choosing the number of principal components is 'AIC_R', which uses the regression AIC criterion, selecting the number of principal components based on the linear relationship between predictor and response (if FIT = -1, the number of principal components of X and Y are jointly selected by 'AIC_R'). If the primary target of the analysis is the estimation of the regression parameter function or surface, suggested settings are FIT = 0 and 'BIC1' for 'selection_k' in param_X. For optimal prediction, it is better to use the default options. For other default values, see setOptions(). param_Y: an object that is returned by setOptions(), that sets the input arguments for FPCA() of the Y (response) functions (default setting is param_Y = []). The default method of choosing the number of principal components for the response trajectories is 'BIC1'. See setOptions() for more details. When y is a scalar, this object will be ignored. FIT: an indicator with values 0 or -1: FIT = 0: Fitting a functional linear regression (default) through decomposition into simple linear regressions between the FPC scores of Y (response) and X (predictor). This approach is denoted as 'YPC' in the following. FIT = -1: Prediction with a functional linear regression between the actual response observations Y and the FPC scores of X, using weighted** least squares with the covariance surface of Y as the weight matrix, this approach is denoted as 'YCO' in the following. Note: Option FIT = -1 is recommended for optimal prediction in the case of sparsely observed functional responses (i.e. Y is sparse and functional), but it may not yield a good parameter surface estimate. K_x: positive integer, normally not needed. Number of principal components of functional predictor X used in functional regression, must be smaller than or equal to the maximum number of principal components available from FPCA. Default K_x = [], which means that K_x is chosen as the number of components selected by FPCA.m for predictor functions. K_y: positive integer, normally not needed. Has no effect for the case of a scalar response. The number of principal components for the response Y, used for functional regression, must be smaller than or equal to the maximum number of principal components available from FPCA. Default K_y = [], in which case K_y is chosen as the number of components selected by FPCA.m for the response functions. isNewSub: i) 1*n vector of 0s or 1s, where n is the total count of subjects and 1 : the data for the corresponding subject, i.e., x(isNewSub == 1), t_x(isNewSub==1), t_y(isNewSub == 1) are used for prediction only; The count is n-nn for subjects with isNewSub = 1. 0 : the data for the corresponding subject, i.e., x(isNewSub == 0), t_x(isNewSub == 0), y and t_y(isNewSub == 0) are used for estimation only. The count is nn for subjects with isNewSub = 0. This option is convenient for computing leave-one-out prediction if desired. ii) If it is a positive integer, say 5, then the last 5 subjects (in the order from top to bottom within the array x) and their values for t_x, t_y are used for prediction. In other words, when choosing this option, one would append the ``new'' subjects for which one desires prediction of the response to occur at the end of x, t_x and t_y. Then the first nn rows of the arrays x, t_x,t_y and the entire array y (of length nn) are used for estimation and the remainder (last n-nn rows) of x,t_x and t_y (of length n-nn) will be used for prediction. Example: isNewSub = 5; n = length(x) x(1:(n-5)), t_x(1:(n-5)), y, t_y(1:(n-5)) are used for estimation. x((n-4):n), t_x((n-4):n) and t_y((n-4):n) are used for prediction. . This option is convenient for obtaining predictions for a set of new subjects. iii) set isNewSub = [] for the default value, which is no prediction. Note: o nn > n - nn, the number of subjects used for estimation, should generally be larger than the number of subjects used for prediction. o When no prediction is requested, x,t_x, y, t_y will be of length n = nn. o When prediction is requested, x, t_x, t_y will be of length n, which includes nn subjects for estimation and n-nn subjects for prediction. Here y is always of length nn (< n). o When prediction is requested and y is of length n < nn, a warning message will be given and y will be truncated to length nn. This assumes that only nn among the n available data for y are used for estimation and the remaining n-nn observations will be ignored. alpha: Optional, the level of the confidence bands. alpha = 0.05 if is left empty. No confidence bands will be produced if the alpha is left out or not within the range of (0, 1). Details: i) There are no default values for the first 3 arguments, that is, they are always part of the input arguments. Note, t_y can be set to [] when y is a scalar response; ii) Any unspecified or optional arguments can be set to "[]" for default values; iii) FPCA() calls PCA(), so setOptions() sets the input arguments for PCA() and the returned object contains all values returned by PCA(); iv) Names of objects can be retrieved through names() function i.e., names(xx), names(yy), names(res) and the actual values can be retrieved through the getVal() function, example: getVal(res, 'BETA'), getVal(res, 'newy') etc. v) When isNewSub is set to be [], no prediction will be performed, and the number of subjects in x is the same as the number of subjects in y; when isNewSub is either a vector of 0's and 1's or a positive integer, the number of subjects in x is larger than the number of subjects of y, since the former contains new subjects for prediction only, whose response will not be used in the model estimation process. Vi)Default: no prediction will be performed. ======= Output: ======= res: an aggregated object that contains BETA, b, newx, newtx, newy, new_ty, r2, Q, fitted_y, K_x, K_y. 1) BETA: an object that contains beta, grid_x and grid_y. i) beta: a ngrid_x*ngrid_y matrix; estimated regression parameter surface or function, evaluated at given grid points. ii) grid_x : 1*ngrid_x vector, contains ngrid of distinct time points for the x function (predictor function), where ngrid is defined in setOptions(). iii) grid_y : 1*ngrid_y vector, contains ngrid of distinct time points for th y function (response function), where ngrid is defined in setOptions(). when y is a scalar, grid_y is []. only calculated when FIT = 0; when FIT = -1, BETA = []. 2) b: a K_x*K_y matrix of estimated coefficients for the regression parameter function in the eigenbasis representation, where K_x is the number of selected FPC for X and K_y is the number of selected FPC for Y; alternatively interpreted as the regression slope coefficients of the response FPC scores on predictor FPC scores. 3) newx: 1*numNewSub cell array that contains measurements for new x (predictor) functions. 4) new_tx: 1*numNewSub cell array that contains time points corresponding to newx. 5) newy: 1*numNewSub cell array that contains predicted measurements for corresponding newx. 6) new_ty: 1*numNewSub cell array that contains time points corresponding to newy. 7) newcb: when y is functional, 2*numNewSub cell array, newcb{1,i} is the lower confidence bound for the i-th new subject, while newcb{2,i} is the upper confidence bound; when y is scalar, 2*numNewSub vector, newcb(1,i) is the lower confidence bound for the i-th new subject, while newcb(2,i) is the upper confidence bound. 8) r2: functional R-square. 9) Q: Quasi R-square; a measure of the fraction of the variation of the responses that is explained by the regression. If Y is a scalar, Q = 1-sum((Y_i-Yhat_i)^2)/sum((Y_i-mean(Y))^2). If Y is a function, Q=1-sum((Y_i-Yhat_i)'*(Y_i-Yhat_i)/n_i)/sum((Y_i-mean(Y))'*(Y_i-mean(Y))/n_i). (See Reference [20] Eq. (20)). 10) fitted_y: when y is functional, 1*nn (same length of y) cell array containing fitted measurements for corresponding y that were used for estimation; when y is scalar, it is a 1*nn vector. 11) K_x: number of principal components used in regression for predictor X. 12) K_y: number of principal components used in regression for response Y. xx: an aggregated object that contains the returned values from FPCA(x,t_x,param_X), see PCA() or pcaHELP.txt for more details. Use getVal() to retrieve the individual values. yy: an aggregated object that contains the returned values from FPCA(y,t_y,param_Y) or returns "[]" if y is a scalar response (see PCA() or pcaHELP.txt for more details). Use getVal() to retrieve the individual values. See o example0.m for an example of the sparse irregular data case, where both predictors and responses are functional. o example2.m for an example of the regular data case, where both predictors and responses are functional. o example_scal for an example of the regular data case, where the predictor is functional and the response is scalar. See also PCA, FPCA, FPCdiag, Rtest. Use Rtest.m to conduct a hypothesis testing of Quasi R-square Q: Q=0, based on bootstrapping. Diagnostics of the fitted functional linear regression model is available only for the case FIT = 0. For more information about diagnostics, consult the help file of FPCdiag.