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Description:
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This is the main function to implement Functional Quadratic Regression (QuadReg), where the
predictor is a function X(t_x) and the response Y is a scalar. 

Reference: Yao, F. and M\"uller, H.G. (2009). Functional Quadratic Regression.
Biometrika.


This model can be used for both sparsely and densely sampled trajectories. 
Gaussian assumptions are required for the coefficient estimates to be 
consistent in the sparse case. Please refer to the above paper for more details.  

Includes the following steps:

1) FPCA using the PACE method for X(t_x).

2) Computation of the quadratic regression model components

3) Prediction of response for given predictor values

4) Computation of (generalized) R-squared.

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Usage:
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[res, xx] = FPCQuadReg(x, t_x, y, param_X, K, isNewSub)

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Input:
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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 new subjects whose response is to be predicted,
         while not contributing to model fitting; this is controlled by 
         "isNewSub" which is either a vector consisting of 0's and 1's, 
         according to whether a subject is used for prediction (0) or 
         estimation (1), or is controlled by 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 more 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 vector for scalar response y, then 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:
            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 1*n, it will be
            truncated to 1*nn according to the isNewSub definition below.
            (see "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 (for default, set 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 (Note that if FIT = -1, the number of principal
         components of X and Y are jointly selected by 'AIC_R'). 
         If the main goal 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().

K:     positive integer, not required. Number of principal components
         of predictor x used in regression, must be smaller than or equal to the maximum
         number of principal components available from FPCA.  Default K = [], in this case
         K is the number of components selected by FPCA.m for predictor functions.


isNewSub: i) 1*n vector of 0s or 1s, where

             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
           number of subjects used for prediction.

       o   When no prediction is requested, x,t_x, y, will be of
           length n = nn.

       o   When prediction is requested, x, t_x 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.

Details: i) There are no default values for the first 3 arguments, that
            is, they are always part of the input arguments.
        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(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.

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Output:
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  res:  an aggregated object that contains coeff, coeffFun, design, newx, new_tx, newy, 
       Q, fitted_y, K.

        1) coeff: an object that contains alpha, beta and gamma.
                 i)  alpha: the estimate of the intercept.
                 ii) beta: a K_x*1 vector of estimated coefficients for the 
                     linear term in the eigenbasis representation, where K_x
                     is the number of included FPC components.
                 iii) gamma: a {K*(K+1)/2}*1 vector of estimated
                 coefficients for the quadratic terms in the eigenbasis
                 representation. The sequence is gamma_{11}, gamma_{21}, gamma_{22},
                  gamma_{31},... ,gamma_{KK}, as described in the
                  reference.
        2) coeffFun: an object that contains betafun, gammafun, and grid.
                 i) betafun: a 1*ngrid vector; estimated regression
                 function for the linear term, evaluated at given grid points. 
                 ii) gammafun: a ngrid*ngrid vector; estimated regression
                 surface for the quadratic term. 
                 iii) grid: 1*ngrid vector, contains ngrid of distinct
                 time points for the x function, where ngrid is defined
                 in setOptions().
        3) design: n*{(K+1)*(K+2)/2} design matrix for the regression model. 
         4) newx: 1*numNewSub cell array that contains measurements for new x
                 (predictor) functions.

        5) new_tx: 1*numNewSub cell array that contains time points corresponding
                   to newx.

        6) newy:   1*numNewSub vector that contains predicted measurements for
                   corresponding newx.

        7) Q:      Quasi R-square; a measure of the fraction of the variation
                   of the responses that is explained by the regression.
                      Q = 1-sum((Y_i-Yhat_i)^2)/sum((Y_i-mean(Y))^2).

        9) fitted_y: 1*nn (same length of y) vector containing fitted
                     measurements for corresponding y values that were used for
                     estimation.
       
        10) K:   number of principal components used in regression for predictor X.
   

 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.


   o    See exampleQuadReg.m for an example of the Functional Quadratic
        Regression.

See also PCA, FPCA, FPCreg
