wtls_manual.tex

加权总体最小二乘matlab工具箱

A complete list of the \matlab\ code is given in Appendix~\ref{app}. It could be consulted for the implementation details.

\section{Overview of commands}\la{s2}

The package has three main groups of functions: transformations, misfit computations, and approximations. 

The transformation functions convert a given representation of a model to an equivalent one. The considered representations are image, kernel, and input/output, so that there are in total 6 transformation functions among them, see Figure~\ref{f1}. In addition a kernel or an image representation might not be minimal, so that functions that convert a given kernel or image representation to a minimal one are added. The transformation functions are summarized in Table~\ref{t3}.

\begin{figure}[!ht]
$$
\xymatrix
@C=1.75cm
@R=2cm
@M=2mm
{ 
\B=\ker(R) 
\ar@{<->}[rr]^{RP=0} 
\ar@<1mm>[rd]^{X^{\top}=-R_{\tout}^{-1}R_{\tin}} 
&& 
\B=\colspan(P)
\ar@<-1mm>[ld]_{X^{\top}=P_{\tout}P_{\tin}^{-1}} 
\\ &
\B=\B_\io(X)
\ar@<1mm>[ul]^{R = [ X^{\top} \  -I ] } 
\ar@<-1mm>[ur]_{P^{\top} = [ I \ \ X ] } 
}
$$
\caption{Transformations among kernel, image, and input/output representation of a model $\B\in\calL_{\ttm,0}^\ttd$.} \la{f1}
\end{figure}

\begin{table}[htb!]
\centering
\caption{Transformation functions}\la{t3}
\vskip.15cm
\begin{tabular}{|l|cl|}
\hline
Functions & \multicolumn{2}{|c|}{Description}\\
\hline
{\tt x2r}  & $X\mapsto R$ & from input/output to kernel representation\\
{\tt x2p}  & $X\mapsto P$ & from input/output to image representation\\
{\tt r2p}  & $R\mapsto P$ & from kernel to image representation\\
{\tt p2r}  & $P\mapsto R$ & from image to kernel representation\\
{\tt r2x}  & $R\mapsto X$ & from kernel to input/output representation\\
{\tt p2x}  & $P\mapsto X$ & from image to input/output representation\\
{\tt minr} & $R\mapsto R_{\min}$ & minimal kernel representation\\
{\tt minp} & $P\mapsto P_{\min}$ & minimal image representation\\
\hline
\end{tabular}
\end{table}

The misfit computation functions are used for validation: they allow the user to verify how well a given model fits given data in terms of a certain misfit function. Since the model can be specified by one of the three alternative representations---kernel, image, or input/output---all misfit functions have three versions. The following naming convention is adopted: misfit computation functions begin with {\tt m} (for misfit), followed by the name of the approximation problem (which identifies the type of misfit to be computed), followed by a letter, indicating the model representation: {\tt r} for kernel, {\tt p} for image, and {\tt x} for input/output. Instead of a model~$\B$ an approximating matrix~$\hat D\in\R^{\ttd\times N}$ can be used for the misfit computation. In this latter case the last letter of the function name is {\tt dh}.

The considered misfit functions are TLS, GTLS, GTLS2, WTLS, and FWTLS. The element-wise versions of the GTLS, GTLS2, and WTLS misfits are specified by the size of the given weight matrices: if vectors are given in {\tt mgtls\{r,p,x,dh\}} and {\tt mgtls2\{r,p,x,dh\}} instead of square weight matrices, then the EWGTLS and EWGTLS2 misfits are computed instead of the GTLS and GTLS2 ones. Similarly, if an $\ttd\times N$ matrix is given instead of an $\ttd\times\ttd\times N$ tensor in {\tt mwtls\{r,p,x,dh\}}, then the EWTLS misfit is computed instead of the WTLS one. The general FWTLS misfit is computed by the functions {\tt mwtls\{r,p,x,dh\}} if the weight matrix is of size $\ttd N\times\ttd N$. The misfit computation functions are summarized in Table~\ref{t4}.

The approximation functions compute a WTLS approximation of the data. The special WTLS problems are called by special functions that are more efficient, see Table~\ref{t5}. As in the misfit computation, the element-wise versions of the functions are recognized by the dimension of the weight matrices. The function {\tt wtls} uses the quasi-Newton optimization algorithm that seems to outperform the alternatives described in Section~\ref{s1}. The alternative methods can be called by the corresponding functions, see Table~\ref{t6}.

\begin{table}[htb!] 
\begin{minipage}{.65\textwidth}
\centering
\caption{Misfit computation functions}\la{t4}
\vskip.15cm
\begin{tabular}{|llll|l|}
\hline
\multicolumn{4}{|c|}{Function} & \multicolumn{1}{|c|}{Description}\\
\hline
{\tt mtlsr}   & {\tt mtlsp}   & {\tt mtlsx}   & {\tt mtlsdh}      & TLS misfit  \\
{\tt mgtlsr}  & {\tt mgtlsp}  & {\tt mgtlsx}  & {\tt mgtlsdh}     & GTLS misfit \\
{\tt mgtls2r} & {\tt mgtls2p} & {\tt mgtls2x} & {\tt mgtls2dh}    & GTLS2 misfit\\
{\tt mwtlsr}  & {\tt mwtlsp}  & {\tt mwtlsx}  & {\tt mwtlsdh}	  & WTLS misfit \\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{.35\textwidth}
\centering
\caption{Approximation functions}\la{t5}
\vskip.15cm
\begin{tabular}{|l|l|}
\hline
Function & \multicolumn{1}{|c|}{Description}\\
\hline
{\tt tls}         & TLS approximation\\
{\tt gtls}	  & GTLS approximation\\
{\tt gtls2}	  & GTLS2 approximation\\
{\tt wtls}	  & WTLS approximation\\
\hline
\end{tabular}
\end{minipage}
\end{table}

\begin{table}[htb!]
\centering
\caption{Auxiliary functions}\la{t6}
\vskip.15cm
\begin{tabular}{|l|l|}
\hline
Function & \multicolumn{1}{|c|}{Description}\\
\hline
{\tt wtlsini}	  & initial approximation for the WTLS approximation functions\\
{\tt wtlsap}	  & WTLS approximation by alternating projections\\
{\tt wtlsopt}     & WTLS approximation by classical optimization methods\\
{\tt qncostderiv} & cost function and gradient for the quasi Newton methods\\
{\tt lmcostderiv} & cost function and Jacobian for the Levenberg--Marquardt method\\
{\tt wtlspr}      & WTLS approximation by the algorithm of~\cite{MRPKV02}\\
\hline
\end{tabular}
\end{table}

%\section{Examples}\la{s3}

\section*{Acknowledgments}

{\small Dr. Sabine Van Huffel is a full professor at the Katholieke Universiteit Leuven, Belgium. Research supported by
Research Council KUL: GOA-Mefisto 666, IDO /99/003 and /02/009 (Predictive computer models for medical classification problems using patient data and expert knowledge), several PhD/postdoc \& fellow grants;  Flemish Government: o FWO: PhD/postdoc grants, projects, G.0078.01 (structured matrices), G.0407.02 (support vector machines), G.0269.02 (magnetic resonance spectroscopic imaging), G.0270.02 (nonlinear Lp approximation), research communities (ICCoS, ANMMM); o IWT: PhD Grants;  Belgian Federal Science Policy Office IUAP P5/22 (`Dynamical Systems and Control: Computation, Identification and Modelling');  EU: PDT-COIL, BIOPATTERN, ETUMOUR.
}

\bibliographystyle{alpha}
\bibliography{bib,mypapers}

\twocolumn
\appendix
\section{Source code}\la{app}
\subsection*{Transformations}
{\scriptsize
\funinput{x2r.m}
\funinput{x2p.m}
\funinput{r2p.m}
\funinput{p2r.m}
\funinput{r2x.m}
\funinput{p2x.m}
\funinput{minr.m}
\funinput{minp.m}
}
\subsection*{TLS and GTLS misfit computation}
{\scriptsize
\funinput{mtlsr.m}
\funinput{mtlsp.m}
\funinput{mtlsx.m}
\funinput{mtlsdh.m}
\funinput{mgtlsr.m}
\funinput{mgtlsp.m}
\funinput{mgtlsx.m}
\funinput{mgtlsdh.m}
\funinput{mgtls2r.m}
\funinput{mgtls2p.m}
\funinput{mgtls2x.m}
\funinput{mgtls2dh.m}
}
\subsection*{TLS and GTLS approximation}
{\scriptsize
\funinput{tls.m}
\funinput{gtls.m}
\funinput{gtls2.m}
}
\subsection*{WTLS misfit computation}
{\scriptsize
\funinput{mwtlsr.m}
\funinput{mwtlsp.m}
\funinput{mwtlsx.m}
\funinput{mwtlsdh.m}
}
\subsection*{WTLS approximation}
{\scriptsize
\funinput{wtlsini.m}
\funinput{wtlsap.m}
\funinput{wtlsopt.m}
\funinput{qncostderiv.m}
\funinput{lmcostderiv.m}
\funinput{wtlspr.m}
}

\end{document}