.TH " monolr" 2 " March 9, 1998" "Fractales Group" "Scilab Function"
.so ../sci.an
.SH NAME
 monolr -  monovariate linear regression
.sp
Author: Christophe Canus
.sp
This C_LAB routine provides six different algorithms to proceed linear
regression on monovariate data:  least square, weighted  least square,
penalized least square, multiple least square, maximum likelyhood and
Lepskii's adaptive procedure least square, in one sole routine.
.sp
.sp
.SH Usage
\f(CR[\fPa_hat,\f(CR[\fPb_hat,y_hat,e_hat,sigma2_e_hat,optvarargout\f(CR]\fP=
monolr(x,y,\f(CR[\fPlrstr,optvarargin\f(CR]\fP)
.SH Input parameters


.RS

.TP
o 
\fBx\fP : real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fPJ,1\f(CR]\fP 
Contains the abscissa.

.TP
o 
\fBy\fP : real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fPJ,1\f(CR]\fP 
Contains the ordinates to be regressed.

.TP
o 
\fBlrstr\fP : string 
Contains the  string which specifies  the type of linear regression to
be used.

.TP
o 
\fBoptvarargin\fP : 
Contains optional variable input arguments. Depending on the choice of
linear regression, the fourth parameter can be


.RS

.TP
o 
\fBw\fP : strictly positive real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fPJ,1\f(CR]\fP 
If weighted least square is chosen, contains the weights.

.TP
o 
\fBI\fP : strictly positive real (integer) scalar 
If penalized least square is chosen, contains the number of iterations.

.TP
o 
\fBsigma2_j\fP : strictly positive real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fPJ,1\f(CR]\fP 
If Lepskii's  adaptive procedure least square  is chosen, contains the
sequence of variances.
.RE

The fifth parameter can be


.RS

.TP
o 
\fBm\fP : real scalar 
If penalized least square is chosen, contains the mean of the normal weights.

.TP
o 
\fBK\fP : strictly positive real scalar 
If Lepskii's adaptive procedure least square is chosen, contains the
confidence constant.
.RE

The sixth parameter can be


.RS

.TP
o 
\fBs\fP : strictly positive real scalar  
If penalized  least  square is  chosen, contains  the variance  of the
normal weights.
.RE

.RE

.SH Output parameters


.RS

.TP
o 
\fBa_hat\fP : real scalar or vector \f(CR[\fP1,J\f(CR]\fP 
Contains the estimated slope.

.TP
o 
\fBb_hat\fP : real scalar or vector \f(CR[\fP1,J\f(CR]\fP 
Contains the estimated ordimate at the origin.

.TP
o 
\fBy_hat\fP : real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fP1,(J+2)*(J-1)/2\f(CR]\fP 
Contains the regressed ordinates.

.TP
o 
\fBe_hat\fP : real vector \f(CR[\fP1,J\f(CR]\fP or
\f(CR[\fP1,(J+2)*(J-1)/2\f(CR]\fP 
Contains the residuals.

.TP
o 
\fBsigma2_e_hat\fP : real scalar  
Contains the residuals' variance (that is, the mean square error).

.TP
o 
\fBoptvarargout\fP : 
Contains optional  variable output  arguments.  If  Lepskii's adaptive
procedure least square is chosen, the parameters are


.RS

.TP
o 
\fBK_star\fP : strictly positive real scalar 
Contains the optimal confidence constant.

.TP
o 
\fBj_hat\fP : strictly positive real (integer) scalar 
Contains the selected index.

.TP
o 
\fBI_c_j_min\fP : real vector \f(CR[\fP1,J\f(CR]\fP 
Contains the minimum bounds of the confidence intervals.

.TP
o 
\fBI_c_j_max\fP : real vector \f(CR[\fP1,J\f(CR]\fP 
Contains the maximum bounds of the confidence intervals.

.TP
o 
\fBE_c_j_hat_min\fP : real scalar  
Contains the minimum bound of the selected intersection interval.

.TP
o 
\fBE_c_j_hat_max\fP : real scalar  
Contains the maximum bound of the selected intersection interval.
.RE

.RE

.SH Description
.SH Parameters 
The abscissa \fIx\fP and the ordinate \fIy\fP  to be regressed with must
be of the same size \f(CR[\fP1,J\f(CR]\fP or \f(CR[\fPJ,1\f(CR]\fP.
The linear regression string \fIlrstr\fP  specifies the type of  linear
regression used.  It can be '\fIls\fP' for least square, '\fIwls\fP' for
weighted least square, '\fIpls\fP' for penalized least
square,   '\fImls\fP' for multiple   least square  (that is for  \fIj\fP
varying from  \fI1\fP  to \fIJ\fP),  '\fIml\fP' for  maximum
likelyhood,  '\fIlapls\fP'  for   Lepskii's adaptive   procedure  least
square.  The default value for \fIlrstr\fP is '\fIls\fP'.
The weights \fIw\fP or the sequence  of variances \fIsigma2_j\fP must be
strictly positive and of size \f(CR[\fP1,J\f(CR]\fP or \f(CR[\fPJ,1\f(CR]\fP.
For   the   meaning  of  the     variable  optional input   parameters
\fIsigma2_j\fP  and  \fIK\fP,   see  \fIlepskiiap\fP (Lepskii's  Adaptive
Procedure) C_LAB routine's help.
The number of iterations \fII\fP must be \fI>=2\fP.
The variance of the normal weights \fIs\fP must be strictly positive.
If multiple least square,   maximum likelyhood or Lepskii's   adaptive
procedure  least square is chosen,  the estimated slope \fIa_hat\fP and
the ordinate   at   the   origin  \fIb_hat\fP  are   vectors  of   size
\f(CR[\fP1,J\f(CR]\fP, resp.  the  regressed  ordinates  \fIy_hat\fP  and the
residuals \fIe_hat\fP vectors  are  of size \f(CR[\fP1,(J+2)*(J-1)/2\f(CR]\fP
(as they contains results for multiple  linear regression, be aware of
that  when vizualising  them :-),  see examples),  otherwise there are
scalars,   resp.  vectors   of size     \f(CR[\fP1,J\f(CR]\fP.  For  maximum
likelyhood, multiple least square  linear regressions are proceeded in
order to obtain  variance estimates.   Then maximum likelyhood  linear
regression is     proceeded   (corresponding results   are    found in
\fIa_hat(1)\fP,  \fIb_hat(1)\fP, \fIy_hat(1:J)\fP,  \fIe_hat(1:J)\fP   and
\fIsigma2_e_hat(1)\fP, see examples).
For  the  meaning   of   the   variable  optional   output  parameters
\fIK_star\fP,    \fIj_hat\fP,         \fII_c_j_min\fP,    \fII_c_j_max\fP,
\fIE_c_j_max\fP, and   \fIE_c_j_max\fP,  see  \fIlepskiiap\fP  (Lepskii's
Adaptive Procedure) C_LAB routine's help.
.SH Algorithm details
For   the   details  of  the      Lepskii's  adaptive procedure,   see
\fIlepskiiap\fP (Lepskii's Adaptive Procedure) C_LAB routine's help.
.SH Examples
.SH Matlab
.sp
.ft CR
.nf
J=32;
x=1+linspace(0,1,J);
% Wiener process
W=randn(1,J); 
epsilon=.1;
y=x+epsilon*W;
% least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y);
plot(x);hold on;plot(y);plot(y_hat,'kd');
plot(epsilon.*W);hold on;plot(e_hat); 
title('least square');
disp('type return');
pause;
clf;
% weighted least square
epsilon=linspace(.05,.5,J);
y=x+epsilon.*W;
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'wls',1./epsilon);
plot(x);hold on;plot(y);plot(y_hat,'kd');
plot(epsilon.*W);hold on;plot(e_hat); 
title('weighted least square');
disp('type return');
pause;
clf;
% penalized least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'pls',30);
plot(x);hold on;plot(y);plot(y_hat);
title('penalized least square');
disp('type return');
pause;
clf;
% multiple least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'mls');
plot(x);hold on;plot(y)
start_j=0;
hold on;
for j=2:J
  plot([1:j],y_hat(start_j+1:start_j+j),'k');
  disp(['estimated slope a_hat =',num2str(a_hat(j))]);
  disp('type return');
  pause;
  start_j=start_j+j;
  j=j+1; 
end
clf
% maximum likelyhood
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'ml');
plot(x);hold on;plot(y_hat(1:J),'kd');
plot(epsilon.*W);hold on;plot(e_hat(1:J));
clf;
% Lespkii's adaptive procedure
epsilon=.01;
y(1:16)=x(1:16)+epsilon*W(1:16);
y(16:32)=2*x(16:32)+epsilon*W(16:32);
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat,K_star,j_hat,I_c_j_min,I_c_j_max,E_c_j_hat_min,E_c_j_hat_max]=monolr(x,y,'lapls');
plot(a_hat);
hold on;
plot(I_c_j_max,'r^');
plot(I_c_j_min,'gV');
title('LAP: estimator vs. index');
xlabel('index: j');
ylabel('estimator: \theta_j');
plot(j_hat,E_c_j_hat_min,'ko');
plot(j_hat,E_c_j_hat_max,'ko');
.fi 
.ec
.ft P
.sp
.SH Scilab
.sp
.ft CR
.nf
// 
.fi 
.ec
.ft P
.sp
.SH References
To be published..SH See Also
\fBlepskiiap\fP (C_LAB routine).