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<DIV><FONT face=Arial>Paul,</FONT></DIV>
<DIV><FONT face=Arial></FONT> </DIV>
<DIV><FONT face=Arial>If you have a look at <A
href="http://www.mathworks.com/moler/leastsquares.pdf">http://www.mathworks.com/moler/leastsquares.pdf</A> you
will find not only a vectorized method, but considerations on what to do
when the data/experient is ill conditioned. Section 5.5 shows the easy way
to calculate the matrix of sums of powers. One consideration is to not
calculate the determinant to see if there is a solution. This calc is very
expensive. Let the algorythm detect singularity.</FONT></DIV>
<DIV><FONT face=Arial></FONT> </DIV>
<DIV><FONT face=Arial>Ray</FONT></DIV>
<BLOCKQUOTE dir=ltr
style="PADDING-RIGHT: 0px; PADDING-LEFT: 5px; MARGIN-LEFT: 5px; BORDER-LEFT: #000000 2px solid; MARGIN-RIGHT: 0px">
<DIV style="FONT: 10pt arial">----- Original Message ----- </DIV>
<DIV
style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B>
<A title=paul.carrico@esterline.com
href="mailto:paul.carrico@esterline.com">Carrico, Paul</A> </DIV>
<DIV style="FONT: 10pt arial"><B>To:</B> <A title=users@lists.scilab.org
href="mailto:users@lists.scilab.org">users@lists.scilab.org</A> </DIV>
<DIV style="FONT: 10pt arial"><B>Sent:</B> Friday, August 27, 2010 5:04
AM</DIV>
<DIV style="FONT: 10pt arial"><B>Subject:</B> [scilab-Users] advice on least
square method</DIV>
<DIV><BR></DIV>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>Dear all,</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>The current email is intend as (an) advice(s) request
regarding the least square method calculations.</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>I'm currently using the function herebellow to
calculate the coefficients of a 2nd order polynomial using thousands of
experimental data (furthermore the function is inserted in a loop =>
thousands of loop = billiards of calculations).</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>So the main time cost is these calculation
...</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>I'm working to improve the code to be faster and in
parallele I would like to know if these a more relevant instruction in
Scilab that performe the calculation faster (maybe more accurate)
?</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>I had a look in the leastsq instruction ... but the
later is strongly influenced by the initial parameters guess X0 (isn't
it).</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>Any advice is welcome</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>Regards / have a good WE</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>Paul</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>PS : A give a basic exampel in
attached</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>PS 2 : additional question (but i'm a newby :
calculations to be parallelized ???)</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>###################################################"</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><IMG
alt="" hspace=0 src="cid:012b01cb45ec$036b8e90$0402a8c0@HOUC35439"
align=baseline border=0></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010>##########################################################</SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">function
R_square = fct_MMC(p,experience)</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// Nota
:</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// p =
polynomial degree</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">//
experience = experimental data</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><FONT face=Arial size=2><SPAN
class=332474908-27082010></SPAN></FONT></SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"></SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><FONT face=Arial
size=2></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><FONT face=Arial
size=2></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// step
1 : "linear" least square method</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">//
matrix size<BR>[n,c] = size(experience);</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// calculation
of S<BR>S = zeros(2*p,1);<BR>for k = 1 : (2*p)<BR> for i = 1 :
n<BR> S(k) = S(k) +
experience(i,1)^k;<BR> end<BR>end</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// calculation
of W<BR>W = zeros(p+1,1);<BR>for k = 1 : (p+1)<BR> for i = 1 :
n<BR> W(k) = W(k) +
experience(i,2)*experience(i,1)^(k-1);<BR> end<BR>end</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// calculation
of A<SPAN class=332474908-27082010> = S matrix</SPAN><BR>A =
zeros(p+1,p+1);<BR>for i = 1 : (p+1)<BR> for j = 1 :
(p+1)<BR> if (i == 1 & j == 1) then<BR> A(1,1)
= n;<BR> else<BR> A(i,j) =
S(i+j-2);<BR> end<BR> end<BR>end</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><BR></SPAN><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// calculation
of B<SPAN class=332474908-27082010> = W matrix</SPAN><BR>B =
zeros(p+1,1);<BR>for j = 1 : (p+1)<BR> B(j) = W(j);<BR>end</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><BR></SPAN><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">// calculation
of the coefficients<BR>if (det(A) == 0) then<BR> printf('
#######################################################################################\n')<BR> printf('
####
Singular matrix => no
solution
####\n')<BR> printf('
#######################################################################################\n')</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">else<BR> <SPAN
class=332474908-27082010>//</SPAN>C = inv(A)*B;</SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"><SPAN
class=332474908-27082010></SPAN>C<SPAN class=332474908-27082010> =
lsq(A,B); // a bit faster than the previous calculation / same
results</SPAN><BR> // calculation of R_square<BR> // formula :
R_square = 1 - ( sum(F_fit - F_measured)^2 / sum(F_mean - F_measured
)<BR> F_fit = zeros(n,1);<BR> R_square = 0;<BR> F_mean =
0;<BR> <BR> for i = 1 : n<BR> // Calcul de
F_fit<BR> for k = 1 :
(p+1)<BR> F_fit(i) = F_fit(i) +
C(k)*experience(i,1)^(k-1) ;<BR> end<BR> end</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"> calculation
of <SPAN style="mso-spacerun: yes"> </SPAN>F_mean<BR> F_mean =
mean(experience(:,2));</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"> //
SSE = sum (F_fit - F_mean)<BR> // SSTE = sum (F_mean -
F_measured)<BR> SSE = 0;<BR> SST = 0;<BR
style="mso-special-character: line-break"><BR
style="mso-special-character: line-break"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"> for
i = 1 : n<BR> SSE = SSE + (F_fit(i) - experience(i,2))^2
;<BR> SST = SST + (F_mean - experience(i,2))^2
;<BR> end</SPAN><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB"> </SPAN><SPAN
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial">R_square = 1 - (SSE /
SST);</SPAN><o:p></o:p></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"> <o:p></o:p></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"> <o:p></o:p></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">end</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">clear
A<BR>clear B<BR>clear S<BR>clear W<BR>clear SSE<BR>clear SST</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><BR></SPAN><SPAN lang=EN-GB
style="FONT-SIZE: 10pt; FONT-FAMILY: Arial; mso-ansi-language: EN-GB">endfunction</SPAN><SPAN
lang=EN-GB style="mso-ansi-language: EN-GB"><o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"> <o:p></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0cm 0cm 0pt"><SPAN lang=EN-GB
style="mso-ansi-language: EN-GB"><o:p> </o:p></SPAN></P><PRE>--------------------------------------------------------------------------------
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