TUdataLSq3D TCL UTILITIES USER MANUAL TUdataLSq3D
SYNOPSIS
Perform a 3D least squares fit to a set of data.
PACKAGE
TCLUTILS
NAME
TUdataLSq3D
USAGE
set rV  [TUdataLSq2D  X Y Z V nP nOrder Mode CoeF {Var} {SigV}]
INPUT DEFINITIONS
X - Input array of X values
Y - Input array of Y values
Z - Input array of Z values
V - Input array of V values
nP - The number of input (X,Y,Z,V) data values
nOrder - The order of the fitting polynomial
Mode - The weighting mode to use
-N : Weight data as (X*X + Y*Y)-N/2.0. (X*X + Y*Y) = 0 has a weighting function of 1.
0 : No weighting applied. All data have weighting function of 1.
1 : Used the supplied weighting function SigV.
CoeF - The return fit coefficients.
Var - Optional input which indicated if the varience to fit should be computed and returned. Default is Yes.
0 : No
1 : Yes
SigV - Optional array of weighting values to use in the fit. Used only if wMode is set to 1.
RETURN DEFINITION
rV - This is a list containing the number of coefficients used in the fit and the varience of the fit to the data. The latter is included in the list only if Var is 1.
DESCRIPTION
TUdataLSq3D performs a least squares 3D fit to a set of data set using the polynomial expression 
      Z = A(0) + A(1)*X + A(2)*Y + A(3)*Z + 
          A(4)*X2 + A(5)*X*Y + A(6)*X*Z + A(7)*Y2 + A(8)*Y*Z + A(9)*Z2 ...
returning the coefficients A. Both the number of coefficients and the varience (if computed) are returned through the function.
ERRORS
None Generated
C BACKING
No
EXAMPLE(S)
EXAMPLE 1: 
# FIT a noisy spherical surface data set.  Data is scattered about a 
#   surface centered on 0 with a radius of 4.  Use the fit to find
#   the best radius

# MAKE up the data

set nP 0
for { set I 0 } { $I < 72 } { incr I } {
   set Phi [expr ($I * 4.0) / $RtoD]
   for { set J 0 } { $J <= 36 } { incr J ; incr nP } {
      set Theta [expr ($J * 5.0) / $RtoD]
      set X($nP) [expr cos($Phi) * sin($Theta)]
      set Y($nP) [expr sin($Phi) * sin($Theta)]
      set Z($nP) [expr cos($Theta)]
      set R [expr 4.0 + [TUdataRnd1 V 1 PN .5]] 
      set V($nP) [expr $R * $R] 
   }
}

# SOLVE for the surface

set rV [TUdataLSq3D X Y Z V $nP 2 0 A 1]

# PRINT out the data

puts stderr "Number Of Coefficients: [lindex $rV 0]"
puts stderr "Varience   : [format "%.4f" [lindex $rV 1]]"
puts stderr "A(0) Const: [format "%.4f" $A(0)]"
puts stderr "A(1) X    : [format "%.4f" $A(1)]"
puts stderr "A(2) Y    : [format "%.4f" $A(2)]"
puts stderr "A(3) Z    : [format "%.4f" $A(3)]"
puts stderr "A(4) X*X  : [format "%.4f" $A(4)]"
puts stderr "A(5) X*Y  : [format "%.4f" $A(5)]"
puts stderr "A(6) X*Z  : [format "%.4f" $A(6)]"
puts stderr "A(7) Y*Y  : [format "%.4f" $A(7)]"
puts stderr "A(8) Y*Z  : [format "%.4f" $A(8)]"
puts stderr "A(9) Z*Z  : [format "%.4f" $A(9)]"

# FIND the average radius of the sphere from the fit

set nC [lindex $rV 0]
set R 0.0
for { set I 0 } { $I < $nP } { incr I } {
   set R [expr $R + [TUpolyExp3D $X($I) $Y($I) $Z($I) $nC A 2]]
}
puts stderr "Average Radius:  [format "%.4f" [expr sqrt($R / double($nP))]]"

> Number Of Coefficients: 10
> Varience   : 2.3567
> A(0) Const: -640.2834
> A(1) X    : -0.0738
> A(2) Y    : -0.0096
> A(3) Z    : -0.0444
> A(4) X*X  : 656.3884
> A(5) X*Y  : -0.1245
> A(6) X*Z  : -0.0950
> A(7) Y*Y  : 656.7266
> A(8) Y*Z  : -0.0206
> A(9) Z*Z  : 656.3282
> Average Radius:  4.0202
Sept 17, 2006