**roskar@jhunix.HCF.JHU.EDU (Veljko Roskar)** (10/05/90)

Since only one person replied to my question about fitting a surface to 3-dimensinal data, I am asking again for help. This time I will be more general: Using Mathematica, is there an easy procedure to plot both 3-dimensional points and a surface that gives a best fit to these points? The data is linear in one dimension and has a polynomial dependency in the other, but a cubic-spline fit would give the best fit in both dimensions. Because of my time constraint, I cannot code the procedures from Numerical Recipes in Fortran. Besides it's the 3D plot I'm after, not an equation to model the data. Below is a summary (long) of the correspondence I've exchanged with Jerry who was my only help so far: This is what my data looks like, the third variable is a function of the first two: ^L ... My data: co1=List[{.807,1.0,22.84},{.575,1.0,12.36},{.417,1.0,7.43},{.267,1 .0,3.44},{0,1.0,1.23}] co2=List[{.807,1.1,24.41},{.575,1.1,13.82},{.417,1.1,8.40},{.267,1 .1,3.85},{0,1.1,1.27}] co3=List[{.807,1.2,25.97},{.575,1.2,15.29},{.417,1.2,9.37},{.267,1 .2,4.26},{0,1.2,1.31}] co4=List[{.807,1.3,27.54},{.575,1.3,16.75},{.417,1.3,10.3},{.267,1 .3,4.67},{0,1.3,1.35}] Note that the first variable repeats itself in every Rco#S set, and that the second one is constant in each set. ... +++++++++++++++ From: IN%"trantow%csd4.csd.uwm.edu%rutgers@APLCEN.BITNET" "Jerry J Trantow" 2-OCT-1990 10:06:11.71 To: roskar%jhunix%aplcen@mimsy Subj: RE: Fitting surfaces (planes) to (x,y,f(x,y)) data:Mathematica I don't think Fit[] will do what you want. What you really want is simple multiple regression. I just had to do this for a HW assignment and I spent a few extra minutes doing the graphics you also requested. I'm not sure how this will cut/paste, but here goes Problem 1.4.8 Multiple Regression Multiple Regression fits two-dimensional data by a plane y=C+Dt+Ez, instead of one-dimensional data by a line. If we are given (t,z,b) Write down the 4 equations in the 3 unknowns C,D,E. What is the least squares solution from the normal equations? givendata={{0,0,2},{0,1,2},{1,0,1},{1,1,5}} {{0, 0, 2}, {0, 1, 2}, {1, 0, 1}, {1, 1, 5}} A=Table[{1,givendata[[i,1]],givendata[[i,2]]},{i,1,4}];MatrixForm[A] 1 0 0 1 0 1 1 1 0 1 1 1 b=Table[{givendata[[i,3]]},{i,1,4}] {{2}, {2}, {1}, {5}} Aprime=Transpose[A].A;bprime=Transpose[A].b {{10}, {6}, {7}} solution=LinearSolve[Aprime,bprime] {{1}, {1}, {2}} yplane[w_]=Transpose[solution].w {{1, 1, 2}} . w ypoly=Polygon[{{0,0,yplane[{{1},{0},{0}}][[1,1]]}, {0,1,yplane[{{1},{0},{1}}][[1,1]]}, {1,0,yplane[{{1},{1},{0}}][[1,1]]}, {1,1,yplane[{{1},{1},{1}}][[1,1]]}}] Polygon[{{0, 0, 1}, {0, 1, 3}, {1, 0, 2}, {1, 1, 4}}] abc=Table[Graphics3D[Point[givendata[[i]]]],{i,1,4}] Show[Graphics3D[ypoly],ViewPoint->{2,1,2}] Show[Graphics3D[PointSize[.02]],abc,ViewPoint->{2,1,2}] Show[Graphics3D[PointSize[.02]],abc,Graphics3D[ypoly],ViewPoint->{2,1,2}] I will also pop a copy of this into snail mail for you. Good luck. .... ************ ^L My reply: You mailed me: ... Problem 1.4.8 Multiple Regression Multiple Regression fits two-dimensional data by a plane y=C+Dt+Ez, instead of one-dimensional data by a line. If we are given (t,z,b) Write down the 4 equations in the 3 unknowns C,D,E. What is the least squares solution from the normal equations? ... Did you mean y=Cb+Dt+Ez? There is also the problem that in one dimension my data is linear (the third variable w/respect to the second--from co1 to co4), and in the second dimension it best fits a polynomial of the 2nd order (the third variable with respect to the first--for each co# set). So my surface will look like a bent sheet of paper that's angled about an axis perpendicular to the "axis" of the bend. Any ideas? ... *************** ^L From: IN%"trantow@csd4.csd.uwm.EDU" "Jerry J Trantow" 2-OCT-1990 22:19:44.39 To: ROSKAR@JHUVMS.BITNET, trantow@csd4.csd.uwm.EDU Subj: RE: Fitting a plane to 3D data The example I gave was correct as sent. In one dimension y=-mx+b with x being the variable In two dimensions, you are fitting y=C+Dt+Ez with t and z being the variables. I didn't pick the variable names they were given in the book. If your data requires a second order polynomial, I suspect you could set up the equation to be y=C+Dt+Ez+Fx^2 the same technique as I gave should work. I mailed it off to you today. Find a linear algebra or statistic book that shows how to do multiple regression. Your problem does not seem very bad. **************** End of included text. -- Veljko Roskar | roskar@jhuvms.bitnet Department of Chemical Engineering | roskar@jhuvms.hcf.jhu.edu The Johns Hopkins University, Baltimore | uunet!mimsy!aplcen!jhunix!roskar