Difficulty in fitting two lines with polynomials

Hi, I want to fit data which resembles two piecewise linear functions using polynomials.

Y=30000X for X<=0.001;

Y=27+3000X for 0.001<X<0.003.

When I use polyfit or cftool, the fitted curve has ups and downs (change of sign in curvature) which I don't want. I'm getting smooth curve when I use polynomials of the order 20 which is really high.

I would prefer a fitted curve without change in sign of curvature even at the expense of some accuracy (with lower orders of polynomials).

How can I do it?

Thanks, Deepesh.

[Image Analyst]

Attach your data (so we can try things with it) and attach a screenshot so we can see what you're seeing.

Answer 1

[dpb]

Piecewise linear regression is fairly easy to code anyway...the algebra to add the condition to match the two at the breakpoint is

y = a1 + b1 x, x<=c,
y = a2 + b2 x, x>c.

Match breakpoint, or a1 + b1 c = a2 + b2 c. Rearrange this to isolate (say) a2 as

a2 = a1 + b1 c - b2 c --> a1 + c(b1-b2) = aprime

Now have

y = a1     + b1 x, x<=c,
y = aprime + b2 x, x>c.

This is easy-peasy to code in Matlab and use as target for nlinfit --

function y=piecewise(coef,x)
% return piecewise linear fit given a1,b1,c,b2 as coefficient array and vector x
% c is breakpoint, a1 is intercept of first section, b1,b2 are two segment slopes

a1=coef(1); b1=coef(2);  % reduce parentheses clutter....
c=coef(3); b2=coef(4);
ix=x>c;                  % location > breakpoint
y(ix)=[a1+c*(b1-b2)+b2*x(ix)];
y(~ix)=[a1+b1*x(~ix)];
y=y(:);

Use this as

coeff=nlinfit(X,Y,@piecewise,coeff0);

Example: for Hayden dataset

www.fs.usda.gov/rds/fedora/objects/RDS:RDS-2007-0004/datastreams/RDS-2007-0004/content

>> coeff=nlinfit(X,Y,@piecewise,[0 .01 1.5 0.1])
coeff =
 -0.0077    0.0155    1.6804    0.1067
>>

Answer 2

[dpb]

Classic piecewise regression problem...use some of the File Exchange tools...

http://www.mathworks.com/matlabcentral/fileexchange/24443-slm-shape-language-modeling http://www.mathworks.com/matlabcentral/fileexchange/40913-piecewise-linear-least-square-fit

[Deepesh upadrashta]

HI,

My problem is approximating two piecewise lines with polynomial.

After reading SLM help, I don't think it can do that. It is able to fit the data with polynomials upto 3 degree between knots. Thanks.

[dpb]

A cubic spline is a polynomial, just a particular form of one... :)

It would appear that the other link is a linear piecewise polynomial w/ specified knot location.

Do you want/need continuous first/second derivatives or simply only two slopes?