ross <[EMAIL PROTECTED]> wrote in message news:<[EMAIL PROTECTED]>... > Assume some 'ordinary' data - a set of (x,y) data points, ordered on x, > with x & y real. Assume we draw a boundary somewhere in the middle of > the x domain > (x = k) to divide it into 2 adjacent subdomains. I want to fit 2 > regression lines of the form y = ax + b, one to each of the subdomains, > with a continuity constraint. > > In other words I want to satisfy these criteria: > (C1) the total sum of squared deviates over both intervals is minimised > (C2) the two fitted lines intersect the boundary x = k at the same > point. > > The sums of squared deviates on the two intervals independently would > be: > s1 = S1(y - a1*x - b1)^2 [S1 = sum over n1 points in > subdomain #1] > s2 = S2(y - a2*x - b2)^2 [S2 = sum over n2 points in > subdomain #2] > requiring the 4 parameters a1, b1, a2, b2. > > but these fits are not independent because of criterion #2, which says > that a1*k + b1 = a2*k + b2. This constraint means that we can replace > one of the parameters, eg, b2 = a1*k + b1 - a2*k. > > The total sum to minimise is: > s = s1 + s2 > = S1[ (y - a1*x - b1)^2 ] + S2[ (y - a2*x - b2)^2 ] > = S1[ (y - a1*x - b1)^2 ] + S2[ (y - a2*x - (a1*k + b1 - a2*k))^2 ] > > Then the solution to these 3 simultaneous partial deriv equations: > ds/d(a1) = ds/d(b1) = ds/d(a2) = 0 ..... eq.1 > is the solution to the problem, for a given k. > > Expanding eq.1, I get: > > S1(x^2) +k^2.n2 S1(x) +k.n2 k.S2(x) - k^2.n2 > A = S1(x) + k.n2 n1 + n2 S2(x) - k.n2 > k.S2(x) - k^2.n2 S2(x) - k.n2 S2(x^2) - 2k.S2(x) + k^2.n2 > > S1(xy) + k.S2(y) > C = S1(y) + S2(y) > S2(xy) - k.S2(y) > > and solution B = A \ C where B = column (a1, b1, a2) > > When I program it and plot the results, the fit is clearly not correct. > Is my expansion of eq.1 wrong? Am I introducing the constraint in > the wrong way?
Your expansion seems ok. I programmed it, and the plots don't look bad to me. What is it that looks wrong to you? . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
