Data fitting and slop

Hi Im interested to plot XY which is a curve also Im interested to fit a best fit curve through data which should follow the data trend as is and then the point from where curve is changing or having maximum curvature, (X-value at that point) and a best fit line from the trend where values are straight to get intercept, After getting intercept I would like to use intercept and slop to draw another model on same graph. please see the attached figure for detailed idea.

Guide me please Thank you

Answer 1

[dpb]

That's what is known as "piecewise" or "broken-stick" regression model. To incorporate the breakpoint into the model is often called a "change point" regression. It's relatively simple to do, but is a nonlinear problem.

The change point model starts with the broken-stick model, i.e.

Y=b0 + b1(X) + b2(X-C) + e

where

Y is the response variable,
X is the covariate, and
CP is the change point, i.e. where the break occurs.
e is the error term

To formulate a changepoint model into a functional form that can use to estimate the coefficients b and C, write something like

CP=b(4);
if (X < CP) then
  Y = b(1) + b(2)*X;
else
  Y = b(1) + b(2)*X + b(3)*(X-CP);
end

as the functional and use Matlab lsqnonlin

I left CP in as the changepoint to make the code read a little more easily while Matlab uses the array for all the coefficients.

ADDENDUM

That is, the "real" function would look more like--

if (X < b(4)) then    % b4 is the estimated breakpoint value
  Y = b(1) + b(2)*X;
else
  Y = b(1) + b(2)*X + b(3)*(X-b(4));
end