Linear-logarithmic regression in MATLAB: 2 Input-Parameters

Question

Look at the following plot and ignore the solid lines please (just look at the dotted/dashed ones).

For each curve, g is between [0, 255] (thus always positive), concave, bijective. I know from the process that lies behind the measures, that by increasing V, the corresponding curve flattens.

The different curves result when varying V. The orange curve at the top is for like V=100, the bottom curve (red/magenta) results for V=180.

I have measured data with a lot more data points in the following form:

T[1]  V[1]  g[1]
T[2]  V[1]  g[2]
T[3]  V[1]  g[3]
...   V[1]  g[4]
T[N]  V[1]  g[5]
.......
T[1]  V[N]  g[1]
T[2]  V[N]  g[2]
T[3]  V[N]  g[3]
...   V[N]  g[4]
T[N]  V[N]  g[5]

Now I want a regression like this:

g = g(V, T)

which would yield the curve for a fixed V-value:

g = g(T), V=Vfix

Which regression-funktion in MATLAB do you think would work out the best way? And how to assume a "model" here? I only know (from the process itself AND obviously from the plots), that its some sort of linear curve at the beginning, pass over into a logarithmic curve, but I dont know how the value of V inferfers with it!?

Thanks a lot in advance: for any advice..

Answer 1

@bjoern, for each fixed V, it seems that your curve is concave and only has positive values. Therefore, my first choice is to assume that Y=A X^r. The easiest way to estimate this is to apply log in both sides to get the linear regression log Y = log A + r log X (you probably will find 0<r<1). Therefore, for each value of V, I would use the function regress in matlab applied to the values log Y and log X in order to estimate the parameters A and r. This function is called Cobb-Douglas and it is very useful in economics: http://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function.

For most curves, it seems that the effect of V is well behaved, but the behavior of the blue curve, which is very strange. I would say that in general the effect of V is to translate the points.

If the behavior of V is really linear, maybe you can estimate Y=A V X^r. Therefore, you have to estimate logY = log A + log V + r log X. In this case, your dependent variable is log Y and your independent variables log X and log V.

In both cases, I think that the function regress of matlab does not automatically include the constant of the regression (A for us). So remember to include a vector of ones with the size of your sample as an independent variable as well,

Furthermore, if you really want to test if the behavior of V is linear, just estimate logY = log A + slog V + r log X ehich is equivalent to Y=A V^s X^r

I hope it helps.