Looking for the Concept About All In One Curve Fitting

Question

I know that there are some Technic for finding curve fitting like polyfit() or pinv(). so we can get a some polynomial equation. it's OK good.

I can get successively a polynomial equation for each curve by using polyfit() or pinv()..it easy. no problem.

But my problems is here.

Actually I want to know a concept to make one equation which from all polynomial equations.

for example, if I get 3 equations(curve #1's equation ,curve #2's equation ,curve #3's equation) then I want to make all in one equation.\

Is this possible? what kinds of Technic is exist?

UPDATE:

I've got 4 polynomials

1) 3.1734x^3−5.8952x^2+3.8194x−0.0377

2) 2.1489x^3−3.9026x^2+2.8169x−0.0160

3) 1.0796x^3−1.9013x^2+1.8500x+0.0008

4) x

But I'm not sure how to apply your way about extracting the corresponding parameters. Would you please help me a small bit more ?

Answer 1

You can always augment the matrices to do so.

Let's assume the first model is given by:

$$ {y}_{1} = {H}_{1} * {\theta}_{1} $$

The second model is given by:

$$ {y}_{2} = {H}_{2} * {\theta}_{2} $$

The third model is given by:

$$ {y}_{3} = {H}_{3} * {\theta}_{3} $$

If we assume the number of parameters of the model are the same, namely $ {\theta}_{1}, {\theta}_{2}, {\theta}_{3} \in {\mathbb{R}}^{d} $ and $ {y}_{i} \in {\mathbb{R}}^{{n}_{i}} $ by the following augmentation:

$$ y = H \theta $$

Where $ y = {\left[ {{y}_{1}}^{T}, {{y}_{2}}^{T}, {{y}_{3}}^{T} \right]}^{T} $, $ H = {\left[ {{H}_{1}}^{T} \mid {{H}_{2}}^{T} \mid {{H}_{3}}^{T} \right]}^{T} $ and $ y = {\left[ {{\theta}_{1}}^{T}, {{\theta}_{2}}^{T}, {{\theta}_{3}}^{T} \right]}^{T} $ you can get:

$$ \hat{\theta} = {\left( {H}^{T} H \right)}^{-1} {H}^{T} y $$

Extracting the the corresponding parameters is easy.

If the parameters vectors are not the same dimension, you can use zeros to make everything work.