Describe how to obtain nonlinear least squares estimates of the parameters in the model y=αx^β + e.

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sbernabel | Student, Graduate | (Level 1) Adjunct Educator

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Suppose you want to fit m data points to your model y = α xβ + e.

Your m data points are (x1,y1),...,(xm,ym). And your initial guess for the parameters is α0,β0,e0.

Let λ = (

α β e

), a 3x1 matrix holding your parameters.

Define d Ui = yi - (α x

β i

+ e) as the residue values. The linearized estimate of these values is

d U i = ∂y(xi) ∂α d α + ∂y(xi) ∂β d β + ∂y(xi) ∂e d e

In our model, the partial derivatives are easily computed,

∂y ∂α = x β , ∂y ∂β = x β log x , ∂y ∂e = 1

Then we can write this in matrix form,

If we let A be the lefthand side matrix and d λ the righthand side, then

d U = A d λ

Multiplying both sides by AT using linear algebra we can solve the following equation for d λ,

a d λ = b

where a≡ATA and b≡ATd U.

This small change d λ is added to our original guess λ0. With that we recompute d U, and solve the linear algebra problem again. The process continues until d λ<ε, where ε is a limit we wish to reach.

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