Question 1.5.3: Fitting a Power Function with a Known Exponent Fit the power...

Fitting a Power Function with a Known Exponent

Fit the power function y = b x^{m} to the data y_{i} . The value of m is known.

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The least-squares criterion is
J = \sum^{n}_{i=1}{(bx^{m}  −  y_{i})^{2}}
To obtain the value of b that minimizes J , we must solve ∂J/∂b = 0.
\frac{∂J}{∂b} = 2  \sum^{n}_{i=1} {x^{m}_{i} (b x^{m}_{i}  −  y_{i}) = 0}
This gives
b = \frac{\sum^{n}_{i=1}  {x^{m}_{i} y_{i}}}{\sum^{n}_{i=1}  {x^{2m}_{i}}}                          (1)

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