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In a calculus textbook by late James Stewart I encountered an exercise to find a power function, as a mathematical model approximately representing data:

The table shows the number N of species of reptiles and
amphibians inhabiting Caribbean islands and the area A of
the island in square miles. 

a) Use a power function to model N as a function of A.

Island      A        N 
Saba        4        5
Monserrat   40       9
Puerto Rico 3,459    40
Jamaica     4,411    39
Hispaniola  29,418   84
Cuba        44,218   76

The confusing thing is that in the previous pages of the textbook there were no explanation of how to do it, except using special software like Mathematica© and only for a linear model.

The correct answer from the textbook is: $N = 3.1046 A^{0.308}$

Could anyone explain to me, please, how this answer was obtained?

I myself found another answer $N = A^{-2.47295*10^{-6} * A + 0.504732}$

by finding $log_A N$ for every item in the table (except the first one, which looked far out of order) and then finding the linear regression for these values with Mathematica©'s tool Fit (which I took as the exponent value for my function):

Island      A        N(real)   N(the textbook's func.) N(my func.)
Saba        4        5         4.75817                 2.01314
Monserrat   40       9         9.6703                  6.43358
Puerto Rico 3,459    40        38.1946                 57.0098
Jamaica     4,411    39        41.1645                 63.0608
Hispaniola  29,418   84        73.848                  85.1851
Cuba        44,218   76        83.724                  68.6738

The right answer definitely looks more close to the reality, so could anyone, please, explain to me, how it was obtained?

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  • 1
    $\begingroup$ This one is even better $N=4.238 A^{0.281}-2.269$ look at the results $$ \begin{array}{l|l} A & N \\ \hline 4 & 3.98762 \\ 40 & 9.68023 \\ 3459 & 39.5724 \\ 4411 & 42.5308 \\ 29418 & 74.0867 \\ 44218 & 83.3508 \\ \end{array}$$ It's a function of Mathematica called NonlinearModelFit $\endgroup$ – Raffaele Oct 27 '17 at 20:45
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You have to log-linearize your relation, i.e.

$N = \alpha A^\beta \iff \ln N =\ln\alpha +\beta \ln A$

If one considers a $6\times2$ matrix $\boldsymbol{X}$ which horizontally stacks a $6\times1$ vector of $1$s with a $6\times1$ vector of $\ln A$, know that computing $(\boldsymbol{X}^{'}\boldsymbol{X})^{-1}\boldsymbol{X}^{'}\ln N$ yields the following $2\times1$ hyper-coefficient $(\ln \alpha, \beta)'$. In matrix terms,

$\ln N = \boldsymbol{X}(\ln \alpha, \beta)' + \boldsymbol{\varepsilon}$

where $\boldsymbol{\varepsilon}$ is a $6\times1$ vector of errors ($0$-centered thanks to the implication of $\ln \alpha$ in the estimation. Doing so yields $e^{\ln \alpha} = \alpha = 3.10462040170919$ and $\beta=0.308044235476563$.


Note that $\boldsymbol{\varepsilon}$ is conceptually not due to the quality of your approximation approach, but is due to the intrinsic stochastic nature of the phenomenon of study. A contrario the vector of residuals, $\widehat{\boldsymbol{\varepsilon}}$, as it stands in $\widehat{\ln N} - \boldsymbol{X}\widehat{(\ln \alpha, \beta)'} = \widehat{\boldsymbol{\varepsilon}}$, entails both $\boldsymbol{\varepsilon}$ and something more which depends on the estimation procedure.

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$$N=\alpha A^{\beta}$$

We want to figure out what is $\alpha$ and $\beta$. By taking logarithm. We obtain

$$\ln N = \ln \alpha + \beta \ln A$$

Regress $\ln N$ on $\ln A$. The slope is $\beta$ and the intercept $c=\ln \alpha$.

$$\alpha = \exp(c)$$

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