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58 lines
2.2 KiB
R
58 lines
2.2 KiB
R
# Goal: To do nonlinear regression, in three ways
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# By just supplying the function to be fit,
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# By also supplying the analytical derivatives, and
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# By having him analytically differentiate the function to be fit.
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#
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# John Fox has a book "An R and S+ companion to applied regression"
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# (abbreviated CAR).
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# An appendix associated with this book, titled
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# "Nonlinear regression and NLS"
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# is up on the web, and I strongly recommend that you go read it.
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#
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# This file is essentially from there (I have made slight changes).
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# First take some data - from the CAR book --
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library(car)
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data(US.pop)
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attach(US.pop)
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plot(year, population, type="l", col="blue")
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# So you see, we have a time-series of the US population. We want to
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# fit a nonlinear model to it.
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library(stats) # Contains nonlinear regression
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time <- 0:20
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pop.mod <- nls(population ~ beta1/(1 + exp(beta2 + beta3*time)),
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start=list(beta1=350, beta2=4.5, beta3=-0.3), trace=TRUE)
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# You just write in the formula that you want to fit, and supply
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# starting values. "trace=TRUE" makes him show iterations go by.
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summary(pop.mod)
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# Add in predicted values into the plot
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lines(year, fitted.values(pop.mod), lwd=3, col="red")
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# Look at residuals
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plot(year, residuals(pop.mod), type="b")
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abline(h=0, lty=2)
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# Using analytical derivatives:
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model <- function(beta1, beta2, beta3, time) {
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m <- beta1/(1+exp(beta2+beta3*time))
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term <- exp(beta2 + beta3*time)
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gradient <- cbind((1+term)^-1,
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-beta1*(1+term)^-2 * term,
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-beta1*(1+term)^-2 * term * time)
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attr(m, 'gradient') <- gradient
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return(m)
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}
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summary(nls(population ~ model(beta1, beta2, beta3, time),
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start=list(beta1=350, beta2=4.5, beta3=-0.3)))
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# Using analytical derivatives, using automatic differentiation (!!!):
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model <- deriv(~ beta1/(1 + exp(beta2+beta3*time)), # rhs of model
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c('beta1', 'beta2', 'beta3'), # parameter names
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function(beta1, beta2, beta3, time){} # arguments for result
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)
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summary(nls(population ~ model(beta1, beta2, beta3, time),
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start=list(beta1=350, beta2=4.5, beta3=-0.3))) |