Using R for Introductory Statistics : John Verzani

Using R for introductory statistics 336The initial parameter guesses are often found by doing some experimental plots. Thesecan be done quickly using the curve () function with the argument add=TRUE, asillustrated in the examples. When we model with a given function, it helps to have ageneral understanding of how the parameters change the graph of the function. Forexample, the parameters in the exponential model, writtenmaybe interpreted by t 0 being the place where we want time to begin counting, N the initialamount at this time, and r the rate of decay. For this model, the mean of the data decaysby 1/e, or roughly 1/3 in 1/r units of time.Some models have self-starting functions programmed for them. These typically startwith SS. A list can be found with the command apropos("SS"). These functions do notneed starting values.■ Example 12.5: Yellowfin tuna catch rate The data set yellowf in (UsingR)contains data on the average number of yellowfin tuna caught per 100 hooks in thetropical Indian Ocean for various years. This data comes from a paper by Myers andWorm (see ?yellowf in) that uses such numbers to estimate the decline of fish stocks(biomass) since the advent of large-scale commercial fishing. The authors fit theexponential decay model with some threshold to the data.We can repeat the analysis using R. First, we plot (Figure 12.2).> plot(count ~ year, data=yellowfin)A scatterplot is made, as the data frame contains two numeric variables. The countvariable does seem to decline exponentially to some threshold. We try to fit the modelY=N(e −r(t−1952) (1−d)+d)+ε.(Instead of β i we give the parameters letter names.)To fit this in R, we define a function for the mean> f = function(t, N, r, d) N*(exp(-r*(t-1952))*(l-d)+d)We need to find some good starting points for nls (). The value of N=7 seems about right,as this is the starting value when t=1952. The value r is a decay rate. It can be estimatedby how long it takes for the data to decay by roughly 1/3. We guess about 10, so we startwith r=1/10. Finally, d is the percent of decay, which seems to be .6/6 = .10.We plot the function with these values to see how well they fit.> curve(f(x, N=6, r=1/10, d=0.1), add=TRUE)The fit is good (the solid line in Figure 12.2), so we expect nls() to converge with thesestarting values.> res.yf = nls(count ~ f(year, N, r, d),start=c(N=6,r=1/10, d=.1),+ data=yellowfin)> res.yf