[R] non-linear plot parameters

[Philippe Grosjean]

On 26/08/10 19:48, David Winsemius wrote:
>
> On Aug 26, 2010, at 1:35 PM, Marlin Keith Cox wrote:
>
>> I need the parameters estimated for a non-linear equation, an example
>> of the
>> data is below.
>>
>>
>> # rm(list=ls()) I really wish people would add comments to destructive
>> pieces of code.
>
>> Time<-c( 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4,
>> 4, 4, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8)
>> Level<-c( 100, 110, 90, 95, 87, 60, 65, 61, 55, 57, 40, 41, 50,
>> 47,
>> 44, 44, 42, 38, 40, 37, 37, 35, 40, 34, 32, 20, 22, 25, 27,
>> 29)
>> plot(Time,Level,pch=16)
>
> You did not say what sort of "non-linear equation" would best suit, nor
> did you offer any background regarding the domain of study. There must
> be many ways to do this. After looking at the data, a first pass looks
> like this:
>
>  > lm(log(Level) ~Time )
>
> Call:
> lm(formula = log(Level) ~ Time)
>
> Coefficients:
> (Intercept) Time
> 4.4294 -0.1673
>
>  > exp(4.4294)
> [1] 83.88107
>  > points(unique(Time), exp(4.4294 -unique(Time)*0.1673), col="red", pch=4)
>
> Maybe a Weibull model would be more appropriate.
>

... and to continue David's analysis:

Time <- c(0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4,
     4, 4, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8)
Level <- c(100, 110, 90, 95, 87, 60, 65, 61, 55, 57, 40, 41, 50,
     47, 44, 44, 42, 38, 40, 37, 37, 35, 40, 34, 32, 20, 22, 25, 27, 29)
plot(Time, Level, pch = 16)

# First look at log-transformed Level (what David did)
plot(Time, log(Level), pch = 16)
lmod <- lm(log(Level) ~ Time)
summary(lmod)
abline(lmod)
# It is not that clear if log transformation stabilizes variance...

# Given these results, I would tend to use something like this:
plot(Time, Level, pch = 16)
nlmod <- nls(Level ~ exp(a * Time + b) + c,
	start = list(a = coef(lmod)[2], b = coef(lmod)[1], c = 0))
summary(nlmod)
Time2 <- seq(0, 8, by = 0.1)
lines(Time2, predict(nlmod, newdata = list(Time = Time2)), col = "red")

You don't tell us enough information about what you study to help 
further. Choosing a regression model should also be dependent upon a 
priori knowledge about what you study. This looks like an exponential 
decay until a non-null value (c) estimated here around 20.6. Do you know 
it in advance? On the contrary, are you looking for this value? If yes, 
it would be nice, perhaps, to measure 'Level's also for a little bit 
longer 'Time's (until 10 or 12 here).

Hope this helps
Best,

Philippe