[R] fitting a sinus curve

[David Winsemius]

On Jul 28, 2011, at 1:07 PM, Hans W Borchers wrote:

> maaariiianne <marianne.zeyringer <at> ec.europa.eu> writes:
>
>> Dear R community!
>> I am new to R and would be very grateful for any kind of help. I am  
>> a PhD
>> student and need to fit a model to an electricity load profile of a
>> household (curve with two peaks). I was thinking of looking if a  
>> polynomial
>> of 4th order,  a sinus/cosinus combination or a combination of 3  
>> parabels
>> fits the data best. I have problems with the sinus/cosinus  
>> regression:
>
> time <- c(
> 0.00, 0.15,  0.30,  0.45, 1.00, 1.15, 1.30, 1.45, 2.00, 2.15, 2.30,  
> 2.45,
> 3.00, 3.15, 3.30, 3.45, 4.00, 4.15, 4.30, 4.45, 5.00, 5.15, 5.30,  
> 5.45, 6.00,
> 6.15, 6.30, 6.45, 7.00, 7.15, 7.30, 7.45, 8.00, 8.15, 8.30, 8.45,  
> 9.00, 9.15,
> 9.30, 9.45, 10.00, 10.15, 10.30, 10.45, 11.00, 11.15, 11.30, 11.45,  
> 12.00,
> 12.15, 12.30, 12.45, 13.00, 13.15, 13.30, 13.45, 14.00, 14.15,  
> 14.30, 14.45,
> 15.00, 15.15, 15.30, 15.45, 16.00, 16.15, 16.30, 16.45, 17.00,  
> 17.15, 17.30,
> 17.45, 18.00, 18.15, 18.30, 18.45, 19.00, 19.15, 19.30, 19.45,  
> 20.00, 20.15,
> 20.30, 20.45, 21.00, 21.15, 21.30, 21.45, 22.00, 22.15, 22.30,  
> 22.45, 23.00,
> 23.15, 23.30, 23.45)
> watt <- c(
> 94.1, 70.8, 68.2, 65.9, 63.3, 59.5, 55, 50.5, 46.6, 43.9, 42.3,  
> 41.4, 40.8,
> 40.3, 39.9, 39.5, 39.1, 38.8, 38.5, 38.3, 38.3, 38.5, 39.1, 40.3,  
> 42.4, 45.6,
> 49.9, 55.3, 61.6, 68.9, 77.1, 86.1, 95.7, 105.8, 115.8, 124.9,  
> 132.3, 137.6,
> 141.1, 143.3, 144.8, 146, 147.2, 148.4, 149.8, 151.5, 153.5, 156,  
> 159, 162.4,
> 165.8, 168.4, 169.8, 169.4, 167.6, 164.8, 161.5, 158.1, 154.9,  
> 151.8, 149,
> 146.5, 144.4, 142.7, 141.5, 140.9, 141.7, 144.9, 151.5, 161.9,  
> 174.6, 187.4,
> 198.1, 205.2, 209.1, 211.1, 212.2, 213.2, 213, 210.4, 203.9, 192.9,  
> 179,
> 164.4, 151.5, 141.9, 135.3, 131, 128.2, 126.1, 124.1, 121.6, 118.2,  
> 113.4,
> 107.4, 100.8)
>
>> df<-data.frame(time,  watt)
>> lmfit <- lm(time ~ watt + cos(time) + sin(time),  data = df)
>
> Your regression formula does not make sense to me.
> You seem to expect a periodic function within 24 hours, and if not  
> it would
> still be possible to subtract the trend and then look at a periodic  
> solution.
> Applying a trigonometric regression results in the following  
> approximations:
>
>    library(pracma)
>    plot(2*pi*time/24, watt, col="red")
>    ts  <- seq(0, 2*pi, len = 100)
>    xs6 <- trigApprox(ts, watt, 6)
>    xs8 <- trigApprox(ts, watt, 8)
>    lines(ts, xs6, col="blue", lwd=2)
>    lines(ts, xs8, col="green", lwd=2)
>    grid()
>
> where as examples the trigonometric fits of degree 6 and 8 are used.
> I would not advise to use higher orders, even if the fit is not  
> perfect.

Thank you ! That is a real gem of a worked example. Not only did it  
introduce me to a useful package I was not familiar with, but there  
was even a worked example in one of the help pages that might have  
specifically answered the question about getting a 2nd(?) order trig  
regression. If I understood the commentary on that page, this method  
might also be appropriate for an irregular time series, whereas  
trigApprox and trigPoly would not?

This is adapted from the trigPoly help page in Hans Werner's pracma  
package:

  A <- cbind(1, cos(pi*time/24), sin(pi*time/24), cos(2*pi*time/24),  
sin(2*pi*time/24))
ab <- qr.solve(A, watt)
ab
# [1] 127.29131 -26.88824 -10.06134 -36.22793 -38.56219
ts <- seq(0, pi, length.out = 100)
xs <- ab[1] + ab[2]*cos(ts) +
       ab[3]*sin(ts) + ab[4]*cos(2*ts) +ab[5]*sin(2*ts)
plot(pi*time/24, watt, col = "red", xlim=c(0, pi), ylim=range(watt),
            main = "Trigonometric Regression")
lines(ts, xs, col="blue")

Hans:  I corrected the spelling of "Trigonometric", but other than  
that I may well have introduced other errors for which I would be  
happy to be corrected. For instance, I'm unsure of the terminology  
regarding the ordinality of this model. I'm also not sure if my pi/24  
and 2*pi/24 factors were correct in normalizing the time scale,  
although the prediction seemed sensible.



>
> Hans Werner
>
>> Thanks a lot,
>> Marianne


-- 
David Winsemius, MD
West Hartford, CT