[R] fitting a sinus curve

[Hans W Borchers]

David Winsemius <dwinsemius <at> comcast.net> writes:

> 
> 
> 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?


That's true. For the moment, the trigPoly() function works correctly
only with equidistant data between 0 and 2*pi.


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


The error I made myself was to take the 'time' variable literally, though
obviously the numbers after the decimal point were meant as minutes. Thus

  time <- seq(0, 23.75, len = 96)

would be a better choice.
The rest in your adaptation is absolutely correct.

  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))
  # [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.


And yes, this curve is the best trigonometric approximation you can get
for this order(?). You will see the same result when you apply and plot

  xs1 <- trigApprox(ts, watt, 1)

But I see your problem with the term 'order' I will have a closer look 
at this and clarify the terminology on the help page.

[All this reminds me of an article in the Mathematical Intelligencer some
 years ago where it was convincingly argued that the universal constant \pi 
 should have the value 2*pi (in today's notation).]

Thanks, Hans Werner


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