One of the doctors brought by a data set that showed the average volume of business
(number of beds filled) in a month for 28 consecutive months starting in January 2005. Here
is the data:
17,21,18,16,10, 6, 7, 9,14,16,12,12,
24,28,21,13, 9, 8, 8, 8,12,12,14,19,
29,29,27,12
Here is a plot of the 28 consecutive values.
and here is the plot aligning the months in a given year (5=2005, 6=2006, 7=2007).
Notice that the number of beds filled is highest in the wintertime and lowest in the
summertime. Also notice that there is slight upward trend over time. If you were trying to
estimate the magnitude of this slight upward trend, you would need to account for the
cyclical pattern as well. A simple way to estimate a cyclical pattern is to use a bit of
trigonometry. Do you remember the sine function from your trig class in high school? Here is
an example of a sine wave.
Notice that it has a cyclical trend with regular peaks and troughs, but this is not the
sine wave that we want to fit to our data. To fit a sine wave properly, you need to estimate
three quantities: the amplitude, the period/frequency, and the phase.
Amplitude is the distance from the middle of the sine curve to the peak (or equivalently,
the distance from the middle to the trough).
Period is the distance that a sine wave travels before it repeats itself. Period is
inversely related to frequency. You can define the frequency as the number of times
(sometimes a fractional number of times) that a sine wave repeats in the interval 0 to 2pi.
Phase is the amount that the sine wave is shifted from the origin.
In our example, we know the period (12 months), but we need to calculate the amplitude and
the phase. Problems where the period/frequency is unknown represent a special challenge and
use approaches like Fourier analysis. I want to skip the complexities of Fourier analysis for
now, because estimating period/frequency and phase is already more than enough challenge. One
formula that will produce a sine wave with a period of 12 and variable amplitude and phase is
This is a nonlinear model, because the term phi is inside the sin function. We can find a
more convenient form that does not require the use of nonlinear regression by remembering the
formula for the sine of a sum of two angles
If you apply this formula, the equation changes to
where
and
With a bit of work, you can show that the amplitude and phase are simple functions of B
and C.
The code in R to fit a linear trend in the presence of cyclical variation is
t <- seq(0,27)
sint <- sin(2*pi*t/12)
cost <- cos(2*pi*t/12)
lm.0 <- lm(x~t+sint+cost)
The results of this fit are
Coefficients:
(Intercept) t sint cost
12.5009 0.1550 1.7619 7.6939
and you can compute the amplitude as
sqrt(1.7619^2+7.6939^2)
[1] 7.89306
This means that the peak season has about 8 more filled beds than the overall yearly
average and the slow season has about 8 fewer filled beds than the overall yearly average.
The phase is computed as
> atan(-7.6939/1.7619)*12/(2*pi)
[1] -2.570058
Notice that the sine wave starts to the left of zero, but since it repeats every 12
months, you can compute an alternate starting point as
> 12-2.570058
[1] 9.429942
This tells you that the start of the sine curve is in mid-September. This starting point
is the halfway marker between the trough and the peak. You can find the location of the
trough and peak by simply subtracting and adding three months, respectively. Thus, the low
point for number of beds occurs in mid-June (6.429942) and the high point occurs in
mid-January (12.429942 or equivalently 0.429942).
The linear trend term is 0.1550 which tells you that the number of filled beds increases
on average by about 0.155 per month (or equivalently 1.86 filled beds per year).
Here is a plot of the data and the estimated equation.
The model used here is overly simplistic for two reasons. First, it fails to take
advantage of the fact that counts often followed a Poisson distribution. Second, it fails to
exploit the tendency for consecutive months values to exhibit a positive autocorrelation.
When I have time, I will illustrate how to produce a more complex model that accounts for
these facts.
07/08/2008.
