Having run the routine again on sine waves of various amplitudes I have plotted boxplots and kernal density plots of the ucl and lcl optimised multipliers for sine waves of periods n=10 to 50 inclusive, using R. Visual inspection of the plots suggest that, in general* (see note below), there are no real differences between the distributions and a typical plot is shown above (again this is for a sine wave of period n=20). The black plots are
- optimised multiplier for ucl of sine wave with amplitude of 1
- optimised multiplier for lcl of sine wave with amplitude of 1
- optimised multiplier for ucl of sine wave with variable amplitude
- optimised multiplier for lcl of sine wave with variable amplitude
Obviously just looking at plots is not statistically robust, so the next thing I did was to run Kolmogorov-Smirnov tests in R to see if the null hypothesis that the four separate distributions come from the same distribution can be rejected. For this purpose I used a p-value of 0.05 for the significance level. The comparative test runs were
- ucl sine wave amplitude 1 and lcl sine wave amplitude 1
- ucl sine wave variable amplitude and lcl sine wave variable amplitude
- ucl sine wave amplitude 1 and ucl sine wave variable amplitude
- lcl sine wave amplitude 1 and lcl sine wave variable amplitude
if A=B and C=D and A=C and B=D then A=B=C=D.
The actual test in R that was used was the Bootstrap Kolmogorov-Smirnov package ks.boot(), the R documentation description for which is quoted below:-
"This function executes a bootstrap version of the univariate Kolmogorov-Smirnov test which provides correct coverage even when the distributions being compared are not entirely continuous. Ties are allowed with this test unlike the traditional Kolmogorov-Smirnov test"
It is obvious from the discrete nature of the raw data being tested that this test is more appropriate than the traditional Kolmogorov-Smirnov test, but more importantly the symmetrical nature of a sine wave almost guarantees that there are likely to be ties in the data, hence the choice of this test. The number of bootstraps performed for each test was n=1000. The p-value was sufficiently low to reject the null hypothesis for the following periods (numbered bullet points correspond to tests identified above)
- Periods 16 ,24, 32, 40 and 48
- Periods 16, 24, 27, 32, 40, 45, 48, 49
- Periods 38 and 49
- Periods 38 and 49
*The exceptions to the general pattern mentioned above will be the subject of a later blog entry.