In the above section, we saw that the tests using Fisher’s G-statistic and Bartlett’s method perform very well. However, when running the simulations to compute the results, we noticed that there were specific values that consistently caused all the tests to incorrectly fail to reject the null hypothesis under the alternative. When analysing these values, it was found that they were multiples of the period or a fraction of a multiple of the period. We consider these two cases separately.
5.5.1 Multiples of the Period
Consider a periodic function with period 3. This means that the function repeats itself on intervals of length 3. Thus, it will also repeat itself on intervals of length 6 and 9. More specifically, it will repeat itself on any interval of length equal to a multiple of 3. Then if we run the tests on this function with hypothesised periods equal to any multiple of 3, the tests will fail to reject the null hypothesis. This is because permuting with a multiple of 3 will not alter the shape of the function and thus the periodogram values will not average out. Due to this limitation, the power of the tests diminish.
CHAPTER 5. RESULTS & COMPARISON OF METHODS
Power Function
For the given problem we are considering a test with a two-sided alternative hypothesis. For a general test with a two-sided alternative hypothesis, the power function has a shape similar to the one in Figure5.19. In this figure, we see that under the null hypothesis the power of the function drops to the significance level α, which can be considered the power of the test under the null hypothesis.
Figure 5.19: Illustrates the general structure of a power function of a test with a two-sided alter-native hypothesis.
In Figure5.20, we have plotted the power function of the test using Fisher’s G-statistic for cosine functions with period 5/2. Note this figure cannot generally be computed; we can only plot them because we can simulate several data sets. For this plot, the data set were of size 150 and 1000 Monte Carlo iterations were used for permuting method 1. In this figure, we see that the power of the test drops to 0 at every multiple of 5/2. In Section5.3we saw that the fraction we reject under the null is zero, therefore, α ≈ 0 for the test using Fisher’s G-statistic. Thus, the power function drops to α at every multiple of the period 5/2 making the power function periodic as well.
Comparing Figures5.19 and5.20we notice that this is not the typical behaviour seen for tests.
This limitation stems from the definition of periodicity and is difficult to eliminate. However, there are ways to work around it. These approaches are discussed in section5.5.3.
Figure 5.20: Shows the periodicity in the power for the test using Fisher’s G-statistic for the cosine function with Period 5/2. Note this figure cannot generally be computed; we can only plot them because we can simulate several data sets.
CHAPTER 5. RESULTS & COMPARISON OF METHODS
5.5.2 Fractions of a Multiple of the Period
In Chapter4we defined three ways to permute the observations without altering the period of the function. Method 1 only permutes blocks of length p. Therefore, if p is a multiple of τ0, the shape of the function under permutations will not change. Due to this, the test will incorrectly fail to reject the null hypothesis if p is a multiple of the period, regardless of the value of q. This means that if the period of the function is 5/2, the test will fail to reject the value of 15/4. Clearly, 15/4 is not a multiple of 5/2, however, method 1 only permutes blocks of length 15 which is a multiple of 5/2. Therefore, the test fails to reject the null hypothesis based on the previous limitation.
For method 2 and 3 we see the same behaviour. This stems from the fact that we are sam-pling the data per time unit. Therefore, sub-blocks or observations are permutable only if the distance between them is a multiple of p. Consider the example in Chapter4. For convenience we repeat Figure4.1 as Figure 5.21. In the figures, we see that the position of the observations in the sub-blocks are only the same when the sub-blocks are in the same relative position in their respective blocks. Permuting sub-blocks that do not satisfy this criterion will alter the period of the function. Similarly, the position of an observation is only the same every p time units.
Therefore, we are again only permuting observations whose distance differs by a multiple of p.
Thus, we get the same result as for method 1 where the tests fail to reject the null hypothesis if p is a multiple of τ0.
Figure 5.21: Example from chapter 4. Used to illustrate that observations and sub-blocks can only be permuted if there are p time units between them.
Note that this problem only arises when the period of the function or the hypothesised functions is a fraction. In case that both are integers, this issue reduces to the first limitation as τ0= p.
5.5.3 Remarks for Possible Solutions
In the case that these limitations are solved the tests can become very powerful. For example, the test using Fisher’s G-statistic will be near perfect, as in almost all cases the values that cause it to incorrectly fail to reject the null hypothesis under the alternative hypothesis are either a multiple of the function’s period or fractions of a multiple of the period.
The first limitation arises from the definition of periodicity. If a function is periodic with pe-riod τ it is also pepe-riodic with any multiple of τ . Therefore, it is understandable that the test fails to reject the null hypothesis for these cases. One possible solution is to consider a composite null hypothesis which tests whether τ0 ∈ A where A is the set of all multiples of the period of the function. However, this poses the issue of identifiability of the period.
CHAPTER 5. RESULTS & COMPARISON OF METHODS
To overcome this issue, it is necessary to test the data for fractions of an accepted period. How-ever, this is only possible if we can avoid the second limitation. The second limitation will cause the test to accept all fractions with the numerator equal to this accepted period. For example, if the tests accept the period 9, due to the second limitation it will also accept the periods 9/2 and 9/4. One way of avoiding the second limitation is to consider only integer periods. Then if the test accepted period 9, it will only have to test period 3 which is not affected by the second limitation as 3 is not a multiple of 9. Moreover, this reduces the number of additional values that need to be tested as we only consider integer fraction of the period.
If we assume that we can avoid the second limitation, we can analyse the process of testing the fractions of the accepted period. We see that there is a natural stopping point for this process due to the Nyquist frequency. The stopping point is when the fraction is less than two. More-over, the periods tested must be rational. Therefore, if the test accepted period τ0 for a given data set, an additional bτ0/2 − 1c values need to be tested in the worst case. These values are τ0/2, τ0/3, . . . , 2τ0/τ0. Although the concept of this process is simple, it will take time to run the test for all these values. Moreover, the assumption that we can avoid the second limitation is quite lenient.
The second limitation arises from the design of the permuting methods which is based on the sampling rate. Therefore, looking into ways of solving this limitation is suggested for future research. A good starting point could be analysing the sampling rate used.