In the book, Economic Cycles: There Law and Cause By Henry Ludwell Moore, he plots Periodogram of rainfall of Ohio valley. He uses 72 years data (1839-1910) and tries to find the most dominant cycle in that time frame.
I was trying to calculate those Periodogram values, but have failed to do so. After applying the Fourier Analysis formula I was getting right coefficient values for cycles which completely divided 72, such as 3, 4, 6, 8, 9, 12, 18, 24 and 36. For all other cycles, I was getting wrong values.
Formula I am using to calculate coefficients (in excel):
a = (2/N)*(R)*Cos(2πt/T)
b = (2/N)*(R)*Sin(2πt/T)
T = no. of cycle from 3 to 36, N = total observations in T cycle, R = rainfall data, t = number ranging from 0 to 71.
I asked this question on another site, where someone was kind enough to suggest me this site. He also stated that the problem I was facing was due to boundary effect. Therefore, I should discard, as an example, data 71 and 72 for period 7. His solution worked and I was able to get coefficient values close to Moore's values.
However, when data points increase from 72 to lets say 500, and T increases from 36 to 150 or 200. I am facing the same problem. The Periodogram values are repeating like waves and I am unable to spot the dominant cycle.
I have been reading a lot about Fourier Analysis and Periodogram for the past few days, but because I don't have good understanding of higher mathematics it has not yielded any result. I am still struggling to find the right answer. Hoping someone could help me here.