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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 have been trying to calculate those Periodogram values, but have failed to do so. The formula used by him is of Fourier Analysis. After applying the formula I am getting right coefficient values for cycles which completely divide 72, such as 3, 4, 6, 8, 9, 12, 18, 24 and 36. For all other cycles, with an exception of 5, I am getting wrong values.

I am not good at higher Mathematics. Please, can someone tell me what mistake I am making? What should I add or subtract from other coefficients to get right Periodogram values?

Book is freely available on the internet. Rainfall Data is on page 32 of that book and calculated Periodogram values is on page 33. All the information regarding calculation is in Chapter II (from page 4 to 34).

Formula I am using to calculate coefficients:

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 have been reading a lot about Fourier Analysis and Periodogram for the past few days, but because I don't have good understanding of relative math and all this information is a century old, it has not yielded any result. I am still struggling to find the right answer. Hoping someone could help me here.

Thanks

Snapshot of Rainfall Data and Periodogram Values

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    $\begingroup$ this is not really suitable for this site (which is for research in mathematics, rather than for problems involving math); you might try math.stackexchange.com --- since you get the right results for commensurate periods, the discrepancy you are finding is presumably a boundary effect (for example, for a period of 7 I might want to discard data number 71 and 72). $\endgroup$
    – Carlo Beenakker
    Commented Feb 7, 2018 at 9:07
  • $\begingroup$ My mistake I am new to this site so didn't knew. However, I applied your suggestion and my problem is solved. The final data is not an exact match, but its very close to Moore's values. Thanks very very much. If I encounter any other problem related to this topic, I will surely go to the other site you mentioned. Thanks again, you were of great help. $\endgroup$
    – Ausar
    Commented Feb 7, 2018 at 13:55

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