Facing problem with Math of Harmonic Analysis

thread: Facing problem with Math of Harmonic Analysis

  1. #1
    Junior Member Gerardomoreno84's Avatar
    22

    Facing problem with Math of Harmonic Analysis

    Hi,

    Few days ago I read one old book: Economic Cycles: There Law and Heal From Henry Ludwell Moore (Publicly available on the Net ). Moore in his book used Harmonic Analysis (Fourier Series) to ascertain the dominant cycle in Ohio Valley rainfall data. Moore utilizes 72 years (1839-1910) of Ohio valley rainfall data to forecast the future trend of rainfall in that region. Moore shares rainfall data and formulation for calculating coefficients. He then shares final values following calculation and plots a chart of those values.

    Moore writes that his results have been based on creation of Joseph Fourier, but the perfection of Fourier's egy is possible due to work with Professor Arthur Schuster. After hunting about Arthur Schuster I found he's well-known for Periodogram Analysis. Then, I started reading about Periodogram Analysis, but because of my lack of understanding of higher math I am unable to loe the right way forward and thus decided to ask for assistance.

    Moore calculated coefficient values for 3 to 36 cycle lengths. After applying the formula I am getting right values for bicycles which completely split 72, for example 3, 4, 6, 8, 9, 12, 18, 24 and 36. For all other cycles, with an exclusion of 5, I am getting erroneous values.

    I am minding the excel sheet with all the pertinent data. Formula is figured in the sheet. I am also attaching the ebook'Economic Cycles: There Legislation and Heal From Henry Ludwell Moore'. The rainfall data and periodogram values are given in page 32 and 33 of the book. Chapter two of the book (page 6 to 36) deals with calculation procedure. If someone who is great in math or possess an understanding of the method can be kind enough to help me in this, I would be genuinely grateful. I really don't need to compute specific values, but even if closure values are calculated that's enough.

    Thanks for reading
    https://www.cliqforex.com/attachment...207802016.xlsx
    https://www.cliqforex.com/attachment...1462919107.pdf

  2. #2
    Junior Member Gerardomoreno84's Avatar
    22
    I didn't read the book and that I didn't reverse engineer the excel sheet but that I believe your results are correct. Fourier decomposition is about regular time show implicitly infinite in length. The formulation assumes that your data includes a significant number of cycles so that it can be repeated infinitely.
    Thanks PipMeUp for your reply. I know that according to Fourier Series my formulation is right, but like I stated Periodogram Analysis of Schuster further refines the Fourier method, which is what Moore says in his publiion. Moore does something with the coefficients, of the periods that don't divide 72, to find different readings compared to what I'm getting. My short reading of Periodogram Analysis informs me that it had been utilized by Schuster to ascertain dominant sunspot cycle. I'm unable to comprehend the math involved in calculating Periodogram values.

    To understand this only examine the sin(x) below. It had been generated on the range 0-8 that is not a integral number of cycles (0-2Number 960;-RRB-. The formulation rubber stamps implicitly. The discontinuity creates artificial frequencies.
    Going by your reply, I assume Schuster was able to get rid of artificial frequencies. Am I right? I'm not good at higher level math, but you seem to have much more understanding of the topic than I have. If you could please go through the applicable pages of the book and take a look at the excel sheet, then perhaps you could suggest some changes in the formulation. Expecting you'll help. Thanks.

  3. #3
    This book was written nearly a century ago. Signal processing theory has evolved somewhat since that time. I really don't see exactly what you mean by Periodigram Analysis of Schuster further refines the Fourier system. Can you stage the webpage? The Fourier transform has nothing to become refined. It's just exactly what it is: a dual representation of a signal in the frequency domain name.

    You'll never be able to get rid of the frequencies introduced by the discontinuities. It is possible to try to limit their effect with windowing. This technique multiplies the signal with a function which compels the beginning and the conclusion of the sign to be zero. Multipliion in the time domain is a convolution (blurring) in the frequency domain name. The peaks become lumps. =gt; search google for Fourier windowing. You'll mainly find DSP and electronics related stuff.

    A couple of things that you need to know whether it's not already true:
    - The principal cycle isn't necessarily the one with the most value in the periodigram. Harmonics may have higher energies. This problem arises for example in speech recognition.
    - The sampling frequency can create phantom frequencies. This is called aliasing. See the picture below.
    - You require a series spanning several cycles to detect them. Cycles come and go so fast in financial time series that the time you obtained enough samples that the cycle is gone and replaced with another.
    - you can get interested in Goertzel algorithm. It's like Fourier but for a single frequency.

    BTW I also attempted to measure the cycles in the price and I completly failed. (https://www.cliqforex.com/trading-sy...-based-ma.html)

  4. #4
    Junior Member Gerardomoreno84's Avatar
    22
    I do not understand exactly what you mean by Periodigram Analysis of Schuster further refines the Fourier method. Can you point the webpage?
    This was my mistake to use the expression'refines', I should have used'improves for analyzing probability'. Initially line of Page 8, Moore writes,'' The perfection of this process whereby the findings could possibly be subjected to this evaluation of probability is that the work of Professor Arthur Schuster of Manchester.

    You can never eliminate the frequencies introduced by the discontinuities. You can try to restrict their effect with windowing. This technique multiplies the signal with a function which compels the start and the conclusion of the signal to be zero.
    I am completely new to this concept. Thanks very much for introducing me to'Fourier Windowing'. By the time that I read your article, I have been studying about it. From what I have gathered, there are several techniques for windowing. And what I am unable to determine is what values to use in which function of my formula to make sure the data at the edges are zero, so there's absolutely not any discontinuity. I believe I have understood in theory what needs to be done, but I don't have any idea what changes has to be made in the formula. That's because of my lack of understanding of Calculus.

    I read your article in'Cycle dimension: the easy way' you have deep knowledge of this topic. From what I got you're also trying to find the dominating cycle in a price movement. Moore also attempts this and combines more than 1 cycle to forecast the probable future. Now, I really don't understand about calling future and I definitely do not want to get in the virtues of a egy I haven't tested. I stumbled upon an older publiion, found its information interesting, put some hours to make an excel sheet and finally realized I am in a soup. I only need to work out this issue, but I also think there is some worth to it.

    I am really grateful for all the help you're providing. Also, I'll research about Goertzel algorithm, just like you suggested, however, will keep searching for the remedy of this issue. I'll need couple more hours of research on the things you have mentioned, before I can actually come-up with a proper query. Again, I am very thankful for all your guie.

  5. #5
    I Needed to Look about Arthur Schuster, probabilities and Fourier. I understand now what you mean by improvement.
    that which I'm unable to determine is what values to use in which purpose of my formulation to be certain that the data in the borders are zero, thus there's not any discontinuity.
    You merely extend the window so that it crosses precisely your samples. The signal multiplies by the window. The consequent transaformed information is what you plug in the formulation. For instance with Hann window you receive this below. I replied the samples times. Because the rainfall data is quite noisy you won't really see a huge difference.

  6. #6
    Junior Member Gerardomoreno84's Avatar
    22
    You just extend the window so it crosses precisely your samples. You multiply the signal by the window. The resulting transaformed information is what you plug in the formula. For instance with Hann window you receive this under. I replied the samples three times. Since the rain data is quite noisy you won't really see a difference.
    I think I have solved the issue. The difference in values was a result of boundary impact. So, I discarded additional values in the problematic series and worth I'm getting are very near Moore's values.

    Thanks very much PipMeUp for helping me.

  7. #7
    Junior Member Andreagd14's Avatar
    1
    Hello Dilmesh!!
    I browse the kind of problem that you were facing and its naice to know that you were able to solve the problem. I'm facing exactly the same problem as you're and for a reason I'm unable to calculate appropriate values for your cycles which are not completely divisible by 72. The only exception being bicycle #11 in which it matches the characters derived by Mr Moore. I'm unable to get correct figures for Cycle #3 which is totally divisible by 72.
    Its been over 5 weeks I'm stuck in this problem and I could nt appear to get an answer.
    I dont have a strong mathematical background. I could nt fathom the significance of your last post which says the gap in values resulted from the boundary effect so you discarded additional values in the problematic series.
    You've got an comprehension of the problem and it would serve me immensly in the event that you can help me resolve the problem partner. I'd be grateful for your help.
    Thank you.

    Attached is my excel sheet on the problem and calculation.
    https://www.cliqforex.com/attachment...401861095.xlsx

  8. #8
    Junior Member Gerardomoreno84's Avatar
    22
    I dont even have a strong mathematical background. I could nt fathom the significance of the last post which states the gap in values resulted from the border effect so you lost additional values in the debatable series.
    Hi Mercury63,

    I visitcliqforexin openings. Though I did came many times in past 2 months, but didn't see your post. I'm quite late in responding, I apologize, I hope this answer clears your doubt.

    Suppose you're calculating for 10th cycle, subsequently 72 divided by 10 leaves 2 as rest. Here, 2 is border effect, so you take just 70 values, don't add last 2 worth. For 15th cycle, rest is 12, which means you don't add last 12 worth since they are additional worth. You don't get the precise worth of Moore, but the chart you may create will be quite similar. We all only need to loe the dominant cycle and that goal is served readily.

    I haven't seen your excel file, but I have explained my solution, I hope you will be able to solve your problem.

    Next time while answering please quote me. It will make it much easier for me to understand your reply.

  9. #9
    I didn't read the book and I didn't reverse engineer the excel sheet but I believe your results are right.
    Fourier decomposition is all about regular time show implicitly unlimited in length. The formula assumes that your data contains a significant number of cycles so it could be repeated infinitely. Your data is 72 samples so the formula works with the cycles of the periods which divide 72. To know this just look at the sin(x) below. It had been generated on the range 0-8 which isn't a integral number of cycles (0-2Number 960;-RRB-. The formula rubber stamps implicitly. The discontinuity generates artificial frequencies.

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