Fourier Transform / Series

[DevilSolution]

I was reading a book on timeseries and all the obvious candidates were there but the fourier was briefly mentioned. Now i dont have a great deal of knowledge of calculus or geometry but i was just curious as to identifying the coefficients. I have what i think is the general formula but the guide i used starts using complex numbers as proof and seems to overcomplicate the equation.

 

Apart from the coefficients do you think the transform would have use in a timeseries? I'm talking straight X and Y discrete data, from which there is no obvious or absolute "periods", although i would like to know if it could dissect that part for me.

 

 

Regards,

[studiot]

I was reading a book on timeseries and all the obvious candidates were there but the fourier was briefly mentioned. Now i dont have a great deal of knowledge of calculus or geometry but i was just curious as to identifying the coefficients. I have what i think is the general formula but the guide i used starts using complex numbers as proof and seems to overcomplicate the equation.

 

Apart from the coefficients do you think the transform would have use in a timeseries? I'm talking straight X and Y discrete data, from which there is no obvious or absolute "periods", although i would like to know if it could dissect that part for me.

 

 

Regards,

 

What do you mean a timeseries and what is it for?

 

Well you need to organise your question in the light of the information below.

 

Fourier series and transforms are a way of representing the original function of time as the sum of a series of functions of frequency because they may be easier to work on than the original function.

 

In the case of the FT you also need the inverse transform when you have done calculating.

 

Fourier series only apply to periodic functions

Fourier transforms can also apply to non periodic functions

 

There is also the fast fourier transform (or finite fourier transform) , FFT, which is different again and is used to work with XY data as opposed to analytical formulae.

 

Over to you

[DevilSolution]

Yeh it was the fast fourier that was mentioned but then i did some further research, the document im reading is regarding data mining and identifying patterns in data. The data is Forex and stocks, which i know have no seasonality or periods persay i was just curious as to whether the FFT could dissect discrete data into frequencies. I suppose the series is related to electrical circuits and breaking down carrier waves etc.

 

Could it have any application or am i wasting my time?

[studiot]

I believe it has been tried, though I don't know the result as I don't know much about the maths of economics.

The applications of the FFT are much wider than just electrical engineering, I last used it to analyse the vibration frequencies of a spinning aircraft propellor, from the wind noise generated.

 

The FFT requires the data to be present in a specific format for analysis.

There are several schemes.

[DevilSolution]

What do you mean when you say "schemes"? i presume i'd have to apply data smoothing using maybe multiple regression so the data is of curved nature rather than jagged?

[studiot]

 

What do you mean when you say "schemes"? i presume i'd have to apply data smoothing using maybe multiple regression so the data is of curved nature rather than jagged?

 

 

Whatever you do, don't smooth it.

 

By schemes I was referring to how you divide up the x axis into segments.

This is related to the number of terms in the series you want to take, given that you are truncating a series.

 

Computer programs have all that built in and provide instructions as to how to enter the data.

 

My comment about the prop was similar to your question.

The prop noise sounded like white noise to the ear and I was looking for patterns or periodicities in the noise.

 

Are you attempting this by hand or by computer?

 

Hand methods have largely been superceeded, but were called harmonic analysis.

[DevilSolution]

Computer, C or any derivative.

So subdividing the x axis into equal sized arrays? (and maybe permutating the sizes?)

Heres the general formula i have: [math]xt=a_0+\sum[a_k \cos(\lambda_k t) + b_k \sin(\lambda_k t)] [/math]

for(k = 1: k <= q; k++){ 
    xt(k);
}

[math]\lambda = 2 \pi \nu_k[/math] And [math]\nu_k = k/q[/math]


Not sure if the a and b coefficients are the same formula i have.

[math]a_0 = \frac{1}{T} \int^{T}_{0} f(t) dt [/math]

[math]a_m= \frac{2}{T} \int^{T}_{0} f(t) \cos( \frac{2 \pi mt}{T})dt [/math]

[math]b_n= \frac{2}{T} \int^{T}_{0} f(t) \sin( \frac{2 \pi nt}{T})dt[/math]


Wow getting the Latex working was a nightmare it kept inserting /span and /font tags mid expression...weird, oh and i guess the [math]a_m[/math] and [math]b_n[/math] are [math]a_k[/math] and [math]b_k[/math] But they're from different sources, as i mentioned earlier about the coefficients.

Does any of this look right? i Think the general formula is a form of multiple regression, it goes on to say how to use the results in a periodogam and how to refine the periodogram for leakage using padding, tapering and smoothing.


Sorry i can already tell im off the mark with this, the coefficients must be figured out but my equations aren't a general formula (i dont think).

In simply regression the coefficient is the slope angle of [math]\frac{cov}{\sum^{n}_{i = 1}(x_i - \bar{x})^2 } [/math] but im not sure how this relates to the a and b coefficients. Any idea how they're derived???

Edited November 16, 2015 by DevilSolution