Generating 1/f^b colored time series

[Jon Norberg]

Please excuse (and ignore) this if this is not the right place to ask this. I am an ecologists and need to generate a time series with a specific color or frequency spectra. I never learned how Fourier transforms work in class and while I get the gist from reading there are so many subtleties that makes it hard to get it right unless one spends considerable time understanding it all. So I hope to get some help here.

I want to be able to define minFreq, maxFreq, Nfreq, Variance and color. I do not want to generate a discrete time series but instead have a function that provides me with a continuous function over time, even if it is evaluated at a specific time. Also, I'd prefer to make this in the sine domain instead of complex.

I want to get two vectors (amplitude, A and phase, P) by providing minFreq, maxFreq, Nfreq, Variance (of the time series) and color such that I can call

T(t,A,P)=sum(A*sin(w*t+P))

with the output having the given variance and color

I tried this code but does not have the properties I need and I probably made many errors in ignorance.

N=64
k=1:N
maxP=365
minP=1
k=linspace(minP,maxP,N)
w=2*pi*k'./maxP
b=1
p=rand(N,1)*2*pi
a=ones(N,1).*(1./w.^(b/2))
for t=1:1000
T(t)=sum(a.*sin(t*w+p))./sum(a);
end

*slight edit*

Any hints are greatly appreciated

P.S. is it possible to define the vectors such that the generated time serie's variance is independent of the choice of Nfreq and color?

 

[jvicens]

Hello there,

Thank you for reaching out to us for help with your time series generation. I am a scientist and I would be happy to assist you with this task.

Firstly, to address your concern about not fully understanding Fourier transforms, I would recommend taking some time to read up on the basics of Fourier analysis and the properties of Fourier transforms. This will help you better understand the code you have written and make any necessary changes to achieve your desired outcome. There are many online resources available that can help you with this, such as tutorials, videos, and textbooks.

Moving on to your code, I can see that you have defined the parameters minFreq, maxFreq, Nfreq, and Variance, which is a good start. However, there are a few things that need to be corrected to achieve your desired result.

Firstly, the code you have written is generating a discrete time series, which is not what you want. To generate a continuous function, you need to use a continuous time variable, such as t=linspace(0,1,1000) to generate 1000 time points between 0 and 1. You can then use this time variable in your sine function as t*w instead of just w.

Secondly, you have defined the phase vector p as a random number between 0 and 2*pi. This will result in a random phase for each frequency, which is not what you want. Instead, you should define a single phase value for the entire time series, which will give you a continuous time series with a fixed phase.

Next, you have defined the amplitude vector a as 1/w^(b/2), which will give you a decreasing amplitude with increasing frequency. To achieve a constant variance, you should define the amplitude as a constant value rather than a function of frequency.

Lastly, you have used the sum function in your code, which will give you a sum of the sine waves rather than a continuous function. To get a continuous function, you should use the dot product (.*), which will give you the product of the sine waves.

Taking all these corrections into account, your code should look something like this:

N=64;
minFreq=1;
maxFreq=365;
Nfreq=1000;
Variance=1;
color='red';

t=linspace(0,1,Nfreq); % generate 1000 time points between 0 and 1
w=2*pi*linspace(minFreq,max