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The Mathematica documentation center provides an example of how to add noise to a process: White Noise Process.

\[ScriptCapitalP] = TransformedProcess[
             Cos[t/8] + noise[t], 
             noise \[Distributed] WhiteNoiseProcess[UniformDistribution[{-1/5, 1/5}]],
             t];

data = RandomFunction[\[ScriptCapitalP], {0, 200}];
ListPlot[data, Filling -> Axis]

Mathematica graphics

I'm also able to do the same with Gaussian noise, in which case I use NormalDistribution[0, 1] instead of UniformDistribution[{-1/5, 1/5}].

I have now two problems:

  • The first one is that I want the standard derivation to be a function of time, so NormalDistribution[0, f[t]], but this doesn't work.
  • The second problem is that I always get nothing when I choose the variance to be zero, which should effectively correspond to no noise.

Can someone help?

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  • $\begingroup$ When I have no noise, so zero mean and zero variance, I should get the original process back again, which I don't. Was only curious about that. $\endgroup$ Nov 23, 2015 at 12:41
  • $\begingroup$ Now I see, sorry I wasn't paying close enough attention $\endgroup$
    – sebhofer
    Nov 23, 2015 at 15:33
  • $\begingroup$ Mathematica can't handle the (singular) case of $\sigma\rightarrow 0$. Try PDF[NormalDistribution[0, 0], x], which throws the error NormalDistribution::posprm: "Parameter 0 at position 2 in NormalDistribution[0,0] is expected to be positive.". $\endgroup$
    – sebhofer
    Nov 23, 2015 at 17:40

2 Answers 2

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The nice thing about NormalDistribution is that it scales:

In[104]:= TransformedDistribution[sigma*u, 
 u \[Distributed] NormalDistribution[0, 1]]

Out[104]= NormalDistribution[0, Abs[sigma]]

So NormalDistribution[0, f[t]] is equivalent to f[t]*NormalDistribution[0,1], assuming f[t] is positive.

Then just adjust the transformation in TransformedProcess:

proc = 
  TransformedProcess[Cos[t/8] + Sqrt[t]*noise[t], 
  noise \[Distributed] WhiteNoiseProcess[NormalDistribution[0, 1]],t];
data = RandomFunction[proc, {0, 200}];
ListPlot[data, Filling -> Axis]

with the adjusted noise

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  • $\begingroup$ Thanks for your answer. How can I increase the number of points? When I'm using RandomFunction[proc,{0,200,0.2}] I always get an error... $\endgroup$ Nov 23, 2015 at 11:14
  • $\begingroup$ That is bc WhiteNoiseProcess in M is a discrete time process and RandomFunction can only output sample on integers. What you can do is to specify the length of the sample and then use TimeSeriesRescale to rescale time stamps to the MinimumTimeIncrement of your choice. $\endgroup$
    – Gosia
    Nov 23, 2015 at 21:51
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I don't have any experience with TransformedProcess or RandomFunction, but coulddn't you just make your own noise and add it to the data?

data = Cos[#/8] + 
     RandomVariate[NormalDistribution[0, # .001 + .05]] & /@ 
   Range[0, 200];
ListPlot[data, Filling -> Axis]

enter image description here

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  • $\begingroup$ Thanks for your answer, I will try to implement this later! :-) $\endgroup$ Nov 20, 2015 at 15:39

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