Choosing Wavelet Coefficients by Minimizing Residual Standard Deviation

Summary

The document considers how to combine coefficient arrays from repeated wavelet decompositions of a time series. The stated aim is to represent the frequency component while leaving residual noise with the smallest standard deviation. The question notes that arithmetic averaging and a proposed norm-based combination were tried, but no justification was established for either approach.

The answer recommends treating the task as numerical optimization: define the residual standard deviation as an objective and search over a coefficient vector to minimize it. This offers a general computational direction, but the response is very brief and does not specify how the residual is constructed, what constraints or regularization should apply, or how to avoid overfitting. It also gives no evidence that the proposed optimization yields a unique or statistically meaningful coefficient combination. The suggestion is therefore a starting point, not a complete wavelet estimation procedure.

Key ideas

• The task is framed as finding coefficients that minimize the standard deviation of the residual noise.
• A numerical optimizer can search coefficient values against a defined standard deviation objective.
• The document does not establish that arithmetic averaging or norm-based combination is optimal.
• A useful implementation needs a precise residual definition and suitable constraints or regularization.
• The proposed optimization alone does not establish statistical validity or protection against overfitting.

Tags

# Proper way to combine wavelet coefficients from multiple rounds of analysis


# Proper way to combine wavelet coefficients from multiple rounds of analysis












I am doing signal analysis for a time series and the assumption of signal is

S = F + e

Where S is the original signal, F is the frequency component and e is white noise (auto-regressive time series with moving average = 0). I have a large number of samples, say, N samples; and I can do wavelet decomposition for all them to obtain N number of coefficients arrays.

Now my question is what's the proper way to combine all those coefficients to achieve an "average" of the coefficients, so that e would have smallest standard deviation? I tried to do arithmetic average or average norm times average coefficients vector by cross product; but I couldn't prove either of them is a proper way.

Thanks in advance for your help.

## Answer by emcor (score 1, accepted)

https://quant.stackexchange.com/a/14353

It sounds like this is a numerical problem without analytic solution. So I would suggest to use a numerical optimizer to minimize the standard deviation, e.g. MATLABs FMINCON() function can minimize virtually any expression that can be calculated.

So for your problem, I recommend to calculate:

$$\text{Fmincon}(\sigma(x))$$

where $x$ being your coefficient vector.

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