Optimal Filter Coefficients: Correlation versus Least Squares

In summary, the paper discusses a claim about finding filter coefficients without proof. The claim states that maximizing the correlation coefficient over filter coefficients is equivalent to minimizing the integral of the difference between two vectors, defined by an inner product. This can be seen intuitively and was initially attempted to be proven using the convolution theorem, but did not yield the correct equations. The connection between the two can be seen through the polarization identity, which states that the correlation coefficient is maximal when the angle between the two vectors is close to 90 degrees or when the sum of the two vectors is maximal and their difference is minimal, both of which depend on the length of the sum.
  • #1
Squatchmichae
12
0
I found a claim in a paper (BSSA, Vol 81, No. 6: "A Waveform Correlation Method for Identifying Quarry Explosions", By D.B. Harris) concerning finding filter coefficients. The statement is given without proof. I cannot locate a reference or theorem for the following, and have not been able thus far to justify this claim quantitatively.

Suppose:

\begin{equation}
\mathbf{v}(t) = \displaystyle \sum_{k=1}^{N} \int_{-T}^{T} \! a_{k}(t-s) \mathbf{u}_{k}(s) \, ds,
\end{equation}

Then maximizing the correlation coefficient over filter coefficients a:

\begin{equation}
\rho(a) = \max_{a} \frac{\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle} { \sqrt{\left\langle {\mathbf{u}(t), \mathbf{u}(t) } \right\rangle \, \left\langle {\mathbf{v}(t), \mathbf{v}(t) } \right\rangle}}
\end{equation}

Is equivalent to:

\begin{equation}
\min_{a} \int_{-T}^{T} \! \parallel \mathbf{u}(t) - \mathbf{v}(t) \parallel^{2} \, dt,
\end{equation}

Where the inner product is defined by:

\begin{equation}
\left\langle {\mathbf{u}(t), \mathbf{v}(t) } \right\rangle = \int_{-T}^{T} \! {\mathbf{u}(t)}^{T} \mathbf{v}(t) \, dt,
\end{equation}

Qualitatively, this makes sense, of course. I initially attempted to prove this in the frequency domain by making use of the convolution theorem, to reduce the problem into one that looks similar to a Rayleigh quotient. This effort did not yield the correct equations.
 
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  • #2
I can only see a connection via the polarization identity: $$4 \langle \mathbf{u},\mathbf{v} \rangle = ||\mathbf{u}+\mathbf{v}||^2- ||\mathbf{u}-\mathbf{v}||^2 = 4 \cdot ||\mathbf{u}||\cdot ||\mathbf{v}||\cos \sphericalangle ( \mathbf{u},\mathbf{v})$$ which says that ##\rho(a)## is maximal if the angle is close to ##90°## or if ##||\mathbf{u}+\mathbf{v}||^2## is maximal and ##||\mathbf{u}-\mathbf{v}||^2## minimal, and vice versa. Both conditions depend also on the length of the sum.
 

1. What is the difference between correlation and least squares in determining optimal filter coefficients?

Correlation is a measure of the linear relationship between two variables, while least squares is a method for finding the best fit line between a set of data points. In terms of optimal filter coefficients, correlation is used when the goal is to maximize the similarity between the input and output signals, while least squares is used when the goal is to minimize the squared error between the input and output signals.

2. How do correlation and least squares affect the accuracy of the filter coefficients?

Correlation tends to result in more accurate filter coefficients when the input and output signals have a high degree of linear correlation. However, least squares can be more accurate when there is a lot of noise in the signals or when the relationship between the input and output is not strictly linear.

3. Can correlation and least squares be used together to determine optimal filter coefficients?

Yes, correlation and least squares can be used together to determine optimal filter coefficients. This is known as the Wiener-Hopf equation, which combines the two methods to find filter coefficients that both maximize correlation and minimize squared error.

4. What are the limitations of using correlation and least squares for determining optimal filter coefficients?

One limitation is that correlation and least squares assume a linear relationship between the input and output signals, which may not always be the case in real-world scenarios. Additionally, both methods can be affected by outliers in the data, which can result in inaccurate filter coefficients.

5. Are there any alternative methods for determining optimal filter coefficients?

Yes, there are other methods such as the Kalman filter and the maximum likelihood estimation, which can also be used to find optimal filter coefficients. These methods may be more suitable for non-linear relationships between input and output signals or when dealing with highly noisy data.

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