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What is the meaning of wavelet coefficients and scaling coefficients? E.g. for a sequence I obtained the following wavelet coeffients.

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How am I supposed to interpret them? I used wavelets package in R. The sequence concers price time series of 21 days. I used the Haar wavelet.

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  • $\begingroup$ The Haar wavelet yields a kind of crude derivative at different scales. Here, there is only 18 coefficients, not enough to be a genuine invertible wavelet transformation. Why are you studying such short data, and why with Haar wavelet? $\endgroup$ – Laurent Duval Nov 6 at 20:02
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Hope this helps:...

The fast wavelet transform (FWT) is a mathematical transform which results in a series of orthogonal coefficients of varying resolution.

Assume an input signal or time series $S$ of length $n=2^{j}$. The length of $S$ must be equal to some power of 2, that is, $n$ equal to 8, 16, 32, 64, 128, 256, 512, 1024, etc. To transform a signal into wavelets, at each resolution the signal is convolved using a series of scaling coefficients, $a_0$, $a_1$, $a_2$, $\ldots$ representing a low pass filter and a sequence of $d_0$, $d_1$, $d_2$, $\ldots$ of a high-pass filter.

Scaling function coefficients, $\phi$. At the $j-1$th resolution, the convolution is taken over the signal's original length, $2^j$, in the form \begin{equation} \phi_k = a_{-1}s_{2k+1} + a_0 s_{2k}, \end{equation} where $k=0,1,2,\ldots,2^{j-1}-1$ are the indices of each scaling function, $s_0, s_1, \ldots, s_{2^{j}-1}$ are the original input signal elements, and $a_{-1}$ and $a_0$ are constants from the Haar scaling function. An example of expanding the notation for the $j-1$ resolution is \begin{equation} \begin{split} \phi_0 &= a_{-1}s_1 + a_0 s_0\\ \phi_1 &= a_{-1}s_3 + a_0 s_2\\ \vdots\\ \phi_{2^{j-1}-1} &= a_{-1}s_{(2^j-1)} + a_0 s_{(2^j-2)}.\\ \end{split} \end{equation} Consider a signal $S$ of length $n=1024=2^{10}$ $(j=10)$, and assume Haar scaling function values of $a_{-1}=a_0=0.5$. Although the length of $S$ is 1024, the notation used to represent the series of values is $s_0$, $s_1$, $s_2$, $\ldots$, $s_{1023}$. The first convolution of $S$ will result in 512 ($2^{10-1}=2^9$) scaling function coefficients, which are obtained using the relationship \begin{equation} \phi_k = 0.5 s_{2k+1} + 0.5 s_{2k} \end{equation} where $k=1,2,\ldots,511$. This translates to \begin{equation} \begin{split} \phi_0 &= 0.5 s_{1} + 0.5 s_{0}\\ \phi_1 &= 0.5 s_{3} + 0.5 s_{2}\\ \vdots\\ \phi_{511} &= 0.5 s_{1023} + 0.5 s_{1022}\\ \end{split} \end{equation} It is easily noticed that the scaling function coefficients are the average of each neighboring pair of signal elements.

Wavelet coefficients, $\psi$. The wavelet coefficient is essentially based on the difference between each neighboring pair of signal elements. At the $j-1$ resolution, these are \begin{equation} \begin{split} \psi_0 &= -d_{-1}s_1 + d_0 s_0\\ \psi_1 &= -d_{-1}s_3 + d_0 s_2\\ \vdots\\ \psi_{2^{j-1}-1} &= -d_{-1}s_{(2^{j-1}-1)} + d_0 s_{(2^{j-1}-2)}\\ \end{split} \end{equation} where $d_{-1}$ and $d_0$ are both 0.5 based on the Haar wavelet.

For our first convolution of signal $S$ of length 1024 at resolution $j-1$, the 512 wavelet coefficients are \begin{equation} \begin{split} \psi_0 &= -0.5 s_{1} + 0.5 s_{0}\\ \psi_1 &= -0.5 s_{3} + 0.5 s_{2}\\ \vdots\\ \psi_{511} &= -0.5 s_{1023} + 0.5 s_{1022}\\ \end{split} \end{equation}

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