Dara wrote:

> first draft of some detailed answers:

> [Sanity Check for Wavelets](http://files.lossofgenerality.com/sanityWLT.pdf)

Unfortunately this pdf doesn't answer my questions. Here are my questions, phrased more clearly I hope. Without answers to these it'll be hard for me to finish this blog article. Can you please answer them here on the forum?

1. What is the formula for the continuous wavelet transforms being used here?

1. What is a "frequency 4" Gabor wavelet?

1. Why are you using a frequency 4 Gabor wavelet?

1. Why are you using the 4th derivative of a Gaussian?

1. Why is there almost no power at periods shorter than 9.5 months?

1. Did you "denoise" the time series before feeding them into these wavelet transforms?

1. If so, how?

I think I can answer question 1 using the Mathematica page [Continuous Wavelet Transform](http://reference.wolfram.com/language/ref/ContinuousWaveletTransform.html). Namely:

The continuous wavelet transform of a sequence of numbers $x_1, x_2, \dots , x_n$ is

$$ w(u,s) = \frac{1}{\sqrt{s}} \sum_{k = 1}^n x_k \psi(\frac{\Delta (k-u)}{s}) $$

Here we think of $x_k$ as the value of some function $x(t)$ defined at all times, via

$$ x_k = x(t_0 + k \Delta) $$

The function $\psi$ is something we get to choose. We could choose it to be a "Gabor wavelet of frequency 4" - but I don't know what this function is, because it's not explained [on the Mathematica page](http://reference.wolfram.com/language/ref/GaborWavelet.html). Or we could choose it to the 4th derivative of a Gaussian, as explained [here](http://reference.wolfram.com/language/ref/GaborWavelet.html).

(In all these links you need to click "Details" to get the information I need.)

The _idea_ behind the continuous wavelet transform is evident to me from the formula above, so if that's what you're using I can explain it. However, I don't see why you chose the frequency 4 Gabor wavelet and the 4th derivative of a Gaussian. These are functions with about 4 wiggles in them; I see why you want some wiggles, but not why you want 4. If there's nothing special about "4", fine, just say so.

> first draft of some detailed answers:

> [Sanity Check for Wavelets](http://files.lossofgenerality.com/sanityWLT.pdf)

Unfortunately this pdf doesn't answer my questions. Here are my questions, phrased more clearly I hope. Without answers to these it'll be hard for me to finish this blog article. Can you please answer them here on the forum?

1. What is the formula for the continuous wavelet transforms being used here?

1. What is a "frequency 4" Gabor wavelet?

1. Why are you using a frequency 4 Gabor wavelet?

1. Why are you using the 4th derivative of a Gaussian?

1. Why is there almost no power at periods shorter than 9.5 months?

1. Did you "denoise" the time series before feeding them into these wavelet transforms?

1. If so, how?

I think I can answer question 1 using the Mathematica page [Continuous Wavelet Transform](http://reference.wolfram.com/language/ref/ContinuousWaveletTransform.html). Namely:

The continuous wavelet transform of a sequence of numbers $x_1, x_2, \dots , x_n$ is

$$ w(u,s) = \frac{1}{\sqrt{s}} \sum_{k = 1}^n x_k \psi(\frac{\Delta (k-u)}{s}) $$

Here we think of $x_k$ as the value of some function $x(t)$ defined at all times, via

$$ x_k = x(t_0 + k \Delta) $$

The function $\psi$ is something we get to choose. We could choose it to be a "Gabor wavelet of frequency 4" - but I don't know what this function is, because it's not explained [on the Mathematica page](http://reference.wolfram.com/language/ref/GaborWavelet.html). Or we could choose it to the 4th derivative of a Gaussian, as explained [here](http://reference.wolfram.com/language/ref/GaborWavelet.html).

(In all these links you need to click "Details" to get the information I need.)

The _idea_ behind the continuous wavelet transform is evident to me from the formula above, so if that's what you're using I can explain it. However, I don't see why you chose the frequency 4 Gabor wavelet and the 4th derivative of a Gaussian. These are functions with about 4 wiggles in them; I see why you want some wiggles, but not why you want 4. If there's nothing special about "4", fine, just say so.