For the LTE formulation along the equator, the analytical solution reduces to *g(f(t))*, where *g(x)* is a periodic function. Without knowing what *g(x)* is, we can use the frequency-domain entropy or spectral entropy of the Fourier series mapping an estimated *x*=*f(t)* forcing amplitude to a measured climate index time series such as ENSO. The frequency-domain entropy is the sum or integral of this mapping of *x* to *g(x)* in reciprocal space applying the Shannon entropy –*I(f)^{.}ln(I(f))* normalized over the

*I(f)*frequency range, which is the power spectral (frequency) density of the mapping from the modeled forcing to the time-series waveform sample.

This measures the entropy or degree of disorder of the mapping. So to maximize the degree of order, we minimize this entropy value.

This calculated entropy is a single scalar metric that eliminates the need for evaluating various cyclic* g(x) *patterns to achieve the best fit. Instead, what it does is point to a highly-ordered spectrum (top panel in the above figure), of which the delta spikes can then be reverse engineered to deduce the primary frequency components arising from the the LTE modulation factor *g(x)*.

The approach works particularly well once the spectral spikes begin to emerge from the background. In terms of a physical picture, what is actually emerging are the principle standing wave solutions for particular wavenumbers. One can see this in the LTE modulation spectrum below where there is a spike at a wavenumber at 1.5 and one at around 10 in panel **A** (isolating the sin spectrum and cosine spectrum separately instead of the quadrature of the two giving the spectral intensity). This is then reverse engineered as a fit to the actual LTE modulation *g(x)* in panel **B**. Panel **D** is the tidal forcing *x=f(t)* that minimized the Shannon entropy, thus creating the final fit *g(f(t))* in panel **C** when the LTE modulation is applied to the forcing.

The approach does work, which is quite a boon to the efficiency of iterative fitting towards a solution, reducing the number of DOF involved in the calculation. Prior to this, a guess for the LTE modulation was required and the iterative fit would need to evolve towards the optimal modulation periods. In other words, either approach works, but the entropy approach may provide a quicker and more efficient path to discovering the underlying standing-wave order.

I will eventually add this to the LTE fitting software distro available on GitHub. This may also be applicable to other measures of entropy such as Tallis, Renyi, multi-scale, and perhaps Bispectral entropy, and will add those to the conventional Shannon entropy measure as needed.

The I(f) corresponding to the best fit — notice the 2 strong delta functions corresponding to the LTE modulation.

LikeLiked by 1 person

An important observation is that since the square-wave-like tidal forcing is very similar across the tidal indices, then the implied LTE spectrum could be easily characterized for AMO, IOD, PNA, etc once the ENSO forcing is established.

Each spectrum’s delta spikes will be slightly different based on the distinct standing wave mode for the geographical index. Amazing if that works out.

LikeLiked by 1 person

Pingback: Overfitting+Cross-Validation: ENSO→AMO | GeoEnergy Math

Pingback: Inverting non-autonomous functions | GeoEnergy Math