Given any time series of historical data, the prediction of the future values in the sequence is a computational task which can increase in complexity depending on the dimensionality of the data. For simple scalar data a predictive model based on differentials and expected continuation is perhaps the easiest. The order to which the series can be analysed depends quite a lot on numerical precision.
The computational complexity can be limited by using the local past to limit the size of the finite difference triangle, with the highest order assumption of zero or Monti Carlo spread Gaussian. Other predictions based on convolution and correlation could also be considered.
When using a local difference triangle, the outgoing sample to make way for the new sample in the sliding window can be used to make a simple calculation about the error introduced by “forgetting” the information. This could be used in theory to control the window size, or Monti Carlo variance. It is a measure related to the Markov model of a memory process with the integration of high differentials multiple times giving more predictive deviation from that which will happen.
This is obvious when seen in this light. The time sequence has within it an origin from differential equations, although of extream complexity. This is why spectral convolution correlation works well. Expensive compute but it works well. Other methods have a lower compute requirement and this is why I’m focusing on other methods this past few days.
A modified Gaussian density approach might be promising. Assuming an amplitude categorization about a mean, so that the signal (of the time series in a DSP sense) density can approximate “expected” statistics when mapped from the Gaussian onto the historical amplitude density given that the motion (differentials) have various rates of motion themselves in order for them to express a density.
The most probable direction until over probable changes the likely direction or rates again. Ideas form from noticing things. Integration for example has the naive accumulation of residual error in how floating point numbers are stored, and higher multiple integrals magnify this effect greatly. It would be better to construct an integral from the local data stream of a time series, and work out the required constant by an addition of a known integral of a fixed point.
Sacrifice of integral precision for the non accumulation of residual power error is a desirable trade off in many time series problems. The inspiration for the integral estimator came from this understanding. The next step in DSP from my creative prospective is a Gaussian Compander to normalize high passed (or regression subtracted normalized) data to match a variance and mean stabilized Gaussian amplitude.
Integration as a continued sum of Gaussians would via the central limit theorem go toward a narrower variance, but the offset error and same sign square error (in double integrals, smaller but no average cancellation) lead to things like energy amplification in numerical simulation of energy conservational systems.
Today’s signal processing piece was sparseLaplace for finding quickly for some sigma and time the integral going toward infinity. I wonder how the series of the integrals goes as a summation of increasing sections of the same time step, and how this can be accelerated as a series approximation to the Laplace integral.
The main issue is that it is calculated from the localized data, good and bad. The accuracy depends on the estimates of differentials and so the number of localized terms. It is a more dimensional “filter” as it has an extra set of variables for centre and length of the window of samples as well as sigma. A few steps of time should be all that is required to get a series summation estimate. Even the error in the time step approximation to the integral has a pattern, and maybe used to make the estimate more accurate.