Anomalous Signal Characterization Using Kalman Filter-Based Spectral Quantification and Bayesian Statistical Diagnostics

Preprocessed electromagnetic signals in the form of frequency spectral groups are constantly acquired from electrooptical platforms where anomalous frequency structure is often hidden.These anomalies can be detrimental to trusted systems holding important information.There is a need to obtain vital anomalous behavioural statistical information which can be transformed into empirically driven predictive models for pattern-of-life estimation supporting trusted system protection.A two-tier algorithmic approach is employed to accomplish this using Kalman filter-based spectral quantification and Bayesian modelling.Kalman filter-based spectral deviation quantification is developed to estimate the average spectral deviation for an ensemble of frequency spectra comprising a series of spectral groups.The spectral-band based quantification of anomalous spectral group behavior over time supports the development of second stage algorithms aimed at parameterizing the statistical structure of the average spectral deviation as a random process.Algorithms here are based on Bayesian statistical estimation of mean and variance of sequentially estimated spectral deviation values for spectral groups, the application of analysis of variance (ANOVA) to mean spectral deviation values, and Markovian modelling of the mean and variance of spectral deviation values.The development of algorithms supporting anomalous spectral deviation quantification and Bayesian diagnostics is based on the analysis of hyperspectral imagery (HSI) data.A HSI cube of land and buildings was broken up into fourteen 100 X 100 pixel image chips which were used as a generator of frequency spectral groups.A typical HSI pixel spectral signature for dirt with added noise was extracted from the first image chip representing the mode of the data set and used in a Euclidean metric for measurement of anomalous spectra within image chips.HSI spectral signatures exceeding the metric threshold of 0.2 were flagged as anomalous spectra for each image chip and Kalman filtration used to characterize flagged spectral signals in each image chip.The Kalman filter applied to frequency spectra uses a series of frequency spectral energy measurements over a finite bandwidth containing random noise to produce a statistically optimal estimate of spectral band deviation from the mode spectral signal.The spectral deviation energy estimated from the filter was averaged over the full spectral bandwidth of a flagged spectrum and then over each image chip.The array of 14 image chips were then cycled over 20 times providing a 280-point average spectral deviation time series.For large numbers of frequency spectra in a single image chip, the average spectral deviation has a probability distribution that is log normal in shape.This is not surprising given that energy deviation is what is measured by the Kalman filter algorithm.Further insight, corroboration, and modelling of the statistical generation process for spectral deviation change was accomplished using analysis of variance (ANOVA) and Bayesian statistical modelling.Twenty F-statistic values for groups of fourteen image chips comprising the data domain were all close to 1 corroborating that the average spectral deviation values for all image chips or spectral groups do emanate from the same statistical process.Bayesian recursive estimation of the mean and variance for the average spectral deviation associated with each image chip was performed to produce a 280-point time series for each of these quantities.Periodicity of the mean spectral deviation was evident along with changes in the uncertainty intervals suggesting possible statistical structure linking mean and variance.Hidden Markov modelling, where the average spectral deviation is the state variable and the accompany variance is the observation variable, was performed to explore this idea.Preliminary analysis suggests that based on limited data, high anomalous spectral deviation structure tends to have a lower uncertainty which is useful information for the synthesis of anomalous spectral behavior characteristic of this underlying random process.

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