Empirically Derived Distance Measures for the Z-Plane

In this post, I test the effects of infinitesimal, orthogonal movements of a single complex pole on mean squared error in residual estimation. The goal is to find and validate a distance metric that relates residual estimation to estimation of eigenfrequencies. My hypothesis, derived from informal inspection of large related datasets, is that such a distance metric will correspond to that of the Poincare disc model.

A specific machine learning algorithm has been developed to characterize the surface vibrations of arbitrary media. In previous experiments, such as 1, 2, and 3, a mean squared error measure was derived for the assessment of residual estimation accuracy. However, the case of modal estimation accuracy remained unexplored, due to the lack of a satisfactory distance measure. Evaluating the modal estimation accuracy for a large number of simulations is an essential step in the overall validation of the machine learning algorithm. Moreover, a conformally invariant distance metric such as that which is afforded by the Poincare disc model would be advantageous for a number of reasons, which will be the subject of a future publication.

Methods

The experiment, whose source code may be viewed here examined a model consisting of complex white noise fed through a single complex pole. The modal estimation component of the algorithm, wherein the poles are located as eigenvalues of the prediction matrix, was not used. Instead, this experiment focused on the inverse filtering component, wherein the information about the poles affords the reconstruction of the complex white noise vector. A distance metric, previously used in validation tests on the total system, is defined as

E \left[ (x_0 - x_0 a_0 - x_1 a_1 ... x_n a_n) (x_0 - x_0 b_0 - x_1 b_1 ... x_n b_n)^* \right]

where x_n is a random independent variable, a_n is a target filter coefficient corresponding to the forward (IIR) and inverse (FIR) filter applied in series, ie
( 1, \lambda_t-\lambda_{\epsilon}, \lambda_t^2 -\lambda_{\epsilon}\lambda_t, \lambda_t^3 - \lambda_{\epsilon}\lambda_t^2 ... \lambda_t^n - \lambda_{\epsilon}\lambda_t^{n-1} ) and b_n is a coefficient corresponding to the same forward filter (1, \lambda_t, \lambda_t^2 ... \lambda_t^n), with a varying infinitesimal orthogonal "error" signal \epsilon added to the inverse filter applied in series, ie \lambda_{\epsilon} = \lambda_t + \epsilon. These values of \epsilon consisted of small increments added to the pure real or pure imaginary directions of the complex zero belonging to the inverse filter. The result of this distance metric is referred to as the "Round-Trip Residual" RTR distance metric for a single complex pole.

The RTR distance was evaluated for 200 different models consisting of complex white noise and a single complex pole, each with a unique, 10000 component complex noise vector. There were 20 different pole positions, each tested 10 times with fresh complex noise. The 20 pole positions \lambda were divided into 2 angles: zero and \pi/2, each with 10 radii ranging from 0 to 0.9. Each test consisted of a control, where \epsilon = 0 and therefore \lambda_{\epsilon} = \lambda_t, a real case, where \epsilon = 10e-6 and therefore \lambda_{\epsilon} = \lambda_t + 10e-6, and an imaginary case, where \epsilon = 10e-6 \sqrt{-1} = i*10e-6 and therefore \lambda_{\epsilon} = \lambda_t + i*10e-6.

To test the hypothesis that the RTR distance metric is correlated with the Poincare Disc Model (PDM) distance metric, the same two complex points \lambda_t and \lambda_e were used to compute the following metric:

d(\lambda_t, \lambda_{\epsilon}) = acosh \left( 1 + \left(2 \frac{|\lambda_t - \lambda_{\epsilon}|^2 }{(1-|\lambda_t|^2)(1-|\lambda_{\epsilon}|^2)}\right) \right)

The result shows a 98% or greater correlation between the PDM and RTR distances. The variance is explained by the pseudorandom complex white noise vector.

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