A PROBABILITY-WEIGHTED MODIFICATION OF THE AKIMA INTERPOLATION METHOD

Abstract

Interpolation of time series and experimental signals is one of the most important problems in signal processing, data analysis, and dynamic systems modeling. However, in many cases, the existing data are distorted by noise, non-uniform sampling, and sudden transitions. These factors greatly limit the effectiveness of standard approaches. Even though the Akima interpolation technique is well-known due to its local character and relatively small oscillations of the interpolated function compared to classical splines, the pure geometric construction of the latter increases its vulnerability to outliers and sudden jumps. This work describes a new probability-based modification of the Akima interpolation technique designed to increase its robustness and performance under stochastic noise. The idea of the approach lies in taking into account the probability of state transitions in the computation of local slopes. Compared to other robust or adaptive spline interpolation algorithms, the probability-based weighting system described in this study does not require any manually tuned parameters and additional signal pre-processing. Transition probabilities are computed via an automatic estimation procedure based on the correction of normal distributions using the Edgeworth approximation. Thus, skewness and kurtosis of distribution are easily accounted for using minimum calculations. The resulting values of probabilities express the typicality of certain state transitions: frequent states have high weights, whereas uncommon changes receive lower ones, decreasing their impact on the final result. The algorithm preserves the main advantages of the Akima technique, namely locality, simplicity, and computational efficiency while enhancing its potential to be applied to stochastic and dynamic processes. The study provides an algorithmic description of the new interpolation technique, including the probability-based slope evaluation step. An implementation algorithm based on the use of transition probabilities for the calculation of local slopes and interpolation segment construction is provided. A set of numerical experiments is conducted using artificial test scenarios with various degrees of noise and outliers in the input signal. Mean absolute and mean squared error are used as quantitative measures of accuracy of the interpolation procedures. It was found that both techniques provide almost identical performance in terms of absolute and mean square errors if the input signal is noiseless. However, in the presence of random distortion, the probability-based method produces much better results. This work presents a new interpolation algorithm based on probability-based weighted computation of local slopes of the Akima spline. The developed approach is especially applicable for the analysis of stochastic data and time series.

Keywords: probability-weighted interpolation, Akima interpolation, time series, Edgeworth expansion, stochastic processes, robustness.