Near-optimal smoothing using a maximum entropy criterion

Abstract

Digital filters have become very common for smoothing and Altering of noisy chemical data. Many papers have appeared on the use of Alters for optimizing slgnal-to-nolse or for optimizing signal shape retention during the process of noise removal. In contrast, little work has been aimed at optimizing the Alter to retain as much of the Information present In the data while removing as much of the noise as possible. This paper reports a simple approach for optimizing digital Alters of several types. The optimization criterion used here Is based on the Informational entropy of the smoothed signal. It Is demonstrated here that, for many forms of data, a filter optimized by maximizing the Informational entropy, but without knowledge of the true signal shape, performs as well as an equivalent, optimal finer optimized with the aid of prior knowledge of the signal shape. Filers optimized by maximizing the Informational entropy have applications where accurate reconstruction must be accompanied by significant Increases In slgnal-to-nolse ratios of noisy, but unknown data. The maximum entropy optimization process Is demonstrated on several different dlgHal fitters.

Signal Processing in Electrochemistry

“…To test the efficiency of the signal processing and the reconstruction of the curve shape other numerical indicators were also applied. They are the sum of squared errors SSE [28,29] and the linear correlation coefficient r [30]. SSE is expressed by the formula:…”

“…The most commonly used technique to remove periodic noise is based on Fourier transformation (FT) [29,54,55]. By means of FT equation, the time function x(t) can be expressed as a function of frequency f:…”

Removing Spikes from Voltammetric Curves in the Presence of Random Noise

“…They are: the percentage of removed spikes, the sum of squares error SSE and the linear correlation coefficient r [13]. The sum of the squared errors [5,6] is defined as:…”

“…The entropy is minimum and equal to zero when the signal is flat, and increases when it has some fluctuations. The criterion of maximizing the informational entropy was applied to describe the level of the arrangement in the curve after signal processing [6]. Also the evaluation criterion [7] was used for the set of curves prepared for calibration.…”

“…Then the applications of SSE coefficient [5] as well as calculation percentage of removed spikes is possible. For a single experimental curve also the informational entropy [6] helps to evaluate the performance of the proposed algorithm. Usually, experimental curves are measured with purpose of define relation such as calibration line (for quantitative determinations), dependence on scan rate, pH, ionic strength of supporting electrolyte or influence of other experimental parameters.…”