How should one quantify the strength of association between two random variables without bias for relationships of a specific form? Despite its conceptual simplicity, this notion of statistical "equitability" has yet to receive a definitive mathematical formalization. Here we argue that equitability is properly formalized by a self-consistency condition closely related to Data Processing Inequality. Mutual information, a fundamental quantity in information theory, is shown to satisfy this equitability criterion. These findings are at odds with the recent work of Reshef et al. [Reshef DN, et al. (2011) Science 334(6062):1518-1524], which proposed an alternative definition of equitability and introduced a new statistic, the "maximal information coefficient" (MIC), said to satisfy equitability in contradistinction to mutual information. These conclusions, however, were supported only with limited simulation evidence, not with mathematical arguments. Upon revisiting these claims, we prove that the mathematical definition of equitability proposed by Reshef et al. cannot be satisfied by any (nontrivial) dependence measure. We also identify artifacts in the reported simulation evidence. When these artifacts are removed, estimates of mutual information are found to be more equitable than estimates of MIC. Mutual information is also observed to have consistently higher statistical power than MIC. We conclude that estimating mutual information provides a natural (and often practical) way to equitably quantify statistical associations in large datasets.

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Equitability, mutual information, and the maximal information coefficient

We propose in this paper the use of Wavelet transform (WT) to detect human falls using a ceiling mounted Doppler range control radar. The radar senses any motions from falls as well as nonfalls due to the Doppler effect. The WT is very effective in distinguishing the falls from other activities, making it a promising technique for radar fall detection in nonobtrusive inhome elder care applications. The proposed radar fall detector consists of two stages. The prescreen stage uses the coefficients of wavelet decomposition at a given scale to identify the time locations in which fall activities may have occurred. The classification stage extracts the time-frequency content from the wavelet coefficients at many scales to form a feature vector for fall versus nonfall classification. The selection of different wavelet functions is examined to achieve better performance. Experimental results using the data from the laboratory and real inhome environments validate the promising and robust performance of the proposed detector.

Doppler Radar Fall Activity Detection Using the Wavelet Transform

Review of soft sensor methods for regression applications

In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and cross-covariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-of-the-art methods.

A Systematization of the Unscented Kalman Filter Theory

Epileptic seizure detection is commonly implemented by expert clinicians with visual observation of electroencephalography (EEG) signals, which tends to be time consuming and sensitive to bias. The epileptic detection in most previous research suffers from low power and unsuitability for processing large datasets. Therefore, a computerized epileptic seizure detection method is highly required to eradicate the aforementioned problems, expedite epilepsy research and aid medical professionals. In this work, we propose an automatic epilepsy diagnosis framework based on the combination of multi-domain feature extraction and nonlinear analysis of EEG signals. Firstly, EEG signals are pre-processed by using the wavelet threshold method to remove the artifacts. We then extract representative features in the time domain, frequency domain, time-frequency domain and nonlinear analysis features based on the information theory. These features are further extracted in five frequency sub-bands based on the clinical interest, and the dimension of the original feature space is then reduced by using both a principal component analysis and an analysis of variance. Furthermore, the optimal combination of the extracted features is identified and evaluated via different classifiers for the epileptic seizure detection of EEG signals. Finally, the performance of the proposed method is investigated by using a public EEG database at the University Hospital Bonn, Germany. Experimental results demonstrate that the proposed epileptic seizure detection method can achieve a high average accuracy of 99.25%, indicating a powerful method in the detection and classification of epileptic seizures. The proposed seizure detection scheme is thus hoped to eliminate the burden of expert clinicians when they are processing a large number of data by visual observation and to speed-up the epilepsy diagnosis.

Automatic Epileptic Seizure Detection in EEG Signals Using Multi-Domain Feature Extraction and Nonlinear Analysis

A common problem found in statistics, signal processing, data analysis and image processing research is the estimation of mutual information, which tends to be difficult. The aim of this survey is threefold: an introduction for those new to the field, an overview for those working in the field and a reference for those searching for literature on different estimation methods. In this paper comparison studies on mutual information estimation is considered. The paper starts with a description of entropy and mutual information and it closes with a discussion on the performance of different estimation methods and some future challenges.

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Estimation of Mutual Information: A Survey

Many learning algorithms rely on distance metrics to receive their input data Research has shown that these metrics can improve the performance of these algorithms Over the years an often popular function is the Euclidean function In this paper, we investigate a number of different metrics proposed by different communities, including Mahalanobis, Euclidean, Kullback-Leibler and Hamming distance Overall, the best-performing method is the Mahalanobis distance metric

Comparative study of distance functions for nearest neighbors

In this paper we present a new algorithm using a merger of Independent Component Analysis and Translation Invariant Wavelet Transform. The efficacy of this algorithm is evaluated by applying contaminated EEG signals. Its performance was compared to three fixed-point ICA algorithms (FastICA, EFICA and Pearson-ICA) using Mean Square Error (MSE), Peak Signal to Noise Ratio (PSNR), Signal to Distortion Ratio (SDR), and Amari Performance Index.
Experiments reveal that our new technique is the most accurate separation method.

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A new approach to denoising EEG signals - merger of translation invariant wavelet and ICA

These signals are however contaminated with artifacts which must be removed to have pure EEG signals. These artifacts can be removed by Independent Component Analysis (ICA). In this paper we studied the performance of three ICA algorithms (FastICA, JADE, and Radical) as well as our newly developed ICA technique. Comparing these ICA algorithms, it is observed that our new techniques perform as well as these algorithms at denoising EEG signals.

Performance comparison of known ICA algorithms to a wavelet-ICA merger

For years the Extended Kalman Filter (EKF) has been the algorithm for non-linear systems due to its simplicity and suitability to real time implementations. Because of its shortfalls, however, the Unscented Kalman Filter (UKF) was introduced to be an algorithm which was more accurate. Since then researches have been conducted to investigate the suitability of both algorithms in different areas. This paper presents a comparison of the estimation quality for the two algorithms when applied to a real time system - Electroencephalography (EEG).

Janett Walters-Williams

Papers

Estimation of Mutual Information: A Survey

Comparative study of distance functions for nearest neighbors

A new approach to denoising EEG signals - merger of translation invariant wavelet and ICA

Performance comparison of known ICA algorithms to a wavelet-ICA merger

Comparison of Extended and Unscented Kalman Filters applied to EEG signals