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The wavelet transform is a popular analysis tool for non-stationary data, but in many cases, the choice of the mother wavelet and basis set remains uncertain, particularly when dealing with physiological data. Furthermore, the possibility exists for combining information from numerous mother wavelets so as to exploit different features from the data. However, the combinatorics become daunting given the large number of basis sets that can be utilized. Recent work in evolutionary computation has produced a subset selection genetic algorithm specifically aimed at the discovery of small, high-performance, subsets from among a large pool of candidates. Our aim was to apply this algorithm to the task of locating subsets of packets from multiple mother wavelet decompositions to estimate cardiac output from chest wall motions while avoiding the computational cost of full signal reconstruction. We present experiments which show how a continuous assessment metric can be extracted from the wavelets coefficients, but the dual-objective nature of the algorithm (high accuracy with small feature sets) imposes a need to restrict the sensitivity of the continuous accuracy metric in order to achieve the small subset size desired. A possibly subtle tradeoff seems to be needed to meet the dual objectives.

Wavelet analysis has become one of the most commonly used digital signal processing tools, with applications in data compression, image processing, time series data filtering, material detection and de-noising [

Wavelet analysis can be viewed as a transformation into the time-frequency domain, and involves a series of convolution operations on the dataset against a particular filter set, called the Mother Wavelet, at various positions and time/magnitude scales. The process separates high frequency components from low frequency components and allows inspection of the data through a small window, in order to detect small features over the full analyzed spectrum [

Genetic Algorithms (GAs) and Wavelets have been combined recently in image processing for fault detection [

An example of where this approach could be utilized is Cardiac Output (CO) monitoring. While various invasive methods have been developed to measure CO directly, all present significant complications, such as blood stream infection, need for medications, decreased hemodynamics, and high cost [

We seek to estimate SV from a seismocardiogram recording, which is obtained by recording chest wall acceleration at the xiphoid process [

Output Monitoring device (NICOM, Cheetah Medical Inc). The NICOM has achieved some acceptance [

Eshelman’s CHC GA [

When it becomes clear that further crossovers are unlikely to advance the search, a soft restart is performed, using mutation to introduce substantial new diversity, but also retaining the best individual chromosome in the population.

The initial GA population is generated randomly using a uniform distribution. In CHC two initial populations are produced and the chromosomes are evaluated and the more fit chromosomes from both populations are selected to become the next population. For all subsequent generations, the pairs of parents (randomly mated) produce two offspring and the selection operator produces the next parent generation by taking the best from the combined parents and offspring using simple deterministic ranking.

Understanding the chromosome structure provides an understanding of the connection between the feature- genes and the Sub-Set-Size (SSS) gene. A chromosome is defined as set of genes, and in our approach, the first gene represents the SSS, that is, the number of genes that are expressed when a chromosome is evaluated (

The expressed genes in a chromosome represent the magnitudes of a subset of wavelet packets. The mathematics of the wavelet transform may be found elsewhere [

called details. Subsequent levels of decomposition are performed on the approximation coefficients; again separating the low frequency components into approximations and details. This process is repeated with entropy, energy, and/or a cost function being computed after each level of decomposition as a means of optimizing the decomposition process. In our application, the acceleration data may include numerous high and low frequencies not associated with cardiac activity. High energy at the low frequency is likely to be associated with breathing and whole body motion, while high frequency components may be associated with vocalization. Since our goal was to identify those components providing the best correlation with SV, the full signal frequency spectrum was investigated regardless of its computation cost, energy, or entropy.

We performed full tree decompositions, that is, was performed on the details and approximation coefficients of each branch using one Mother wavelet (

The goal of utilizing the subset selection GA was to identify the minimal subset of features capable of accurately estimating the NICOM reported SVs. The NICOM provides thirty-second averages of SV and so we performed wavelet decomposition on each thirty seconds of recoded acceleration data. Eighty-five thirty-second averaged measurements were taken sequentially using the NICOM, the ECG, and chest accelerations, from a single subject during both resting and exercising. There were five exercise periods for one hundred and fifty seconds at the same intensity and five resting periods of two hundred and seventy seconds. We started to collect data while the subject was at rest, in an upright position for four hundred and fifty seconds. Multivariate regression was used to correlate the expressed chromosome genes ‘packets energy’ to the averaged NICOM SV measurements. The R^{2} value of the regression line was used as the chromosome fitness value. The higher the R^{2} value, the better the gene set predicts the NICOM SV.

In the CHC GA, the more fit chromosomes remain in the population until they are replaced by even more fit offspring. The fitness function returns a two-vector, where one is the R^{2} value, and the other is the SSS. The vector selection process works by comparing two chromosomes, a parent, A and an offspring B, if R^{2}(A) > R^{2}(B), than A is more fit (and vice versa). However, if R^{2}(A) = R^{2}(B), then the chromosome with the smaller SSS is more fit. If the SSS’s are also equal, the parent is not replaced.

The crossover operator is responsible for offspring reproduction. It consists of three operators: Incest Prevention which decides if the two parents can mate; Index Gene Crossover which is responsible for inheritance of both parents’ genes to the offspring; SSS Recombination crossover which is responsible for setting the SSS gene of the offspring based on both parents’ SSS genes.

The crossover operator is applied to each random pair of parents. The first step is to check the pair for incest prevention. Parents who are too closely related are prevented from mating. The distance between two chromosomes is simply the number of unique genes in the leading portion of the chromosomes out to the furthest genes an offspring might inherit (the larger value of SSS genes from the two chromosomes).The initial value for the incest threshold is half of the maximum SSS, but it is decremented whenever a generation occurs in which no offspring survive. When the incest threshold drops to zero, any chromosome may mate with any other, including a clone of itself. The incest threshold dropping to zero is one of the criteria used by CHC for halt and restart decisions. This incest prevention algorithm has been shown to effectively defeat genetic drift [

GA research has shown that “respect” is an important property for a crossover operator [

The last step in crossover is to set the values for the SSS genes in the offspring. In our representation, the SSS gene is the left-most gene in the chromosome. This operation uses the “blend crossover” or BLX [

The interval is first set to that bounded by the parental values, and then extended by fifty percent in the direction of the more fit parent. In the example illustrated in

To evaluate this approach, we performed a series of experiments to test each aspect of the algorithm; these experiments are described in sequential order. All experiments used seismocardiogram data from a single subject obtained at rest and while undergoing mild exercise (light bike pedaling in an upright position with back support).

Four levels of wavelet decomposition were performed on successive thirty-second time intervals. Six mother wavelets were utilized: Daubechies, Symlets, discrete Meyer, Coiflet, Biorthogonal, and reverse Biorthogonal. A “ground truth” SV value was obtained for each thirty-second interval from the NICOM. This produced a data set with 96 features (6 × 16), and a “true” SV for each of the 85 intervals that were measured. We chose to set the maximum value of SSS to 32 assuming the GA could obtain results with a subset much smaller than this. Thus, the chromosome contained 33 genes, one for SSS and 32 packet indexes. For fitness to maximize, we used the R^{2} from a linear regression of the packets energy to SV. The population size was one hundred, the number of soft restarts was set to ten, with maximum zero accepts (restart condition) set to three.

The first experiment was directed toward achieving a maximum R^{2} value, but showed little evidence of convergence. ^{2} value at or near 0.988. Thus, the GA was unable to distinguish any features as any better than any others, and so used the maximum number of features it was permitted (32). The GA discovered many combinations of features that were able to predict SV nearly perfectly. In the example experiment shown

Failure of convergence in experiment one caused us to verify the algorithm. We elected to embed a perfect solution in the data, just to test the algorithm’s ability to discover it. We selected a set of five features and “doc-

tored” their values so that together they have perfect SV correlation. These features we gave indexes of 4, 31, 67, 80, and 92. (i.e. widely distributed among the pool of features). The “doctored” features emerged as the only genes left in the population after about one hundred generations (

preferred by evolution, but even the non-doctored features were each sampled several hundred times while the GA sorted through the combinations to locate the good one. Thus, we observed that the algorithm can work as expected when there is one perfect solution among a sea of poor ones.

We then challenged the algorithm by perturbing the data with Gaussian noise, where each feature is the original value plus twenty percent Gaussian noise. Again we saw the characteristic pattern of convergence failure (^{2} value. We hypothesized that the problem might be the sensitivity of the original algorithm’s hierarchical selection scheme on any difference in the first dimension of fitness (R^{2}), no matter how small. Selection for small subset size was never triggered because ties on R^{2} virtually never occurred. This feature of our problem makes it different from previous applications of this algorithm that were on classification tasks, where the fitness was usually to reduce classification errors or some similar metric. These errors, being modest discrete integers, often resulted in ties.

To test the influence of R^{2} on convergence, we reduced the number of significant digits in the value of R^{2} reported by the regression to the GA. By setting this to two significant figures, we essentially declared that chromosomes that differ in R^{2} by less than 0.01 should be considered equivalent, thereby allowing for ties and enabling the second level of the hierarchical fitness selection to kick in. One may also think of this as an admission that an R^{2} estimated from a sample of cases must of necessity contain a certain amount of noise (sampling noise rather than measurement noise); allowing the GA to over-exploit noise provides no benefit. This strategy resulted in a return of effective performance even though the problem is now more difficult because of the noise perturbation (

included in the regression. However, other features 21 and 26 (plus their noise) provided better results and were chosen by the GA. The end result provided four genes 21, 26, 67, and 80 with final R^{2} of about 0.98.

Having an indication that over-precision was precluding convergence in the presence of noise, we reran the original dataset with R^{2} reduced to two significant digits. We observed the patterns that indicate successful learning, and this time without the presence of doctored data. Now SSS evolves, first to 22 packets (in the first convergence, and the next eight soft restarts) and finally to 21 and 22 in the last two soft restarts (^{2} reached about 0.97 (

The CHC genetic algorithm with the MMX_SSS crossover operator has previously been applied to take out feature selection in bioinformatics classification tasks. We provide evidence that this algorithm may also be applicable to feature subset selection tasks in time series data processing, but the use of a high-precision first fitness metric such as regression R^{2}, seems to require a judicious reduction in significant digits provided to the GA in order to induce ties so that the second metric (SSS) may become active. In classification tasks, ties are common since counts of classification errors have a limited dynamic range. This work seems to show that a tradeoff may be needed between sensitivity to small improvements in accuracy and the desire for small subsets.

We are optimistic that this algorithm can be applied to selecting high-performance, small sets of signal features that can be combined to yield accurate metrics of some signal content, proving the data processing community with a powerful new tool. Finding specific mother wavelet packets that can be combined at the energy level without full waveform reconstruction can enable computationally inexpensive ways to extract information from time series data.

Ohad Bar SimanTov,J. DavidSchaffer,Kenneth J.McLeod, (2015) Developing an Evolutionary Algorithm to Search for an Optimal Multi-Mother Wavelet Packets Combination. Journal of Biomedical Science and Engineering,08,458-470. doi: 10.4236/jbise.2015.87043