cultural relativism thesis statement

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The difference between the o r i g i n a l measured s i g n a l and the estimated s i g n a l i s considered to be additive o u t l i e r content see Section 4.


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The o u t l i e r content i s then processed see Section 4. The objective of the cleaning process i s to provide an estimate of the o r i g i n a l s i g n a l without the AO content. Note that with the s c a l i n g given 4. Hampel's three part redescending p s i function [43] was used i n Equation 4. A GM1 estimate i s based on the estimated cleaned time ser i e s and a GM2 estimate i s a further i t e r a t i o n where the parameters from GM1 are used i n the cleaner to provide a t h e o r e t i c a l l y improved estimate of the cleaned time s e r i e s.

Through the simulation studies described i n Section 3. Each segment i s modeled at the order expected for i d l e task EEG and hence, reducing the a b i l i t y of the model to account for active task information i n the EEG see Section 4. The s i g n a l from each segment i s then cleaned using the estimated model parameters i n the cleaner described i n Section 4. The o u t l i e r s are then calculated by taking the diff e r e n c e between the o r i g i n a l and cleaned signals see Figure 4. This t e s t confirmed that the extraction process had some d i s t i n c t a b i l i t y to recover the o u t l i e r content from the simulated s i g n a l.

Patchy contamination was used since i t was expected that i n the case of r e a l EEG the additive event r e l a t e d p o t e n t i a l s would be correlated. As suggested by Martin and Zeh [51] , the correlated v a l u e s f o r v. This procedure r e s u l t s i n a correlated o u t l i e r series which has roughly the same variance as i n the independent case [51].

It was found i n t h i s a p p l i c a t i o n that the tuning parameters for the p s i function given i n Equation 4. Figures 4. OOO It i s clear from these examples that the process performs better with GM2 estimates than with GM estimates and much better than with LSQ estimates.

Since these t e s t s revealed three c l e a r l y discernable jumps i n performance i n using LSQ, GM, and GM2 parameter estimates, i t was decided that subsequent studies using o u t l i e r detection i n t h i s t h e s i s work would be r e s t r i c t e d to those three estimation methods. The o u t l i e r pattern i s then smoothed by convolving i t with a 16 point tapered smoothing window which i s based on a minimum-bias sp e c t r a l window suggested by Papoulis [52].

Time Series Talk : ARMA Model

As w e l l , these studies were instrumental i n e s t a b l i s h i n g appropriate EEG segment lengths and a procedure for the s e l e c t i o n of the AR model order. Since the term S e f applies to the residuals which are i n theory white, the r e s u l t i n g power density function of the residuals should be f l a t and t h e r e f o r e S e f w i l l be a constant independent of frequency.

Id e a l l y , the value of t h i s constant noting that the mean of the residuals i s zero w i l l be proportional to the variance of the residuals [A6]. Hence, the f i n a l expression for the conventional AR spe c t r a l estimate i s obtained by r e p l a c i n g S f i n A. Single t r i a l AR sp e c t r a l estimates from adjacent one second segments demonstrated that considerable change i n si g n a l c h a r a c t e r i s t i c s could occur over t h i s span of two seconds.

An example of t h i s i s provided i n Figure A. I t contains four consecutive AR spe c t r a l p l o t s , each derived from a one second segment of 61 Figure 4. As was discussed i n Section 3.

This was an attempt to trade o f f the need for short segments because of the r e l a t i v e l y r a pid changing signal c h a r a c t e r i s t i c s with the desire to r a i s e the segment length above the lower bound for purposes of improving the parameter estimation e f f i c a c y. It was found that s e l e c t i n g the model order v i a conventional methods such as Akaike's Information C r i t e r i a AIC does not work well with these short segments [53].

Conclusions were s i m i l a r to Jansen [32] i n that the s e l e c t i o n of an appropriate model order requires some t r i a l and error and, i f possi b l e , some a - p r i o r i knowledge of expected r e s u l t s. I t was found useful to t r y a number of orders within a reasonable range for a sample rate of 64Hz, somewhere between 8 to 25 , following the trend of the estimate as the model order was increased. Features were i d e n t i f i e d that seemed reasonable based on both the a - p r i o r i knowledge of the condition under which the EEG was c o l l e c t e d and a conventional FFT based estimate.

The order was sequentially increased, expecting the features to become better defined, u n t i l spurious peaks began to occur. The appropriate model order was then selected to be two or three below that value. T y p i c a l l y , model orders were selected i n the range of 12 to 14 from subjects during the i d l e task and i n the range of 18 to 22 from subjects during the active task.

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The r e s i d u a l s i g n a l can be thought of as a whitened si g n a l because the information that can be represented by an AR model has been subtracted r e s u l t i n g i n a si g n a l with a much f l a t t e r spectrum. When the FFT i s applied to t h i s prewhitened signal the inherent drawback of leakage i s gre a t l y reduced. A p p l i c a t i o n of conventional leakage c o n t r o l , such as Blackman windowing, serves to further reduce t h i s problem.

The prewhitened AR estimation method, therefore, combines the spe c t r a l information from both the AR model and the re s i d u a l FFT spe c t r a l estimate. Some i n s i g h t into the a b i l i t y of the AR model to represent short segments of EEG was gained by pursuing studies using prewhitened AR spe c t r a l estimates. These studies demonstrated that when an appropriate model order was u t i l i z e d the conventional AR spe c t r a l estimates were reasonably good compared to the prewhitened AR estimate which makes use of information retained i n the residuals see Birch et a l.

This indicates that the AR model, although not perfect, does represent much of the information contained i n a short segment of EEG. An example of both a conventional and a prewhitened 12th order AR spe c t r a l estimate of i d l e task EEG i s given i n Figure 4. Figure A. Motor p o t e n t i a l a c t i v i t y i n the active case should occur, approximately, during the f i r s t three seconds of the epoch, noting that the actual thumb movement began one second into the epoch.

These pl o t s demonstrate that the conventional averaging technique reveals some d i s t i n c t motor a c t i v i t y i n the o r i g i n a l active case raised l e v e l of p o s i t i v i t y i n the averaged s i g n a l during the f i r s t three seconds with a peak at about two seconds. Results provided i n the following sections demonstrate that the information i n these o u t l i e r patterns i s r e l a t e d to the thumb movements. As w e l l , for comparison purposes, a p l o t of the conventional average for the active case i s also included i n t h i s f i g u r e.

Abstract/Summary

The fact that the average active case patterns maintain a general shape s i m i l a r to the s i n g l e t r i a l patterns, strongly indicates that there i s information r e l a t e d to the thumb movement that i s consistent from t r i a l to t r i a l. The conventional average of active t r i a l s shows that with N's of 6 and 15 the motor p o t e n t i a l information i s quite l i m i t e d and the "smearing" e f f e c t of event r e l a t e d information that i s discussed above for the active case o u t l i e r patterns would also be occurring i n these conventional averages.

Hence, with the conventional averaging method, even with much greater N's as i n the case of the Grunewald study see Section A. A LSQ Active O u t l i e r Patterns Degrading with Higher Model Orders It would also be expected, based on the neurological premise, that the single t r i a l processing method would perform best when the AR model order was selected to best f i t the i d l e case. As the model order i s increased the AR model would be expected to gain some improved a b i l i t y to represent the motor r e l a t e d a c t i v i t y i n the active task EEG.

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Hence, the performance of the sin g l e t r i a l method should begin to degrade since the cleaning process, which u t i l i z e s the higher order AR model, would lose some of i t s effectiveness i n 78 detecting motor r e l a t e d o u t l i e r s. A p a i r of averaged active o u t l i e r pattern p l o t s using LSQ parameters for model order 12 generally appropriate for the i d l e case and model order 22 generally appropriate for the active case are shown i n Figure 4.

These p l o t s demonstrate that the performance does degrade, i n terms of both the amplitude and the d e t a i l of features i n the averaged o u t l i e r pattern, when the model order i s better matched to the active case. The patterns, although unique from t r i a l to t r i a l , do seem to posses a generally consistent waveform which contains features that appear r e l a t e d to events i n the thumb movements. The features are described below and are shown i n Figure 4. Feature 1: Time from epoch onset to the point when the thumb movement f i r s t reaches the "on target" p o s i t i o n.

Feature 2: Time from epoch onset to the point when the thumb movement f i r s t leaves the "on target" p o s i t i o n.

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Feature A: Time from epoch onset to the f i r s t negative peak i n the o u t l i e r pattern a f t e r feature 3, that has a minimum of 5 units magnitude peak-to-trough d i f f e r e n c e. Feature 5: Time from epoch onset to the next p o s i t i v e peak i n the o u t l i e r pattern a f t e r feature A, that has a minimum of 20 units magnitude peak-to-trough diffe r e n c e on both sides of the peak.

There was an expectation r e s u l t i n g from the e a r l i e r conventional study by Grunewald and Grunewald-Zuberbier [7] and from observations taken from Figure A. The sample c o r r e l a t i o n c o e f f i c i e n t s between a l l of the features from Subject 1 were calculated and are summarized i n Table A.

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Hence, t h i s demonstrates that there i s a strong consistent r e l a t i o n s h i p between features i n the thumb movement and features i n the sing l e t r i a l o u t l i e r pattern. In p a r t i c u l a r , the r e l a t i o n s h i p between features i n the o u t l i e r pattern and i n the thumb movement was examined using the z-test for the difference between c o r r e l a t i o n s calculated on dependent samples see Steiger [5A].


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The r e s u l t s from these t e s t s are also summarized i n Table A. As expected, the c o r r e l a t i o n between feature 5 and feature 2 was larger than that between feature 5 and feature 1, but t h i s difference achieved only a marginal l e v e l of s i g n i f i c a n c e. These i n i t i a l i n v e s t i g a t i o n s , revealed that the use of dynamic time warping DTW provided the best quantitative measure of performance for the s i n g l e t r i a l processing method compared to the other previous a n a l y s i s.