Experiments

Here, the type of dictionary that is the best choice for decomposing the EEG signal will be identified, and a simple method will be introduced to achieve consistent results.

3.7.1. EEG Signal Recording and Pre-processing

A 64-channel EEG signal recording session was conducted within an EM-insulated chamber in NCTU Brain Research Centre (BRC) in July 2009. During the recording session, the test subject was asked to listen to a randomized sequence of rising or falling tones and was instructed to press a button immediately after hearing the rising tone but do nothing otherwise. The entire session lasted 40 minutes and produced total 1600 auditory-motor ERP epochs.

The signals from all 62 scalp channels were sampled into single-precision (16 bit) data at 1 KHz sampling rate and passed through ideal 1 Hz – 75 Hz FFT filters. They were then down sampled to 200 samples per second and fed through the artefact removal procedures. Only 266 epochs of the no bottom pressing event and 108 epochs of the bottom pressing ones remained after the procedure. Each epoch contain 290 samples that cover a period from

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40ms before to 1405ms after each Event On-Set (EOS) moment. Independent component analysis was then applied to all the epochs to produce their corresponding ICA components.

Average ERP signals of the two events were generated by removing the DC components of individual epochs and then averaging their signal samples. Similar process was used to produce the average ICA components.

3.7.2. Experiment Configurations

The experiment goal in this early phase of the research period is to identify correctness and guarantee stability of the decomposition result, and select the most promising dictionary that can appropriately decompose the EEG signal with the smallest amount of atoms required for later use.

Let the signal elapses for 𝒯 and has the bandwidth of ℬ, the resolution in time, angular frequency and scale is respectively Δ𝑡 Δ𝜔 and Δ𝑠 (scaling factors 𝑠 and Δ𝑠 have the same unit as time shifts 𝑡₀ and Δ𝑡, both are time units). The following algorithm is implemented to decompose the signal in a trivial way (the restriction of scaling factor 𝒮_min and 𝒮_max is usually problem dependent, it’s common for EEG signal having 𝒮_min 0 1𝑠 and 𝒮max 1𝑠):

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Algorithm

E

M

P

(EMP)

(

G

)

Input: Signal 𝑓(𝑡)

Output: A list of coefficients and waveform parameters (𝑎𝑛 𝛾_𝑛) Initialization: 𝑅𝑓₁ ← 𝑓(𝑡);

While 𝑛 1 to Iteration Limit and Stopping Condition not met 𝑎_max ← −∞; 𝛾_max ← 𝑁𝑢𝑙𝑙

For 𝑡₀ 0 to 𝒯 step Δ𝑡 For 𝜔₀ 0 to ℬ step Δ𝜔

For 𝑠 𝒮_min to 𝒮_max step Δ𝑠 𝛾 ← ,𝑡₀ 𝜔₀ 𝑠₀-; 𝑎 ← 〈𝑅𝑓_𝑛 𝑔_𝛾〉 If |𝑎| > 𝑎_max then

(𝑎_max 𝛾_max) ← (𝑎 𝛾) # the most correlated atom End If

End For End For End For

𝛾_𝑚𝑎𝑥^′ ← 𝑪𝑴𝑨𝑬𝑺(𝛾_max) # local optimization procedure with pruning rules 𝑎_𝑚𝑎𝑥^′ ← 〈𝑅𝑓_𝑛 𝑔_{𝛾𝑚𝑎𝑥}^′ 〉 # use the updated correlation value

(𝑎_n 𝛾_n) ← (𝑎_max^′ 𝛾_max^′ ) 𝑅𝑓_𝑛+1 ← 𝑅𝑓_𝑛− 𝑎_𝑛𝑔_𝛾𝑛 End While

Table 7 Exhaustive Matching Pursuit Algorithm (Gabor example)

The objective function is simple, given the parameter set 𝛾; the object function returns the value |〈𝑅𝑓_𝑛 𝑔_𝛾〉|. For the wavelets, simply remove the frequency scanning loop of 𝜔₀.

3.7.3. Experiment Results

This experiment was suggested to use the event related potential of ICA component 20 and channel 17 as our input. (Signal elapse 𝒯 1 45, Bandwidth ℬ 75Hz).

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Figure 9 Comparison of Convergence Rate of Different Dictionaries over ICA Component #20

Figure 10 Gabor Energy Map (ICA #20) Figure 11 Chirplet Energy Map (ICA #20)

# Time

Energy Map in Decibels, Base -25.00dB

Time (s)

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Figure 12 Mex. Hat Energy Map (ICA #20) Figure 13 Morlet Energy Map (ICA #20)

# Time Shift Scale Weight (%)

Table 11 Morlet Atom List (ICA #20)

Figure 14 CB-Spline Energy Map (ICA

#20)

Figure 15 Mixing Energy Map (ICA #20)

# Time Shift Scale Weight (%) Table 12 CB-Spline Atom List (ICA #20) Table 13 Mixing Atom List (ICA #20)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Energy Map in Decibels, Base -25.00dB

Time (s)

Energy Map in Decibels, Base -25.00dB

Time (s)

Energy Map in Decibels, Base -25.00dB

Time (s)

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Figure 16 Comparison of Convergence Rate of Different Dictionaries over Event Related Potential of Channel #17

Figure 17 Gabor Energy Map (Ch. #17) Figure 18 Chirplet Energy Map (Ch. #17)

# Time

100 Energy Map in Decibels, Base -20.00dB

Time (s)

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Table 16 Gabor Atom List (Ch. #17)

# Time Shift Scale Weight (%)

Table 17 Chirplet Atom List (Ch. #17)

Figure 21 CB-Spline Energy Map (ICA

#20)

Figure 22 Mixing Energy Map (ICA #20)

# Time Shift Scale Weight (%)

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3.7.4. Result Analyses

From Figure 9 and Figure 16, the result is quite apparent that the Gabor atoms out-perform the other dictionaries during the decomposition process. In the following stages, Gabor atoms will be used as the primary decomposition dictionary in the research.

First, the decomposition sequence should be examined. The following Figure 23 shows the weights of the decomposed atoms and the decays of residue of ICA component #20 during the pursuit processes, generally, the amount of energy extracted by the exhaustive MP is higher than Durka’s Stochastic MP. This phenomenon is expected since the CMA-ES optimization procedure breaks the parameter restrictions of integral resolution units, where atoms can be found to be generated by parameters in the field of real numbers ℝ.

Figure 23 Residue and Atom Comparison between Exhaustive MP & Durka’s SMP

# Tme Table 20 Gabor Atom List Generated

by Exhaustive MP (ICA #20)

Table 21 Gabor Atom List Generated by Durka’s Stochastic MP (ICA #20)

0

EMP Atom Wgh. SMP Atom Wgh. EMP Residue SMP Residue

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Then, the correctness and precision of the decomposition results should be examined.

Figure 24 and Figure 25 present the projection of the objective function on the axis of each parameter. For example, the “Time Shift Scanning” sub-graphs represent the changes of fitness values corresponding to the changes of time centers, while the other parameters are fixed during the scanning process.

Figure 24 EMP on ICA #20 Chirplet Atom 1 in Table 9

(Desired Result)

Figure 25 EMP on ICA #20 Chirplet Atom 2 in Table 9

(Typical Failure)

In Figure 24, it shows a perfectly optimized result of CMA-ES, each variable are closely lying beside their optimal values, which maximize the inner product of the atom waveform and the signal and consequentially, position themselves on the top of concaves with derivatives close to zero in their vicinities.

However Figure 25 presents another circumstances, which is a typical failure of insufficient sampling on large scaling, although the searched parameters are in close

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time Shift Scanning (centre: 0.507227 sec.)

Time Shift, Found-Real Difference: -0.000750 sec.

Inner Product Value

Frequency Scanning (centre: 3.247493 Hz)

Frequency Shift, Found-Real Difference: -0.015923 Hz.

Inner Product Value

Chirp Rate Scanning (centre: -0.002813)

Chirp Rate, Found-Real Difference: 0.000025

Inner Product Value

Time Shift Scanning (centre: 0.249099 sec.)

Time Shift, Found-Real Difference: 0.023762 sec.

Inner Product Value

Frequency Scanning (centre: 0.010090 Hz)

Frequency Shift, Found-Real Difference: -1.257953 Hz.

Inner Product Value

Chirp Rate Scanning (centre: 0.000000)

Chirp Rate, Found-Real Difference: -0.050000

Inner Product Value

IP Curve Real Found

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proximity of the optimal value only with differences in only a small amount of samples (the resolutions are Δ𝑡 0 05 s and Δ𝜔 0 689 Hz. Therefore, the time difference in the number of Δ𝑡 is approximately 0 024𝑠 ≅ 0 48 Δ𝑡, and the frequency difference in the number of Δ𝜔 is 1 26 𝐻𝑧 ≅ 1 824 Δ𝜔). Notice that the real optimal value for scaling is quite large and locate on the edge of the objective curve. This is a classical degenerated case usually occurs when the scaling factor is too large in this case. The objective value can change drastically when modifying other parameters such as frequency and time centers due to the dilation and shrinkage of the resolution respectively. In the future design of the search algorithm, the dependencies of parameter should be considered in the design.

Still, the decompositions are precise enough to achieve its goal, but the heavily reliance on the exhaustive sampling in the 3-dimensional parameter space is the primary drawback.

During the exhaustive scanning process, the number of samples in the EEG signal is 290, it requires 290 × 145 × 290 12 194 500 iteration to acquire the densely sampled grid, and takes about 4 to 6 minutes for each Gabor atom in the table even on a 2.4GHz, Quad-core computer with 4GB RAM even written in C.

In the following section, we’ll discover the mathematical properties of the Gabor atoms and used the derived result to decrease the cost of computation during the decompositions.

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Chapter 4. Natural Gradient Search over