## 3 Quantitative Assessment of Parkinsonian Tremor
### Least-Square-Estimation Model and Signal Flow Diagram  
Because accelerometers and gyroscopes are used to measure linear and rotational movement, respectively, for some tremor tasks the accelerometers present a higher correlation, while the gyroscopes perform a higher correlation during other tremor assessment tasks. Thus, a multiple linear regression model is used to fit the clinician ratings (UPDRS tremor score) and the peak powers during each tremor task [29,31,35,36].  
Then the linear regression model can be expressed as:  
\[R=R_{0}+\ln\left(\sum_{i-1}^{3}b_{i}\cdot PA_{i}+\sum_{i-1}^{3}c_{i}\cdot PG_{ i}\right) \tag{4}\]  
where \(R\) is the predicated tremor score; \(R_{0}\), \(b_{i}\), and \(c_{i}\) are the regression coefficients. \(i=1\), 2, and 3 denotes \(x\), \(y\), and \(z\) axis, respectively; \(PA_{i}\) and \(PG_{i}\) are the single-axis peak power for the triple-axis accelerometer and triple-axis gyroscope, respectively [20]. \(R_{0}\) and the three scaling factors (\(b_{i}\) and \(c_{i}\)) are different for the three tremors (rest, posture, and action) [37]. These three regression coefficients (scaling factors) were obtained according to Stevens' power law in psychophysics [38].  
The flowchart of the signal processing for the tremor quantification in this study is shown in Figure 4. The sensor data are from the glove part, which is based on the IMU board and command module. In the computer part, the sensor data are band-pass filtered from 3.25 to 12 Hz (tremor band) before tremor amplitude quantification, in order to remove gravitational acceleration and dyskinesia components.  
The signals in ten seconds are used for the PSD estimation at the end of the ten-second tremor-assessment task. Initially, the signals from the gyroscope and the accelerometer are analyzed separately. The single-sided, scaled, and auto power spectra of the signal of each gyroscope axis and each accelerometer axis, of which length is set to ten seconds, are separately computed to get the four-channel power spectra in real-time. The dominant frequencies and the estimated peak powers of these frequencies are thus obtained. Then, the estimated peak powers are multiplied by scaling factors. The channel with the highest peak power is the dominant channel, and its dominant frequency is the dominant frequency (\(F\)) of all channels. Other channels are needed to perform PSD estimations again with the dominant frequency, and multiply the peak powers by the scaling factors again. The power estimations in all axes around the same dominant frequency (\(F\)) are calculated and sum up to \(P^{\prime}\).  
Figure 4: Signal processing for the tremor detection and quantification (six channels).  
Then, the sum of peak powers, which is regarded as the tremor amplitude, and the dominant frequency can be displayed in the recoding list.  
At the same time, the power spectrum of every axis in a certain period (three seconds) is calculated continuously. The dominant frequency and peak power can be obtained as soon as the tremor occurs. When the dominant frequency is in the range of 3.5-7.5 Hz and the sum of peak powers is more than a threshold, the tremor is reported.