Stationary myoelectric spectra from a nonparametric, low bias, and low variance estimator
Proceedings of 17th International Conference of the Engineering in Medicine and Biology Society
We explain Thomson's crude multiple window method (MWM), a nonparamet& spectral estimator with low bias and variance. We then compare its output to Hamming-windowed periodogrm and autoregressive (AR) method outputs for a sample myoelectric signal. Our goal is a speclral estimator to embed in a myoelectric controller. hlroductim. Myoelectric signal spectrum estimation requires a method that (1) works well on small (and hence relatively stationary) windows of a single realization of the signal,
... on of the signal, (2) does not suffer fiom bias and variance that obscures the spectral features we are trying to detect; and (3) incorporates no incorrect assumptions &out the structure of the signal. The periodogram suffers from substantial bias and wiance [l]. Autoregressive (AR) model-based estimators [l],  have lower variance than the periodogram, but their all-pole assumption matches certain motion artifacts better than the underlying physiological signal, Thus, motion-specific detail can be lost as the model converges on motion artifact. Here, we test a more recently develciped nonparametric spectral estimator, Thomson's multipile window method (MWM) , which gives lower bias and variance than the periodogram without introducing apotentially misleading model , 141. Methods. Instead of one ad hoc window such as the Hamnung window as is often used in the periodogram, we apply a set of N-point windows to the data that are (1) mutually orthogonal and (2) optimally concentrated in frequency. The windows -discrete prolate spheroidal sequaices (DPSSs) -are the inverse discrete time Fourier transfims of the eigenfunctions of a cascaded band-liting (to [-W,v,wl) and time-limiting (to [O,N) operation . The crudest MWM power spectral density estimate on a length4 data window is where, x,(n are the eigenspectra 2 ~xku>lz = ril n=O x[nlvk[nle+* I . (2) v, is the kth DPSS window and h ,is its corresponding eigenvalue. W is the resolution of the estimate. We use only the largest K = 2 M eigenvalues and their corresponding eigenfunctions in the estimate; K is the tradeoff between the bias and additional spectral information contributed by the higher order DPSSs. Larger K gives smaller variance but larger W and bias. Figure 1 shows the K=4 case computations. Results and Discussion. Figure 2 compares the Hamming-windowed periodogram, the AR (model order set at 5 by Akaike Information Criterion [l]), and the crude Thomson's estimates for a surfacemeasured Abductor Pollicis Longus and Extensor Pollicis Brevis monopolar myoelectric signal (6" recessed wet electrode, bandpass filmed 3-300 Hz, sampled at 1000 Hz). The sample contains some motion artifact, as we measured it during thumb abduction. Canclusions. Thomson's MWM estimates more closely match what we expect from the physiological processes producing the myoelectric signal  than the periodogram and AR estimam. Crude MWM estimates have significantly lower bias and variance than the periodogram, and lower bias than the AR estimate m the presence of noise or poor model fit Averages over many trials of the same motion contrast the periodogram-MWM variance (Figure 3 ). Thomson developed a lower bias version of this spectral estimation approach, the adaptive MWM, as well as tests for embedded harmonics that, in the myoelectric analysis application, give insights into motion artifact . A G k t l D W l W n .