#### Sponsor

Portland State University. Department of Electrical and Computer Engineering

#### Advisor

James McNames

#### Date of Award

Summer 7-24-2015

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.) in Electrical and Computer Engineering

#### Department

Electrical and Computer Engineering

#### Physical Description

1 online resource (xii, 182 pages)

#### Subjects

Spline theory, Periodic functions, Interpolation, Electrocardiography

#### DOI

10.15760/etd.2458

#### Abstract

Semiperiodic signals possess an underlying periodicity, but their constituent spectral components include stochastic elements which make it impossible to analytically determine locations of the signal's critical points. Mathematically, a signal's critical points are those at which it is not differentiable or where its derivative is zero. In some domains they represent characteristic points, which are locations indicating important changes in the underlying process reflected by the signal.

For many applications in healthcare, knowledge of precise locations of these points provides key insight for analytic, diagnostic, and therapeutic purposes. For example, given an appropriate signal they might indicate the start or end of a breath, numerous electrophysiological states of the heart during the cardiac cycle, or the point in a stride at which the heel impacts the ground. The inherent variability of these signals, the presence of noise, and often, very low signal amplitudes, makes accurate estimation of these points challenging.

There has been much effort in automatically estimating characteristic point locations. Approaches include algorithms operating in the time domain, on various transformations of the data, and using different models of the signal. These methods apply a wide variety of techniques ranging from simple thresholds and search windows to sophisticated signal processing and pattern recognition algorithms. Existing approaches do not explicitly use prior knowledge of characteristic point locations in their estimation.

This dissertation first develops a framework for an efficient parametric representation of semiperiodic signals using splines. It then implements an instance of that framework to optimally estimate locations of characteristic points, incorporating prior knowledge from manual annotations on training data. Splines represent signals in a piecewise manner by applying an interpolant to constraint points on the signal known as knots. The framework allows choice of interpolant, objective function, knot initialization algorithm, and optimization algorithm. After initialization it iteratively modifies knot locations until the objective function is met.

For optimal estimation of characteristic points the framework relies on a Bayesian objective function, the *a posteriori* probability of knot locations given the observed signal. This objective function fuses prior knowledge, the observed signal, and its spline estimate. With a linear interpolant, knot locations after optimization serve as estimates of the signal's characteristic points.

This implementation was used to determine locations of 11 characteristic points on a prospective test set comprising 200 electrocardiograph (ECG) signals from 20 subjects. It achieved a mean error of -0.4 milliseconds, less than one quarter of a sample interval. A low bias is not sufficient, however, and the literature recognizes error variance to be the more important factor in assessing accuracy. Error variances are typically compared to the variance of manual annotations provided by reviewers. The algorithm was within two standard deviations for six of the characteristic points, and within one sample interval of this criterion for another four points.

The spline framework described here provides a complementary option to existing methods for parametric modeling of semiperiodic signals, and can be tailored to represent semiperiodic signals with high fidelity or to optimally estimate locations of their characteristic points.

#### Persistent Identifier

http://archives.pdx.edu/ds/psu/15882

#### Recommended Citation

Guilak, Farzin G., "A Spline Framework for Optimal Representation of Semiperiodic Signals" (2015). *Dissertations and Theses.* Paper 2460.

http://pdxscholar.library.pdx.edu/open_access_etds/2460

10.15760/etd.2458