Extraction of a local linear trend from physiological time series

Abstract

We discuss methods for robust signal extraction from noisy physiological time series as measured in intensive care. The aim is a method which allows a fast and reliable de-noising of the data and separation of artifacts from relevant changes in the patients condition¹. For approximating local linear trends we use robust regression estimators. We examine the performance of the L₁ regression, the repeated median² and the least median of squares³ for this task.

Introduction

In intensive care, clinical information systems acquire and store physiological variables online at least every minute. Reliable automatic monitoring systems are needed to process these data in real time and to support decision-making at the bedside in time critical situations. Clinically relevant changes such as sudden level shifts and trends need to be detected and to be distinguished from noise and irrelevant artifacts. Median filtering is frequently applied for signal extraction from a noisy time series¹. However, the performance of the median worsens markedly when a trend occurs⁴. Therefore, we use robust regression estimators as these are able to adapt to local trends⁵. Our main interests are the reproduction of a linear trend, the detection of level shifts, the detection of trend changes, and the computational demands.

Methods

Robust regression.

For robust extraction of a local linear trend we apply robust estimators designed for the simple linear regression model y = μ + βx + e. An intuitive approach to robust estimation of μ and β is to replace squared by absolute distances using ^L1 regression. The finite sample replacement breakdown point denotes the minimal percentage of outliers which suffices to carry the estimate beyond all bounds. In our situation, for L₁ regression it is about 29.3%. The repeated median² and the least median of squares³ (LMS) both share the breakdown point of the median, which is approximately 50%. For calculation of the exact LMS solution⁶ we use an algorithm which has a computational complexity of O(n⁴). Hence, the time needed for computation of the LMS is much larger than for the other methods and increases rapidly with the window width.

Simulation model.

In order to compare the regression estimators in a single time window we generate data from the model Y_t = μ + β t + e_t, t=−m, ... , m, with μ=0 and for several slopes β. The error e_t is simulated from an autoregressive model of order one [AR(1)] e_t = φ e _t−1 + u_t with mean zero, φ in (−1,1) and Gaussian innovations u_t with unit variance σ²=1. Estimates of μ and β allow to approximate the level and the slope of the signal at the center t = 0 of the time window. The resulting time delay m is determined by the required stability (m large) and the time delay possible in an online application (m small). We consider the cases m=10, 15, 25.

Results and Conclusions

We find that the L₁ regression offers little advantage in comparison to the repeated median. The LMS is the least influenced by outliers if φ is close to zero, but this advantage disappears as |φ| increases. Moreover, the LMS is very variable and computationally demanding. The repeated median does also resist some outliers, is computationally much less expensive and does not show severe instabilities. Hence, we consider it to be a promising candidate for robust signal extraction when computation time is critical.