II. Time series: Overview

A. Definitions

Although most psychophysiological data are presented in terms of mean levels within or across subjects, the sequential pattern, on which the mean is based, may contain important information. Time series statistics provide methods to describe and evaluate these patterns. A set of sequential observations, such as the circumference of the chest sampled twice a second or the time intervals between sequential heart beats, constitutes a time series. Although there are various types of time series, the unifying dimension is that they are all indexed by time. Mathematically, a time series may be described as a string of variables that are sequentially indexed, for example: Xt, Xt+1, Xt+2,.........., Xt+n. In this example, the index t represents time.

A time series is continuous when observations are made continuously in time. An analog signal which changes over time, characteristic of many physiological processes, is a continuous process. The term continuous is used for series even when the measured variable can only take a fixed set of values. A time series is discrete when the observations are taken only at specific times, usually equally spaced. Most time series methods assume that data represent discrete samples of a continuous time series sampled at equally spaced intervals. In practice researchers are usually dealing with discrete time series, although the underlying physiological process is continuous. For example, the analog-to-digital converter in a laboratory computer transforms a continuous analog signal into digital representations at discrete time intervals. Problems often arise with discrete time series, like inter-beat intervals, which are not equally spaced in time. To deal with the problem of equally spaced samples, one may assume that sequential heart periods are a discrete manifestation of a continuous process (e.g., neural inputs to the heart). Thus, interpolations are possible to adjust sampling into equal time intervals.

Time series have other characteristics. For example, a time series may be characterized by observations which can take one of only two values (e.g., O and 1). This type of time series is known as a binary process. Binary processes are common in communication theory and in the modelling of neuronal activity. In these examples, a neuron is viewed as a switch which may be either "on" or "off, and can be coded as a one or a zero. Other time series are characterized by the sequential time intervals between events. Unlike the EEG and EKG which are clearly continuous processes or the binary process which has two states, the heart period series is a series of inter-event intervals triggered by the heart beat. This type of time series is known as a point process. Point processes are time series in which a series of events occur randomly in time and the duration of the event is assumed to be instantaneous.

B. Statistical characteristics of physiological time series

Much of statistical theory is concerned with random samples of independent observations. The special feature of time-series analysis is that sequential observations are usually not independent and that the analysis must take into account the time order of the observations. This time ordered dependency may be assessed by calculating an autocorrelation (see below). There is a very special case of a time series in which the sequential observations are independent. This is a string of identically and independently distributed random variables (IID). In the IID case, knowledge of any one random variable does not influence the distribution of any other. Thus, the expected value for any one time sample is the same as any other sample. Physiological time series of healthy alert subjects are never IID. Physiological time series may approach IID in situations when the nervous system input to peripheral organs is removed via surgery or drugs.

Random variables in most time series take on other probabilities and may be viewed on a continuum of dependency. When sequential observations are dependent, future values may be predicted from past observations. If a time series can be predicted exactly from past observations, it is said to be deterministic. Since all physiological and behavioral processes are influenced by unknown factors, it may not be possible to describe behavioral and bio-behavioral processes with a totally deterministic model. Most time series are stochastic and the future is only partly determined by past values. In fact, time series actually may be expressed as the sum of two uncorrelated processes, one purely deterministic and one purely nondeterministically stochastic. This theorem is known as the Wold Decomposition Theorem.

C. Parameters of a time series

Figure 1a illustrates a sine wave indexed by time. Note that the duration of one complete cycle of the sine wave is 5 seconds. The duration of a sine wave defines the period. The reciprocal of the period defines the frequency. In our example, the sine wave with a 5 sec period has a frequency of 0.2 cycles per second or 0.2 Hz. This frequency was selected because it is similar than the frequency of spontaneous breathing in adult subjects. In this example, the sampling interval was 500 msec. This could be stated as a sampling rate of 2 Hz. Thus, if we sample twice a second (i.e., 2 Hz), a 0.2 Hz sine wave would require 10 of these samples for one complete cycle.

A number of parameters are necessary to describe a periodic process. In Figure 1a the sine wave takes on values from -100 to +100. The peak of the periodic process is called the amplitude. In Figure 1a the amplitude is 100. In psychophysiological research we are often interested in quantifying the variance of the signal. In conceptualizing variance of a periodic process, it is useful to recall the arithmetic relationship between amplitude and variance. The square of the amplitude divided by two is equivalent to the variance [var = A(squared)/2]. Thus, decomposition of physiological processes into sine waves would provide a method of describing component variances of the different periodic processes. This method of decomposition is the basis for spectral analysis and will be described in the section on frequency domain analyses.

The phase of a periodic process relates the onset of the process to the time at the origin. If the process starts at the origin (i.e., O on the ordinate and O on the abscissa), the phase is zero. The phase can be described in degrees (from O to 360), in radians (from 0 to 2 Pi), or in proportion of the sinusoid temporally displaced from the origin (e.g., 1/2 cycle). The same temporal displacement could be described as a phase shift of 180 degrees, Pi radians, or 1/2 cycle. In most situations the periodic process does not originate at the origin and exhibits some displacement.

D. Methods to analyze periodic activity

There are two basic approaches that may be used to describe and analyze the periodic components of a time series. A time series may be represented and analyzed in the time domain or in the frequency domain. Time domain representations plot data as a function of time. The time domain methods that are most relevant to the study of periodic processes are based on the autocorrelation function. Autocorrelation techniques are mathematical extensions of traditional correlation techniques. An autocorrelation is the correlation of one time series with a time-shifted version of itself. Frequency domain techniques are those based on the spectral density function. The procedure for estimating the spectral densities at various frequencies is called spectral analysis. Spectral analysis decomposes a time series into sinusoidal components at different frequencies and different amplitudes. Both time domain and frequency domain methods have provided valuable tools to describe periodic phenomena. In the following sections, examples of time domain and frequency domain methods will be described and evaluated.