Extrapolation of exponential time series is not enhanced by having more data points

Abstract

H. TIMMERS Institute for Perception Research, Eindhoven, Holland W. A. WAGENAAR Institu te for Perception TNO, Soesterberg, Holland Extrapolation of exponential time series is not enhanced by having more data points curves are worse than descending curves (Timmers & Wagenaar, 1977). The value of ~ was not affected by a variable expected to influence the second stage: year-by-year extrapolation vs. extrapolation in one large step. The year-by-year extrapolation showed that subjects use the same multiplier aef3b in all successive steps; this was the most direct evidence that subjects use the correct rule but the wrong parameters. The present study is designed to investigate another variable that may influence the first stage. This variable is the number of data points in the starting series. The paradigm is simple. In previous studies, subjects were always presented with five successive data points, e.g., 3, 7, 20, 55, 148 (y = ex, x = 1, 2 ... , 5). The trajectory 3 through 148 can also be covered by three data points (3, 20, 148) or seven data points (3, 5, 10, 20, 39, 76, 148). These three representations describe the same exponential function, but with a different sampling frequency. The number series can be mathematically described by second-, fourth-, or sixth-order polynomials. Since such functions increase faster when more high-order terms are added, it is to be expected that subjects, fitting the simplest polynomial, would produce higher extrapolations when more data points are available. This is a prediction put forward by Jones (1977; see also Wagenaar, 1977). The example presented in Table 1 may serve to illustrate the point. In contrast, Timmers and Wagenaar (1977) suggested that subjects do no such thing as fitting a polynomial, but rather that they intuitively estimate a growth factor by inspecting a few initial differences between successive data points. This would explain why descending curves are better extrapolated. Obviously, differences between successive data points become smaller the more data points are presented. Accordingly, it may be expected that extrapolations based on these differences decrease when more data points are presented; in short, the more data points, the worse the extrapolations. This phenomenon has been observed before with non numerical presentations of exponential growth (Wagenaar & Timmers, 1978, in press). In the present experiment, the prediction was tested with numerically presented data points. (1) y = ebx, Subjective extrapolation of time series can formally be described as a two-stage process. In the first stage, some properties of the time series are identified; whether the series is stationary or not, whether there is an increase or decrease, acceleration or deceleration, whether or not there are periodicities, and so on. In the second stage, the rule previously discovered is applied in generating further elements of the series. Both stages are open to error; acceleration in a nonlinear series can be overlooked (Stage 1), or the subject may produce the third future element instead of the fifth (Stage 2). Whenever subjects fail to correctly extrapolate a time series, one would like to know whether either one or both stages contribute to the inaccuracy. Previous experiments on the extrapolation of exponential processes showed that subjects tend to make mistakes in Stage 1 rather than in Stage 2. The evidence was that subjective extrapolations in a variety of conditions could adequately be described by a simple model which assumed that the subject applied a constant, but underestimated, growth factor. If the time series is described by: