Chapter 8 links diffusion models to linear system models of visual encoding, which leads to models with time-varying drift rates. These models apply to near-threshold detection (Yes/No accuracy) and simple reaction time tasks in which decisions are based on the outputs of transient and sustained mechanisms that respond either to stimulus onset and offset transients or to steady-state intensity levels, respectively. These mechanisms are represented by diffusion processes with drift rates that either increase to a maximum and then decrease to zero for transient mechanisms or increase to an asymptote for sustained mechanisms. The chapter evaluates the response time and accuracy predictions of these kinds of models, together with their associated hazard functions, in simple reaction time and temporal integration tasks and contrasts them to the predictions of a simple, constant-drift diffusion model in which a proportion of stimuli are not detected because the drift rate is negative or zero.

This chapter introduces the power spectrum. The power spectrum is the Fourier Transform of the autocovariance function, and the autocovariance function is the (inverse) Fourier Transform of the power spectrum. As such, the power spectrum and autocovariance function offer two complementary but mathematically equivalent descriptions of a stochastic process. The power spectrum quantifies how variance is distributed over frequencies and is useful for identifying periodic behavior in time series. The discrete Fourier transform of a time series can be summarized in a periodogram, which provides a starting point for estimating power spectra. Estimation of the power spectrum can be counterintuitive because the uncertainty in periodogram elements does not decrease with increasing sample size. To reduce uncertainty, periodogram estimates are averaged over a frequency interval called the bandwidth. Trends and discontinuities in time series can lead to similar low-frequency structure despite very different temporal characteristics. Spectral analysis provides a particularly insightful way to understand the behavior of linear filters.