In a visual search experiment, the subject must find a target item hidden in a display of other items, and their performance is measured by their reaction time (RT). Here I look at how visual search reaction times are correlated with past reaction times. Target-absent RTs (i.e. RTs to displays that have no target) are strongly correlated with past target-absent RTs and, treated as a time series, have a 1/*f* power spectrum. Target-present RTs, on the other hand, are effectively uncorrelated with past RTs. A model for visual search is presented which generates search RTs with this pattern of correlations and power spectra. In the model, search is conducted by matching search items up with “categorizers,” which take a certain time to categorize each item as target or distractor; the RT is the sum of categorization times. The categorizers are drawn at random from a pool of active categorizers. After each search, some of the categorizers in the active pool are replaced with categorizers drawn from a larger population of unused categorizers. The categorizers that are not replaced are responsible for the RT correlations and the 1/*f* power spectrum.

*f*power spectrum (Farrell, Wagenmakers, & Ratcliff, 2006; Gilden, 2001; Gilden, Thornton, & Mallon, 1995). The power spectrum of a time series is calculated by treating the sequence of RTs as an evenly spaced set of signal values, and taking the Fourier transform of them. A 1/

*f*power spectrum has power at frequency f proportional to 1/

*f*. The finding of a 1/

*f*spectrum is interesting for two reasons. First, 1/

*f*power spectra are found in many diverse phenomena, such as the fluctuation of light intensity from quasars, current noise in resistors, semiconductors, and thermionic tubes, sea level fluctuations, music, intervals between heartbeats, and errors in time interval estimation, among others (Dutta & Horn, 1981; Milotti, 2002; Press, 1978; Voss & Clarke, 1975). The widespread occurrence of 1/

*f*power spectra, and power spectra close to 1/

*f,*suggests that there may be a universal mechanism behind all these phenomena. The second reason 1/

*f*spectra are interesting is because, despite many attempts, no one has so far convincingly put forward any such universal mechanism.

*f*power spectra can be appreciated by comparing 1/

*f*power spectra to two other spectra which we do know the causes of. If visual search RTs were independent of each other, their power spectrum would be flat (i.e. “white noise”), and the process generating RTs would have no memory for past RTs. On the other hand, if the RTs were generated by a random walk process, moving up or down by random amounts each trial, then they would show a 1/

*f*

^{2}power spectrum (“Brownian noise”). In this case, the process retains everything that happened to it in the past. A 1/

*f*power spectrum is intermediate between the flat and the Brownian spectra, suggesting that some aspects of past RTs are retained for a very long time, while other aspects decay quickly.

*f*power spectrum discovered in visual search RTs was, however, found in a rather atypical search task: Gilden et al. (1995) used a complete report task where the subject had to say how many targets were present, from zero to four, in a display that had only four items. The 1/

*f*power spectrum was found in the sequence of RT residuals, obtained by subtracting the mean RT for the particular search display (zero to four items) from the RTs. It is not clear whether the results obtained in this search experiment generalize to RTs obtained from ordinary search; in particular, whether they have a 1/

*f*power spectrum.

*f,*or not.

*f*power spectrum. On the other hand, the target-present RTs have little correlation with past RTs, and their power spectrum is nearly flat, so target-present RTs appear to be similar to a white-noise process.

*f*power spectrum, this would account for the difference between target-present and target-absent power spectra. This deadline hypothesis was tested in a second experiment, in which searches of very different speed were interleaved, and it cannot account for the results obtained. Thus the correlations found between successive target-absent RTs cannot be explained by a deadline model of search.

*f*power spectrum? The last part of this paper describes a model which answers these questions. In this model, display items are serially examined by a pool of categorizers, which categorize them as either target or distractor. Search ends when either the target item is found or no items remain (that is, Serial Self-Terminating Search, or SSTS). The categorizers in the pool have a distribution of speeds. After each search has ended, the pool of categorizers is refreshed by discarding the slowest one, and discarding the fastest one (which, by the speed-accuracy tradeoff, is the least accurate), and replacing them by selecting from a large population of dormant categorizers. Those categorizers which stay in the active pool for a long time are responsible for the correlations between RTs. Computer simulations show that the sequence of RTs generated by this model have a 1/

*f*power spectrum, and the pattern of correlations between RTs closely parallels the pattern found in real visual search RTs. The exact reason why this model produces a 1/

*f*power spectrum is not known, but may be because the refresh process produces a wide range of residence times for the categorizers in the active pool, with those categorizers whose speed is close to the median speed staying in the pool longer than those whose speed is far from it. Matlab code to generate 1/

*f*power spectra via a simplified version of the model is included as 1.

*formal*difficulty in applying time series methods to merely ordered (rather than regularly spaced) sequences, and the methods will be valid if the data come from an event-based process (that is, a process that changes as events occur, rather than over continuous time), and approximately correct if the process is time-based, since visual search trials typically occur at approximately regular intervals.

*t*is the correlation between all data in the sequence that are

*t*timesteps apart (Chatfield, 1996). The autocorrelation can be visualized by imagining the sequence of data duplicated, then shifted, and computing the correlation between all pairs of data where the two sequences overlap (see Figure 1a). Note that the shifted series is in the past relative to the unshifted series, because the subscript increases with time. If each item in the sequence is independent, the expected serial correlation is zero.

*x*

_{i}represents one kind of response and

*o*

_{j}a different kind, is copied and shifted, and all ordered pairs of the same type are identified; in this case where the first item in the pair is of type

*x*(unshifted) and the second item in the pair is of response type

*o,*from the shifted series. The correlation is computed between the selected pairs.

*amplitude*spectrum, which is the square root of the power spectrum.

- Search for a cross among slanted distractors (Figure 3, left panel). This is a “feature search,” where the target has a feature that the distractors lack. Typically this kind of search is very fast, and the rate found here was 2.2 ms/item;

*timeGetTime,*which has millisecond granularity when appropriately initialized with the Windows XP function

*timeBeginPeriod*. Data was collected in blocks of 100 trials (Experiment 1) or 120 trials (Experiment 2, below). Each block consisted entirely of searches of the same number of items (20, 60, or 100 items), and search type (feature search, conjunction, or missing-feature search). Each combination of number of items and search type was repeated three times by every subject. Thus each block of 100 trials can be uniquely identified by the combination of subject × search type × set size × repetition. In the Results section below, the data is averaged across blocks in various ways. For example, if a data point can be identified by the combination of search type × set size, then this data point comes from an average across subject and repetition.

*f*power spectrum (Figure 6), but there is a lot of white noise which obscures this at frequencies above about 5. This transition from 1/

*f*to white noise is similar to the spectra found by Thornton and Gilden (2005). The target-present RT sequence is more similar to white noise, consistent with the small serial correlations shown in Figures 4c and 4d.

*f*power spectrum shows that the target-absent RT series is being generated by a process with a long memory. However, the serial correlations (Figure 4a) appear to die out after about 10 trials. There is, however, no real contradiction here. Provided the serial correlations do not tend to zero too quickly, the amount of correlation between an RT and the aggregate of RTs some time in the past can be quite high, even if the correlation between an RT and a single RT some time in the past is not.

*f*power spectrum in the target-absent RTs. One of the subjects showed a spectra similar to 1/

*f*

^{α}, with

*α*less than 1. The last subject showed a more Brownian-like spectrum (see supplementary material). The reason for the slightly different results from two of the subjects is unknown. They could be simply random deviations, since the entire power spectrum of these two subjects is highly variable. Possible structural reasons for the differences will be discussed in the section on modeling below.

*f*power spectrum found in the RTs. This is demonstrated in computer simulations, described next. Intuitively, though, the cause of the 1/

*f*power spectrum produced by this model can be identified. It is well-known that 1/

*f*power spectra can be trivially produced by a system whose output is a sum of perturbations which have a wide variation in half-life or duration (Halford, 1968; Hausdorff & Peng, 1996). The problem with describing 1/

*f*power spectra in this way is that there is no mechanism for the wide variation in duration of the events; it is simply assumed. In the pool of categorizers model proposed here, the mechanism for producing a wide range of durations is the selection process. Those categorizers whose categorization time is close to the pool median will tend to reside in the pool for a long time, whereas those categorizers whose categorization time is far from the median will tend to be removed rather quickly, if not immediately. Thus, the lifetime of the categorizer in the pool is determined by how close the categorizer speed is to the median speed in the pool.

- There are
*N*categorizers in the active pool, and an effectively infinite number in the dormant population. [The number of categorizers in the active pool has some effect on the power spectrum.] - Search is performed using a limited capacity parallel model. In this model, there are
*S*memory ‘slots’, where*S*is a small number. If there is a free memory slot, an unexamined item is selected randomly from the display and placed in the free slot. Then an unused categorizer is selected from the active pool and assigned to categorizing the item. The slot is occupied until the categorizer finishes. The number of slots*S*must be smaller than the number of categorizers*N*. [A pure serial model has*S*= 1. However, a serial model does not adequately model the coefficient of variation found in the RTs here. A limited capacity parallel model can. The number of slots*S*has little influence on the power spectrum.] - The RT is the time from the start of search until the last categorizer finishes, plus a base reaction time
*B*. If one of the items is a target, the “last categorizer” is the one assigned to the target item, since once the target is found, ongoing categorizations in the other free slots can be ignored. [The base reaction time is the sum of fixed delays e.g. motor movements, and is responsible for the RT when there is only one item.] - The time for categorizer
*i*to complete its task is exponentially distributed with mean*m*_{i}. [The exponential distribution was chosen because it is skew and has been used successfully in other models of search, e.g. Bundesen (1990). This distribution has no effect on the 1/*f*power spectrum, but does affect the distribution of RTs; for example if the categorizer times instead had a Gaussian distribution, the overall RT would also be Gaussian.] - The mean categorizer times
*m*_{i}are also exponentially distributed with median*M*and hence mean*M*/log(2). [The exponential distribution was used again to keep things simple. However, many positively valued distributions with the same median will work equally well. Some distribution of mean categorizer times is vital to generate 1/*f*power spectrum from the selection process.] - After each search, the pool of categorizers is refreshed in the following way: With probability
*P,*the slowest categorizer (i.e. the one with the largest*m*_{i}) is replaced with one drawn randomly from the dormant pool. With probability*P,*the fastest categorizer is likewise replaced with one drawn randomly from the dormant pool. Note that this replacement process changes the distribution of categorizers in the pool, but the categorizer distribution will stabilize at some point. [The probability*P*has an effect on the power spectrum.] - Search terminates when a categorizer finds the target, or all items have been examined.

*N, M, P, B,*and

*S*. Parameters

*M, B,*and

*S*mostly affect the fit of the model to RT distributions, and parameters

*N*and

*P*mostly affect the RT correlations and the power spectrum. The model was only fitted to the “missing-feature” and “conjunction” experiments, since its essentially serial nature is a poor fit to the feature search experiments.

*N*= 10,

*M*= 56,

*P*= 0.3,

*B*= 460, and

*S*= 4.

*f*at low frequencies, transitioning to white noise at higher frequencies. The transition to white noise is caused by the assumption that categorizer times are randomly distributed, since when categorizer times do not vary about their mean, a 1/

*f*power spectrum holds to high frequencies (as in the code given in 1). However, the transition to white noise at high frequencies could equally be caused by white noise in the base reaction time

*B*(Gilden, 2001).

*f*power spectrum in the target-absent RTs, though all showed some correlation. The above model may be able to produce power spectra different from 1/

*f*. If for example the replacement probability

*P*is reduced, the pool of categorizers changes less between trials and more correlation is observed, yielding a power spectrum steeper than 1/

*f*. If the number of categorizers

*N*in the pool is reduced, a greater proportion get replaced after each trial, reducing the correlation between successive RTs, and whitening the power spectrum.

*f*power spectrum.

*f*power spectrum is frequently accounted for by supposing that the system producing the 1/

*f*power spectrum is the result of a sum of processes which have a wide range of correlation decay times (Halford, 1968; Hausdorff & Peng, 1996; Press, 1978). If the amplitude and lifetimes of the component processes are properly chosen, the power spectrum of the resultant system is 1/

*f*over a wide range of frequencies. Explanations of this type have been invoked to account for 1/

*f*spectra in human cognition (Hausdorff & Peng, 1996; Wagenmakers, Farrell, & Ratcliff, 2004). However, this kind of explanation just “transfers the mystery” (Press, 1978) of 1/

*f*spectra to the hypothesized wide range of lifetimes of the component processes. If this kind of explanation is accepted, one must find a cause for the postulated wide range of lifetimes. Extremal dynamics (Miller, Miller, & McWhorter, 1992) is one method for generating processes or perturbations with a sufficiently wide range of lifetimes. Extremal dynamics is, however, rather abstract and it isn't clear how it can be applied to the case of visual search. The categorizer replacement model suggested here is another way of generating a system that is a sum of processes with a wide range of lifetimes. In this model, the lifetimes of the categorizers are dependent on how similar the categorizer speed is to the median speed, because selection quickly replaces categorizers whose speed is far from the median.

*f*power spectra. Self-organized criticality (Bak, Tang, & Wiesenfeld, 1987) is an influential idea in statistical physics which was claimed to generate 1/

*f*dynamics, though in fact it may not (Jensen, Christensen, & Fogedby, 1989). Van Orden, Holden, and Turvey (2003) have suggested a variety of processes that together might give rise to 1/

*f*power spectra in RT series, but these suggestions are more descriptive than explanatory.

*f*power spectra are observed in many different places, not just visual search RTs, what implications does the model presented here have for these other 1/

*f*processes? For some other psychological processes, a similar model could be constructed. For example, to match time intervals (Gilden et al., 1995), one could imagine a pool of interval estimators, where the extreme estimators are replaced at regular intervals. The particular time interval produced by the subject would be a summary statistic (perhaps a sample mean) drawn from the estimator pool. This model, being functionally identical to the visual search model suggested here, would readily generate a 1/

*f*power spectrum.

*f*power spectra in other domains, the model is less likely to be directly applicable. Nonetheless, one may discern a general principle which is embodied in the model. In the visual search model, the output is related to the sum of the speeds of the categorizers in the active pool. The lifetime of the categorizer in the pool is related to its speed: extreme speeds do not last as long as speeds close to the median. In a more general setting, the lifetime of a process could be related to its magnitude, with processes of small magnitude lasting longer than those with large or extreme magnitudes.

*f*power spectrum. Here, I give Matlab code for a simplified version of the model which produces a very clean 1/

*f*power spectrum from the lowest to very high frequencies. For the given parameter set, 1/

*f*power spectra result. The 1/

*f*power spectrum is stable over a wide range of parameter values, but different spectra can result if the parameter values are changed markedly from those given. Small values of n (around 5) or large values of trim (trim >

*n*/4) generate spectra closer to white noise. To obtain a Brownian power spectrum, reduce the value of pr_replace to approx. 0.05 or less and/or increase

*n*to more than 100.

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