To explore the relation between ๐1and ๐, I use the datasets of Wolfe et al. (2010).
First, I obtain a naive estimate of ๐1for each set size level by solving Equation
(4.3) for ๐1, whereby ๐ is an estimate obtained from the empirical miss rates. If ๐1
is inversely proportional to ๐, then the product of the naive estimates of ๐1and ๐
must be approximately constant. However, this product declines systematically. Even if we substitute ๐ by ๐, the result only changes to a negligible degree. This observation indicates that an inverse proportional relation underestimates the speed at which ๐1declines with ๐.
Considering that the speculation of an inversely proportional relation essen- tially describes a power law, I plot the log-transformed naive estimates of ๐1
as a function of the log-transformed set size ๐. On the log-log plot, the naive estimates of ๐1clearly form a straight line (Figue 4.4). This is observed for both
conjunction search and spatial configuration tasks, with only minimal deviation especially for conjunction search tasks. If the log-transformed of a variable can be described by a straight-line of the log-transformed of another variable, i.e., log(๐ฃ1) = ๐ + ๐ log(๐ฃ2), then the former itself can be described by a monomial of
the latter, i.e., ๐ฃ1= ๐๐๐ฃ2๐. Thus, this observation leads me to hypothesize that the
relation between ๐1and ๐ follows a power law, i.e.,
๐1= ๐1๐โ๐, whereby ๐1 > 0, ๐ > 0. (4.13) I restrict the power of ๐ to be negative because ๐1should decline as ๐ increases, as
concluded before. The slope of both straight-lines in Figure 4.4 is slightly smaller than โ1 (โ1.28 and โ1.15, respectively), which is in line with the speculation that ๐1declines at a higher speed than an inversely proportional would predict.
The observation of a power law relation here has not been documented in the literature. Although both data sets demonstrate patterns that are highly consistent with a power law, it should be born in mind that this observation is based on data for only four set size levels. Although the error rates of these data sets are supposed to be reliable because each single value is calculated based on about 5,000 trials of each task type, they reflect the aggregated data of 10 or 9 participants. When the log-log plot is created using individual data (each value is based on ca. 500 trials), a straight line is observed for only about a half of the participants (6 out of 10 for conjunction search, 4 out of 9 for spatial configuration). However, zero or extraordinarily low false alarm rates (< 0.005) for some of the set size levels in case of the other participants can explain that no straight line is observed. They lead to missing values or outliers in the estimate of ๐1(when the false alarm rate is zero, ๐1is estimated at 0 according to Equation
4.12 and its logarithm is not defined). In sum, a large amount of empirical data is needed to answer the question of how universal this observation is for visual search.
Furthermore, this discovery is an ad hoc analysis and not sufficient to conclude a convincing theoretical explanation. Power laws are observed in numerous natural and social phenomena of all kinds of contents but not fully
4.4. IMPERFECT PROCESSING 107 -7.0 -6.5 -6.0 -5.5 -5.0 1.5 2.0 2.5
log(set size)
log(es
timated
p1)
-7.0 -6.5 -6.0 -5.5 1.5 2.0 2.5log(set size)
log(es
timated
p1)
Figure 4.4: Log-log plot of the naive estimates of ๐1as a function of the set size ๐
for conjunction search (top) and spatial configuration (bottom) in the study by Wolfe et al. (2010).
well understood. There are many mechanisms that can lead to a power law but so far no theorems that prove the necessary and sufficient conditions for power laws to be found. Although some power laws in physics point to specific mechanisms conclusively, it is not the case for most phenomena in other disciplines. Therefore, even if the power law can be found for various visual search data generally, additional evidence is required to allow justified inferences on the underlying mechanism.
Nevertheless, there are several possibilities worth considering. One might think of Stevensโ power law and speculate that changes in ๐1may result from
a distortion of the magnitude of the set size. In fact, set size must be an estimate based on our ability to roughly enumerate larger number of items (Dehaene, 2011; Krueger, 1984). But this speculation is not consistent with the
data. According to Krueger (1984), magnitude estimation is roughly โ ๐0.8. It is more difficult to discriminate large set sizes. That would lead to the prediction that ๐1 will decline slower than inversely proportional to ๐, contradicting the
observation. Considering that there may be some optimization principles driving the discrimination process of visual stimuli, another thinkable connection could be that many complex systems optimizing certain quantities generate power laws4. For example, maximizing entropy (Mandelbrot, 1953) and maximizing a risk-neutral utility function under certain constraints (e.g., Carlson & Doyle, 1999).
Considered from a signal detection perspective, the observation of a power law may indicate an adaptive mechanism to maintain detection performance in a background wit irregular and cluttered noise. Recall that ๐1is conceived as the
false alarm probability of discriminating a single stimulus and its dependence on set size ๐ is interpreted as an adaptation of a discrimination threshold. What could be the reason for keeping a flat false alarm rate in the decision regarding multiple stimuli by such an adaptation? In engineering, the property of maintaining an approximately constant false alarm rate does belong to an ideal binary classifier that is adaptive in general contexts. This property requires a threshold adaptation following a power law in some circumstances. For instance, Constant False Alarm Rate (CFAR) is an important principle for designing detection scheme according to which the radar decides whether or not the target is present. It is applied in an environment of varying background noise and clutter. If the power threshold is set constant, the radarโs performance will be affected by the background noise level, resulting in increased number of false alarms in the presence of strong noise or interference. By adapting the power threshold above which the echo signal is classified as originating from the target, the radarโs performance can be maintained regardless of the level of noise or interference. When the background is an even, regular surface, the background noise is often modeled as Gaussian noise. However, when the background is a rough, irregular, or cluttered surface, Gaussian noise does not provide an accurate model. Rather, the background noise is commonly modeled by a (generalized) Pareto distribution (e.g., Newman, 2005). Since the tail of a Pareto distribution (i.e., the survival function) follows 4However, the power laws in those contexts mostly refer to power law distribution of variables.
4.4. IMPERFECT PROCESSING 109
a power law5, a threshold adaptation that follows a power law is necessary to achieve CFAR.
It is conceivable that the set size ๐ correlates with the level of background noise, degree of clutter or interference, which can be characterized by a Pareto distribution. Observers adapt their discrimination threshold in a way such that their detection performance remains independent of the level of background noise, similar to the CFAR principle for radars. Perhaps it is even not necessary that a Pareto distribution describes the background noise in the display in a visual search experiment accurately. After all, it is not a typical environment in which humans perform visual search. Human observers are confronted with a visual search task primarily in natural, continuous, noisy and cluttered scenes. It is possible that a Pareto distribution characterizes the background noise in such a scene accurately, so that observers, learned or programmed, stick to the detection scheme that works well in everyday life even when facing an environment that is better characterized by another distribution (e.g., Gaussian noise).