Bayesian approximations to the theory of visual attention (TVA) in a foraging task

Abstract

Foraging as a natural visual search for multiple targets has increasingly been studied in humans in recent years. Here, we aimed to model the differences in foraging strategies between feature and conjunction foraging tasks found by Á. Kristjánsson et al. Bundesen proposed the theory of visual attention (TVA) as a computational model of attentional function that divides the selection process into filtering and pigeonholing. The theory describes a mechanism by which the strength of sensory evidence serves to categorise elements. We combined these ideas to train augmented Naïve Bayesian classifiers using data from Á. Kristjánsson et al. as input. Specifically, we attempted to answer whether it is possible to predict how frequently observers switch between different target types during consecutive selections (switches) during feature and conjunction foraging using Bayesian classifiers. We formulated 11 new parameters that represent key sensory and bias information that could be used for each selection during the foraging task and tested them with multiple Bayesian models. Separate Bayesian networks were trained on feature and conjunction foraging data, and parameters that had no impact on the model’s predictability were pruned away. We report high accuracy for switch prediction in both tasks from the classifiers, although the model for conjunction foraging was more accurate. We also report our Bayesian parameters in terms of their theoretical associations with TVA parameters, $π_{j}$ (denoting the pertinence value), and $β_{i}$ (denoting the decision-making bias).

Keywords

Visual attention foraging TVA Bayesian classifier attentional switching

Introduction

At any given moment, people are faced with a massive amount of visual information from the environment, and because of limited processing resources, we need to choose the items that we wish to attend to. Despite decades of research, the study of visual attention is still a thriving area of cognitive science, and one of the most common ways of studying such processing has been the well-known visual search task (Á. Kristjánsson & Egeth, 2020; Wolfe, 2000). Typical visual search tasks involve searching for one target object while varying the number or complexity of distractors (Treisman, 1986; Wolfe, 1998). Other experiments involve multiple target types, but observers still search for one target of many types on a given trial (Horowitz & Wolfe, 2001; Metlay et al., 1970).

Real-world visual tasks can involve complex combinations of multiple objects and elements that require attentional selection of relevant stimuli and suppression of irrelevant information (Chun et al., 2001; Posner et al., 1980). Despite the enormous influence of the visual search experimental design, several authors have speculated that more dynamic visual tasks can provide a more realistic picture of how objects in the visual field are selected (Á. Kristjánsson & Draschkow, 2021). One such task that researchers have recently used to study how visual attention is allocated in the visual world is visual foraging, where observers are asked to select many targets, often of different types. This task may provide a better analogy for real-world selection than single-target tasks as we may encounter many possible targets during our daily interactions with the environment (Fougnie et al., 2015; Wolfe, 2013).

Investigations of foraging as a natural search process were initially performed on animals (Bukovinszky et al., 2017; Mallott et al., 2017; Pyke et al., 1997). These foraging studies often involve multiple preys of different types, and several studies suggest that predators choose a specific type of prey in a nonrandom sequence (Bond, 1983; Dawkins, 1971; Punzalan et al., 2005). Typically, these studies show that when there are multiple types of easily detectable prey (targets), predators will forage for multiple targets simultaneously during the hunt. But if the prey is difficult to detect, they will typically select only one target type repeatedly until all targets of this type have been selected and only then turn to other target types. This difference in the difficulty of searching for prey was thought to reflect that animals have limited attentional capacities and cannot search for multiple complex objects (e.g., if the prey has a masking colour, i.e., imitative colouration; Johnstone et al., 2002).

The results of computerised foraging tasks on humans have revealed similar selection patterns but have also demonstrated that most humans can adapt to target difficulty through changes in search strategies in the environment to optimise their hit rate (Cain et al., 2012; Á. Kristjánsson et al., 2014; T. Kristjánsson et al., 2018).

Á. Kristjánsson et al. (2014) investigated foraging using a classic feature/conjunction design from the visual search literature (Á. Kristjánsson & Egeth, 2020; Treisman & Gelade, 1980). In feature search, the target was distinguished by a colour that was not shared with distractors, with all objects on the screen sharing the same shape (circles). A typical visual search finding is that distinguishable targets based on a single feature will pop out during single-target visual search tasks and can be found quickly (Treisman, 1986), and the search is therefore not affected by display size. In conjunction search, the target does not have any unique features but is defined by a conjunction between targets and distractor features. For example, the targets could be red circles and green squares, and distractors could be red and green circles. During this task, search times tend to be longer as the number of distractors increases, showing that the target does not pop out among distractors (Treisman, 1986; Wolfe, 1998).

However, during foraging, where targets disappear once selected, when only one target remains, the task effectively changes to a single-target search task. According to the literature, this final target should pop out. However, T. Kristjánsson et al. (2020) showed that the reaction times for the last target were very high, in contrast to the original findings from the single-target visual search literature. These results may argue for a lessened emphasis on preattentive versus attentive processing in theories of visual search and visual foraging and visual attention more generally.

Á. Kristjánsson et al. (2014) found that observers typically behaved very differently for different foraging tasks. There were large differences in selection sequences between feature and conjunction foraging. For example, subjects were likely to switch between target types when they foraged based on a single feature (feature foraging). However, during conjunction foraging, they mostly preferred to select all instances of one target type and only switch to different target types once those were all finished. This strategy during conjunction foraging led to low switch rates where most observers switched between two types of targets only once per trial. In particular, the Á. Kristjánsson et al. (2014) study showed that during feature foraging, switches were 32.1% of the total number of selections, and runs tests revealed that the selections were more-or-less random, while for conjunction foraging, they were only 7.3% of the total number of selections and were not random according to a runs test.

Switches during foraging may be an important measure of attentional function as they can provide information about observers’ capacity for storing different target templates simultaneously and the difficulty of switching between target types during the trial. In fact, switches during conjunction foraging led to far higher switch costs where the intertarget times (the time that elapsed between selections) were much larger when observers switched between target types during conjunction than feature foraging. This is consistent with the broader literature on attentional switching and is thought to reflect the extra cost of inhibiting repetition of the previous task (Arbuthnott & Frank, 2000).

In the feature foraging condition, targets differed from distractors on a single feature, and the data in Á. Kristjánsson et al. (2014) indicated that observers could keep more than one template in their working memory simultaneously. But Kristjánsson and Kristjánsson (2018) then showed that keeping two or more items in working memory during feature foraging nevertheless led to costs in intertarget selection times over the case when observers only needed to forage for one target category. So, in the feature task, while observers seem to be able to switch between target types randomly, this is probably not quite effortless.

But we can nevertheless speculate that random selection sequences are mostly determined in a bottom-up way. Note, however, that Maljkovic and Nakayama (1994) showed that in a pop-out search task, observers respond faster to a search target with the same colour as on a previous trial than pop-out targets of another colour. The previous target primes the next target, and Wolfe et al. (2003) speculated that this process could be thought of as a form of implicit top-down guidance. If storing several templates entails costs, as Kristjánsson and Kristjánsson (2018) showed, priming may also influence performance, reflected in the increased intertarget times. At the same time, we may speculate that priming becomes dominant in the conjunction foraging condition, creating conditions where observers repeat their last target selections, using priming to overcome attentional limitations.

Another exciting issue is that complex scenes and tasks can have large numbers of objects competing for attention. Although theory of visual attention (TVA) treats all objects equally, it may be more likely that bias and salience parameters might be influenced by the distance of an object to the location of the current gaze. Hulleman and Olivers (2017), for example, suggested that all items within, what they called the functional visual field (FVF), are processed and that the FVFs are an essential parameter or unit in visual search (Young & Hulleman, 2013). The FVF can be defined as “an index of the total visual field area from which target characteristics can be acquired when eye and head movements are precluded” (Scialfa et al., 1987, p. 14). In our analyses, we speculate that each selection applied to a foraging subset may be analogous to the FVF.

Models of foraging

Bayesian optimal foraging models have been proposed in an attempt to quantify human strategies in multiple-target search and to explore whether observers adapt their strategies to complex object distribution statistics (Cain et al., 2012; Najemnik & Geisler, 2005). But it is essential to note an important difference in that these models have typically focused on foraging paradigms where the main measure of interest has been when observers leave the current area of selection (or patch) and switch to a new patch. These authors modelled ideal searches according to optimal foraging theory (OFT), previously developed to explain animal behaviour. In short, an optimal forager should stay in a place where the quantity of food in the patch is larger than the average quantity of food in the entire environment and leave when the average food quantity drops below the average in the environment (so-called patch leaving; Hutchinson et al., 2008; A. Kristjánsson et al., 2020). In other words, observers should adapt their performance according to the marginal value theorem (MVT), which states that foragers will leave a food source when the collection rate drops below the average collection rate in the environment they are in Charnov (1976). Cain and colleagues (2012) compared the performance of humans with that of modelled ideal observers. They found that observers tend to stop searching for additional targets when target presence is unlikely but persevered longer when the probability of target presence was higher. It is important to note that in Cain et al., the number of targets on trial could vary from 0 to 12, which differs from the methods by Á. Kristjánsson et al. (2014), whose data we use in this study, where a fixed number of targets was present on each trial.

Hutchinson and colleagues (2008) also tried to model a foraging paradigm where observers could move on to a new display before all targets had been finished. During the experiment, the subjects performed a computerised task of catching fish in ponds with three different distributions of the number of fish on the screen. They could move to another pond at any time, but this took some time. The result of their experiment showed that observers fished for longer in ponds where they had caught a lot of fish but preferred to stay longer than optimal, delaying their switches to a new pond far longer than optimal. However, the high success rate in the previous patch reduced the dwell time, and subjects seemed to have learned to leave earlier during the last two thirds of the experiment. In addition, Á. Kristjánsson et al. (2020) found that foraging in humans deviated from the predictions of optimal foraging concepts and that participants continued to forage within the same area for longer than expected. These findings show that in some cases, people do not act according to OFT and adapt their strategies depending on the particular situation.

But, for the current purposes, it is important to note that these models are aimed at accounting for patch-leaving behaviour in foraging and do not apply well to selection patterns within trials, as is the main focus of many measures in foraging tasks that involve two or more target types.

A more recent model of foraging performance (Clarke et al., 2021) uses a generative Bayesian model to simulate selection runs as a sampling process without replacement. While this particular model does include a parameter for the proximity of the next selection, it does not include all the other variables tested for our model. Our aim here is to investigate ways of modelling such tasks to provide insights into how visual attention operates as these tasks are performed, with particular emphasis on whether observers stick with the same target type or switch between target types on each selection.

The theory of visual attention

Our current study is aimed at investigating the mechanisms underlying foraging using the TVA proposed by Bundesen (1990). This theory was proposed as a computational system that divides the attentional selection process into two mechanisms: filtering and pigeonholing, with sensory input from the (front-end) visual system and control parameters from a high-level executive system, such as the prefrontal cortex (Bundesen et al., 2005; Duncan & Owen, 2000). TVA is based on the biased competition principle (Desimone & Duncan, 1995), and according to this idea, possible visual categorisations that assign features to objects compete (race) to be encoded in capacity-limited visual short-term memory (VSTM) before it is filled. The possible categorisations are supported by sensory evidence; however, the competition is also influenced by perceptual biases (red circle, green square, etc.) and attentional weights (important objects are weighted highly, such as the features of the objects you are searching for). TVA thus represents a mathematical formalisation of the so-called biased competition principle (Desimone & Duncan, 1995).

A central assumption in TVA is that visual recognition and attentional selection consist of making perceptual categorisations of the form “object $x$ belongs to category $i$ ” or “element $x$ has feature $j$ ,” where object x is a perceptual object, category $i$ is a perceptual category (e.g., the red circle category), and feature $j$ is a perceptual feature (colour, shape, etc.) (Bundesen & Habekost, 2008). Two fundamental equations describe this process, the rate equation (Online Supplementary Material, Equation 1) and the weight equation (Online Supplementary Material, Equation 2). Feature $j$ is assumed to have a pertinence value π_j $π_{j}$ , reflecting the current importance of attending to objects belonging to feature $j$ .

Bundesen (1990) divided visual recognition and attentional selection into filtering, which involves the selection of elements, and pigeonholing, which involves the selection of categories. The filtering mechanism works in the following way: for instance, if perceptual category $j$ (such as a red disc or green disc in the feature foraging task in Kristjánsson et al., described above) is a target category, elements belonging to feature $j$ (such as a red colour or green colour in Á. Kristjánsson et al. (2014)) will be more likely to be selected as the pertinence value of target feature $j$ will be high compared with pertinence values of other categories. The filtering mechanism increases the likelihood that a target element will be perceived (selected) without changing the probability that the element belongs to any particular category. A change in the strength of sensory evidence for a category causes a shift in the distribution of attention weights among elements in the visual field.

The pigeonholing mechanism can be described as a situation where perceptual category $i$ (such as the red disc or green square in the conjunction foraging task in Á. Kristjánsson et al. (2014)) is relevant for action. Categorising any element $x$ as belonging to category $i$ is preferred over other categorisations if the bias parameter associated with category $i$ is higher than the bias parameter related to other categories. If the bias parameter increases, the value of the categorisation that $x$ belongs to the target category increases. However, other values of the categorisation are not affected.

Considering specifically the foraging task in Á. Kristjánsson et al. (2014), in the feature condition, targets were defined based on the colour feature or the category of the conjunction of colour and shape features. For example, in the feature foraging condition, one subject might recognise a target as a red disc by pigeonholing the collection of physical attributes into the category red disc (conjunction of features red and disc), but another subject might filter the visual input based on feature $j$ , which has no necessary relationship with the category. In such cases, one subject would be biased towards making target categorisations (and perhaps more biased towards some over others), but another would tend to filter the input based on colour (e.g., red items are pertinent targets, blue ones are irrelevant distractors). So, the feature (red) would have a high pertinence value when determining the category (target). Observers would filter objects based on the colour of the stimuli that have a direct relationship with the category.

As for the conjunction foraging condition, the target category was defined only by a conjunction of colour and shape, so the filtering mechanism became irrelevant. If an observer filtered based on the shape feature, the weights would increase for half the targets and half of the distractors. The same would happen if the observer filtered based on colour. Assuming that these properties contributed similar to the strength of sensory evidence values, the weight equation would yield the same weight for all elements in the display. So, for conjunction foraging, the pigeonholing mechanism, therefore, plays a key role from the perspective of TVA.

TVA has been found to apply well to a large number of single stimulus recognition tasks from multielement displays (see Bundesen & Habekost, 2008, 2014, for overviews). However, the application of this theory is not limited to visual attention, and TVA has been extended to areas such as memory and executive control (Logan, 2002; Logan & Gordon, 2001; Nosofsky & Palmieri, 2008). Many classic results in the literature on visual cognition have been found to be well explained by TVA, including effects of the number and spatial position of targets in studies of divided attention (Posner et al., 1978; Sperling, 1960, 1967) and the effects of consistent practice in search (Kyllingsbæk et al., 2001; Schneider & Fisk, 1982).

Overview of current goals

The goal of this work was to train a Bayesian classifier to predict switch events during feature and conjunction foraging using data from Á. Kristjánsson et al. (2014) as input. In our modelling, the Bayesian classifier determines whether a switch to a new target type occurs or whether the same target type as on the last selection is chosen. We hypothesised that TVA could provide a good mathematical explanation of behaviour during foraging and that a Bayesian classifier could serve as an approximation for key parameters of TVA. Namely, we proposed and formulated new parameters to represent key sensory and bias information that could be used for each selection during the foraging task. We chose switches as our primary aim for prediction as this measure can provide information about the storage capacity of templates and the difficulty of switching between target types.

Methods

Observers and stimuli

The data provided by Á. Kristjánsson et al. (2014) were used as input for the classifiers. Sixteen students at the University of Iceland (22–39 years old, 9 females, 5 males; M = 28.3 years, SD = 4.6 years) participated in the experiment. Additional data for seven subjects from a pilot study with an identical experimental design were also included in the input data. As switches were relatively rare events, we used this additional data to augment the data reported in Á. Kristjánsson et al. (2014) to represent these events better (see further discussion below).

During the experiment, observers had to find two target types among two types of distractors and tap them on an IPad with a display resolution of 1,024 × 768 pixels. There were 20 targets of each type and 20 distractors of each type on the screen on each experimental trial (80 items in total on the screen; Figure 1). The task was to select all targets as quickly and accurately as possible, and once selected, the targets disappeared. The observers had to finish 20 trials of each condition before the experiment finished. As explained above, the difficulty of the experiment also varied through the use of a feature/conjunction manipulation. One condition consisted of discs with different colours, so-called feature-based targets (red and green discs were target objects; blue and yellow discs were distractors or vice versa). Another condition consisted of discs and squares with different colours, the so-called conjunction task (red discs and green squares were targets; green discs and red squares were distractors or vice versa). All stimulus categories were randomised and counterbalanced between participants.

Augmented Bayesian models

In this study, we used Bayesian classifiers. A Bayesian network is a graphical model that takes the structure of a directed acyclic graph (DAG). This structure contains a probabilistic representation of the dependencies (edges) between variables (nodes) of interest (Pearl, 2011) and as a graphical model has more explanatory value than other solutions like neural networks. Naïve Bayesian classifier also show better predictive ability with small data sets as input, compared with traditional methods like logistic regression (Ng & Jordan, 2001). In addition, the augmented Naïve Bayesian classifier learns connections between the feature variables to account for possible dependence between them, conditional on the class variable (BayesFusion LLC, 2017).

First, we prepared the data obtained by Á. Kristjánsson et al. (2014) and used it as input for two augmented Bayesian networks trained separately on data from the feature and conjunction tasks. These Bayesian networks were trained to predict when observers would switch between the two types of targets that would occur during the foraging task, considering that switches are rare events and are influenced by bottom-up and top-down factors. Augmented Bayesian classifiers were built using the GeNIe 3.0 academic software. Each training example from the data involved a single selection of one target from those still available at that point within the trial. As dictated by the experimental design, there were 40 targets, and therefore 40 selections per trial. Trials, where distractors were mistakenly selected, were not included (in the original study in Á. Kristjánsson et al. (2014), trials were terminated when observers selected distractors and the trials had to be repeated).

For complex visual tasks like foraging, TVA does not define which scene features influence the salience and bias parameters of the TVA formula. Therefore, data preparation involved proposing and quantifying parameters potentially useful for a predictive model from the raw selection data, and this set of discrete parameters was calculated as input data for building and training the augmented Bayesian models. Initial models were trained using the full set of parameters to predict switches (Table 1). However, not all parameters contributed equally to the prediction accuracy, and a model quantification procedure was therefore carried out to identify the most relevant parameters by comparing the Akaike information criteria (AIC) for the parameters (Vrieze, 2012).

Table 1.

Set of all tested parameters for feature and conjunction foraging.

Feature foraging	Conjunction foraging
Distance between the selected and the previously selected target	Distance between the selected and the previously selected target
Distance to the closest same	Distance to the closest same
Distance to the closest other	Distance to the closest other
Distance to the closest distractor (Type 1)	Distance to the closest same colour distractor
Distance to the closest distractor (Type 2)	Distance to the closest same shape distractor
The ratio of remaining targets of one type (same type as currently selected) to the other target type	The ratio of remaining targets of one type (same type as currently selected) to the other target type
Run-length (number of the selected same type of targets)	Run-length (number of the selected same type of targets)
Grouping around the selected target	Grouping around the selected target
Grouping around the closest another type of target (according to the selected target)	Grouping around the closest another type of target (according to the selected target)
Grouping around the closest another type of target (according to the selected target)	Grouping around the closest another type of target (according to the selected target)
Is there a target of the other type between the selected target and the closest same? (Strict line)	Is there a target of the other type between the selected target and the closest same? (Strict line)

Shaded parameters were pruned from further use as inputs and were not used in the final models.

All calculated parameters were based on the position of the most recently selected target. For example, if a participant had just selected a red circle at some location, that red circle became the location from where the parameters for the next potential target were calculated. Some of the parameters are self-explanatory, but others (like grouping) relied on calculations that concern multiple nearby potential targets.

Basic parameters were based on the location of other nearby objects, such as the distance to the nearest target/distractor (Figure 2A). Other parameters were more global in nature, for example, the ratio of remaining targets of one type (same type as the current selection) to the alternate type of target. As one target type was selected and depleted, the ratio of targets of the other type grew. Similarly, run-length involved measuring the recent history of selections and was calculated as the number of selected targets of the same type without interruption from an alternate selection of the other target type. Spatial grouping involved using a local area of interest around the selected target and calculating the number of alternate target types within the area of interest of the current selection (Figure 2B). Our calculation fixed the area of interest at a radius of one.

After defining our parameters, we faced the problem that switching as a parameter is a rare event during foraging, particularly during the conjunction task. For feature foraging, switches were 32.1% of the total selections, whereas for conjunction foraging, they were 7.3% of the total selections. Classification of rare events is a well-known challenging problem and can often result in biased classifiers that skew the predictions in favour of the more common class to increase the overall model accuracy (Barandela et al., 2004). To address this, we artificially increased the training data with a focus on increasing switching events. All switching events in the data from the feature and conjunction tasks were selected and repeated once for the feature foraging data and three times for the conjunction foraging data. The additional data amounted to approximately 40% for the feature task and approximately 23% for the conjunction task. We chose the oversampling method for addressing the problem of rare events as a number of studies showed that oversampling is suitable for working with Naïve Bayesian classifiers (Maloof, 2003; Mohammed et al., 2020; Putri & Frieyadie, 2017). Our approach was to initially train the models on oversampled data, and following this training, test them on the actual data.

Results

Our goal was to predict switches during foraging for multiple targets in Á. Kristjánsson et al. (2014), using the previously described parameters as input for the Bayesian classifier. To evaluate the model, the accuracy of the model and expectation–maximisation (EM) log-likelihood were used. The software used in this study allows us to represent the resulting model in a graphical form for both feature (Figure 3A) and conjunction foraging (Figure 3B). We report results from models trained on the raw, original data to establish a baseline and then for models trained on the data set augmented by oversampling the rare switches.

Before the quantification and oversampling stages, the model was trained by 10-fold cross-validation on the data. For feature foraging, the overall accuracy for switch and nonswitch events combined was 76.2%. For repeats of the same type of target selection, the accuracy was 88%, and for switch events, the accuracy was 51.3%. For conjunction foraging, the overall accuracy was 95.2%, for repeats, 99.4%, and for switch events, accuracy was 46.7%. Although the overall model accuracy was high, it was nevertheless important to improve the accuracy of the more frequent nonswitch events. As this first training stage, we decided to prune the feature foraging model (Table 2) and conjunction foraging model (Table 3) to remove parameters that did not improve the fit of the model. Parameters were removed if the difference in AIC was less than 2.0, but only if the removal also did not have a large impact on the model accuracy.

Table 2.

Comparison of parameters for the feature foraging model based on the AIC (values above 25 lead to overfitting of the model).

Parameters removed	Log-likelihood	Accuracy	Repeats accuracy	Switches accuracy	AIC
None	−174,425	0.762626	0.880481	0.513664	271,073,815
Distance between the selected and the previously selected target	−158,716	0.730847	0.868918	0.439178	246,660,451
Grouping around the selected target	−165,631	0.783994	0.919863	0.496977	257,406,371
Grouping around the closest same type of target (according to the selected target)	−165,322	0.786713	0.926617	0.491173	256,926,185
Grouping around the closest other type of target (according to the selected target)	−164,848	0.76216	0.881168	0.510762	256,189,588
Is there a target of the other type between the selected target and the closest same? (Strict line)	−171,123	0.786092	0.926503	0.48948	265,940,948
Ratio of remaining targets of one type (coinciding with the selected) to another type of target	−154,362	0.769852	0.943102	0.403869	239,894,328
Run-length (number of selected same type of targets)	−148,279	0.772883	0.929708	0.441596	230,441,336
Distance to the closest same	−162,231	0.709635	0.871437	0.367836	252,122,766
Distance to the closest other	−156,974	0.766356	0.929823	0.42104	24,395,338
Distance to the blue distractor	−155,177	0.76589	0.894906	0.493349	241,160,839
Distance to the yellow distractor	−155,359	0.785315	0.924327	0.491657	241,443,667

AIC: Akaike information criterion.

Distance to the closest same parameter was an exception because of the reduced accuracy of the model. Highlighted lines indicate parameters removed from the final model.

Table 3.

Comparison of parameters for the conjunction foraging model based on the AIC (values above 25 lead to the model overfitting).

Parameters removed	Log-likelihood	Accuracy	Repeats accuracy	Switches accuracy	AIC
None	−174,059	0.952303	0.994699	0.467793	2,713,272,543
Distance between the selected and the previously selected target	−157,653	0.948484	0.996382	0.347413	2,457,532,538
Grouping around the selected target	−166,570	0.954329	0.993605	0.461457	2,596,524,043
Grouping around the closest same type of target (according to the selected target)	−164,759	0.955031	0.993605	0.470961	2,568,295,534
Grouping around the closest other type of target (according to the selected target)	−166,727	0.950666	0.994446	0.401267	2,598,971,242
Is there a target of the other type between the selected target and the closest same? (Strict line)	−171,943	0.954797	0.993352	0.470961	2,680,274,336
Ratio of remaining targets of one type (coinciding with the selected) to another type of target	−160,369	0.945133	0.99857	0.274551	2,499,867,508
Run-length (number of selected same type of targets)	−142,839	0.931806	0.993268	0.160507	2,226,623,022
Distance to the closest same	−158,932	0.951524	0.995709	0.397043	2,477,468,631
Distance to the closest other	−157,262	0.953082	0.993773	0.44245	2,451,437,924
Distance to the blue distractor	−157,047	0.955031	0.993521	0.472017	2,448,086,665
Distance to the yellow distractor	−156,698	0.951056	0.994783	0.402323	2,442,646,715

AIC: Akaike information criterion.

Grouping around the closest another type of target parameter became an exception because of the dropped accuracy of the model. Highlighted lines indicate parameters removed from the final model.

After the pruning, the switching rate was again trained by 10-fold cross-validation on the data. For feature foraging, the overall accuracy was 78.5%, for repeats, accuracy was 92.2%, and for switches, accuracy was 49.3%. For conjunction foraging, the overall accuracy was 95.4%, for repeats, accuracy was 99.3%, and for switches, accuracy was 45.8%. Since after quantification of both models, the accuracy of predicting the switching event for feature and conjunction foraging models was less than chance (less than 50%), we decided to oversample the data (as explained above).

After oversampling, the model was again trained by 10-fold cross-validation on the oversampled data set and tested on the original data set. For the feature task, the overall accuracy was 69.5%, with a repeat accuracy of 60.7%, and for the switches, accuracy was 88.1%. For the conjunction task, the overall switching rate prediction was 92.1%. For repeats, the accuracy was 93.5%, and for switches, it was 75.7%. The models trained with oversampled switch events did drop slightly in overall accuracy, but the results were much more balanced with both switch and nonswitch events predicted at well above chance. Interestingly, for feature foraging, the accuracy for switches was higher than the more frequent nonswitches. Graphical representations of the final models are shown in Figure 3.

Discussion

Several studies have shown that selection patterns during foraging for two or more targets change as the difficulty of distinguishing targets from distractors increases, and switches may even be minimised to the point of exhaustive search within one category (Jóhannesson et al., 2016; Á. Kristjánsson et al., 2014; T. Kristjánsson et al., 2020; Ólafsdóttir et al., 2019). These studies show how run-length and the number of switches can be used to assess foraging behaviour and foraging strategies for different tasks. The prediction of switches could therefore play a key role in understanding attentional mechanisms behind visual foraging, and attentional switching and switch costs more generally (Á. Kristjánsson et al., 2020).

In this project, we tried to assess whether it is possible to predict switches during feature and conjunction foraging with Bayesian classifiers. The Bayesian networks were trained separately on feature and conjunction foraging data. In addition, the models were pruned according to the predictive value of the model parameters. The overall accuracy of the final model for feature foraging was 69.5%, and 92.1% for conjunction foraging. We also found that the accuracy of predictions of rare switching events was greater than chance (88.1% for feature foraging and 75.7% for conjunction foraging). Predicting rare events is a common problem in many data sets, but by using oversampling, we were able to improve the results to produce an accurate and less biased model of switching behaviour.

Bundesen (1990) suggested separating visual recognition and selective attention into filtering and pigeonholing. TVA was formulated based on these parameters to describe how feature information influences categorisation and selection. Our Bayesian model defined a set of explicit parameters from the foraging task that were inspired by TVA to test which parameters had the largest influence on feature and conjunction foraging. According to TVA, the filtering mechanism increases the likelihood of perceiving objects from the target category without changing the probability that they belong to a specific category. This mechanism is likely to be reflected in the selection patterns when observers perform the feature foraging task. Participants seem to select targets throughout the foraging task regardless of which colour category the previous target belongs to (e.g., they easily switch between green and red targets). In this case, it can be assumed that the “pertinence values” for these two target categories are higher than for other categories. Therefore, objects belonging to both target categories are more likely to be selected. We propose that this mechanism can be described by considering such feature foraging model parameters as the distance to the closest target of the same type, distance to the closest target of the other type, distance to the distractor (Type 1), and distance to the distractor (Type 2) as these parameters are associated with features that determine whether a candidate item belongs to one category or the other. We assume that the type of target category is not essential, and in this case, the distance between targets strongly affects whether observers decide to switch (see Tagu & Kristjánsson, 2020).

As for the conjunction condition, the most commonly observed foraging pattern was that observers stuck to a particular category throughout a foraging trial; we speculated that this corresponded to the pigeonholing mechanism as a certain category was vital for this behaviour. The pigeonholing mechanism explains objects’ categorisations determined by a decision-making bias parameter—if certain types of categorisations are relevant, then the corresponding bias values should be high, so the required types of categorisations are likely to be made. We associated this mechanism with such parameters as the ratio of targets against distractors and run-length. The ratio parameter changed depending on the number of remaining targets of the same type and the remaining targets of another type. If you imagine a situation where 10% of targets remain of the type most recently selected and 90% to 100% of targets of a different type, then the likelihood of switching during foraging increases. The value of this parameter depended on the degree to which participants preferred to select objects belonging to the same category type (the more frequent, the higher the value). This may indicate higher bias values for categories where more extended selection sequences are preferred.

Future directions

The idea that a foraging subset may be analogous to an FVF may involve a promising avenue for future research as research on FVFs has typically been carried out with eye-tracking (Hulleman & Olivers, 2017; Young & Hulleman, 2013). In a recent study, Tagu and Kristjánsson (2020) manipulated target crypticity (difficulty) during feature versus conjunction foraging and the type of effector for target selection (mouse or gaze foraging). In contrast to the results of Jóhannesson and colleagues’s study (2017), who found no differences in gaze foraging performance between feature and conjunction foraging, Tagu and Kristjánsson found effects of target crypticity and effector type on foraging strategies. They also showed that different strategies are comparable with oculomotor dynamics. Thus, feature foraging was associated with focal exploration with long fixations and short-amplitude saccades, whereas conjunction foraging was in contrast associated with ambient exploration, with short fixations and high-amplitude saccades. These results may serve as a base for additional investigation regarding FVFs to expand future models for a better understanding of foraging.

Regarding future developments based on the current work, it is important to remember that there were no parameters directly related to time. Although our primary predicted variable was a change or switch in the selection, these choices to switch were modelled as independent of other previous choices. One of the critical parameters in foraging has proved to be switch costs which denote the slowdown in target selection that occurs when observers select a different target type than they did on the last trial. A. Kristjánsson et al. (2020) found that switch costs were higher during conjunction foraging than feature foraging, which means that switching between different target types takes much longer. We plan to use a dynamic Bayesian classifier to add a temporal component to foraging models. This model’s transformation will significantly contribute to the model’s depth and more accurately implement TVA as, according to Bundesen (1990), the time component is an essential part of TVA.

Our model could also be considered in light of another influential theoretical approach for explaining foraging, the so-called OFT (Hills, 2006; Pyke et al., 1997, see also discussion above). OFT is based on the MVT, which suggests that if a predator aims to maximise the energy gain from foraging in an environment consisting of different patches with prey or food, it should leave a particular patch when the expected profit from staying drops to the expected profit from travelling, and start selecting from the next patch (Bartumeus & Catalan, 2009; Charnov, 1976). In other words, foragers should stay longer in a particular patch if travelling to the following location costs more in terms of energy or time.

These ideas can enable modelling of the optimal time that subjects spend in a patch depending on the number of objects within it and measure the cost of switching to another patch. Some early studies supported such ideas (Kamil et al., 1988; Pyke, 1978). However, recent data have suggested that humans are not optimal foragers (Hutchinson et al., 2008; A. Kristjánsson et al., 2020; Pierce & Ollason, 1987) and that foraging behaviour involves biases and factors not considered by OFT. In addition, Wolfe (2013) used a patch-leaving design, manipulating patch quality (number of berries, berry value, see also Tagu & Kristjánsson, 2022), finding differences from MVT predictions, and it has been suggested that foraging strategy may depend on the task itself. Thus, it seems that foraging behaviour is more dynamic and flexible than OFT and MVT imply. But, for the current purposes, it is important to note that these models are aimed at accounting for patch-leaving behaviour in foraging and do not apply well to selection patterns within trials, as is the main focus of many measures in foraging tasks that involve two or more target types. Our aim here has been to investigate ways of modelling such tasks to provide insights into how visual attention operates as these tasks are performed (see Clarke et al., 2021, for a recently proposed Bayesian model of foraging).

Our current data do not involve measures of patch leaving, and such paradigms would seem to be an important future direction of study. Although our data looked at switches within a single display, or “patch,” there may be potential overlaps in how single-screen foraging can be envisioned. For example, complex single screens may be treated strategically by participants where they consider regions or quadrants as separate patches to explore, and OFT might be able to reveal this strategy. Particularly interesting might be to apply OFT to FVFs to determine whether OFT might predict short saccades within an FVF versus longer saccades to locations outside the FVF.

It is also worth noting that individual differences were found in Á. Kristjánsson’s et al.’s (2014) work, in particular, observers who switched frequently in both feature and conjunction conditions (so-called super foragers). However, Kristjánsson et al. (2018) showed that setting a time limit for each trial leads to more frequent switches regardless of task difficulty. Tagu and Kristjánsson (2022) also showed that during gaze-conjunction foraging “super foragers” had a higher number of runs than normal foragers and intermediate foragers (behaving like super foragers during mouse foraging but acting as normal foragers during gaze foraging) but they made a higher number of errors than intermediate foragers, who made more errors than normal foragers. So, the jury is still out on the role of individual differences, and we believe it is premature to try to address this issue with the current models of foraging. Clarke and colleagues (2021), however, have had success in discovering parameters for individual foragers in their recent model.

Conclusion

In this study, we highlighted several key contributions to the understanding of foraging from a TVA perspective. We defined a set of parameters that might best approximate the filtering and pigeonholing mechanisms in TVA and tested how well these parameters predicted target switching using Bayesian classifiers. We created separate Bayesian classifiers for feature and conjunction foraging using data from Á. Kristjánsson et al. (2014) and achieved high accuracy in predicting rare switching events. Given that the roles of pigeonholing and filtering should differ across feature and conjunction search, we tested the degree to which our parameters differed for these two tasks. The Bayesian approach to foraging seems to be a promising start for a deeper understanding of underlying mechanisms of attentional selection.

Supplemental Material

sj-docx-1-qjp-10.1177_17470218221094572 – Supplemental material for Bayesian approximations to the theory of visual attention (TVA) in a foraging task

Supplemental material, sj-docx-1-qjp-10.1177_17470218221094572 for Bayesian approximations to the theory of visual attention (TVA) in a foraging task by Sofia Tkhan Tin Le, Árni Kristjánsson and W Joseph MacInnes in Quarterly Journal of Experimental Psychology

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Sofia Tkhan Tin Le

Árni Kristjánsson

Supplementary material

The supplementary material is available at .

References

Arbuthnott

Frank

(2000). Executive control in set switching: Residual switch cost and task-set inhibition. Canadian Journal of Experimental Psychology = Revue Canadienne de Psychologie Experimentale, 54(1), 33–41. https://doi.org/10.1037/h0087328

Barandela

Valdovinos

R. M.

Sánchez

J. S.

Ferri

F. J.

(2004). The imbalanced training sample problem: Under or over sampling? In Fred

Caelli

T. M.

Duin

R. P. W.

Campilho

A. C.

de Ridder

(Eds.), Structural, Syntactic, and Statistical Pattern Recognition (pp. 806–814). Springer. https://doi.org/10.1007/978-3-540-27868-9_88

Bartumeus

Catalan

(2009). Optimal search behavior and classic foraging theory. Journal of Physics A: Mathematical and Theoretical, 42(43), Article 434002. https://doi.org/10.1088/1751-8113/42/43/434002

BayesFusion LLC. (2017). GeNIe modeler: User manual.

Bond

A. B.

(1983). Visual search and selection of natural stimuli in the pigeon: The attention threshold hypothesis. Journal of Experimental Psychology. Animal Behavior Processes, 9(3), 292–306. https://doi.org/10.1037/0097-7403.9.3.292

Bukovinszky

Rikken

Evers

Wäckers

F. L.

Biesmeijer

J. C.

Prins

H. H. T.

Kleijn

(2017). Effects of pollen species composition on the foraging behaviour and offspring performance of the mason bee Osmia bicornis (L.). Basic and Applied Ecology, 18, 21–30. https://doi.org/10.1016/j.baae.2016.11.001

Bundesen

(1990). A theory of visual attention. Psychological Review, 97(4), 523–547. https://doi.org/10.1037/0033-295x.97.4.523

Bundesen

Habekost

(2008). Principles of visual attention: Linking mind and brain. Oxford University Press.

Bundesen

Habekost

(2014). Theory of visual attention (TVA). In Nobre

A. C.

Kastner

(Eds.), The Oxford handbook of attention (pp. 1095–1121). Oxford University Press.

10.

Bundesen

Habekost

Kyllingsbaek

(2005). A neural theory of visual attention: Bridging cognition and neurophysiology. Psychological Review, 112(2), 291–328. https://doi.org/10.1037/0033-295X.112.2.291

11.

Cain

M. S.

Vul

Clark

Mitroff

S. R.

(2012). A Bayesian optimal foraging model of human visual search. Psychological Science, 23(9), 1047–1054. https://doi.org/10.1177/0956797612440460

12.

Charnov

E. L.

(1976). Optimal foraging, the marginal value theorem. Theoretical Population Biology, 9(2), 129–136. https://doi.org/10.1016/0040-5809(76)90040-X

13.

Chun

M. M.

Wolfe

J. M.

Goldstein

E. B.

(2001). Visual attention. In Goldstein

(Ed.), Blackwell handbook of perception (pp. 272–310). Blackwell.

14.

Clarke

Hunt

A. R.

Hughes

(2021). Building Bayesian cognitive models of visual foraging. PsyArXiv. https://doi.org/10.31234/osf.io/zacr9

15.

Dawkins

(1971). Shifts of “attention” in chicks during feeding. Animal Behaviour, 19(3), 575–582. https://doi.org/10.1016/S0003-3472(71)80114-8

16.

Desimone

Duncan

(1995). Neural mechanisms of selective visual attention. Annual Review of Neuroscience, 18(1), 193–222. https://doi.org/10.1146/annurev.ne.18.030195.001205

17.

Duncan

Owen

A. M.

(2000). Common regions of the human frontal lobe recruited by diverse cognitive demands. Trends in Neurosciences, 23(10), 475–483. https://doi.org/10.1016/S0166-2236(00)01633-7

18.

Fougnie

Cormiea

S. M.

Zhang

Alvarez

G. A.

Wolfe

J. M.

(2015). Winter is coming: How humans forage in a temporally structured environment. Journal of Vision, 15(11), Article 1. https://doi.org/10.1167/15.11.1

19.

Hills

(2006). Animal foraging and the evolution of goal-directed cognition. Cognitive Science, 30, 3–41. https://doi.org/10.1207/s15516709cog0000_50

20.

Horowitz

T. S.

Wolfe

J. M.

(2001). Search for multiple targets: Remember the targets, forget the search. Perception & Psychophysics, 63(2), 272–285. https://doi.org/10.3758/BF03194468

21.

Hulleman

Olivers

C. N. L.

(2017). On the brink: The demise of the item in visual search moves closer. The Behavioral and Brain Sciences, 40, Article e163. https://doi.org/10.1017/S0140525X16000364

22.

Hutchinson

J. M. C.

Wilke

Todd

P. M.

(2008). Patch leaving in humans: Can a generalist adapt its rules to dispersal of items across patches? Animal Behaviour, 75(4), 1331–1349. https://doi.org/10.1016/j.anbehav.2007.09.006

23.

Jóhannesson

Ó. I.

Kristjánsson

Á.

Thornton

I. M

. (2017). Are foraging patterns in humans related to working memory and inhibitory control? Japanese Psychological Research, 59(2), 152–166. https://doi.org/10.1111/jpr.12152

24.

Jóhannesson

Ó. I.

Thornton

I. M.

Smith

I. J.

Chetverikov

Kristjánsson

. (2016). Visual foraging with fingers and eye gaze. I-Perception, 7(2), Article 637279. https://doi.org/10.1177/2041669516637279

25.

Johnstone

R. A.

Dall

S. R. X.

Dukas

(2002). Behavioural and ecological consequences of limited attention. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 357(1427), 1539–1547. https://doi.org/10.1098/rstb.2002.1063

26.

Kamil

A. C.

Yoerg

S. I.

Clements

K. C.

(1988). Rules to leave by: Patch departure in foraging blue jays. Animal Behaviour, 36(3), 843–853. https://doi.org/10.1016/S0003-3472(88)80167-2

27.

Kristjánsson

Á.

Björnsson

A. S.

Kristjánsson

. (2020). Foraging with Anne Treisman: Features versus conjunctions, patch leaving and memory for foraged locations. Attention, Perception & Psychophysics, 82(2), 818–831. https://doi.org/10.3758/s13414-019-01941-y

28.

Kristjánsson

Á.

Draschkow

. (2021). Keeping it real: Looking beyond capacity limits in visual cognition. Attention, Perception & Psychophysics, 83(4), 1375–1390. https://doi.org/10.3758/s13414-021-02256-7

29.

Kristjánsson

Á.

Egeth

. (2020). How feature integration theory integrated cognitive psychology, neurophysiology, and psychophysics. Attention, Perception & Psychophysics, 82(1), 7–23. https://doi.org/10.3758/s13414-019-01803-7

30.

Kristjánsson

Á.

Jóhannesson

Ó. I.

Thornton

I. M

. (2014). Common attentional constraints in visual foraging. PLOS ONE, 9(6), Article e100752. https://doi.org/10.1371/journal.pone.0100752

31.

Kristjánsson

Á.

(2018). Foraging through multiple target categories reveals the flexibility of visual working memory. Acta Psychologica, 183, 108–115. https://doi.org/10.1016/j.actpsy.2017.12.005

32.

Kristjánsson

Thornton

I. M.

Chetverikov

Kristjánsson

. (2020). Dynamics of visual attention revealed in foraging tasks. Cognition, 194, Article 104032. https://doi.org/10.1016/j.cognition.2019.104032

33.

Kristjánsson

Thornton

I. M.

Kristjánsson

. (2018). Time limits during visual foraging reveal flexible working memory templates. Journal of Experimental Psychology. Human Perception and Performance, 44(6), 827–835. https://doi.org/10.1037/xhp0000517

34.

Kyllingsbæk

Schneider

W. X.

Bundesen

(2001). Automatic attraction of attention to former targets in visual displays of letters. Perception & Psychophysics, 63(1), 85–98. https://doi.org/10.3758/BF03200505

35.

Logan

G. D.

(2002). An instance theory of attention and memory. Psychological Review, 109(2), 376–400. https://doi.org/10.1037/0033-295X.109.2.376

36.

Logan

G. D.

Gordon

R. D.

(2001). Executive control of visual attention in dual-task situations. Psychological Review, 108(2), 393–434. https://doi.org/10.1037/0033-295X.108.2.393

37.

Maljkovic

Nakayama

(1994). Priming of pop-out: I—Role of features. Memory & Cognition, 22(6), 657–672. https://doi.org/10.3758/BF03209251

38.

Mallott

E. K.

Garber

P. A.

Malhi

R. S.

(2017). Integrating feeding behavior, ecological data, and DNA barcoding to identify developmental differences in invertebrate foraging strategies in wild white-faced capuchins (Cebus capucinus). American Journal of Physical Anthropology, 162(2), 241–254. https://doi.org/10.1002/ajpa.23113

39.

Maloof

M. A.

(2003). Learning when data sets are imbalanced and when costs are unequal and unknown. ICML-2003 workshop on learning from imbalanced data sets II, 2, 8. https://www.site.uottawa.ca/~nat/Workshop2003/maloof-icml03-wids.pdf

40.

Metlay

Sokoloff

Kaplan

I. T.

(1970). Visual search for multiple targets. Journal of Experimental Psychology, 85(1), 148–150. https://doi.org/10.1037/h0029519

41.

Mohammed

Rawashdeh

Abdullah

(2020, April 7–9). Machine learning with oversampling and undersampling techniques: Overview study and experimental results [Conference session]. 20 11th International Conference on Information and Communication Systems (ICICS, Irbid, Jordon. https://doi.org/10.1109/ICICS49469.2020.239556

42.

Najemnik

Geisler

W. S.

(2005). Optimal eye movement strategies in visual search. Nature, 434(7031), 387–391. https://doi.org/10.1038/nature03390

43.

Jordan

M. I.

(2001). On discriminative vs. generative classifiers: A comparison of logistic regression and naive Bayes. NIPS.

44.

Nosofsky

R. M.

Palmieri

T. J.

(2008). “An exemplar-based random walk model of speeded classification”: Correction to Nosofsky and Palmeri (1997). Psychological Review, 115(2), 446–446. https://doi.org/10.1037/0033-295X.115.2.446

45.

Ólafsdóttir

I. M.

Gestsdóttir

Kristjánsson

Á.

(2019). Visual foraging and executive functions: A developmental perspective. Acta Psychologica, 193, 203–213. https://doi.org/10.1016/j.actpsy.2019.01.005

46.

Pearl

(2011). Bayesian networks. https://escholarship.org/uc/item/53n4f34m

47.

Pierce

G. J.

Ollason

J. G.

(1987). Eight reasons why optimal foraging theory is a complete waste of time. Oikos, 49(1), 111–118. https://doi.org/10.2307/3565560

48.

Posner

M. I.

Nissen

M. J.

Ogden

W. C.

(1978). Attended and unattended processing modes: The role of set for spatial location. Modes of Perceiving and Processing Information, 137(158), Article 2.

49.

Posner

M. I.

Snyder

C. R.

Davidson

B. J.

(1980). Attention and the detection of signals. Journal of Experimental Psychology: General, 109(2), 160–174. https://doi.org/10.1037/0096-3445.109.2.160

50.

Punzalan

Rodd

F. H.

Hughes

K. A.

(2005). Perceptual processes and the maintenance of polymorphism through frequency-dependent predation. Evolutionary Ecology, 19(3), 303–320. https://doi.org/10.1007/s10682-005-2777-z

51.

Putri

Frieyadie

(2017, August 8–10). Combining integreted sampling technique with feature selection for software defect prediction [Conference session]. 2017 5th International Conference on Cyber and IT Service Management (CITSM), Denpasar, Indonesia. https://doi.org/10.1109/CITSM.2017.8089264

52.

Pyke

G. H.

(1978). Optimal foraging: Movement patterns of bumblebees between inflorescences. Theoretical Population Biology, 13(1), 72–98. https://doi.org/10.1016/0040-5809(78)90036-9

53.

Pyke

G. H.

Pulliam

H. R.

Charnov

E. L.

(1997). Optimal foraging: A selective review of theory and tests. The Quarterly Review of Biology, 52(2), 137–154. https://doi.org/10.1086/409852

54.

Schneider

Fisk

A. D.

(1982). Degree of consistent training: Improvements in search performance and automatic process development. Perception & Psychophysics, 31(2), 160–168. https://doi.org/10.3758/BF03206216

55.

Scialfa

C. T.

Kline

D. W.

Lyman

B. J.

(1987). Age differences in target identification as a function of retinal location and noise level: Examination of the useful field of view. Psychology and Aging, 2(1), 14–19. https://doi.org/10.1037/0882-7974.2.1.14

56.

Sperling

(1960). The information available in brief visual presentations. Psychological Monographs: General and Applied, 74(11), 1–29. https://doi.org/10.1037/h0093759

57.

Sperling

(1967). Successive approximations to a model for short term memory. Acta Psychologica, 27, 285–292. https://doi.org/10.1016/0001-6918(67)90070-4

58.

Tagu

Kristjánsson

. (2020). Dynamics of attentional and oculomotor orienting in visual foraging tasks. Quarterly Journal of Experimental Psychology, 75, 260–276. https://doi.org/10.1177/1747021820919351

59.

Tagu

Kristjánsson

. (2022). The selection balance: Contrasting value, proximity and priming in a multitarget foraging task. Cognition, 218, Article 104935.

60.

Treisman

(1986). Features and objects in visual processing. Scientific American, 255(5), 114–125. https://doi.org/10.1038/scientificamerican1186-114B

61.

Treisman

A. M.

Gelade

(1980). A feature-integration theory of attention. Cognitive Psychology, 12(1), 97–136. https://doi.org/10.1016/0010-0285(80)90005-5

62.

Vrieze

S. I.

(2012). Model selection and psychological theory: A discussion of the differences between the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Psychological Methods, 17(2), 228–243. https://doi.org/10.1037/a0027127

63.

Wolfe

J. M.

(1998). What can 1 million trials tell us about visual search? Psychological Science, 9(1), 33–39. https://doi.org/10.1111/1467-9280.00006

64.

Wolfe

J. M.

(2000). Chapter 8: Visual attention. In De Valois

K. K.

(Ed.), Seeing (pp. 335–386). Academic Press. https://doi.org/10.1016/B978-012443760-9/50010-6

65.

Wolfe

J. M.

Butcher

S. J.

Lee

Hyle

(2003). Changing your mind: on the contributions of top-down and bottom-up guidance in visual search for feature singletons. Journal of Experimental Psychology: Human Perception and Performance, 29(2), 483.

66.

Wolfe

J. M.

(2013). When is it time to move to the next raspberry bush? Foraging rules in human visual search. Journal of Vision, 13(3), 10–10. https://doi.org/10.1167/13.3.10

67.

Young

A. H.

Hulleman

(2013). Eye movements reveal how task difficulty moulds visual search. Journal of Experimental Psychology. Human Perception and Performance, 39(1), 168–190. https://doi.org/10.1037/a0028679

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.01 MB