Introduction to Probability with R

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Planning Routes Across Economic Terrains: Maximizing Utility, Following Heuristics

We designed an economic task to investigate human planning of routes in landscapes where travel in different kinds of terrain incurs different costs. Participants moved their finger across a touch screen from a starting point to a destination. The screen was divided into distinct kinds of terrain and travel within each kind of terrain imposed a cost proportional to distance traveled. We varied costs and spatial configurations of terrains and participants received fixed bonuses minus the total cost of the routes they chose. We first compared performance to a model maximizing gain. All but one of 12 participants failed to adopt least-cost routes and their failure to do so reduced their winnings by about 30% (median value). We tested in detail whether participants’ choices of routes satisfied three necessary conditions (heuristics) for a route to maximize gain. We report failures of one heuristic for 7 out of 12 participants. Last of all, we modeled human performance with the assumption that participants assign subjective utilities to costs and maximize utility. For 7 out 12 participants, the fitted utility function was an accelerating power function of actual cost and for the remaining 5, a decelerating power function. We discuss connections between utility aggregation in route planning and decision under risk. Our task could be adapted to investigate human strategy and optimality of route planning in full-scale landscapes

MLDS: Maximum Likelihood Difference Scaling in R

The MLDS package in the R programming language can be used to estimate perceptual scales based on the results of psychophysical experiments using the method of difference scaling. In a difference scaling experiment, observers compare two supra-threshold differences (a,b) and (c,d) on each trial. The approach is based on a stochastic model of how the observer decides which perceptual difference (or interval) (a,b) or (c,d) is greater, and the parameters of the model are estimated using a maximum likelihood criterion. We also propose a method to test the model by evaluating the self-consistency of the estimated scale. The package includes an example in which an observer judges the differences in correlation between scatterplots. The example may be readily adapted to estimate perceptual scales for arbitrary physical continua

Ubiquitous Log Odds: A Common Representation of Probability and Frequency Distortion in Perception, Action, and Cognition

In decision from experience, the source of probability information affects how probability is distorted in the decision task. Understanding how and why probability is distorted is a key issue in understanding the peculiar character of experience-based decision. We consider how probability information is used not just in decision-making but also in a wide variety of cognitive, perceptual, and motor tasks. Very similar patterns of distortion of probability/frequency information have been found in visual frequency estimation, frequency estimation based on memory, signal detection theory, and in the use of probability information in decision-making under risk and uncertainty. We show that distortion of probability in all cases is well captured as linear transformations of the log odds of frequency and/or probability, a model with a slope parameter, and an intercept parameter. We then consider how task and experience influence these two parameters and the resulting distortion of probability. We review how the probability distortions change in systematic ways with task and report three experiments on frequency distortion where the distortions change systematically in the same task. We found that the slope of frequency distortions decreases with the sample size, which is echoed by findings in decision from experience. We review previous models of the representation of uncertainty and find that none can account for the empirical findings

Sub-Optimal Allocation of Time in Sequential Movements

The allocation of limited resources such as time or energy is a core problem that organisms face when planning complex actions. Most previous research concerning planning of movement has focused on the planning of single, isolated movements. Here we investigated the allocation of time in a pointing task where human subjects attempted to touch two targets in a specified order to earn monetary rewards. Subjects were required to complete both movements within a limited time but could freely allocate the available time between the movements. The time constraint presents an allocation problem to the subjects: the more time spent on one movement, the less time is available for the other. In different conditions we assigned different rewards to the two tokens. How the subject allocated time between movements affected their expected gain on each trial. We also varied the angle between the first and second movements and the length of the second movement. Based on our results, we developed and tested a model of speed-accuracy tradeoff for sequential movements. Using this model we could predict the time allocation that would maximize the expected gain of each subject in each experimental condition. We compared human performance with predicted optimal performance. We found that all subjects allocated time sub-optimally, spending more time than they should on the first movement even when the reward of the second target was five times larger than the first. We conclude that the movement planning system fails to maximize expected reward in planning sequences of as few as two movements and discuss possible interpretations drawn from economic theory

MLDS: Maximum Likelihood Difference Scaling in R

The MLDS package in the R programming language can be used to estimate perceptual scales based on the results of psychophysical experiments using the method of difference scaling. In a difference scaling experiment, observers compare two supra-threshold differences (a,b) and (c,d) on each trial. The approach is based on a stochastic model of how the observer decides which perceptual difference (or interval) (a,b) or (c,d) is greater, and the parameters of the model are estimated using a maximum likelihood criterion. We also propose a method to test the model by evaluating the self-consistency of the estimated scale. The package includes an example in which an observer judges the differences in correlation between scatterplots. The example may be readily adapted to estimate perceptual scales for arbitrary physical continua.

Planning multiple movements within a fixed time limit: The cost of constrained time allocation in a visuo-motor task

S.-W. Wu, M. F. Dal Martello, and L. T. Maloney (2009) evaluated subjects' performance in a visuo-motor task where subjects were asked to hit two targets in sequence within a fixed time limit. Hitting targets earned rewards and Wu et al. varied rewards associated with targets. They found that subjects failed to maximize expected gain; they failed to invest more time in the movement to the more valuable target. What could explain this lack of response to reward? We first considered the possibility that subjects require training in allocating time between two movements. In Experiment 1, we found that, after extensive training, subjects still failed: They did not vary time allocation with changes in payoff. However, their actual gains equaled or exceeded the expected gain of an ideal time allocator, indicating that constraining time itself has a cost for motor accuracy. In a second experiment, we found that movements made under externally imposed time limits were less accurate than movements made with the same timing freely selected by the mover. Constrained time allocation cost about 17% in expected gain. These results suggest that there is no single speed–accuracy tradeoff for movement in our task and that subjects pursued different motor strategies with distinct speed–accuracy tradeoffs in different conditions

Very Slow Search and Reach: Failure to Maximize Expected Gain in an Eye-Hand Coordination Task

We examined an eye-hand coordination task where optimal visual search and hand movement strategies were inter-related. Observers were asked to find and touch a target among five distractors on a touch screen. Their reward for touching the target was reduced by an amount proportional to how long they took to locate and reach to it. Coordinating the eye and the hand appropriately would markedly reduce the search-reach time. Using statistical decision theory we derived the sequence of interrelated eye and hand movements that would maximize expected gain and we predicted how hand movements should change as the eye gathered further information about target location. We recorded human observers' eye movements and hand movements and compared them with the optimal strategy that would have maximized expected gain. We found that most observers failed to adopt the optimal search-reach strategy. We analyze and describe the strategies they did adopt.Psycholog

Color and material perception: Achievements and challenges

There is a large literature characterizing human perception of the lightness and color of matte surfaces arranged in coplanar arrays. In the past ten years researchers have begun to examine perception of lightness and color using wider ranges of stimuli intended to better approximate the conditions of everyday viewing. One emerging line of research concerns perception of lightness and color in scenes that approximate the three-dimensional environment we live in, with objects that need not be matte or coplanar and with geometrically complex illumination. A second concerns the perception of material surface properties other than color and lightness, such as gloss or roughness. This special issue features papers that address the rich set of questions and approaches that have emerged from these new research directions. Here, we briefly describe the articles in the issue and their relation to previous work. Keywords: color perception, material perception Citation: Maloney, L. T., & Brainard, D. H. (2010). Color and material perception: Achievements and challenges. Journal of Vision, 10(9):19, 1-6, http://www.journalofvision.org/content/10/9/19, doi:10.1167/10.9.19. Introduction The classic perceptual correlates of object surface properties are lightness and color. Although there are notable exceptions (e.g., Over the past decade, research has increasingly begun to focus on two rich generalizations of the classic "flatmatte" paradigm (for reviews, see In 2004, we edited a special issue in the Journal of Vision entitled "Perception of color and material properties in complex scenes" Characterizing, estimating, and discriminating the light field One of the most important themes in color and material perception is the role of scene illumination. In flat-matte scenes, the placement and directionality of light sources was little emphasized Journal of Vision Complex light fields and surface color/lightness perception In parallel with direct assessment of the perception of the light field, the past decade has seen a slew of papers that study how the visual system achieves color and lightness constancy in the context of spatially complex light fields (e.g., Along these lines in the current issue, Radonjić, Todorović, and Gilchrist (2010) examine surface lightness perception in three-dimensional scenes with directional lighting and show how grouping principles such as adjacency and surroundedness can help organize the empirical phenomena. Olkkonen, Witzel, Hansen, and Gegenfurtner (2010) study color categorization for real surfaces and daylight illuminants. An entire room with controlled illumination served as the laboratory. They find that color categorization was little changed by marked changes in daylight illumination. Surface material perception: Gloss, roughness A very active area of research is the assessment of lighting and environmental conditions that affect the perception of material properties such as gloss and roughness, and how these properties interact with the perception of color and lightness. Important early work includes Surface material perception is represented by a number of papers in this issue; Kim and Anderson Interactions There are several studies that examined interactions among different material properties. In this issue, Giesel and Gegenfurtner (2010) systematically investigate color perception for real objects made of different materials varying in roughness and gloss from smooth and glossy to matte and corrugated. They show that hue is quite stable across their manipulations, but that other attributes interact. Olkkonen and Brainard (2010) study how changes in real-world illumination affect perceived glossiness and lightness with emphasis on testing independence principles. They show, for example, that the effect of geometric changes in the light field on perceived glossiness is independent of the diffuse reflectance component of the surfaces. Novel themes A number of papers in the current issue introduce novel themes. Wolfe and Myers (2010) examine visual search performance when targets and distractors are characterized by surface material. They find that, although it may be easy to discriminate "fur" or "stone," searching for a patch of fur among the stones is difficult and time-consuming. Visual search based on material differences is inefficient. Motoyoshi (2010) examines how the relationship between highlights and shading triggers perception of translucency and transparency. Goddard, Solomon, and Colin (2010) study the adaptable neural mechanisms responsible for surface color constancy. Boyaci, Fang, Murray, and Kersten (2010) also consider mechanism. They report behavioral results showing how lightness across occlusion depends on spatially distant image features and show (using brain imaging) that human early visual cortex responds strongly to occlusion-dependent lightness variations. They conclude that early cortical processing of lightness is affected by three-dimensional scene interpretation