Detection of visual stimuli in correlated noise

The hypothesis of detection by temporal probability summation (TPS) as characterised in Mortensen [(2007). An analysis of visual detection by temporal probability summation, submitted for publication] is critically evaluated, considering data from an experiment of Roufs and Blommaert [(1981). Temporal impulses and step responses of the human eye obtained psychophysically by means of a drift-correcting perturbation technique. Vision Research, 21, 1203-1221], who derived the impulse and the step response for a sustained type of channel. The assumed approximate linearity of the channel is discussed with respect to recent findings from neurophysiological investigations. The data are shown not to be compatible with the TPS-postulate. Further, a model of a cell assembly, consisting of a homogeneous set of neurons, is presented that allows for a natural interpretation of random fluctuations in case of temporal peak detection (TPD). The model may be discussed with respect to TPS as well as to TPD; in any case, the model allows to integrate some results concerning the effects of attentional focussing on the detection process.     

Further considerations of a general d′ in multidimensional space

Thomas (Psychonom. Bull. Rev. 6 (1999) 224) proposed a generalization of d′,d_g′ for multidimensional distributions and demonstrated that it is not equivalent to Euclidean distance as had been assumed in some previous studies. In this note, it is further shown not to be a metric in the general sense as it fails the triangle inequality. Also, a rigorous proof is offered of the claim (found in Thomas, Psychonom. Bull. Rev. 6 (1999) 224) that in order for the definition of d_g′ to correspond to the quantity that is estimated from data in the traditional way (using hits and false alarms) one must assume a distance classifier (proved in the case of equal covariance matrices). If one assumes optimal responding instead, then the estimated d′ corresponds to (the square root of) Mahalanobis distance. This latter observation clears up an apparent paradox between the fact that d_g′ is not a metric and Ashby and Perrin's (Psychol. Rev. 95 (1988) 124) statement relating the weighted Euclidean scaling model and a signal detection model of similarity that would yield a distance classifier.     

Testing for selectivity in the dependence of random variables on external factors

Random variables A and B, whose joint distribution depends on factors (x,y), are selectively influenced by x and y, respectively, if A and B can be represented as functions of, respectively, (x,S_A,C) and (y,S_B,C), where S_A,S_B,C are stochastically independent and do not depend on (x,y). Selective influence implies selective dependence of marginal distributions on the respective factors: thus no parameter of A may depend on y. But parameters characterizing stochastic interdependence of A and B, such as their mixed moments, are generally functions of both x and y. We derive two simple necessary conditions for selective dependence of (A,B) on (x,y), which can be used to conduct a potential infinity of selectiveness tests. One condition is that, for any factor values x,x^′ and y,y^′,

s_xy≤s_xy^′+s_x^′y^′+s_x^′y,     

Binary choice, subset choice, random utility, and ranking: A unified perspective using the permutahedron

The d-permutahedron Π_d−1⊂R^d is defined as the convex hull of all d-dimensional permutation vectors, namely, vectors whose components are distinct values of a d-element set of integers [d]≡{1,2,…,d}. By construction, Π_d−1 is a convex polytope with d! vertices, each representing a linear order (ranking) on [d], and has dimension dim(Π_d−1)=d−1. This paper provides a review of some well-known properties of a permutahedron, applies the geometric-combinatoric insights to the investigation of the various popular choice paradigms and models by emphasizing their inter-connections, and presents a few new results along this line.Permutahedron provides a natural representation of ranking probability; in fact it is shown here to be the space of all Borda scores on ranking probabilities (also called “voters profiles” in the social choice literature). The following relations are immediate consequences of this identification. First, as all d! vertices of Π_d−1 are equidistant to its barycenter, Π_d−1 is circumscribed by a sphere S^d−2 in a (d−1)-dimensional space, with each spherical point representing an equivalent class of vectors whose components are defined on an interval scale. This property provides a natural expression of the random utility model of ranking probabilities, including the condition of Block and Marschak. Second, Π_d−1 can be realized as the image of an affine projection from the unit cube of dimension d(d−1)/2. As the latter is the space of all binary choice vectors describing probabilities of pairwise comparisons within d objects, Borda scores can be defined on binary choice probabilities through this projective mapping. The result is the Young's formula, now applicable to any arbitrary binary choice vector. Third, Π_d−1 can be realized as a “monotone path polytope” as induced from the lift-up of the projection of the cube onto the line segment [0,d]⊂R¹. As the 2^d vertices of the d-cube are in one-to-one correspondence to all subsets of [d], a connection between the subset choice paradigm and ranking probability is established. Specifically, it is shown here that, in the case of approval voting (AV) with the standard tally procedure (Amer. Pol. Sci. Rev. 72 (1978) 831), under the assumption that the choice of a subset indicates an approval (with equal probability) of all linear orders consistent with that chosen subset, the Brams-Fishburn score is then equivalent to the Borda score on the induced profile. Requiring this induced profile (ranking probability) to be also consistent with the size-independent model of subset choice (J. Math. Psychol. 40 (1996) 15) defines the “core” of the AV Polytope. Finally, Π_d−1 can be realized as a canonical projection from the so-called Birkhoff polytope, the space of rank-position probabilities arising out of the rank-matching paradigm; thus Borda scores can be defined on rank-position probabilities. To summarize, the many realizations of a permutahedron afford a unified framework for describing and relating various ranking and choice paradigms.     

Order from chaos? 1/f noise predicts performance on reaction time measures

In repetitive tasks, latencies typically display non-random variation called 1/f noise, but the magnitude of 1/f noise can depend on effort. Between-participant variation in 1/f noise might therefore predict differences in performance for tasks that are sensitive to effort. Feedback was manipulated to suggest that participants had (or had not) shown bias on a preliminary task. Participants then completed a computer-based measure of prejudice. Consistent with an ironic process account, participants who were falsely informed that they had shown bias ultimately did show increased prejudice. But ironic processes should depend on control-related effort. Taking 1/f noise as a proxy for effort, results show that the effect of the feedback was moderated by 1/f noise. Participants demonstrating a low-effort 1/f pattern were unaffected by the manipulation (η_p² < .01), whereas those demonstrating a high-effort 1/f pattern showed larger effects (η_p² = .19). Results thus suggest that 1/f noise can provide novel explanatory power regarding behavior.     

How many response options? A study of remember-know testing procedures

The remember-know procedure can be conducted in one or two steps. The one-step procedure does not include a recognition response (old-new) prior to the remember-know response. It is observed consistently that the one-step procedure leads to a more liberal placement of the response criterion, but it is unclear whether recognition accuracy is affected by the number of procedural steps. However, previous studies used bias-dependent measures of accuracy (A′ and d′). We manipulated the number of steps and confirmed the finding that the response criterion is more liberal with the one-step procedure. More importantly, we employed a signal detection theory bias-free accuracy measure (d_a) to show that varying the number of steps does not affect recognition accuracy, and we demonstrated that this pattern of results does not change when the dual process signal detection model (Yonelinas, 1997) is applied.     

Assessing Parameter Invariance in the BLIM: Bipartition Models

In knowledge space theory, the knowledge state of a student is the set of all problems he is capable of solving in a specific knowledge domain and a knowledge structure is the collection of knowledge states. The basic local independence model (BLIM) is a probabilistic model for knowledge structures. The BLIM assumes a probability distribution on the knowledge states and a lucky guess and a careless error probability for each problem. A key assumption of the BLIM is that the lucky guess and careless error probabilities do not depend on knowledge states (invariance assumption). This article proposes a method for testing the violations of this specific assumption. The proposed method was assessed in a simulation study and in an empirical application. The results show that (1) the invariance assumption might be violated by the empirical data even when the model’s fit is very good, and (2) the proposed method may prove to be a promising tool to detect invariance violations of the BLIM.     

Dimension in latent variable models

Psychologists would like to say that a probability distribution on {0,1}ⁿ is d-dimensional if (1) the distribution can be represented by some smooth d-dimensional latent variable model and (2) the distribution cannot be represented by any smooth d−1 dimensional model. This does not work out because for d>1, every distribution that can be represented by a smooth d-dimensional model can also be represented by a smooth one-dimensional model. A proof and discussion of some implications of this mathematical result follow.     

Temporal summation at the warmth threshold

Threshold levels for warmth aroused by infrared irradiation were measured in sLx Ss at durations between 0.05 and 10 sec. Beyond a critical duration of about a second, the threshold does not depend on duration. Below critical duration time t trades for irradiance ? by the formula ? = kt^?0.82. That these properties do not depend much on areal extent of stimulation was demonstrated by a study that compared temporal summation for two different areas of the same S’s skin. Individual differences in apparent absolute sensitivity were explored under the rubric of the theory of signal detection.     

A discrete time model for detection of randomly presented stimuli

Considered in this paper is a discrete-time Markov decision process with unobservable states, which occurs in equipment checking, medical diagnosis, signal detection, and reaction time studies. The process is characterized by (i) a state S₀ that at some unknown time t, which is determined probabilistically, turns into an absorbing state S₁; (ii) observations that provide fallible information about the true state and which are taken sequentially to determine whether or not the change from S₀ to S₁ has occurred; and (iii) losses for a false alarm and for delays in detecting the change. A response model is proposed, prescribing a terminal decision as soon as the posterior probability that S₁ obtains reaches or exceeds a fixed probability boundary. Predictions are derived concerning the probability of false alarm and several conditional distributions of the number of observations taken.     

Extensions of the significance test for one-parameter signal detection hypotheses

The basic models of signal detection theory involve the parametric measure,d, generally interpreted as a detectability index. Given two observers, one might wish to know whether their detectability indices are equal or unequal. Gourevitch and Galanter (1967) proposed a large sample statistical test that could be used to test the hypothesis of equald values. In this paper, their large two sample test is extended to aK-sample detection test. If the null hypothesisd ₁=d ₂=...=d _K is rejected, one can employ the post hoc confidence interval procedure described in this paper to locate possible statistically significant sources of variance and differences. In addition, it is shown how one can use the Gourevitch and Galanter statistics to testd=0 for a single individual.This paper was written while the author was associated with the Institute of Human Learning at the University of California at Berkeley.     

Binary choice probabilities: on the varieties of stochastic transitivity

Let {P_λ} denote the family of decisiveness relations {$\text{P}{}_{\mathrm{\lambda }}\text{:}\text{1}\text{2}\text{≤ λ < 1}$} with aP_λb if and only if P(a,b) > λ, where P is a binary choice probability function. Families in which all decisiveness relations are of the same type, such as all strict partial orders or all semiorders, are characterized by stochastic transitivity conditions. The conditions used for this purpose differ in various ways from the traditional forms of strong, moderate, and weak stochastic transitivity. The family {P_λ} is then examined from the viewpoint of interval representation models, the most general of which is aP_λb if and only if I(a, λ) > I(b, λ), where the I's are real intervals with I(a, λ) > I(b, λ) if and only if the first interval is completely to the right of the second. With I(a, λ) = [f(a, λ), f(a, λ) + σ(a, λ)], the specializations of the interval model that are discussed include those where the location function f (for left end-points) depends only on the set A of alternatives or stimuli and where the length function σ depends only on A or on λ or neither.     

Statistical manifold as an affine space: A functional equation approach

A statistical manifold M_μ consists of positive functions f such that defines a probability measure. In order to define an atlas on the manifold, it is viewed as an affine space associated with a subspace of the Orlicz space L^Φ. This leads to a functional equation whose solution, after imposing the linearity constrain in line with the vector space assumption, gives rise to a general form of mappings between the affine probability manifold and the vector (Orlicz) space. These results generalize the exponential statistical manifold and clarify some foundational issues in non-parametric information geometry.     

No relevant spectator impact on home advantage in male and female professional volleyball – A longitudinal multilevel logistic model analysis over 25 years

Home advantage (HA) regularly occurs in volleyball (Pollard et al., 2017: men: 56.62%, women: 55.26%). Research to date has investigated primarily small samples of mostly female matches and not looked into the potential impact of spectators on HA. This archival analysis uses multilevel modelling to examine HA in professional German volleyball (men & women) over 25 seasons in all regular and play-off matches (N = 6,833). We analyze how spectators drive HA and whether this projects to the COVID-19 season 2020/21.When intercepts varied between teams (2-level model, ICC = 27%), the winning probability increased when playing at home (men: 57.01%, OR_men = 2.39, d = 0.48; women: 55.39%, OR_women = 2.19, d = 0.43), while controlling for team strength, interaction with gender, and travelling distance. More spectators had a negligible effect on the men’s and women’s chances (|d| < 0.07). Similar trends were observed for the probability of winning sets. Contrary to other team sports (e.g., soccer), there is no HA-development over the last decades.     

Compensation for subliminal timing perturbations in perceptual-motor synchronization

 It is sometimes assumed that limits of temporal discrimination established in psychophysical tasks constrain the timing information available for the control of action. Results from the five perceptual-motor synchronization experiments presented here argue against this assumption. Experiment 1 demonstrates that subliminal (0.8–2%) local changes in interval duration in an otherwise isochronous auditory sequence are rapidly compensated for in the timing of synchronized finger tapping. If this compensation is based on perception of the highly variable synchronization error (SE) rather than of the local change in stimulus period, then it could be based solely on SEs that exceed the temporal order threshold. However, that hypothesis is ruled out by additional analyses of Exp. 1 and the results of Exp. 2, a combined synchronization and temporal order judgment task. Experiments 3–5 further show that three factors that affect the detectability of local deviations from stimulus isochrony do not inhibit effective compensation for such deviations in synchronized tapping. Experiment 5, a combined synchronization and detection task, shows directly that compensation for timing perturbations does not depend on explicit detection. Overall, the results suggest that the automatic processes involved in the temporal control of action have access to more accurate timing information than do the conscious decision processes of auditory temporal judgment. Received: 19 November 1998 / Accepted: 18 March 1999     

Detection of changes in speed and direction of motion: Reaction time analysis

Observers reacted to the change in the movement of a random-dot field whose initial velocity,V _o, was constant for a random period and then switched abruptly to another value,V ₁. The two movements, both horizontally oriented, were either in the same direction (speed increments or decrements), or in the opposite direction but equal in speed (direction reversals). One of the two velocities,V ₀ orV ₁, could be zero (motion onset and offset, respectively). in the range of speeds used, 0–16 deg/sec (dps), the mean reaction time (MRT) for a given value ofV ₀ depended on |V ₁-V ₀| only: MRT ≈r +c(V ₀/|V ₁ ?V ₀|) ^β , where β=2/3,r is a velocity-independent component of MRT, andc(V ₀) is a parameter whose value is constant for low values ofV ₀ (0–4 dps), and increases beginning with some value ofV ₀ between 4 and 8 dps. These and other data reviewed in the paper are accounted for by a model in which the time-position function of a moving target is encoded by mass activation of a network of Reichardt-type encoders. Motion-onset detection (V ₀=0) is achieved by weighted temporal summation of the outputs of this network, the weights assigned to activated encoders being proportional to their squared spatial spans. By means of a “subtractive normalization,” the visual system effectively reduces the detection of velocity changes (a change fromV ₀ toV ₁) to the detection of motion onset (a change from 0 toV ₁-V ₀). Subtractive normalization operates by readjustment of weights: the weights of all encoders are amplified or attenuated depending on their spatial spans, temporal spans, and the initial velocityV ₀. Assignment of weights and weighted temporal summation are thought of as special-purpose computations performed on the dynamic array of activations in the motion-encoding network, without affecting the activations themselves.     

Perceptual learning in contrast detection: presence and cost of shifts in response criteria

Contemporary theoretical accounts of perceptual learning typically assume that observers are either unbiased or stably biased across the course of learning. However, standard methods for estimating thresholds, as they are typically used, do not allow this assumption to be tested. We present an approach that allows for this test specific to perceptual learning for contrast detection. We show that reliable decreases in detection thresholds and increases in hit rates are not uniformly accompanied by reliable increases in sensitivity (d′), but are regularly accompanied by reliable liberal shifts in response criteria (c). In addition, we estimate the extent to which sensitivity could have increased in the absence of these liberal shifts. The results pose a challenge to the assumption that perceptual learning has limited or no impact on response criteria.     

Intensity discrilnination: Performance in three paradigms

Performance in the following auditory intensity-discrimination paradigms was compared: (1) 2IFC, (2) single-interval rating with an intensity cue preceding each observation interval, and (3) single-interval rating with no cue. ROCs obtained in the single-interval conditions could be well approximated by assuming that the underlying distributions were both normal and had equal variance. No large difference was observed between performance in the cue and the no-cue conditions. The normalized separation of the means for 2IFC was approximately twice the corresponding estimate for the single-interval conditions. i.e., d₂ = 1.91 d₁. This is consistent with the hypothesis that the O in intensity-discrimination tasks bases his decisions on the difference between the observed input and that of a noisy stored reference. The familiar result that d₂ = (2)^1/2 d₁ can be predicted for simple detection.     

Predictive probability and analogy by similarity in inductive logic

The-continuum of inductive methods was derived from an assumption, called-condition, which says that the probability of finding an individual having propertyx _j depends only on the number of observed individuals having propertyx _j and on the total number of observed individuals. So, according to that assumption, all individuals with properties which are different fromx _j have equal weight with respect to that probability and, in particular, it does not matter whether any individual was observed having some propertysimilar tox _j (the most complete proof of this result is presented in Carnap, 1980).The problem thus remained open to find some general condition, weaker than the-condition, which would allow for thederivation of probability functions which might be sensitive to similarity. Carnap himself suggested a weakening of the-condition which might allow for similarity sensitive probability functions (Carnap, 1980, p. 45) but he did not find the family of probability functions derivable from that principle. The aim of this paper is to present the family of probability functions derivable from Carnap's suggestion and to show how it is derived.In Section 1 the general problem of analogy by similarity in inductive logic is presented, Section 2 outlines the notation and the conceptual background involved in the proof, Section 3 gives the proof, Section 4 discusses Carnap's principle and the result, Section 5 is a brief review of the solutions which have previously been proposed.