Monotone Majorizable Functionals

Several properties of monotone functionals (MF) and monotone majorizable functionals (MMF) used in the earlier work by the author and van de Pol are proved. It turns out that the terms of the simply typed lambda-calculus define MF, but adding primitive recursion, and even monotonic primitive recursion changes the situation: already Z.Z(1 — sg) is not MMF. It is proved that extensionality is not Dialectica-realizable by MMF, and a simple example of a MF which is not hereditarily majorizable is given.     

Positive Monotone Modal Logic

Studia Logica - Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and . We axiomatise positive monotone modal...     

Detecting latent taxa: Monte Carlo comparison of taxometric, mixture model, and clustering procedures

A Monte Carlo evaluation of four procedures for detecting taxonicity was conducted using artificial data sets that were either taxonic or nontaxonic. The data sets were analyzed using two of Meehl's taxometric procedures, MAXCOV and MAMBAC, Ward's method for cluster analysis in concert with the cubic clustering criterion and a latent variable mixture modeling technique. Performance of the taxometric procedures and latent variable mixture modeling were clearly superior to that of cluster analysis in detecting taxonicity. Applied researchers are urged to select from the better procedures and to perform consistency tests.     

Monotone spline transformations for dimension reduction

Consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.     

Monotone Quantifiers Emerge via Iterated Learning

Natural languages exhibit many semantic universals, that is, properties of meaning shared across all languages. In this paper, we develop an explanation of one very prominent semantic universal, the monotonicity universal. While the existing work has shown that quantifiers satisfying the monotonicity universal are easier to learn, we provide a more complete explanation by considering the emergence of quantifiers from the perspective of cultural evolution. In particular, we show that quantifiers satisfy the monotonicity universal evolve reliably in an iterated learning paradigm with neural networks as agents.     

Monotone regression: Continuity and differentiability properties

Least-squares monotone regression has received considerable discussion and use. Consider the residual sum of squaresQ obtained from the least-squares monotone regression ofy _i onx _i . TreatingQ as a function of they _i , we prove that the gradient Q exists and is continuous everywhere, and is given by a simple formula. (We also discuss the gradient ofd=Q ^1/2.) These facts, which can be questioned (Louis Guttman, private communication), are important for the iterative numerical solution of models, such as some kinds of multidimensional scaling, in which monotone regression occurs as a subsidiary element, so that they _i and hence indirectlyQ are functions of other variables.     

Linguistically invariant inductive logic

Summary Carnap's early system of inductive logic make degrees of confirmation depend on the languages in which they are expressed. They are sensitive to which predicates are, in the language, taken as primitive. Hence they fail to be linguistically invariant. His later systems, in which prior probabilities are assigned to elements of a model rather than sentences of a language, are sensitive to which properties in the model are called primitive. Critics have often protested against these features of his work. This paper shows how to make his systems independent of any choice of primitive predicates or primitive properties.The solution is related to another criticism of inductive logic. It has been noticed that Carnap's systems are too all-embracing. Hisc(h, e) is defined for all sentencesh ande. Yet for manyh ande, the evidencee does not warrant any assessment of the probability ofh. We need an inductive logic in whichc(h, e) is defined only whene really does bear onh. This paper sketches the measure theory of such a logic, and, within this measure theory, provides relativized versions of Carnap's systems which are linguistically invariant.     

Untangling invariant object recognition

Despite tremendous variation in the appearance of visual objects, primates can recognize a multitude of objects, each in a fraction of a second, with no apparent effort. However, the brain mechanisms that enable this fundamental ability are not understood. Drawing on ideas from neurophysiology and computation, we present a graphical perspective on the key computational challenges of object recognition, and argue that the format of neuronal population representation and a property that we term 'object tangling' are central. We use this perspective to show that the primate ventral visual processing stream achieves a particularly effective solution in which single-neuron invariance is not the goal. Finally, we speculate on the key neuronal mechanisms that could enable this solution, which, if understood, would have far-reaching implications for cognitive neuroscience.     

Interactive clustering

An interactive strategy for applying cluster-analytic techniques in behavioral research is presented. The two-part approach stresses the use of on-line computers for both data collection and analysis. In data collection, an extension of multidimensional unfolding to clustering reduces the number of judgments required of subjects by as much as 50%, During data analysis, the interactive procedures described permit the testing of multiple clustering models from an extensive family. With each selection, the goodness of fit of the model to the data can be tested. In addition to improving efficiency, the interactive strategy promoted here combines the advantages of the original nonmetric clustering procedures (e.g., Johnson, 1967) with those of the latest linear additive models (e.g., Sattath & Tversky, 1977; Shepard & Arabie, 1979).     

Monotone mapping of similarities into a general metric space

A method of “maximum variance nondimensional scaling” is described and tested that transforms similarity measures into distances that meet just three conditions: (C1) they exactly satisfy the metric axioms, (C2) they are, as nearly as possible, monotonically related to the similarity measures, (C3) they have maximum variance possible under the two preceding conditions. By achieving an appropriate balance between the last two conditions, one can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre. The method appears to have potential applications, e.g., to studies of stimulus generalization and the structure and processing of semantic information.     

U-statistic hierarchical clustering

A monotone invariant method of hierarchical clustering based on the Mann-Whitney U-statistic is presented. The effectiveness of the complete-link, single-link, and U-statistic methods in recovering tree structures from error perturbed data are evaluated. The U-statistic method is found to be consistently more effective in recovering the original tree structures than either the single-link or complete-link methods.     

Hierarchical clustering schemes

Techniques for partitioning objects into optimally homogeneous groups on the basis of empirical measures of similarity among those objects have received increasing attention in several different fields. This paper develops a useful correspondence between any hierarchical system of such clusters, and a particular type of distance measure. The correspondence gives rise to two methods of clustering that are computationally rapid and invariant under monotonic transformations of the data. In an explicitly defined sense, one method forms clusters that are optimally connected, while the other forms clusters that are optimally compact.I am indebted to R. N. Shepard and J. D. Carroll for many stimulating discussions about this work, and for aid in preparing this paper.     

Subspace K-means clustering

To achieve an insightful clustering of multivariate data, we propose subspace K-means. Its central idea is to model the centroids and cluster residuals in reduced spaces, which allows for dealing with a wide range of cluster types and yields rich interpretations of the clusters. We review the existing related clustering methods, including deterministic, stochastic, and unsupervised learning approaches. To evaluate subspace K-means, we performed a comparative simulation study, in which we manipulated the overlap of subspaces, the between-cluster variance, and the error variance. The study shows that the subspace K-means algorithm is sensitive to local minima but that the problem can be reasonably dealt with by using partitions of various cluster procedures as a starting point for the algorithm. Subspace K-means performs very well in recovering the true clustering across all conditions considered and appears to be superior to its competitor methods: K-means, reduced K-means, factorial K-means, mixtures of factor analyzers (MFA), and MCLUST. The best competitor method, MFA, showed a performance similar to that of subspace K-means in easy conditions but deteriorated in more difficult ones. Using data from a study on parental behavior, we show that subspace K-means analysis provides a rich insight into the cluster characteristics, in terms of both the relative positions of the clusters (via the centroids) and the shape of the clusters (via the within-cluster residuals).     

Tarski's Fixed-Point Theorem And Lambda Calculi With Monotone Inductive Types

The new concept of lambda calculi with monotone inductive types is introduced byhelp of motivations drawn from Tarski's fixed-point theorem (in preorder theory) andinitial algebras and initial recursive algebras from category theory. They are intendedto serve as formalisms for studying iteration and primitive recursion ongeneral inductively given structures. Special accent is put on the behaviour ofthe rewrite rules motivated by the categorical approach, most notably on thequestion of strong normalization (i.e., the impossibility of an infinitesequence of successive rewrite steps). It is shown that this key propertyhinges on the concrete formulation. The canonical system of monotone inductivetypes, where monotonicity is expressed by a monotonicity witness beinga term expressing monotonicity through its type, enjoys strong normalizationshown by an embedding into the traditional system of non-interleavingpositive inductive types which, however, has to be enriched by the parametricpolymorphism of system F. Restrictions to iteration on monotone inductive typesalready embed into system F alone, hence clearly displaying the differencebetween iteration and primitive recursion with respect to algorithms despitethe fact that, classically, recursion is only a concept derived from iteration.     

Variability and detection of invariant structure

Two experiments investigated learning of nonadjacent dependencies by adults and 18–month–olds. Each learner was exposed to three–element strings (e.g., pel–kicey–jic) produced by one of two artificial languages. Both languages contained the same adjacent dependencies, so learners could distinguish the languages only by acquiring dependencies between the first and third elements (the nonadjacent dependencies). The size of the pool from which the middle elements were drawn was systematically varied to investigate whether increasing variability (in the form of decreasing predictability between adjacent elements) would lead to better detection of nonadjacent dependencies. Infants and adults acquired nonadjacent dependencies only when adjacent dependencies were least predictable. The results point to conditions that might lead learners to focus on nonadjacent versus adjacent dependencies and are important for suggesting how learning might be dynamically guided by statistical structure.     

Ultrametric hierarchical clustering algorithms

Johnson has shown that the single linkage and the complete linkage hierarchical clustering algorithms induce a metric on the data known as the ultrametric. Through the use of the Lance and Williams recurrence formula, Johnson's proof is extended to four other common clustering algorithms. It is also noted that two additional methods produce hierarchical structures which can violate the ultrametric inequality.     

Phase congruency: A low-level image invariant

 Phase congruency is a low-level invariant property of image features. Interest in low-level image invariants has been limited. This is surprising, considering the fundamental importance of being able to obtain reliable results from low-level image operations in order to successfully perform any higher level operations. However, an impediment to the use of phase congruency to detect features has been its sensitivity to noise. This paper extends the theory behind the calculation of phase congruency in a number of ways. An effective method of noise compensation is presented that only assumes that the noise power spectrum is approximately constant. Problems with the localization of features are addressed by introducing a new, more sensitive measure of phase congruency. The existing theory that has been developed for 1D signals is extended to allow the calculation of phase congruency in 2D images. Finally, it is argued that high-pass filtering should be used to obtain image information at different scales. With this approach, the choice of scale only affects the relative significance of features without degrading their localization. Received: 20 November 1998 / Accepted: 29 September 1999     

Multidimensional scale types and invariant multivariate statistics

This paper proposes an extension of the definition of one‐dimensional ordinal, interval, ratio, difference and absolute scales to the multidimensional case. The connection between measurement structures and statistical structures suggested in a previous paper is also extended to the multivariate case, and examples of invariance properties of multivariate functions and statistics are discussed.