On Herrnstein's equation and related forms

In 1970, Herrnstein proposed a simple equation to describe the relation between response and reinforcement rates on interval schedules. Its empirical basis is firm, but its theoretical foundation is still uncertain. Two approaches to the derivation of Herrnstein's equation are discussed. It can be derived as the equilibrium solution to a process model equivalent to familiar linear-operator learning models. Modifications of this approach yield competing power-function formulations. The equation can also be derived from the assumption that response strength is proportional to reinforcement rate, given that there is a ceiling on response rate. The proportional relation can, in turn, be derived from a threshold assumption equivalent to Shimp's “momentary maximizing”. This derivation implies that the two parameters of Herrnstein's equation should be correlated, and may explain its special utility in application to internal schedules.     

On rotating to smooth functions

Tucker has outlined an application of principal components analysis to a set of learning curves, for the purpose of identifying meaningful dimensions of individual differences in learning tasks. Since the principal components are defined in terms of a statistical criterion (maximum variance accounted for) rather than a substantive one, it is typically desirable to rotate the components to a more interpretable orientation. Simple structure is not a particularly appealing consideration for such a rotation; it is more reasonable to believe that any meaningful factor should form a (locally) smooth curve when the component loadings are plotted against trial number. Accordingly, this paper develops a procedure for transforming an arbitrary set of component reference curves to a new set which are mutually orthogonal and, subject to orthogonality, are as smooth as possible in a well defined (least squares) sense. Potential applications to learning data, electrophysiological responses, and growth data are indicated.Portions of this research were supported by the National Research Council of Canada, Grant A8615 to the second author. We thank Jagdeth Sheth for supplying his raw data.     

Latent roots of random data correlation matrices with squared multiple correlations on the diagonal: A monte carlo study

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are log (N–1) and log {[n(n–1)/2]–[(i–1)n]}, whereN is the number of observations;n, the number of variables; andi, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.This research was supported by the Office of Naval Research under Contract N00014-67-A-0305-0012, Lloyd G. Humphreys, principal investigator, and by the Department of Computer Science of which Richard G. Montanelli, Jr., is a member.     

Comments on “the measurement of factorial indeterminacy”

Guttman's index of indeterminacy (2² – 1) measures the potential amount of uncertainty in picking the right alternative interpretation for a factor. When alternative solutions for a factor are equally likely to be correct, then the squared multiple correlation ² for predicting the factor from the observed variables is the average correlation _AB between independently selected alternative solutionsA andB, while var ( _AB )=(1 – ²)²/s, wheres is the dimensionality of the space in which unpredicted components of alternative solutions are to be found. When alternative solutions for the factor are not equally likely to be chosen, ² is the lower bound for E( _AB ); however, E( _AB ) need not be a modal value in the distribution of _AB . Guttman's index and E( _AB ) measure different aspects of the same indeterminacy problem.     

Transformation of a three-mode multidimensional scaling solution to indscal form

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.     

A look,by simulation,at the validity of some asymptotic distribution results for rotated loadings

In the last few years, a number of asymptotic results for the distribution of unrotated and rotated factor loadings have been given. This paper investigates the validity of some of these results based on simulation techniques. In particular, it looks at principal component extraction and quartimax rotation on a problem with 13 variables. The indication is that the asymptotic results are quite good.     

农村选民选举行为的结构特征

以云南省两个县八个自然村的 2 62名选民为被试 ,采用心理测量原理与技术探讨了农村选民选举行为的结构特征。结果表明 ,政见取向由“政策效应”、“民生意识”、“村民权益”和“监督机制”4个因素构成 ,非政见取向由“候选人身份”、“候选人德能”、“选民社会关系”、“候选人背景”、“选民自我利益”、“选民自我权利”和“政党意见”7个因素构成。经检验 ,由此构成的农村选举行为问卷具有较高的内部一致性 ,问卷项目具有较高的区分度。文章讨论了农村选民选举行为的结构特征及其意义。     

Speaker identification using multi-modal i-vector approach for varying length speech in voice interactive systems

The development in the interface of smart devices has lead to voice interactive systems. An additional step in this direction is to enable the devices to recognize the speaker. But this is a challenging task because the interaction involves short duration speech utterances. The traditional Gaussian mixture models (GMM) based systems have achieved satisfactory results for speaker recognition only when the speech lengths are sufficiently long. The current state-of-the-art method utilizes i-vector based approach using a GMM based universal background model (GMM-UBM). It prepares an i-vector speaker model from a speaker’s enrollment data and uses it to recognize any new test speech. In this work, we propose a multi-model i-vector system for short speech lengths. We use an open database THUYG-20 for the analysis and development of short speech speaker verification and identification system. By using an optimum set of mel-frequency cepstrum coefficients (MFCC) based features we are able to achieve an equal error rate (EER) of 3.21% as compared to the previous benchmark score of EER 4.01% on the THUYG-20 database. Experiments are conducted for speech lengths as short as 0.25 s and the results are presented. The proposed method shows improvement as compared to the current i-vector based approach for shorter speech lengths. We are able to achieve improvement of around 28% even for 0.25 s speech samples. We also prepared and tested the proposed approach on our own database with 2500 speech recordings in English language consisting of actual short speech commands used in any voice interactive system.