Mathematical psychology
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Mathematical psychology is an approach to
Although psychology, as an independent subject of science, is a more recent discipline than
As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology. Mathematical psychology is therefore closely related to psychometrics. However, where psychometrics is concerned with individual differences (or population structure) in mostly static variables, mathematical psychology focuses on process models of perceptual, cognitive and motor processes as inferred from the 'average individual'. Furthermore, where psychometrics investigates the stochastic dependence structure between variables as observed in the population, mathematical psychology almost exclusively focuses on the modeling of data obtained from experimental paradigms and is therefore even more closely related to experimental psychology, cognitive psychology, and psychonomics. Like computational neuroscience and econometrics, mathematical psychology theory often uses statistical optimality as a guiding principle, assuming that the human brain has evolved to solve problems in an optimized way. Central themes from cognitive psychology (e.g., limited vs. unlimited processing capacity, serial vs. parallel processing) and their implications are central in rigorous analysis in mathematical psychology.
Mathematical psychologists are active in many fields of psychology, especially in psychophysics,
History
Mathematics and psychology before the 19th century
Choice and decision making theory are rooted in the development of probability theory. In the mid 1600s, Blaise Pascal considered situations in gambling and further extended to Pascal's wager.[4] In the 18th century, Nicolas Bernoulli proposed the St. Petersburg Paradox in decision making, Daniel Bernoulli gave a solution and Laplace proposed a modification to the solution later on. In 1763, Bayes published the paper "An Essay towards solving a Problem in the Doctrine of Chances", which is the milestone of Bayesian statistics.
Robert Hooke worked on modeling human memory, which is a precursor of the study of memory.
Mathematics and psychology in the 19th century
The research developments in Germany and England in the 19th century made psychology a new academic subject. Since the German approach emphasized experiments in the investigation of the psychological processes that all humans share and the English approach was in the measurement of individual differences, the applications of mathematics were also different.
In Germany, Wilhelm Wundt established the first experimental psychology laboratory. The math in German psychology is mainly applied in sensory and psychophysics. Ernst Weber (1795–1878) created the first mathematical law of the mind, Weber's law, based on a variety of experiments. Gustav Fechner (1801–1887) contributed theories in sensations and perceptions and one of them is the Fechner's law, which modifies Weber's law.
Mathematical modeling has a long history in psychology starting in the 19th century with
Researchers in astronomy in the 19th century were mapping distances between stars by denoting the exact time of a star's passing of a cross-hair on a telescope. For lack of the automatic registration instruments of the modern era, these time measurements relied entirely on human response speed. It had been noted that there were small systematic differences in the times measured by different astronomers, and these were first systematically studied by German astronomer Friedrich Bessel (1782–1846). Bessel constructed personal equations from measurements of basic response speed that would cancel out individual differences from the astronomical calculations. Independently, physicist Hermann von Helmholtz measured reaction times to determine nerve conduction speed, developed resonance theory of hearing and the Young-Helmholtz theory of color vision.
These two lines of work came together in the research of Dutch physiologist F. C. Donders and his student J. J. de Jaager, who recognized the potential of reaction times for more or less objectively quantifying the amount of time elementary mental operations required. Donders envisioned the employment of his mental chronometry to scientifically infer the elements of complex cognitive activity by measurement of simple reaction time[5]
Although there are developments in sensation and perception, Johann Herbart developed a system of mathematical theories in cognitive area to understand the mental process of consciousness.
The origin of English psychology can be traced to the theory of evolution by Darwin. But the emergence of English psychology is because of Francis Galton, who interested in individual differences between humans on psychological variables. The math in English psychology is mainly statistics and the work and methods of Galton is the foundation of psychometrics.
Galton introduced bivariate normal distribution in modeling the traits of the same individual, he also investigated measurement error and built his own model, and he also developed a stochastic branching process to examine the extinction of family names. There is also a tradition of the interest in studying intelligence in English psychology started from Galton. James McKeen Cattell and Alfred Binet developed tests of intelligence.
The first psychological laboratory was established in Germany by Wilhelm Wundt, who amply used Donders' ideas. However, findings that came from the laboratory were hard to replicate and this was soon attributed to the method of introspection that Wundt introduced. Some of the problems resulted from individual differences in response speed found by astronomers. Although Wundt did not seem to take interest in these individual variations and kept his focus on the study of the general human mind, Wundt's U.S. student James McKeen Cattell was fascinated by these differences and started to work on them during his stay in England.
The failure of Wundt's method of introspection led to the rise of different schools of thought. Wundt's laboratory was directed towards conscious human experience, in line with the work of Fechner and Weber on the intensity of stimuli. In the United Kingdom, under the influence of the anthropometric developments led by Francis Galton, interest focussed on individual differences between humans on psychological variables, in line with the work of Bessel. Cattell soon adopted the methods of Galton and helped laying the foundation of psychometrics.
20th century
Many statistical methods were developed even before the 20th century: Charles Spearman invented factor analysis which studies individual differences by the variance and covariance. German psychology and English psychology have been combined and taken over by the United States. The statistical methods dominated the field during the beginning of the century. There are two important statistical developments: Structural Equation Modeling (SEM) and analysis of variance (ANOVA). Since factor analysis unable to make causal inferences, the method of structural equation modeling was developed by Sewall Wright to correlational data to infer causality, which is still a major research area today. Those statistical methods formed psychometrics. The Psychometric Society was established in 1935 and the journal Psychometrika was published since 1936.
In the United States,
During the war, developments in
Two seminal papers on learning theory in Psychological Review helped to establish the field in a world that was still dominated by behaviorists: A paper by Bush and Mosteller instigated the linear operator approach to learning,[7] and a paper by Estes that started the stimulus sampling tradition in psychological theorizing.[8] These two papers presented the first detailed formal accounts of data from learning experiments.
Mathematical modeling of learning process were greatly developed in the 1950s as the behavioral learning theory was flourishing. One development is the stimulus sampling theory by Williams K. Estes, the other is linear operator models by Robert R. Bush, and Frederick Mosteller.
Signal processing and detection theory are broadly used in perception, psychophysics and nonsensory area of cognition. Von Neumann's book The Theory of Games and Economic Behavior establish the importance of game theory and decision making. R. Duncan Luce and Howard Raiffa contributed to the choice and decision making area.
The area of language and thinking comes into the spotlight with the development of computer science and linguistics, especially information theory and computation theory. Chomsky proposed the model of linguistics and computational hierarchy theory. Allen Newell and Herbert Simon proposed the model of human solving problems. The development in artificial intelligence and human computer interface are active areas in both computer science and psychology.
Before the 1950s, psychometricians emphasized the structure of measurement error and the development of high-power statistical methods to the measurement of psychological quantities but little of the psychometric work concerned the structure of the psychological quantities being measured or the cognitive factors behind the response data. Scott and Suppes studied relationship between the structure of data and the structure of numerical systems that represent the data.[9] Coombs constructed formal cognitive models of the respondent in a measurement situation rather than statistical data processing algorithms, for example the unfolding model.[10][11] Another breakthrough is the development of a new form of the psychophysical scaling function along with new methods of collecting psychophysical data, like Stevens' power law.[12]
The 1950s saw a surge in mathematical theories of psychological processes, including
To better define the field of mathematical psychology, the mathematical models of the 1950s were brought together in sequence of volumes edited by Luce, Bush, and Galanter: Two readings
Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling gravitated towards computational mechanisms and devices. Examples of the latter constitute so called
Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are
Influential mathematical psychologists
- John Anderson
- Richard C. Atkinson
- William H. Batchelder
- Jerome Busemeyer
- Hans Colonius
- C. H. Coombs
- Robyn Dawes
- Adele Diederich
- Ehtibar Dzhafarov
- William Kaye Estes
- Jean-Claude Falmagne
- B. F. Green
- Daniel Kahneman
- Roger E. Kirk
- D. H. Krantz
- D. R. J. Laming
- Michael D. Lee
- Philip Marcus Levy
- R. Duncan Luce
- David Marr
- James L. McClelland
- Jeff Miller
- Jay Myung
- Louis Narens
- Allen Newell
- Robert M. Nosofsky
- Roger Ratcliff
- David E. Rumelhart
- Herbert A. Simon
- Roger Shepard
- Richard Shiffrin
- Philip L. Smith
- Stanley S. Stevens
- George Sperling
- Saul Sternberg
- Patrick Suppes
- John A. Swets
- Joshua Tenenbaum
- James T. Townsend
- Louis L. Thurstone
- Amos Tversky
- Rolf Ulrich
- Dirk Vorberg
- Eric-Jan Wagenmakers
- Elke U. Weber
- Thomas D. Wickens
Important theories and models[16]
Sensation, perception, and psychophysics
- Stevens' power law
- Weber–Fechner law
Stimulus detection and discrimination
- Signal detection theory
Stimulus identification
- Accumulator models
- Diffusion models
- Neural network/connectionist models
- Race models
- Random walk models
- Renewal models
Simple decision
- Cascade model
- Level and change race model
- Recruitment model
- SPRT
- Decision field theory
Memory scanning, visual search
- Push-down stack
- Serial exhaustive search (SES) model
Error response times
- Fast guess model
Sequential effects
- Linear operator model
Learning
- Linear operator model
- Stochastic learning theory
Measurement theory
Developmental psychology
Developmental psychology is concerned not only with describing the characteristics of psychological change over time but also seeks to explain the principles and internal workings underlying these changes. Psychologists have attempted to better understand these factors by using
Mathematical modeling is useful in developmental psychology for implementing theory in a precise and easy-to-study manner, allowing generation, explanation, integration, and prediction of diverse phenomena. Several modeling techniques are applied to development:
Journals and organizations
Central journals are the Journal of Mathematical Psychology and the British Journal of Mathematical and Statistical Psychology. There are three annual conferences in the field, the annual meeting of the Society for Mathematical Psychology in the U.S, the annual European Mathematical Psychology Group meeting in Europe, and the Australasian Mathematical Psychology conference.
See also
- Computational cognition
- Mathematical models of social learning
- Outline of psychology
- Psychological statistics
- Quantitative psychology
References
- ISBN 978-0-08-097087-5.
- ^ S2CID 63723309.
- ISBN 978-0-08-043076-8, retrieved 2019-11-23
- S2CID 216213892, retrieved 2019-11-24
- ^ ISBN 0-13-391764-9.
- ^ ISBN 1-55798-654-1.
- PMID 14883244.
- doi:10.1037/h0058559.
- S2CID 20138712.
- PMID 15417683.
- ^ "PsycNET". psycnet.apa.org. Retrieved 2019-12-09.
- PMID 13441853.
- ^ a b Estes, W. K. (2002). History of the Society
- ^ Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). Readings in mathematical psychology. Volumes I & II. New York: Wiley.
- ^ Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). Handbook of mathematical psychology. Volumes I-III. New York: Wiley. Volume II from Internet Archive
- ISBN 0-19-503642-5.
- ISBN 978-1138951655.
- S2CID 33567516.
- PMID 24911782.
External links
- British Journal of Mathematical and Statistical Psychology
- European Mathematical Psychology Group
- Journal of Mathematical Psychology
- Online tutorials on Mathematical Psychology from the Open Distance Learning initiative of the University of Bonn.
- Society for Mathematical Psychology