Mathematical psychology
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'''Mathematical psychology''' is an approach to [[psychology|psychological]] research that is based on [[mathematical modeling]] of perceptual, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. In practice "quantifiable behavior" is often constituted by "task performance".
As quantification of behavior is fundamental in this endeavor, the [[measurement|theory of measurement]] is a central topic in mathematical psychology.
<!--Such mathematical modeling allows to derive more exact hypotheses and, therefore, stricter empirical validations.-->
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]]/[[psychonomics]]. <!--The models used are mostly probabilistic but there are also discrete ones.--> 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; limited vs. unlimited processing capacity, serial vs. parallel processing, etc., and their implications, are central in rigorous analysis in mathematical psychology. <!--There are many subfields including [[measurement|theory of measurement]].-->
Mathematical psychologists are active in many fields of psychology, especially in [[psychophysics]], [[sensation]] and [[perception]], [[problem solving]], [[decision theory|decision-making]], [[learning]], [[memory]], and [[psycholinguistics|language]], collectively known as [[cognitive psychology]], <!-- and the [[Society for Quantitative Analysis of Behavior|quantitative analysis of behavior]]--> but also, e.g., in [[clinical psychology]], [[social psychology]], and [[music psychology|psychology of music]].
<!--Some mathematical models include but are not limited to the [[matching law]], [[detection_theory|signal detection theory]], [[game theory]], [[behavioral economics]]]], etc.-->
<!--gallery>
Image:Ernst Heinrich Weber.jpg|thumb|Ernst Heinrich Weber, pioneer in the mathematical approach to the study of behavior.
Image:Gustav Fechner.jpg|Gustav Fechner, pioneer in the mathematical approach to the study of behavior.
</gallery -->
==History==
[[Image:Ernst Heinrich Weber.jpg|left|thumb|150px|Ernst Heinrich Weber.]]
[[Image:Gustav Fechner.jpg|left|thumb|150px|Gustav Fechner.]]
Mathematical modeling has a long tradition in psychology. [[Ernst Heinrich Weber|Heinrich Weber]] (1795–1878) and [[Fechner|Gustav Fechner]] (1801–1887) were among the first to apply successful mathematical technique of [[functional equations]] from physics to psychological processes, thereby establishing the fields of [[experimental psychology]] in general, and within that [[psychophysics]] in particular. During that time, in [[astronomy]] researchers 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'' constructed 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. These two lines of work came together in the research of Dutch physiologist [[Donders|F. C. Donders]] and his student [[Johan Jacob de Jaager|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''<ref name="Leahey1987">Leahey, T. H. (1987). ''A History of Psychology''. Englewood Cliffs, NJ: Prentice Hall.</ref>
The first psychological laboratory was established in Germany by [[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. Part of the problems was due to the 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 American 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. Catell soon adopted the methods of Galton and helped laying the foundation of [[psychometrics]]. In the United States, behaviorism arose in opposition to introspectionism and associated reaction time research, and turned the focus of psychological research entirely to learning theory.<ref name="Leahey1987">Leahey, T. H. (1987). ''A History of Psychology''. Englewood Cliffs, NJ: Prentice Hall.</ref> Behaviorism dominated American psychology until then end of the [[Second World War]]. In Europe introspection survived in [[Gestalt psychology]]. Behaviorism largely refrained from inference on mental processes, and formal theories were mostly absent (except for [[visual perception|vision]] and [[audition]]. During the war, developments in [[engineering]], [[mathematical logic]] and [[computability theory]], [[computer science]] and [[mathematics]], and the military need to understand [[human factors|human performance and limitations]], brought together experimental psychologist, mathematicians, engineers, physicists, and economists. Out of this mix of different disciplines mathematical psychology arose. Especially the developments in [[signal processing]], [[information theory]], [[linear systems]] and [[filter theory]], [[game theory]], [[stochastic processes]] and [[mathematical logic]] gained a large influence on psychological thinking.<ref name="Leahey1987">Leahey, T. H. (1987). ''A History of Psychology''. Englewood Cliffs, NJ: Prentice Hall.</ref><ref name="Batchelder2002">Batchelder, W. H. (2002). Mathematical Psychology. In A. E. Kazdin (Ed.), ''Encyclopedia of Psychology'', Washington/New York: APA/Oxford University Press.</ref>
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<ref>Bush, R. R. & Mosteller, F. (1951). A mathematical model for simple learning. ''Psychological Review'', ''58'':313-323.</ref> instigated the linear operator approach to learning, and a paper by Estes<ref>Estes, W. K. (1950). Towards a statistical theory of learning. ''Psychological Review'', ''57'':94-107.</ref> that started the stimulus sampling tradition in psychological theorizing. These two papers presented the first detailed formal accounts of data from learning experiments.
The 1950s saw a surge in mathematical theories of psychological processes, including [[Luce's choice axiom|Luce's theory of choice]], Tanner and Swets' introduction of [[Signal detection theory]] for human stimulus detection, and Miller's approach to information processing.<ref name="Batchelder2002">Batchelder, W. H. (2002). Mathematical Psychology. In A. E. Kazdin (Ed.), ''Encyclopedia of Psychology'', Washington/NY: APA/Oxford University Press.</ref> By the end of the 1950s, the number of mathematical psychologists had increased from a hand full by more than a tenfold, not counting psychometricians. Most of these were concentrated at the University of Indiana, Michigan, Pennsylvania, and Stanford.<ref name="Estes2002">Estes, W. K. (2002). History of the Society [http://www.cogs.indiana.edu/socmathpsych/history.html]</ref><ref name="Batchelder2002"> Batchelder, W. H. (2002). Mathematical Psychology. In A. E. Kazdin (Ed.), ''Encyclopedia of Psychology''. Washington/NY: APA/Oxford University Press.</ref> Some of these were regularly invited by the U.S. Social Science Research Counsel to teach in summer workshops in mathematics for social scientists at Stanford University, promoting collaboration.
To better define 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<ref>Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). ''Readings in mathematical psychology. Volumes I & II.'' New York: Wiley.</ref> and three handbooks<ref>Luce, R. D., Bush, R. R. & Galanter, E. (Eds.) (1963). Handbook of mathematical psychology. Volumes I-III''. New York: Wiley. [http://www.archive.org/download/handbookofmathem017893mbp/handbookofmathem017893mbp.pdf| Volume II from Internet Archive]</ref>. This series of volumes turned out to be helpful in the developmen of the field. In the summer of 1963 the need was felt for a journal for theoretical and mathematical studies in all areas in psychology, excluding work that was mainly factor analytical. An initiative led by [[Richard C. Atkinson|R. C. Atkinson]], [[R. R. Bush]], [[W. K. Estes]], [[R. Duncan Luce|R. D. Luce]], and [[P. Suppes]] resulted in the appearance of the first issue of the ''Journal of Mathematical Psychology'' in January, 1964.<ref name="Estes2002">Estes, W. K. (2002). ''History of the Society'' [http://www.cogs.indiana.edu/socmathpsych/history.html]</ref>
Under the influence of developments in computer science, logic, and language theory, in the 1960s modeling became more in terms of computational mechanisms and devices. Examples of the latter constitute so called [[cognitive architectures]] (e.g., [[production rule system]]s, [[ACT-R]]) as well [[connectionist]] systems or [[neural networks]].
Important mathematical expressions for relations between physical characteristics of stimuli and subjective perception are [[Weber-Fechner law|Weber's law]] (which is now sometimes called Weber-Fechner Law), [[Ekman's Law]], [[Stevens' Power Law]], Thurstone's [[Law of Comparative Judgment]], the [[Signal detection theory|Theory of Signal Detection]] (borrowed from radar engineering), the [[Matching Law]], and [[Rescorla-Wagner model|Rescorla-Wagner rule]] for classical conditioning. While the first three laws are all deterministic in nature, later established relations are more fundamentally stochastic. This has been a general theme in the evolution in mathematical modeling of psychological processes: From deterministic relations as found in classical physics to inherently stochastic models.
==Influential mathematical psychologists==
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* [[Richard C. Atkinson]]
* [[Clyde_Coombs| C. H. Coombs]]
* [[Robyn Dawes]]
* [[William Estes]]
* [[B. F. Green]]
* [[Daniel Kahneman]]
* [[D. H. Krantz]]
<!--* [[Richard Herrnstein]] --- discussion: it is doubtful whether this person can be called 'influential' -->
* [[D. R. J. Laming]]
* [[R. Duncan Luce]]
* [[David Marr]]
* [[James McClelland|James L. McClelland]]
* [[Saul Sternberg]]
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* [[Louis Narens]]
* [[Allen Newell]]
<!-- * [[Howard Rachlin]] discussion: ---- it is doubtful whether this person can be called 'influential' or even 'mathematical psychologist' for that matter -->
* [[David E. Rumelhart]]
* [[Herbert Simon|Herbert A. Simon]]
* [[Stanley Smith Stevens|Stanley S. Stevens]]
* [[P. Suppes|Patrick Suppes]]
* [[John A. Swets]]
* [[Thurstone| Louis L. Thurstone]]
* [[Amos Tversky]]
* [[Thomas D. Wickens]]
{{column}}
{{columns-end}}
==Important theories and models<ref>Luce, R. D. (1986). ''Response Times (Their Role in Inferring Elementary Mental Organization)''. New York: Oxford University Press.</ref>==
===Sensation, Perception, and Psychophysics===
* [[Stevens' Power Law]]
* [[Weber-Fechner Law]]
===Simple detection===
* [[Signal detection theory|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]]
===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
==Journals and organizations==
Central journals are the ''[[Journal of Mathematical Psychology]]'' and the ''[[British Journal of Mathematical and Statistical Psychology]]''. There are two annual conferences in the field, the annual meeting of the ''Society for Mathematical Psychology'' in the U.S, and the annual ''European Mathematical Psychology Group'' (EMPG) meeting.
==External links==
* [http://www.bps.org.uk/publications/journals/bjmsp/bjmsp_home.cfm British Journal of Mathematical and Statistical Psychology]
* [http://sma.uni.lu/empg38/ European Mathematical Psychology Group]
* [http://www.elsevier.com/wps/find/journaldescription.cws_home/622887/description#description Journal of Mathematical Psychology]
<!--* [http://www.elsevier.com/wps/find/journaldescription.cws_home/505565/description#description Mathematical Social Science]-->
* [http://www.mathpsyc.uni-bonn.de/tutorials.htm Online tutorials on Mathematical Psychology] from the Open Distance Learning initiative of the University of Bonn.
* [http://www.cogs.indiana.edu/socmathpsych/ Society for Mathematical Psychology]
==References==
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