Artificial neural networks (ANNs), usually simply called neural networks (NNs), are

Learning representations by back-propagating errors

" ''Nature', 323, pages 533–536 1986. In 1992, max-pooling was introduced to help with least-shift invariance and tolerance to deformation to aid 3D object recognition.J. Weng, N. Ahuja and T. S. Huang,

Cresceptron: a self-organizing neural network which grows adaptively

" ''Proc. International Joint Conference on Neural Networks'', Baltimore, Maryland, vol I, pp. 576–581, June, 1992.J. Weng, N. Ahuja and T. S. Huang,

Learning recognition and segmentation of 3-D objects from 2-D images

" ''Proc. 4th International Conf. Computer Vision'', Berlin, Germany, pp. 121–128, May, 1993.J. Weng, N. Ahuja and T. S. Huang,

Learning recognition and segmentation using the Cresceptron

" ''International Journal of Computer Vision'', vol. 25, no. 2, pp. 105–139, Nov. 1997. adopted a multi-level hierarchy of networks (1992) pre-trained one level at a time by

Learning complex, extended sequences using the principle of history compression

" ''Neural Computation'', 4, pp. 234–242, 1992.

Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition

" ''In 20th International Conference Artificial Neural Networks (ICANN)'', pp. 92–101, 2010. . Between 2009 and 2012, ANNs began winning prizes in ANN contests, approaching human level performance on various tasks, initially in

A learning algorithm of CMAC based on RLS

" Neural Processing Letters 19.1 (2004): 49–61.Ting Qin, et al.

Continuous CMAC-QRLS and its systolic array

" Neural Processing Letters 22.1 (2005): 1–16.

Online

/ref> long short-term memory avoid the vanishing gradient problem and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition, text-to-speech synthesis, and photo-real talking heads; competitive networks such as generative adversarial networks Reinforcement learning, in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game or on deceiving the opponent about the authenticity of an input.

Distributed hierarchical processing in the primate cerebral cortex

" ''Cerebral Cortex'', 1, pp. 1–47, 1991. displaying a wide variety of invariance. WengJ. Weng,

Natural and Artificial Intelligence: Introduction to Computational Brain-Mind

" BMI Press, , 2012. argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

File:Single layer ann.svg, A single-layer feedforward artificial neural network. Arrows originating from $\backslash scriptstyle\; x\_2$ are omitted for clarity. There are p inputs to this network and q outputs. In this system, the value of the qth output, $\backslash scriptstyle\; y\_q$ would be calculated as $\backslash scriptstyle\; y\_q\; =\; K*(\backslash sum(x\_i*w\_)-b\_q)$
File:Two layer ann.svg, A two-layer feedforward artificial neural network.
File:Artificial neural network.svg, An artificial neural network.
File:Ann dependency (graph).svg, An ANN dependency graph.
File:Single-layer feedforward artificial neural network.png, A single-layer feedforward artificial neural network with 4 inputs, 6 hidden and 2 outputs. Given position state and direction outputs wheel based control values.
File:Two-layer feedforward artificial neural network.png, A two-layer feedforward artificial neural network with 8 inputs, 2x8 hidden and 2 outputs. Given position state, direction and other environment values outputs thruster based control values.
File:Cmac.jpg, Parallel pipeline structure of CMAC neural network. This learning algorithm can converge in one step.

PDF

* * * * **created for National Science Foundation, Contract Number EET-8716324, and Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976 under Contract F33615-87-C-1499. * * * * * * * * * * * * *

The Neural Network Zoo

– a compilation of neural network types

The Stilwell Brain

– a Mind Field episode featuring an experiment in which humans act as individual neurons in a neural network that classifies handwritten digits {{DEFAULTSORT:Artificial Neural Network Computational statistics Artificial neural networks, Classification algorithms Computational neuroscience Market research Mathematical psychology Mathematical and quantitative methods (economics)

computing system
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d. ...

s vaguely inspired by the biological neural network
A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Neural circuits interconnect to one another to form large scale brain networks. Biological neural networks have inspired the ...

s that constitute animal brain
A brain is an organ
Organ may refer to:
Biology
* Organ (anatomy)
An organ is a group of Tissue (biology), tissues with similar functions. Plant life and animal life rely on many organs that co-exist in organ systems.
A given organ's tiss ...

s.
An ANN is based on a collection of connected units or nodes called artificial neuron
An artificial neuron is a Function (mathematics), mathematical function conceived as a Mathematical model, model of biological neurons, a neural network. Artificial neurons are elementary units in an artificial neural network. The artificial neuron ...

s, which loosely model the neuron
A neuron or nerve cell is an electrically excitable cell
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Closed spaces
* Monastic cell, a small room, hut, or cave in which a monk or re ...

s in a biological brain. Each connection, like the synapse
In the nervous system
In biology
Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiol ...

s in a biological brain, can transmit a signal to other neurons. An artificial neuron that receives a signal then processes it and can signal neurons connected to it. The "signal" at a connection is a real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

, and the output of each neuron is computed by some non-linear function of the sum of its inputs. The connections are called ''edges''. Neurons and edges typically have a ''weight
In science
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (procedural knowledge ...

'' that adjusts as learning proceeds. The weight increases or decreases the strength of the signal at a connection. Neurons may have a threshold such that a signal is sent only if the aggregate signal crosses that threshold. Typically, neurons are aggregated into layers. Different layers may perform different transformations on their inputs. Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.
Training

Neural networks learn (or are trained) by processing examples, each of which contains a known "input" and "result," forming probability-weighted associations between the two, which are stored within the data structure of the net itself. The training of a neural network from a given example is usually conducted by determining the difference between the processed output of the network (often a prediction) and a target output. This is the error. The network then adjusts its weighted associations according to a learning rule and using this error value. Successive adjustments will cause the neural network to produce output which is increasingly similar to the target output. After a sufficient number of these adjustments the training can be terminated based upon certain criteria. This is known assupervised learning
Supervised learning (SL) is the machine learning task of learning a function that Map (mathematics), maps an input to an output based on example input-output pairs. It infers a function from ' consisting of a set of ''training examples''. In supe ...

.
Such systems "learn" to perform tasks by considering examples, generally without being programmed with task-specific rules. For example, in image recognition, they might learn to identify images that contain cats by analyzing example images that have been manually labeled as "cat" or "no cat" and using the results to identify cats in other images. They do this without any prior knowledge of cats, for example, that they have fur, tails, whiskers, and cat-like faces. Instead, they automatically generate identifying characteristics from the examples that they process.
History

Warren McCulloch
Warren Sturgis McCulloch (November 16, 1898 – September 24, 1969) was an American neurophysiologist and cybernetician, known for his work on the foundation for certain brain theories and his contribution to the cybernetics movement.Ken Aizawa (20 ...

and Walter Pitts
Walter Harry Pitts, Jr. (23 April 1923 – 14 May 1969) was a logician
Logic (from Greek: grc, λογική, label=none, lit=possessed of reason
Reason is the capacity of consciously making sense of things, applying logic
Lo ...

(1943) opened the subject by creating a computational model for neural networks. In the late 1940s, D. O. Hebb created a learning hypothesis based on the mechanism of neural plasticity
Neuroplasticity, also known as neural plasticity, or brain plasticity, is the ability of neural networks#REDIRECT Artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs), are computing syste ...

that became known as Hebbian learning
Hebbian theory is a neuroscientific theory claiming that an increase in synaptic efficacy arises from a presynaptic cell
Chemical synapses are biological junctions through which neuron
A neuron or nerve cell is an electrically excitable ...

. Farley and Wesley A. Clark (1954) first used computational machines, then called "calculators", to simulate a Hebbian network. Rosenblatt (1958) created the perceptron
In machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model ...

. The first functional networks with many layers were published by Ivakhnenko and Lapa in 1965, as the Group Method of Data HandlingGroup method of data handling (GMDH) is a family of inductive algorithms for computer-based mathematical modeling of multi-parametric datasets that features fully automatic structural and parametric optimization of models.
GMDH is used in such field ...

. The basics of continuous backpropagation were derived in the context of control theory
Control theory deals with the control of dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in a Manifold, geometrical space. ...

by Kelley in 1960 and by Bryson in 1961, using principles of dynamic programming
Dynamic programming is both a mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lo ...

.
In 1970, Seppo Linnainmaa
Seppo Ilmari Linnainmaa (born 28 September 1945) is a Finnish
Finnish may refer to:
* Something or someone from, or related to Finland
* Finnish culture
* Finnish people or Finns, the primary ethnic group in Finland
* Finnish language, the national ...

published the general method for automatic differentiationIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

(AD) of discrete connected networks of nested differentiable
In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

functions. In 1973, Dreyfus used backpropagation to adapt parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

s of controllers in proportion to error gradients. Werbos's (1975) backpropagation
In machine learning, backpropagation (backprop, BP) is a widely used algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm ...

algorithm enabled practical training of multi-layer networks. In 1982, he applied Linnainmaa's AD method to neural networks in the way that became widely used. Thereafter research stagnated following and (1969), who discovered that basic perceptrons were incapable of processing the exclusive-or circuit and that computers lacked sufficient power to process useful neural networks.
The development of metal–oxide–semiconductor
The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of insulated-gate field-effect transistor
The field-effect tran ...

(MOS) very-large-scale integration
Very large-scale integration (VLSI) is the process of creating an integrated circuit
An integrated circuit or monolithic integrated circuit (also referred to as an IC, a chip, or a microchip) is a set of electronic circuits on one small f ...

(VLSI), in the form of complementary MOS (CMOS) technology, enabled increasing MOS transistor count
upright=1.4, gate
File:Kebun Raya Bali Candi Bentar IMG 8794.jpg, Candi bentar, a typical Indonesian gate that is often found on the islands of Java and Bali
A gate or gateway is a point of entry to or from a space enclosed by walls. The w ...

s in digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons
The electron is a subatomic particle
In physica ...

. This provided more processing power for the development of practical artificial neural networks in the 1980s.
In 1986 Rumelhart, Hinton and Williams
Williams may refer to:
People
* Williams (surname), a surname English in origin, but popular in Wales, 3rd most common in the United Kingdom
Places Astronomy
* Williams (lunar crater)
* Williams (Martian crater)
Australia
*Williams, Western A ...

showed that backpropagation learned interesting internal representations of words as feature vectors when trained to predict the next word in a sequence.David E. Rumelhart, Geoffrey E. Hinton & Ronald J. Williams ,Learning representations by back-propagating errors

" ''Nature', 323, pages 533–536 1986. In 1992, max-pooling was introduced to help with least-shift invariance and tolerance to deformation to aid 3D object recognition.J. Weng, N. Ahuja and T. S. Huang,

Cresceptron: a self-organizing neural network which grows adaptively

" ''Proc. International Joint Conference on Neural Networks'', Baltimore, Maryland, vol I, pp. 576–581, June, 1992.J. Weng, N. Ahuja and T. S. Huang,

Learning recognition and segmentation of 3-D objects from 2-D images

" ''Proc. 4th International Conf. Computer Vision'', Berlin, Germany, pp. 121–128, May, 1993.J. Weng, N. Ahuja and T. S. Huang,

Learning recognition and segmentation using the Cresceptron

" ''International Journal of Computer Vision'', vol. 25, no. 2, pp. 105–139, Nov. 1997. adopted a multi-level hierarchy of networks (1992) pre-trained one level at a time by

unsupervised learning
Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is the primary way young children learn the machine is forced to build a compact internal representation of its world a ...

and fine-tuned by backpropagation
In machine learning, backpropagation (backprop, BP) is a widely used algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm ...

.J. Schmidhuber.,Learning complex, extended sequences using the principle of history compression

" ''Neural Computation'', 4, pp. 234–242, 1992.

Geoffrey Hinton
Geoffrey Everest Hinton One or more of the preceding sentences incorporates text from the royalsociety.org website where: (born 6 December 1947) is a British-Canadian
Canadians (french: Canadiens) are people identified with the country o ...

et al. (2006) proposed learning a high-level representation using successive layers of binary or real-valued latent variableIn statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a ...

s with a restricted Boltzmann machine
A restricted Boltzmann machine (RBM) is a generative stochastic
Stochastic () refers to the property of being well described by a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in even ...

to model each layer. In 2012, and Dean
Dean may refer to:
People
* Dean (given name)
* Dean (surname), a surname of Anglo-Saxon English origin
* Dean (South Korean singer), a stage name for singer Kwon Hyuk
* Dean Delannoit, a Belgian singer most known by the mononym Dean
Title
...

created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images. Unsupervised pre-training and increased computing power from GPU
A graphics processing unit (GPU) is a specialized electronic circuit
File:PExdcr01CJC.jpg, 200px, A circuit built on a printed circuit board (PCB).
An electronic circuit is composed of individual electronic components, such as resistors, transis ...

s and distributed computing
Distributed computing is a field of computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Comp ...

allowed the use of larger networks, particularly in image and visual recognition problems, which became known as "deep learning #REDIRECT Deep learning#REDIRECT Deep learning
Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be ...

".
Ciresan and colleagues (2010) showed that despite the vanishing gradient problem, GPUs make backpropagation feasible for many-layered feedforward neural networks.Dominik Scherer, Andreas C. Müller, and Sven Behnke:Evaluation of Pooling Operations in Convolutional Architectures for Object Recognition

" ''In 20th International Conference Artificial Neural Networks (ICANN)'', pp. 92–101, 2010. . Between 2009 and 2012, ANNs began winning prizes in ANN contests, approaching human level performance on various tasks, initially in

pattern recognition
Pattern recognition is the automated recognition of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...

and machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data ...

. For example, the bi-directional and multi-dimensional long short-term memory
Long short-term memory (LSTM) is an artificial recurrent neural network
A recurrent neural network (RNN) is a class of artificial neural networks where connections between nodes form a directed graph along a temporal sequence. This allows it to ...

(LSTM) of Graves
A grave is a location where a cadaver, dead body (typically that of a human, although sometimes that of an animal) is burial, buried. Graves are usually located in special areas set aside for the purpose of burial, such as graveyards or cemet ...

et al. won three competitions in connected handwriting recognition in 2009 without any prior knowledge about the three languages to be learned.
Ciresan and colleagues built the first pattern recognizers to achieve human-competitive/superhuman performance on benchmarks such as traffic sign recognition (IJCNN 2012).
Models

ANNs began as an attempt to exploit the architecture of the human brain to perform tasks that conventional algorithms had little success with. They soon reoriented towards improving empirical results, mostly abandoning attempts to remain true to their biological precursors. Neurons are connected to each other in various patterns, to allow the output of some neurons to become the input of others. The network forms adirected
Director may refer to:
Literature
* Director (magazine), ''Director'' (magazine), a British magazine
* The Director (novel), ''The Director'' (novel), a 1971 novel by Henry Denker
* The Director (play), ''The Director'' (play), a 2000 play by Nanc ...

, weighted graph
This is a glossary of graph theory. Graph theory is the study of graph (discrete mathematics), graphs, systems of nodes or vertex (graph theory), vertices connected in pairs by lines or #edge, edges.
Symbols
A
...

.
An artificial neural network consists of a collection of simulated neurons. Each neuron is a node
In general, a node is a localized swelling (a "knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splic ...

which is connected to other nodes via links
Link or Links may refer to:
Places
* Link, West Virginia, an unincorporated community in the US
* Link River, Klamath Falls, Oregon, US
People with the name
* Link (singer) (Lincoln Browder, born 1964), American R&B singer
* Link (surname)
* ...

that correspond to biological axon-synapse-dendrite connections. Each link has a weight, which determines the strength of one node's influence on another.
Components of ANNs

Neurons

ANNs are composed ofartificial neurons
An artificial neuron is a mathematical function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

which are conceptually derived from biological neuron
A neuron or nerve cell is an electrically excitable cell
Cell most often refers to:
* Cell (biology), the functional basic unit of life
Cell may also refer to:
Closed spaces
* Monastic cell, a small room, hut, or cave in which a monk or re ...

s. Each artificial neuron has inputs and produces a single output which can be sent to multiple other neurons. The inputs can be the feature values of a sample of external data, such as images or documents, or they can be the outputs of other neurons. The outputs of the final ''output neurons'' of the neural net accomplish the task, such as recognizing an object in an image.
To find the output of the neuron, first we take the weighted sum of all the inputs, weighted by the ''weights'' of the ''connections'' from the inputs to the neuron. We add a ''bias'' term to this sum. This weighted sum is sometimes called the ''activation''. This weighted sum is then passed through a (usually nonlinear) activation function
In artificial neural network
Artificial neural networks (ANNs), usually simply called neural networks (NNs), are computing systems vaguely inspired by the biological neural networks that constitute animal brain
A brain is an organ (ana ...

to produce the output. The initial inputs are external data, such as images and documents. The ultimate outputs accomplish the task, such as recognizing an object in an image.
Connections and weights

The network consists of connections, each connection providing the output of one neuron as an input to another neuron. Each connection is assigned a weight that represents its relative importance. A given neuron can have multiple input and output connections.Propagation function

The ''propagation function'' computes the input to a neuron from the outputs of its predecessor neurons and their connections as a weighted sum. A ''bias'' term can be added to the result of the propagation.Organization

The neurons are typically organized into multiple layers, especially indeep learning #REDIRECT Deep learning#REDIRECT Deep learning
Deep learning (also known as deep structured learning) is part of a broader family of machine learning methods based on artificial neural networks with representation learning. Learning can be ...

. Neurons of one layer connect only to neurons of the immediately preceding and immediately following layers. The layer that receives external data is the ''input layer''. The layer that produces the ultimate result is the ''output layer''. In between them are zero or more ''hidden layers''. Single layer and unlayered networks are also used. Between two layers, multiple connection patterns are possible. They can be ''fully connected'', with every neuron in one layer connecting to every neuron in the next layer. They can be ''pooling'', where a group of neurons in one layer connect to a single neuron in the next layer, thereby reducing the number of neurons in that layer. Neurons with only such connections form a directed acyclic graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and are known as ''feedforward networks''. Alternatively, networks that allow connections between neurons in the same or previous layers are known as ''recurrent networks''''.''
Hyperparameter

A hyperparameter is a constantparameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whol ...

whose value is set before the learning process begins. The values of parameters are derived via learning. Examples of hyperparameters include learning rate
In machine learning and statistics
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conv ...

, the number of hidden layers and batch size. The values of some hyperparameters can be dependent on those of other hyperparameters. For example, the size of some layers can depend on the overall number of layers.
Learning

Learning is the adaptation of the network to better handle a task by considering sample observations. Learning involves adjusting the weights (and optional thresholds) of the network to improve the accuracy of the result. This is done by minimizing the observed errors. Learning is complete when examining additional observations does not usefully reduce the error rate. Even after learning, the error rate typically does not reach 0. If after learning, the error rate is too high, the network typically must be redesigned. Practically this is done by defining a cost function that is evaluated periodically during learning. As long as its output continues to decline, learning continues. The cost is frequently defined as astatistic
A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population
Population typically refers the number of peop ...

whose value can only be approximated. The outputs are actually numbers, so when the error is low, the difference between the output (almost certainly a cat) and the correct answer (cat) is small. Learning attempts to reduce the total of the differences across the observations. Most learning models can be viewed as a straightforward application of optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
Mathematical optimization (alter ...

theory and statistical estimation
Estimation theory is a branch of statistics that deals with estimating the values of Statistical parameter, parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such ...

.
Learning rate

The learning rate defines the size of the corrective steps that the model takes to adjust for errors in each observation. A high learning rate shortens the training time, but with lower ultimate accuracy, while a lower learning rate takes longer, but with the potential for greater accuracy. Optimizations such as Quickprop are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid oscillation inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an adaptive learning rate that increases or decreases as appropriate. The concept of momentum allows the balance between the gradient and the previous change to be weighted such that the weight adjustment depends to some degree on the previous change. A momentum close to 0 emphasizes the gradient, while a value close to 1 emphasizes the last change.Cost function

While it is possible to define a cost functionad hoc
Ad hoc is a Latin phrase
__NOTOC__
This is a list of Wikipedia articles of Latin phrases and their translation into English.
To view all phrases on a single, lengthy document, see:
* List of Latin phrases (full)
The list also is divided alpha ...

, frequently the choice is determined by the function's desirable properties (such as ) or because it arises from the model (e.g. in a probabilistic model the model's posterior probability
In Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian probability, Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an Event (probability theory), e ...

can be used as an inverse cost).
Backpropagation

Backpropagation is a method used to adjust the connection weights to compensate for each error found during learning. The error amount is effectively divided among the connections. Technically, backprop calculates thegradient
In vector calculus
Vector calculus, or vector analysis, is concerned with differentiation
Differentiation may refer to:
Business
* Differentiation (economics), the process of making a product different from other similar products
* Prod ...

(the derivative) of the cost function associated with a given state with respect to the weights. The weight updates can be done via stochastic gradient descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions ...

or other methods, such as Extreme Learning Machines, "No-prop" networks, training without backtracking, "weightless" networks, and non-connectionist neural networks.
Learning paradigms

The three major learning paradigms aresupervised learning
Supervised learning (SL) is the machine learning task of learning a function that Map (mathematics), maps an input to an output based on example input-output pairs. It infers a function from ' consisting of a set of ''training examples''. In supe ...

, unsupervised learning
Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is the primary way young children learn the machine is forced to build a compact internal representation of its world a ...

and reinforcement learning
Reinforcement learning (RL) is an area of machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. M ...

. They each correspond to a particular learning task
Supervised learning

Supervised learning
Supervised learning (SL) is the machine learning task of learning a function that Map (mathematics), maps an input to an output based on example input-output pairs. It infers a function from ' consisting of a set of ''training examples''. In supe ...

uses a set of paired inputs and desired outputs. The learning task is to produce the desired output for each input. In this case the cost function is related to eliminating incorrect deductions. A commonly used cost is the mean-squared error
In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the expected value, average of the squares of the Error (statistics), errors—that is, the ...

, which tries to minimize the average squared error between the network's output and the desired output. Tasks suited for supervised learning are pattern recognition
Pattern recognition is the automated recognition of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...

(also known as classification) and regression (also known as function approximation). Supervised learning is also applicable to sequential data (e.g., for hand writing, speech and gesture recognition
A gesture is a form of non-verbal communication or non-vocal communication
Communication (from Latin ''communicare'', meaning "to share") is the act of developing Semantics, meaning among Subject (philosophy), entities or Organization, gro ...

). This can be thought of as learning with a "teacher", in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.
Unsupervised learning

Inunsupervised learning
Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is the primary way young children learn the machine is forced to build a compact internal representation of its world a ...

, input data is given along with the cost function, some function of the data $\backslash textstyle\; x$ and the network's output. The cost function is dependent on the task (the model domain) and any ''a priori
''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...

'' assumptions (the implicit properties of the model, its parameters and the observed variables). As a trivial example, consider the model $\backslash textstyle\; f(x)\; =\; a$ where $\backslash textstyle\; a$ is a constant and the cost $\backslash textstyle\; C=E$x - f(x))^2
X, or x, is the twenty-fourth and third-to-last letter in the modern English alphabet and the ISO basic Latin alphabet
The ISO basic Latin alphabet is a Latin-script alphabet
A Latin-script alphabet (Latin alphabet or Roman alphabet) ...

/math>. Minimizing this cost produces a value of $\backslash textstyle\; a$ that is equal to the mean of the data. The cost function can be much more complicated. Its form depends on the application: for example, in compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
*Gas compression
*Compression ratio, of a co ...

it could be related to the mutual information
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual Statistical dependence, dependence between the two variables. More specifically, it quantifies the "Information content ...

between $\backslash textstyle\; x$ and $\backslash textstyle\; f(x)$, whereas in statistical modeling, it could be related to the posterior probability
In Bayesian statistics
Bayesian statistics is a theory in the field of statistics based on the Bayesian probability, Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an Event (probability theory), e ...

of the model given the data (note that in both of those examples those quantities would be maximized rather than minimized). Tasks that fall within the paradigm of unsupervised learning are in general estimation
Estimation (or estimating) is the process of finding an estimate, or approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived ...

problems; the applications include clustering, the estimation of statistical distributions, compression
Compression may refer to:
Physical science
*Compression (physics), size reduction due to forces
*Compression member, a structural element such as a column
*Compressibility, susceptibility to compression
*Gas compression
*Compression ratio, of a co ...

and filtering
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

.
Reinforcement learning

In applications such as playing video games, an actor takes a string of actions, receiving a generally unpredictable response from the environment after each one. The goal is to win the game, i.e., generate the most positive (lowest cost) responses. Inreinforcement learning
Reinforcement learning (RL) is an area of machine learning
Machine learning (ML) is the study of computer algorithms that can improve automatically through experience and by the use of data. It is seen as a part of artificial intelligence. M ...

, the aim is to weight the network (devise a policy) to perform actions that minimize long-term (expected cumulative) cost. At each point in time the agent performs an action and the environment generates an observation and an instantaneous cost, according to some (usually unknown) rules. The rules and the long-term cost usually only can be estimated. At any juncture, the agent decides whether to explore new actions to uncover their costs or to exploit prior learning to proceed more quickly.
Formally the environment is modeled as a (MDP) with states $\backslash textstyle\; \backslash in\; S$ and actions $\backslash textstyle\; \backslash in\; A$. Because the state transitions are not known, probability distributions are used instead: the instantaneous cost distribution $\backslash textstyle\; P(c\_t,\; s\_t)$, the observation distribution $\backslash textstyle\; P(x\_t,\; s\_t)$ and the transition distribution $\backslash textstyle\; P(s\_,\; s\_t,\; a\_t)$, while a policy is defined as the conditional distribution over actions given the observations. Taken together, the two define a Markov chain
A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. A countably infinite sequence, in which the chain moves state at discr ...

(MC). The aim is to discover the lowest-cost MC.
ANNs serve as the learning component in such applications. Dynamic programming
Dynamic programming is both a mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lo ...

coupled with ANNs (giving neurodynamic programming) has been applied to problems such as those involved in vehicle routing, video games, natural resource management
Natural resource management (NRM) is the management of natural resource
Natural resources are resource
Resource refers to all the materials available in our environment which help us to satisfy our needs and wants. Resources can broadl ...

and medicine
Medicine is the science
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge), skills (proced ...

because of ANNs ability to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of control problems. Tasks that fall within the paradigm of reinforcement learning are control problems, game
A game is a structured form of play
Play most commonly refers to:
* Play (activity), an activity done for enjoyment
* Play (theatre), a work of drama
Play may refer also to:
Computers and technology
* Google Play, a digital content serv ...

s and other sequential decision making tasks.
Self learning

Self learning in neural networks was introduced in 1982 along with a neural network capable of self-learning named Crossbar Adaptive Array (CAA). It is a system with only one input, situation s, and only one output, action (or behavior) a. It has neither external advice input nor external reinforcement input from the environment. The CAA computes, in a crossbar fashion, both decisions about actions and emotions (feelings) about encountered situations. The system is driven by the interaction between cognition and emotion. Given memory matrix W =, , w(a,s), , , the crossbar self learning algorithm in each iteration performs the following computation: In situation s perform action a; Receive consequence situation s'; Compute emotion of being in consequence situation v(s'); Update crossbar memory w'(a,s) = w(a,s) + v(s'). The backpropagated value (secondary reinforcement) is the emotion toward the consequence situation. The CAA exists in two environments, one is behavioral environment where it behaves, and the other is genetic environment, where from it initially and only once receives initial emotions about to be encountered situations in the behavioral environment. Having received the genome vector (species vector) from the genetic environment, the CAA will learn a goal-seeking behavior, in the behavioral environment that contains both desirable and undesirable situations.Other

In aBayesian
Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister.
Bayesian () refers to a range of concepts and approaches that are ultimately based on a degree-of-belief interpretation of probability, ...

framework, a distribution over the set of allowed models is chosen to minimize the cost. Evolutionary methods, gene expression programming, simulated annealing, expectation-maximization, non-parametric methods and particle swarm optimization are other learning algorithms. Convergent recursion is a learning algorithm for cerebellar model articulation controller (CMAC) neural networks.Ting Qin, et al.A learning algorithm of CMAC based on RLS

" Neural Processing Letters 19.1 (2004): 49–61.Ting Qin, et al.

Continuous CMAC-QRLS and its systolic array

" Neural Processing Letters 22.1 (2005): 1–16.

Modes

Two modes of learning are available: stochastic gradient descent, stochastic and batch. In stochastic learning, each input creates a weight adjustment. In batch learning weights are adjusted based on a batch of inputs, accumulating errors over the batch. Stochastic learning introduces "noise" into the process, using the local gradient calculated from one data point; this reduces the chance of the network getting stuck in local minima. However, batch learning typically yields a faster, more stable descent to a local minimum, since each update is performed in the direction of the batch's average error. A common compromise is to use "mini-batches", small batches with samples in each batch selected stochastically from the entire data set.Types

ANNs have evolved into a broad family of techniques that have advanced the state of the art across multiple domains. The simplest types have one or more static components, including number of units, number of layers, unit weights and topology. Dynamic types allow one or more of these to evolve via learning. The latter are much more complicated, but can shorten learning periods and produce better results. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. Some of the main breakthroughs include: convolutional neural networks that have proven particularly successful in processing visual and other two-dimensional data;LeCun ''et al.'', "Backpropagation Applied to Handwritten Zip Code Recognition," ''Neural Computation'', 1, pp. 541–551, 1989.Yann LeCun (2016). Slides on Deep LearninOnline

/ref> long short-term memory avoid the vanishing gradient problem and can handle signals that have a mix of low and high frequency components aiding large-vocabulary speech recognition, text-to-speech synthesis, and photo-real talking heads; competitive networks such as generative adversarial networks Reinforcement learning, in which multiple networks (of varying structure) compete with each other, on tasks such as winning a game or on deceiving the opponent about the authenticity of an input.

Network design

Neural architecture search (NAS) uses machine learning to automate ANN design. Various approaches to NAS have designed networks that compare well with hand-designed systems. The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network. Available systems include Automated machine learning, AutoML and AutoKeras. Design issues include deciding the number, type and connectedness of network layers, as well as the size of each and the connection type (full, pooling, ...). Hyperparameters must also be defined as part of the design (they are not learned), governing matters such as how many neurons are in each layer, learning rate, step, stride, depth, receptive field and padding (for CNNs), etc.Use

Using Artificial neural networks requires an understanding of their characteristics. * Choice of model: This depends on the data representation and the application. Overly complex models slow learning. * Learning algorithm: Numerous trade-offs exist between learning algorithms. Almost any algorithm will work well with the correct hyperparameters for training on a particular data set. However, selecting and tuning an algorithm for training on unseen data requires significant experimentation. * Robustness: If the model, cost function and learning algorithm are selected appropriately, the resulting ANN can become robust. ANN capabilities fall within the following broad categories: * Function approximation, or regression analysis, including Time series#Prediction and forecasting, time series prediction, fitness approximation and modeling. * Statistical classification, Classification, including Pattern recognition, pattern and sequence recognition, novelty detection and sequential decision making. * Data processing, including filtering, clustering, blind source separation and compression. * Robotics, including directing manipulators and prosthesis, prostheses.Applications

Because of their ability to reproduce and model nonlinear processes, Artificial neural networks have found applications in many disciplines. Application areas include system identification and control (vehicle control, trajectory prediction, process control,natural resource management
Natural resource management (NRM) is the management of natural resource
Natural resources are resource
Resource refers to all the materials available in our environment which help us to satisfy our needs and wants. Resources can broadl ...

), quantum chemistry, general game playing, pattern recognition
Pattern recognition is the automated recognition of pattern
A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...

(radar systems, Facial recognition system, face identification, signal classification, 3D reconstruction, object recognition and more), sequence recognition (gesture, speech, handwriting recognition, handwritten and printed text recognition), medical diagnosis, finance (e.g. algorithmic trading, automated trading systems), data mining, visualization, machine translation, social network filtering and e-mail spam filtering. ANNs have been used to diagnose several types of cancers and to distinguish highly invasive cancer cell lines from less invasive lines using only cell shape information.
ANNs have been used to accelerate reliability analysis of infrastructures subject to natural disasters and to predict foundation settlements. ANNs have also been used for building black-box models in geoscience: hydrology, ocean modelling and coastal engineering, and geomorphology. ANNs have been employed in Computer security, cybersecurity, with the objective to discriminate between legitimate activities and malicious ones. For example, machine learning has been used for classifying Android malware, for identifying domains belonging to threat actors and for detecting URLs posing a security risk. Research is underway on ANN systems designed for penetration testing, for detecting botnets, credit cards frauds and network intrusions.
ANNs have been proposed as a tool to solve partial differential equations in physics and simulate the properties of many-body open quantum systems. In brain research ANNs have studied short-term behavior of biological neuron models, individual neurons, the dynamics of neural circuitry arise from interactions between individual neurons and how behavior can arise from abstract neural modules that represent complete subsystems. Studies considered long-and short-term plasticity of neural systems and their relation to learning and memory from the individual neuron to the system level.
Theoretical properties

Computational power

The multilayer perceptron is a UTM theorem, universal function approximator, as proven by the universal approximation theorem. However, the proof is not constructive regarding the number of neurons required, the network topology, the weights and the learning parameters. A specific recurrent architecture with Rational number, rational-valued weights (as opposed to full precisionreal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

-valued weights) has the power of a Universal Turing Machine, universal Turing machine, using a finite number of neurons and standard linear connections. Further, the use of Irrational number, irrational values for weights results in a machine with Hypercomputation, super-Turing power.
Capacity

A model's "capacity" property corresponds to its ability to model any given function. It is related to the amount of information that can be stored in the network and to the notion of complexity. Two notions of capacity are known by the community. The information capacity and the VC Dimension. The information capacity of a perceptron is intensively discussed in Sir David MacKay's book which summarizes work by Thomas Cover. The capacity of a network of standard neurons (not convolutional) can be derived by four rules that derive from understanding a neuron as an ADALINE, electrical element. The information capacity captures the functions modelable by the network given any data as input. The second notion, is the VC dimension. VC Dimension uses the principles of measure theory and finds the maximum capacity under the best possible circumstances. This is, given input data in a specific form. As noted in, the VC Dimension for arbitrary inputs is half the information capacity of a Perceptron. The VC Dimension for arbitrary points is sometimes referred to as Memory Capacity.Convergence

Models may not consistently converge on a single solution, firstly because local minima may exist, depending on the cost function and the model. Secondly, the optimization method used might not guarantee to converge when it begins far from any local minimum. Thirdly, for sufficiently large data or parameters, some methods become impractical. The convergence behavior of certain types of ANN architectures are more understood than others. When the width of network approaches to infinity, the ANN is well described by its first order Taylor expansion throughout training, and so inherits the convergence behavior of Linear model, affine models. Another example is when parameters are small, it is observed that ANNs often fits target functions from low to high frequencies. This phenomenon is the opposite to the behavior of some well studied iterative numerical schemes such as Jacobi method.Generalization and statistics

Applications whose goal is to create a system that generalizes well to unseen examples, face the possibility of over-training. This arises in convoluted or over-specified systems when the network capacity significantly exceeds the needed free parameters. Two approaches address over-training. The first is to use cross-validation (statistics), cross-validation and similar techniques to check for the presence of over-training and to select hyperparameters to minimize the generalization error. The second is to use some form of ''regularization (mathematics), regularization''. This concept emerges in a probabilistic (Bayesian) framework, where regularization can be performed by selecting a larger prior probability over simpler models; but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting. Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model. The MSE on a validation set can be used as an estimate for variance. This value can then be used to calculate the confidence interval of network output, assuming a normal distribution. A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified. By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based network) for categorical target variables, the outputs can be interpreted as posterior probabilities. This is useful in classification as it gives a certainty measure on classifications. The softmax activation function is: :$y\_i=\backslash frac$Criticism

Training

A common criticism of neural networks, particularly in robotics, is that they require too much training for real-world operation. Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example, grouping examples in so-called mini-batches and/or introducing a recursive least squares algorithm for cerebellar model articulation controller, CMAC.Theory

A fundamental objection is that ANNs do not sufficiently reflect neuronal function. Backpropagation is a critical step, although no such mechanism exists in biological neural networks. How information is coded by real neurons is not known. Sensory neuron, Sensor neurons fire action potentials more frequently with sensor activation and muscle cells pull more strongly when their associated motor neurons receive action potentials more frequently. Other than the case of relaying information from a sensor neuron to a motor neuron, almost nothing of the principles of how information is handled by biological neural networks is known. A central claim of ANNs is that they embody new and powerful general principles for processing information. These principles are ill-defined. It is often claimed that they are Emergent properties, emergent from the network itself. This allows simple statistical association (the basic function of artificial neural networks) to be described as learning or recognition. Alexander Dewdney commented that, as a result, artificial neural networks have a "something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are. No human hand (or mind) intervenes; solutions are found as if by magic; and no one, it seems, has learned anything". One response to Dewdney is that neural networks handle many complex and diverse tasks, ranging from autonomously flying aircraft to detecting credit card fraud to mastering the game of Go (game), Go. Technology writer Roger Bridgman commented: Biological brains use both shallow and deep circuits as reported by brain anatomy,D. J. Felleman and D. C. Van Essen,Distributed hierarchical processing in the primate cerebral cortex

" ''Cerebral Cortex'', 1, pp. 1–47, 1991. displaying a wide variety of invariance. WengJ. Weng,

Natural and Artificial Intelligence: Introduction to Computational Brain-Mind

" BMI Press, , 2012. argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

Hardware

Large and effective neural networks require considerable computing resources. While the brain has hardware tailored to the task of processing signals through a Graph (discrete mathematics), graph of neurons, simulating even a simplified neuron on von Neumann architecture may consume vast amounts of Random-access memory, memory and storage. Furthermore, the designer often needs to transmit signals through many of these connections and their associated neurons which require enormous Central processing unit, CPU power and time. noted that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by General-purpose computing on graphics processing units, GPGPUs (on Graphics processing unit, GPUs), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before. The use of accelerators such as Field-programmable gate array, FPGAs and GPUs can reduce training times from months to days. Neuromorphic engineering addresses the hardware difficulty directly, by constructing non-von-Neumann chips to directly implement neural networks in circuitry. Another type of chip optimized for neural network processing is called a Tensor Processing Unit, or TPU.Practical counterexamples

Analyzing what has been learned by an ANN is much easier than analyzing what has been learned by a biological neural network. Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful. For example, local vs. non-local learning and shallow vs. deep architecture.Hybrid approaches

Advocates of Hybrid neural network, hybrid models (combining neural networks and symbolic approaches), claim that such a mixture can better capture the mechanisms of the human mind.Gallery

See also

* Large width limits of neural networks * Hierarchical temporal memory * 20Q * ADALINE * Adaptive resonance theory * Artificial life * Associative Memory Base, Associative memory * Autoencoder * BEAM robotics * Biological cybernetics * Biologically inspired computing * Blue Brain Project * Catastrophic interference * Cerebellar Model Articulation Controller (CMAC) * Cognitive architecture * Cognitive science * Convolutional neural network (CNN) * Connectionist expert system * Connectomics * Cultured neuronal networks * Deep learning * Differentiable programming * Encog * Fuzzy logic * Gene expression programming * Genetic algorithm * Genetic programming * Group method of data handling * Habituation * In Situ Adaptive Tabulation * List of machine learning concepts, Machine learning concepts * Models of neural computation * Neuroevolution * Neural coding * Neural gas * Neural machine translation * Neural network software * Neuroscience * Nonlinear system identification * Optical neural network * Parallel Constraint Satisfaction Processes * Parallel distributed processing * Radial basis function network * Recurrent neural networks * Self-organizing map * Spiking neural network * Systolic array * Tensor product network * Time delay neural network (TDNN)References

Bibliography

* * * ** * * * **created for National Science Foundation, Contract Number EET-8716324, and Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976 under Contract F33615-87-C-1499. * * * * * * * * * * * * *

External links

The Neural Network Zoo

– a compilation of neural network types

The Stilwell Brain

– a Mind Field episode featuring an experiment in which humans act as individual neurons in a neural network that classifies handwritten digits {{DEFAULTSORT:Artificial Neural Network Computational statistics Artificial neural networks, Classification algorithms Computational neuroscience Market research Mathematical psychology Mathematical and quantitative methods (economics)