Much of our understanding of human thinking is based on probabilistic models. This innovative book by Jerome R. Busemeyer and Peter D. Bruza argues that, actually, the underlying mathematical structures from quantum theory provide a much better account of human thinking than traditional models. They introduce the foundations for modelling probabilistic-dynamic systems using two aspects of quantum theory. The first, 'contextuality', is a way to understand interference effects found with inferences and decisions under conditions of uncertainty. The second, 'quantum entanglement', allows cognitive phenomena to be modeled in non-reductionist ways. Employing these principles drawn from quantum theory allows us to view human cognition and decision in a totally new light. Introducing the basic principles in an easy-to-follow way, this book does not assume a physics background or a quantum brain and comes complete with a tutorial and fully worked-out applications in important areas of cognition and decision.

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Quantum Models of Cognition and Decision

Metaheuristic design of feedforward neural networks

In recent years the Probabilistic Neural Network (PPN) has been used in a large number of applications due to its simplicity and efficiency. PNN assigns the test data to the class with maximum likelihood compared with other classes. Likelihood of the test data to each training data is computed in the pattern layer through a kernel density estimation using a simple Bayesian rule. The kernel is usually a standard probability distribution function such as a Gaussian function. A spread parameter is used as a global parameter which determines the width of the kernel. The Bayesian rule in the pattern layer estimates the conditional probability of each class given an input vector without considering any probable local densities or heterogeneity in the training data. In this paper, an enhanced and generalized PNN (EPNN) is presented using local decision circles (LDCs) to overcome the aforementioned shortcoming and improve its robustness to noise in the data. Local decision circles enable EPNN to incorporate local information and non-homogeneity existing in the training population. The circle has a radius which limits the contribution of the local decision. In the conventional PNN the spread parameter can be optimized for maximum classification accuracy. In the proposed EPNN two parameters, the spread parameter and the radius of local decision circles, are optimized to maximize the performance of the model. Accuracy and robustness of EPNN are compared with PNN using three different benchmark classification problems, iris data, diabetic data, and breast cancer data, and five different ratios of training data to testing data: 90:10, 80:20, 70:30, 60:40, and 50:50. EPNN provided the most accurate results consistently for all ratios. Robustness of PNN and EPNN is investigated using different values of signal to noise ratio (SNR). Accuracy of EPNN is consistently higher than accuracy of PNN at different levels of SNR and for all ratios of training data to testing data.

Enhanced probabilistic neural network with local decision circles: A robust classifier

http://audition.ens.fr/dp/pdfs/pressnitzer-2006-bistable_AV_Authors_Manuscript.pdf

Temporal dynamics of auditory and visual bistability reveal common principles of perceptual organization

What are quantum fluctuations

The aim of this paper is to investigate storing and recalling performances of embedded patterns on associative memory. The associative memory is composed of quaternionic multistate Hopfield neural network. The state of a neuron in the network is described by three kinds of discretized phase with fixed amplitude. These phases are set to discrete values with arbitrary divide size. Hebbian rule and projection rule are used for storing patterns to the network. Recalling performance is evaluated through storing random patterns with changing the divide size of the phases in a neuron. Color images are also embedded and their noise tolerance is explored.

Quaternionic multistate Hopfield neural network with extended projection rule

This paper explores two types of multistate Hopfield neural networks, based on commutative quaternions that are similar to Hamilton's quaternions but with commutative multiplication. In one type of the networks, the state of a neuron is represented by two kinds of phases and one real number. The other type of the networks adopts the decomposed form of commutative quaternion, i.e., the state of a neuron consists of a combination of two complex values. We have investigated the stabilities of these networks, i.e., the energies monotonically decreases with respect to the changes of the network states.

Commutative quaternion and multistate Hopfield neural networks

Feed forward neural network with random quaternionic neurons

On the Scheme of Quaternionic Multistate Hopfield Neural Network

Associative memory by Hopfield-type recurrent neural networks with quaternionic algebra, called quaternionic Hopfield neural network, is proposed in this paper. The variables in the network are represented by quaternions of four dimensional hypercomplex numbers. The neuron model, the energy function, and the Hebbian rule for embedding patterns into the network are introduced. The properties of this network are analyzed concretely through examples of the network with 3 and 4 quaternion neurons. It is demonstrated that there exist fixed attractors in the network, i.e., the pattern association from test pattern close to a stored pattern is possible in the quaternionic network, as in real-valued Hopfield networks.

Nobuyuki Matsui

Papers

Quaternionic multistate Hopfield neural network with extended projection rule

Commutative quaternion and multistate Hopfield neural networks

Feed forward neural network with random quaternionic neurons

On the Scheme of Quaternionic Multistate Hopfield Neural Network

Fundamental Properties of Quaternionic Hopfield Neural Network