Generalized regression neural network (GRNN) is a variation to radial basis

neural network A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

s. GRNN was suggested by D.F. Specht in 1991. GRNN can be used for

regression Regression or regressions may refer to: Science * Marine regression, coastal advance due to falling sea level, the opposite of marine transgression * Regression (medicine), a characteristic of diseases to express lighter symptoms or less extent ( ...

prediction A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exact ...

, and

classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...

. GRNN can also be a good solution for

online In computer technology and telecommunications, online indicates a state of connectivity and offline indicates a disconnected state. In modern terminology, this usually refers to an Internet connection, but (especially when expressed "on line" or ...

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 an ambient space. Examples include the mathematical models that describe the swinging of a ...

s. GRNN represents an improved technique in the neural networks based on the

nonparametric regression Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...

. The idea is that every training sample will represent a mean to a radial basis

neuron A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...

Mathematical representation

Y(x) = \frac

where: *

Y(x)

is the prediction value of input

x

y_k

is the activation weight for the pattern layer neuron at

k

K(x, x_k)

is the

Radial basis function kernel In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification. The RBF kernel on two s ...

(Gaussian kernel) as formulated below.

Gaussian Kernel In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is n ...

K(x, x_k) = e^, \qquad d_k = (x-x_k)^T(x-x_k)

where

d_k

is the squared euclidean distance between the training samples

x_k

and the input

x.

Implementation

GRNN has been implemented in many computer languages including

MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...

, R- programming language,

Python (programming language) Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming para ...

and

Node.js Node.js is an open-source server environment. Node.js is cross-platform and runs on Windows, Linux, Unix, and macOS. Node.js is a back-end JavaScript runtime environment. Node.js runs on the V8 JavaScript Engine and executes JavaScript code ou ...

. Neural networks (specifically Multi-layer Perceptron) can delineate non-linear patterns in data by combining with generalized linear models by considering distribution of outcomes (sightly different from original GRNN). There have been several successful developments, including Poisson regression, ordinal logistic regression, quantile regression and

multinomial logistic regression In statistics, multinomial logistic regression is a statistical classification, classification method that generalizes logistic regression to multiclass classification, multiclass problems, i.e. with more than two possible discrete outcomes. T ...

that described by Fallah in 2009.

Advantages and disadvantages

Similar to RBFNN, GRNN has the following advantages: * Single-pass learning so no

backpropagation In machine learning, backpropagation (backprop, BP) is a widely used algorithm for training feedforward neural network, feedforward artificial neural networks. Generalizations of backpropagation exist for other artificial neural networks (ANN ...

is required. * High accuracy in the estimation since it uses

Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real constants , and non-zero . It is n ...

s. * It can handle noises in the inputs. * It requires relatively few data to train. The main disadvantages of GRNN are: *Its size can be huge, which would make it computationally expensive. *There is no optimal method to improve it.

References

{{Reflist Neural network architectures Regression analysis