{ "id": "epmcda14q1", "slug": "general-regression-neural-network", "title": "General regression neural network", "updated": "2026-10-15", "topic_path": [ { "id": "technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/topics/technology" }, { "id": "technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing" }, { "id": "technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai" }, { "id": "technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.ml_computation" }, { "id": "technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.neural_deep" }, { "id": "technology.computing.ai.neural_deep.alpha-a-to-h", "label": "Titles A to H", "api_url": "https://www.edgepedia.org/api/v1/topics/technology.computing.ai.neural_deep.alpha-a-to-h" } ], "geo": [ { "id": "geo.nongeo.t1946.technology.computing.ai.neural_deep", "label": "Non-geographic · 1946 to 2000: Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.neural_deep", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.technology", "label": "Technology and the built world", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology" }, { "id": "geo.nongeo.t1946.technology.computing", "label": "Computing and digital systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing" }, { "id": "geo.nongeo.t1946.technology.computing.ai", "label": "Artificial intelligence and data", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai" }, { "id": "geo.nongeo.t1946.technology.computing.ai.ml_computation", "label": "Machine learning and neural computation", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.ml_computation" }, { "id": "geo.nongeo.t1946.technology.computing.ai.neural_deep", "label": "Neural networks and deep learning", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.technology.computing.ai.neural_deep" } ] } ], "excerpt": "A general regression neural network (GRNN) is a memory-based neural network for regression, introduced by Donald F. Specht in 1991 as a neural-network form of kernel regression.", "snippet": "A general regression neural network (GRNN) is a memory-based neural network for regression, introduced by Donald F. Specht in 1991 as a neural-network form of kernel regression.", "node": "technology.computing.ai.neural_deep.alpha-a-to-h", "markdown": "# General regression neural network\n\nA general regression neural network (GRNN) is a memory-based, one-pass learning algorithm for regression that estimates the conditional mean of a continuous target variable from training data, as a neural-network form of Nadaraya–Watson kernel regression (smoothing data by weighted averaging of nearby points).[1](https://doi.org/10.1109/72.97934) It is a variant of the [radial basis function network](https://www.edgepedia.org/radial-basis-function-network) in which one hidden neuron is centered on every training case, and it needs no iterative weight training: the only parameter to fit is a single smoothing width.[2](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf) • [3](https://ar5iv.labs.arxiv.org/html/1805.11236)\n\n| Key fact | Detail |\n|---|---|\n| What it estimates | The conditional mean \\( E[y \\mid X] \\), as a Gaussian-weighted average of training targets[4](https://www.mdpi.com/2072-4292/12/2/248) |\n| Introduced by | D. F. Specht, IEEE Transactions on Neural Networks, 1991[1](https://doi.org/10.1109/72.97934) |\n| Architecture | Four layers: input, pattern (one neuron per training case), summation, output[4](https://www.mdpi.com/2072-4292/12/2/248) |\n| Training | Single pass, no backpropagation; only the smoothing parameter \\( \\sigma \\) is learned[3](https://ar5iv.labs.arxiv.org/html/1805.11236) • [5](https://scialert.net/fulltext/?doi=jai.2009.56.64) |\n| Main cost | Memory and prediction time grow with the number of training samples[6](https://sciendo.com/pdf/10.2478/amcs-2018-0031) |\n| Key weakness | Curse of dimensionality; cannot ignore irrelevant inputs without modification[5](https://scialert.net/fulltext/?doi=jai.2009.56.64) |\n\n## How it works\n\nThe GRNN computes the regression of a scalar on a vector independent variable as a locally weighted average with a kernel as the weighting function; it is an adaptation of the Nadaraya–Watson estimator, which itself rests on Parzen's nonparametric density estimation.[7](https://publications.waset.org/43.pdf) • [8](https://github.com/federhub/pyGRNN) Starting from the conditional mean\n\n\\[ E[y \\mid X] = \\frac{\\int_{-\\infty}^{\\infty} y f(X,y)\\,dy}{\\int_{-\\infty}^{\\infty} f(X,y)\\,dy}, \\]\n\nwhere \\( f \\) is the joint density of input and output, and estimating \\( f \\) with Parzen Gaussian kernels centered on the training pairs gives the network's output:[4](https://www.mdpi.com/2072-4292/12/2/248)\n\n\\[ \\hat{Y} = \\frac{\\sum_{i=1}^{N} Y_i\\, e^{-D_i/2\\sigma^2}}{\\sum_{i=1}^{N} e^{-D_i/2\\sigma^2}}, \\qquad D_i = (X - X_i)^{T}(X - X_i), \\]\n\nwith \\( \\sigma \\) the smoothing parameter and \\( D_i \\) the squared [Euclidean distance](https://www.edgepedia.org/euclidean-distance) from the input to training case \\( i \\).[3](https://ar5iv.labs.arxiv.org/html/1805.11236) The weights sum to one, so the prediction is a weighted average of the training targets \\( \\hat{y} = \\sum_{i} w_i \\cdot y_i \\), where each weight is the Gaussian closeness \\( \\exp(-\\lVert x - x_i \\rVert^2 / 2\\sigma^2) \\).[2](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf) A GRNN can be viewed as a normalized RBF network with a hidden unit at every training case; the only learned weights are the kernel widths.[5](https://scialert.net/fulltext/?doi=jai.2009.56.64)\n\nThe smoothing parameter \\( \\sigma \\) (called SPREAD in MATLAB) is the distance an input must lie from a neuron's weight vector for that neuron's output to be 0.5.[9](https://au.mathworks.com/help/deeplearning/ug/generalized-regression-neural-networks.html) Its value controls the bias–variance trade-off directly. When \\( \\sigma \\) is large, all targets receive small, similar weights and the output approaches the mean of the targets; as \\( \\sigma \\to \\infty \\), \\( \\hat{Y}(X) \\) becomes the average of all \\( Y_i \\) regardless of the input.[2](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf) • [7](https://publications.waset.org/43.pdf) When \\( \\sigma \\) is small, only targets whose patterns are close to the input carry significant weight; as \\( \\sigma \\to 0 \\) the prediction equals the training target for observed cases but may produce unpredictable errors elsewhere.[2](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf) • [4](https://www.mdpi.com/2072-4292/12/2/248)\n\nIn practice \\( \\sigma \\) is chosen by cross-validation.[7](https://publications.waset.org/43.pdf) The pyGRNN package supports grid-search cross-validation and L-BFGS gradient search for per-feature bandwidths.[8](https://github.com/federhub/pyGRNN) The GRNNs R package provides a `findSpread` function for tuning.[10](https://cran.r-project.org/web/packages/GRNNs/vignettes/GRNNs.html) For time series, the tsfgrnn package selects \\( \\sigma \\) by maximizing forecast accuracy under rolling origin evaluation, which gave better accuracy than a fixed-value strategy at higher computational cost.[11](https://ftp2.uib.no/cran/web/packages/tsfgrnn/vignettes/tsfgrnn.html) • [2](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf)\n\n## How it is done\n\nThe standard topology has four layers.[4](https://www.mdpi.com/2072-4292/12/2/248) The input layer passes the vector \\( X \\) fully connected to the pattern layer, which holds one neuron per training pattern and computes \\( p_i = \\exp\\!\\left[-(X - X_i)^{T}(X - X_i)/(2\\sigma^2)\\right] \\).[4](https://www.mdpi.com/2072-4292/12/2/248) The summation layer has two units: \\( S_D = \\sum_i p_i \\) and \\( S_N = \\sum_i Y_i \\cdot p_i \\).[4](https://www.mdpi.com/2072-4292/12/2/248) The output unit divides them, \\( \\hat{Y}(X) = S_N / S_D \\).[4](https://www.mdpi.com/2072-4292/12/2/248)\n\nBuilding the network means storing the training inputs as pattern centers and the targets as output weights; in MATLAB, `newgrnn(P,T)` sets first-layer weights to the transposed inputs, first-layer bias to \\( 0.8326/\\text{SPREAD} \\), and second-layer weights to the targets.[9](https://au.mathworks.com/help/deeplearning/ug/generalized-regression-neural-networks.html) Input features must be normalized, because the network is sensitive to features with larger value ranges and \\( \\sigma \\) must match their scale.[12](https://github.com/itdxer/neupy/blob/master/neupy/algorithms/rbfn/grnn.py)\n\nGRNN uses lazy learning: training only stores the data, and prediction computes the kernel-weighted sum, so there is no iterative weight training of the kind a backpropagation multilayer perceptron requires.[12](https://github.com/itdxer/neupy/blob/master/neupy/algorithms/rbfn/grnn.py) Training is performed in one pass, which makes it the fastest algorithm in a comparative speech-recognition study, though it requires large memory because all training examples are memorized.[7](https://publications.waset.org/43.pdf) The network is consistent: the estimation error approaches zero as the training-set size grows, and it can approximate any arbitrary continuous function.[13](https://www.sciencedirect.com/science/article/abs/pii/S0925231207002160) On standard MATLAB regression datasets, a review reports shorter training time and higher accuracy than back-propagation networks.[3](https://ar5iv.labs.arxiv.org/html/1805.11236) The trade-off is prediction cost: the larger the training dataset, the slower the prediction, so the algorithm is most efficient for small datasets.[12](https://github.com/itdxer/neupy/blob/master/neupy/algorithms/rbfn/grnn.py)\n\n## Origin\n\nThe GRNN was introduced by Donald F. Specht of Lockheed Palo Alto Research Laboratory in \"A general regression neural network\", IEEE Transactions on Neural Networks 2(6), 1991.[1](https://doi.org/10.1109/72.97934) It followed his [probabilistic neural network](https://www.edgepedia.org/probabilistic-neural-network), a related memory-based network published in Neural Networks in 1990.[14](https://doi.org/10.1016/0893-6080%2890%2990049-q) The statistical underpinnings are older: Parzen's nonparametric density estimator (1962)[15](https://doi.org/10.1214/aoms/1177704472) and Nadaraya's kernel regression estimator (1964)[16](https://doi.org/10.1137/1109020) supply the mathematics the network implements. Specht also presented a clustering version of the GRNN in the same 1991 paper to reduce the pattern layer.[6](https://sciendo.com/pdf/10.2478/amcs-2018-0031)\n\n## Variants\n\nSeveral modifications address the pattern-layer cost or the single shared bandwidth. Seng, Khalid, and Yusof proposed an adaptive GRNN for modeling dynamic plants in 1999, with flexible pattern-node add-in and delete-off, dynamic initial sigma assignment, automatic target adjustment, and sigma tuning; it outperformed extended recursive least-squares identification on a linear plant in a noisy environment.[17](https://doi.org/10.1243/0959651991540142) Tomandl and Schober's modified GRNN (MGRNN, 2001) added efficient training algorithms as a data-analysis tool.[18](https://doi.org/10.1016/s0893-6080%2801%2900051-x) Rutkowski generalized the GRNN to time-varying environments in 2004.[19](https://doi.org/10.1109/tnn.2004.826127) Goulermas and colleagues introduced a multiple-bandwidth GRNN with hybrid optimization in 2007.[20](https://doi.org/10.1109/tsmcb.2007.904541) For pattern reduction, later work includes SOM-GRNN, which uses a [self-organizing map](https://www.edgepedia.org/self-organizing-map) to feed only samples from the most similar map units into the pattern layer, and a local-learning k-nearest-neighbor GRNN that reduces operation cost from \\( O(N) \\) to \\( O(k) \\).[6](https://sciendo.com/pdf/10.2478/amcs-2018-0031) The anisotropic GRNN (AGRNN) assigns a different bandwidth to each feature, acting as an embedded feature selector, whereas the isotropic version uses one bandwidth and can serve as a wrapper for feature selection.[8](https://github.com/federhub/pyGRNN) The GRNNs R package extends the network to 22 distance functions including manhattan, canberra, mahalanobis, bray, jaccard, and correlation.[10](https://cran.r-project.org/web/packages/GRNNs/vignettes/GRNNs.html) For time-series forecasting, Martínez, Charte, Frías, and Martínez-Rodríguez published the tsfgrnn package with MIMO and recursive multi-step strategies in 2021.[21](https://doi.org/10.1016/j.neucom.2021.12.028) In 2024, Kechao Xu, Bo Meng, and Zhen Wang published a GRNN-based data-driven [iterative learning control](https://www.edgepedia.org/iterative-learning-control) scheme for nonlinear non-affine discrete-time systems in Expert Systems with Applications.[22](https://doi.org/10.1016/j.eswa.2024.123339)\n\n## Applications\n\nDocumented applications span dynamic-system modeling and control (process modeling and monitoring, wind generation control, helicopter motion control, flapping-wing micro aerial vehicle control), aerodynamic force prediction, solar PV power forecasting, traffic accident prediction, and fault diagnosis and engine management.[3](https://ar5iv.labs.arxiv.org/html/1805.11236) • [13](https://www.sciencedirect.com/science/article/abs/pii/S0925231207002160) In remote sensing, a GRNN-based bidirectional polarization distribution function model reduced average RMSE by 13.4% against the best current BPDF models using a full year of POLDER data.[4](https://www.mdpi.com/2072-4292/12/2/248) In hydrology, GRNN calibration of a SWMM sub-catchment achieved \\( R^2 \\) of 0.9517 (calibration) and 0.7722 (validation) for runoff simulation.[23](https://www.mdpi.com/2073-4441/13/8/1089) In speech recognition, a GRNN system for Arabic digits reduced word error rate by 15.2% over an HMM baseline for male speakers and 9% for female speakers.[7](https://publications.waset.org/43.pdf)\n\n## Limitations and alternatives\n\nThe main drawbacks follow from the memory-based design. The number of pattern-layer neurons is proportional to the number of training samples, so large datasets cause substantial memory growth and calculation slowdown; with large datasets, data reduction by clustering or distance-based methods is essential.[6](https://sciendo.com/pdf/10.2478/amcs-2018-0031) • [3](https://ar5iv.labs.arxiv.org/html/1805.11236) Like kernel methods in general, the GRNN suffers seriously from the curse of dimensionality and cannot ignore irrelevant inputs without major modifications to the basic algorithm.[5](https://scialert.net/fulltext/?doi=jai.2009.56.64) Because it predicts a weighted average of historical values, a plain GRNN cannot forecast a trending series or values outside the range of the training data unless additive or multiplicative transformations are applied.[11](https://ftp2.uib.no/cran/web/packages/tsfgrnn/vignettes/tsfgrnn.html)\n\nAgainst alternatives, the GRNN keeps a simple, static architecture compared with an RBF network, and in wind-farm data it consistently outperformed an RBFN (MAPE 0.48%–10.53% versus a maximum of 25.6% for RBFN).[24](https://discovery.researcher.life/article/performance-comparison-of-generalized-regression-network-radial-basis-function-network-and-support-vector-regression-for-wind-power-forecasting/2ff70f0da2a43bceab34445d99e73185) In short-term [wind power forecasting](https://www.edgepedia.org/wind-power-forecasting), however, support vector regression performed better than GRNN and RBFN in MAPE except when wind power generation was very low. No published head-to-head benchmark compares the GRNN with k-nearest-neighbor regression or [Gaussian process regression](https://www.edgepedia.org/gaussian-process-regression).\n\n## References\n\n1. [D.F. Specht (1991). A general regression neural network. IEEE Transactions on Neural Networks.](https://doi.org/10.1109/72.97934)\n2. [Strategies for time series forecasting with generalized regression neural networks (Neurocomputing, 2022)](https://fcharte.com/assets/pdfs/2022-Neucom-GRNN.pdf)\n3. [Review of Applications of Generalized Regression Neural Networks in Identification and Control of Dynamic Systems (arXiv:1805.11236)](https://ar5iv.labs.arxiv.org/html/1805.11236)\n4. [Modeling Polarized Reflectance of Natural Land Surfaces Using Generalized Regression Neural Networks (Remote Sensing, 2020)](https://www.mdpi.com/2072-4292/12/2/248)\n5. [A Comparison Between Three Neural Network Models for Classification Problems](https://scialert.net/fulltext/?doi=jai.2009.56.64)\n6. [A SOM-based approach to reduce pattern layer size for GRNN (Sciendo, 2018)](https://sciendo.com/pdf/10.2478/amcs-2018-0031)\n7. [Efficient System for Speech Recognition using General Regression Neural Network (WASET)](https://publications.waset.org/43.pdf)\n8. [pyGRNN: Python implementation of General Regression Neural Network](https://github.com/federhub/pyGRNN)\n9. [Generalized Regression Neural Networks, MATLAB & Simulink documentation](https://au.mathworks.com/help/deeplearning/ug/generalized-regression-neural-networks.html)\n10. [GRNNs R package vignette (CRAN)](https://cran.r-project.org/web/packages/GRNNs/vignettes/GRNNs.html)\n11. [Time Series Forecasting with GRNN in R: the tsfgrnn Package (CRAN vignette)](https://ftp2.uib.no/cran/web/packages/tsfgrnn/vignettes/tsfgrnn.html)\n12. [neupy GRNN implementation (source code documentation)](https://github.com/itdxer/neupy/blob/master/neupy/algorithms/rbfn/grnn.py)\n13. [Hardware architecture for a general regression neural network coprocessor (Neurocomputing)](https://www.sciencedirect.com/science/article/abs/pii/S0925231207002160)\n14. [Probabilistic neural networks (Neural Networks, 1990)](https://doi.org/10.1016/0893-6080%2890%2990049-q)\n15. [Emanuel Parzen (1962). On Estimation of a Probability Density Function and Mode. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177704472)\n16. [E. A. Nadaraya (1964). On Estimating Regression. Theory of Probability and Its Applications.](https://doi.org/10.1137/1109020)\n17. [T L Seng, M Khalid, R Yusof (1999). Adaptive general regression neural network for modelling of dynamic plants. Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering.](https://doi.org/10.1243/0959651991540142)\n18. [A Modified General Regression Neural Network (MGRNN) with new, efficient training algorithms as a robust ‘black box’-tool for data analysis (Neural Networks, 2001)](https://doi.org/10.1016/s0893-6080%2801%2900051-x)\n19. [L. Rutkowski (2004). Generalized Regression Neural Networks in Time-Varying Environment. IEEE Transactions on Neural Networks.](https://doi.org/10.1109/tnn.2004.826127)\n20. [J.Y. Goulermas and colleagues (2007). Generalized Regression Neural Networks With Multiple-Bandwidth Sharing and Hybrid Optimization. IEEE Transactions on Systems Man and Cybernetics Part B (Cybernetics).](https://doi.org/10.1109/tsmcb.2007.904541)\n21. [Francisco Martínez and colleagues (2021). Strategies for time series forecasting with generalized regression neural networks. Neurocomputing.](https://doi.org/10.1016/j.neucom.2021.12.028)\n22. [Kechao Xu, Bo Meng, Zhen Wang (2024). Generalized regression neural networks-based data-driven iterative learning control for nonlinear non-affine discrete-time systems. Expert Systems with Applications.](https://doi.org/10.1016/j.eswa.2024.123339)\n23. [Using the General Regression Neural Network Method to Calibrate the Parameters of a Sub-Catchment (Water, 2021)](https://www.mdpi.com/2073-4441/13/8/1089)\n24. [Performance Comparison of Generalized Regression Network, Radial Basis Function Network and Support Vector Regression for Wind Power Forecasting](https://discovery.researcher.life/article/performance-comparison-of-generalized-regression-network-radial-basis-function-network-and-support-vector-regression-for-wind-power-forecasting/2ff70f0da2a43bceab34445d99e73185)\n\n---\n*Topic: Encyclopedia › Technology and the built world › Computing and digital systems › Artificial intelligence and data › Machine learning and neural computation › Neural networks and deep learning › Titles A to H*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/general-regression-neural-network", "markdown_url": "https://www.edgepedia.org/general-regression-neural-network.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"General regression neural network\", Edgepedia (EdgeChat), https://www.edgepedia.org/general-regression-neural-network. Edgepedia Community License 1.0.", "credit_md": "\"[General regression neural network](https://www.edgepedia.org/general-regression-neural-network)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/general-regression-neural-network](https://www.edgepedia.org/general-regression-neural-network). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"General regression neural network\", Edgepedia (EdgeChat), https://www.edgepedia.org/general-regression-neural-network. Edgepedia Community License 1.0.", "speakable": "A general regression neural network is a memory-based neural network for regression, introduced by Donald F. Specht in 1991 as a neural-network form of kernel regression." }