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A generalized regression neural network (GRNN) is often used for function approximation. It has a radial basis layer and a special linear layer.

The architecture for the GRNN is shown below. It is similar to the radial basis network, but has a slightly different second layer.

Here the **nprod** box shown above
(code function `normprod`

) produces *S*^{2} elements
in vector **n**^{2}.
Each element is the dot product of a row of LW^{2,1 }and
the input vector **a**^{1},
all normalized by the sum of the elements of **a**^{1}.
For instance, suppose that

LW{2,1}= [1 -2;3 4;5 6]; a{1} = [0.7;0.3];

Then

aout = normprod(LW{2,1},a{1}) aout = 0.1000 3.3000 5.3000

The first layer is just like that for `newrbe`

networks.
It has as many neurons as there are input/ target vectors in **P**. Specifically, the first-layer weights are
set to **P**`'`

. The
bias **b**^{1} is
set to a column vector of 0.8326/`SPREAD`

. The user
chooses `SPREAD`

, the distance an input vector must
be from a neuron's weight vector to be 0.5.

Again, the first layer operates just like the `newrbe`

radial basis layer described previously.
Each neuron's weighted input is the distance between the input vector
and its weight vector, calculated with `dist`

.
Each neuron's net input is the product of its weighted input with
its bias, calculated with `netprod`

.
Each neuron's output is its net input passed through `radbas`

. If a neuron's weight vector is
equal to the input vector (transposed), its weighted input will be
0, its net input will be 0, and its output will be 1. If a neuron's
weight vector is a distance of `spread`

from the
input vector, its weighted input will be `spread`

,
and its net input will be sqrt(−log(.5)) (or 0.8326). Therefore
its output will be 0.5.

The second layer also has as many neurons as input/target vectors,
but here `LW{2,1}`

is set to `T`

.

Suppose you have an input vector **p** close
to **p*** _{i}*,
one of the input vectors among the input vector/target pairs used
in designing layer 1 weights. This input

A larger `spread`

leads to a large area around
the input vector where layer 1 neurons will respond with significant
outputs. Therefore if `spread`

is small the radial
basis function is very steep, so that the neuron with the weight vector
closest to the input will have a much larger output than other neurons.
The network tends to respond with the target vector associated with
the nearest design input vector.

As `spread`

becomes larger the radial basis
function's slope becomes smoother and several neurons can respond
to an input vector. The network then acts as if it is taking a weighted
average between target vectors whose design input vectors are closest
to the new input vector. As `spread`

becomes larger
more and more neurons contribute to the average, with the result that
the network function becomes smoother.

You can use the function `newgrnn`

to
create a GRNN. For instance, suppose that three input and three target
vectors are defined as

P = [4 5 6]; T = [1.5 3.6 6.7];

You can now obtain a GRNN with

net = newgrnn(P,T);

and simulate it with

P = 4.5; v = sim(net,P);

You might want to try `demogrn1`

. It shows
how to approximate a function with a GRNN.

Function | Description |
---|---|

Competitive transfer function. | |

Euclidean distance weight function. | |

Dot product weight function. | |

Convert indices to vectors. | |

Negative Euclidean distance weight function. | |

Product net input function. | |

Design a generalized regression neural network. | |

Design a probabilistic neural network. | |

Design a radial basis network. | |

Design an exact radial basis network. | |

Normalized dot product weight function. | |

Radial basis transfer function. | |

Convert vectors to indices. |