Package 'neuralnet'

Training of Neural Networks

Description

Training of neural networks using the backpropagation, resilient backpropagation with (Riedmiller, 1994) or without weight backtracking (Riedmiller, 1993) or the modified globally convergent version by Anastasiadis et al. (2005). The package allows flexible settings through custom-choice of error and activation function. Furthermore, the calculation of generalized weights (Intrator O & Intrator N, 1993) is implemented.

Note

This work has been supported by the German Research Foundation
(DFG: http://www.dfg.de) under grant scheme PI 345/3-1.

Author(s)

Stefan Fritsch, Frauke Guenther [email protected],

Maintainer: Frauke Guenther [email protected]

References

Riedmiller M. (1994) Rprop - Description and Implementation Details. Technical Report. University of Karlsruhe.

Riedmiller M. and Braun H. (1993) A direct adaptive method for faster backpropagation learning: The RPROP algorithm. Proceedings of the IEEE International Conference on Neural Networks (ICNN), pages 586-591. San Francisco.

Anastasiadis A. et. al. (2005) New globally convergent training scheme based on the resilient propagation algorithm. Neurocomputing 64, pages 253-270.

Intrator O. and Intrator N. (1993) Using Neural Nets for Interpretation of Nonlinear Models. Proceedings of the Statistical Computing Section, 244-249 San Francisco: American Statistical Society (eds).

See Also

plot.nn for plotting of the neural network.

gwplot for plotting of the generalized weights.

compute for computation of the calculated network.

confidence.interval for calculation of a confidence interval for the weights.

prediction for calculation of a prediction.

Examples

AND <- c(rep(0,7),1)
OR <- c(0,rep(1,7))
binary.data <- data.frame(expand.grid(c(0,1), c(0,1), c(0,1)), AND, OR)
print(net <- neuralnet(AND+OR~Var1+Var2+Var3,  binary.data, hidden=0, 
             rep=10, err.fct="ce", linear.output=FALSE))

XOR <- c(0,1,1,0)
xor.data <- data.frame(expand.grid(c(0,1), c(0,1)), XOR)
print(net.xor <- neuralnet(XOR~Var1+Var2, xor.data, hidden=2, rep=5))
plot(net.xor, rep="best")

data(infert, package="datasets")
print(net.infert <- neuralnet(case~parity+induced+spontaneous,  infert, 
                    err.fct="ce", linear.output=FALSE, likelihood=TRUE))
gwplot(net.infert, selected.covariate="parity")
gwplot(net.infert, selected.covariate="induced")
gwplot(net.infert, selected.covariate="spontaneous")
confidence.interval(net.infert)

Var1 <- runif(50, 0, 100) 
sqrt.data <- data.frame(Var1, Sqrt=sqrt(Var1))
print(net.sqrt <- neuralnet(Sqrt~Var1, sqrt.data, hidden=10, 
                  threshold=0.01))
predict(net.sqrt, data.frame(Var1 = (1:10)^2))

Var1 <- rpois(100,0.5)
Var2 <- rbinom(100,2,0.6)
Var3 <- rbinom(100,1,0.5)
SUM <- as.integer(abs(Var1+Var2+Var3+(rnorm(100))))
sum.data <- data.frame(Var1,Var2,Var3, SUM)
print(net.sum <- neuralnet(SUM~Var1+Var2+Var3, sum.data, hidden=1, 
                 act.fct="tanh"))
prediction(net.sum)

---

Deprecated function

Description

The function compute is deprecated. Please refer to the new function predict.nn.

Usage

compute(x, covariate, rep = 1)

Arguments

x	an object of class nn.
covariate	a dataframe or matrix containing the variables that had been used to train the neural network.
rep	an integer indicating the neural network's repetition which should be used.

Value

compute returns a list containing the following components:

neurons	a list of the neurons' output for each layer of the neural network.
net.result	a matrix containing the overall result of the neural network.

---

Calculates confidence intervals of the weights

Description

confidence.interval, a method for objects of class nn, typically produced by neuralnet. Calculates confidence intervals of the weights (White, 1989) and the network information criteria NIC (Murata et al. 1994). All confidence intervals are calculated under the assumption of a local identification of the given neural network. If this assumption is violated, the results will not be reasonable. Please make also sure that the chosen error function equals the negative log-likelihood function, otherwise the results are not meaningfull, too.

Usage

confidence.interval(x, alpha = 0.05)

Arguments

x	neural network
alpha	numerical. Sets the confidence level to (1-alpha).

Value

confidence.interval returns a list containing the following components:

lower.ci	a list containing the lower confidence bounds of all weights of the neural network differentiated by the repetitions.
upper.ci	a list containing the upper confidence bounds of all weights of the neural network differentiated by the repetitions.
nic	a vector containg the information criteria NIC for every repetition.

Author(s)

Stefan Fritsch, Frauke Guenther [email protected]

References

White (1989) Learning in artificial neural networks. A statistical perspective. Neural Computation (1), pages 425-464

Murata et al. (1994) Network information criterion - determining the number of hidden units for an artificial neural network model. IEEE Transactions on Neural Networks 5 (6), pages 865-871

See Also

neuralnet

Examples

data(infert, package="datasets")
    print(net.infert <- neuralnet(case~parity+induced+spontaneous,  
                        infert, err.fct="ce", linear.output=FALSE))
    confidence.interval(net.infert)

---

Plot method for generalized weights

Description

gwplot, a method for objects of class nn, typically produced by neuralnet. Plots the generalized weights (Intrator and Intrator, 1993) for one specific covariate and one response variable.

Usage

gwplot(x, rep = NULL, max = NULL, min = NULL, file = NULL,
  selected.covariate = 1, selected.response = 1, highlight = FALSE,
  type = "p", col = "black", ...)

Arguments

x	an object of class nn
rep	an integer indicating the repetition to plot. If rep="best", the repetition with the smallest error will be plotted. If not stated all repetitions will be plotted.
max	maximum of the y axis. In default, max is set to the highest y-value.
min	minimum of the y axis. In default, min is set to the smallest y-value.
file	a character string naming the plot to write to. If not stated, the plot will not be saved.
selected.covariate	either a string of the covariate's name or an integer of the ordered covariates, indicating the reference covariate in the generalized weights plot. Defaulting to the first covariate.
selected.response	either a string of the response variable's name or an integer of the ordered response variables, indicating the reference response in the generalized weights plot. Defaulting to the first response variable.
highlight	a logical value, indicating whether to highlight (red color) the best repetition (smallest error). Only reasonable if rep=NULL. Default is FALSE
type	a character indicating the type of plotting; actually any of the types as in plot.default.
col	a color of the generalized weights.
...	Arguments to be passed to methods, such as graphical parameters (see par).

Author(s)

Stefan Fritsch, Frauke Guenther [email protected]

References

Intrator O. and Intrator N. (1993) Using Neural Nets for Interpretation of Nonlinear Models. Proceedings of the Statistical Computing Section, 244-249 San Francisco: American Statistical Society (eds.)

See Also

neuralnet

Examples

data(infert, package="datasets")
print(net.infert <- neuralnet(case~parity+induced+spontaneous, infert, 
		                err.fct="ce", linear.output=FALSE, likelihood=TRUE))
gwplot(net.infert, selected.covariate="parity")
gwplot(net.infert, selected.covariate="induced")
gwplot(net.infert, selected.covariate="spontaneous")

---

Training of neural networks

Description

Train neural networks using backpropagation, resilient backpropagation (RPROP) with (Riedmiller, 1994) or without weight backtracking (Riedmiller and Braun, 1993) or the modified globally convergent version (GRPROP) by Anastasiadis et al. (2005). The function allows flexible settings through custom-choice of error and activation function. Furthermore, the calculation of generalized weights (Intrator O. and Intrator N., 1993) is implemented.

Usage

neuralnet(formula, data, hidden = 1, threshold = 0.01,
  stepmax = 1e+05, rep = 1, startweights = NULL,
  learningrate.limit = NULL, learningrate.factor = list(minus = 0.5,
  plus = 1.2), learningrate = NULL, lifesign = "none",
  lifesign.step = 1000, algorithm = "rprop+", err.fct = "sse",
  act.fct = "logistic", output.act.fct = "logistic",
  linear.output = TRUE, exclude = NULL, constant.weights = NULL,
  likelihood = FALSE)

Arguments

formula	a symbolic description of the model to be fitted.
data	a data frame containing the variables specified in formula.
hidden	a vector of integers specifying the number of hidden neurons (vertices) in each layer.
threshold	a numeric value specifying the threshold for the partial derivatives of the error function as stopping criteria.
stepmax	the maximum steps for the training of the neural network. Reaching this maximum leads to a stop of the neural network's training process.
rep	the number of repetitions for the neural network's training.
startweights	a vector containing starting values for the weights. Set to NULL for random initialization.
learningrate.limit	a vector or a list containing the lowest and highest limit for the learning rate. Used only for RPROP and GRPROP.
learningrate.factor	a vector or a list containing the multiplication factors for the upper and lower learning rate. Used only for RPROP and GRPROP.
learningrate	a numeric value specifying the learning rate used by traditional backpropagation. Used only for traditional backpropagation.
lifesign	a string specifying how much the function will print during the calculation of the neural network. 'none', 'minimal' or 'full'.
lifesign.step	an integer specifying the stepsize to print the minimal threshold in full lifesign mode.
algorithm	a string containing the algorithm type to calculate the neural network. The following types are possible: 'backprop', 'rprop+', 'rprop-', 'sag', or 'slr'. 'backprop' refers to backpropagation, 'rprop+' and 'rprop-' refer to the resilient backpropagation with and without weight backtracking, while 'sag' and 'slr' induce the usage of the modified globally convergent algorithm (grprop). See Details for more information.
err.fct	a differentiable function that is used for the calculation of the error. Alternatively, the strings 'sse' and 'ce' which stand for the sum of squared errors and the cross-entropy can be used.
act.fct	a differentiable function that is used for smoothing the result of the cross product of the covariate or neurons and the weights. Additionally the strings, 'logistic', 'tanh' and 'relu' are possible for the logistic function, tangent hyperbolicus and rectified linear unit.
output.act.fct	activation function used in the output layer (if linear.output = FALSE). Use same functions/strings as in act.fct.
linear.output	logical. If output.act.fct should not be applied to the output neurons set linear output to TRUE, otherwise to FALSE.
exclude	a vector or a matrix specifying the weights, that are excluded from the calculation. If given as a vector, the exact positions of the weights must be known. A matrix with n-rows and 3 columns will exclude n weights, where the first column stands for the layer, the second column for the input neuron and the third column for the output neuron of the weight.
constant.weights	a vector specifying the values of the weights that are excluded from the training process and treated as fix.
likelihood	logical. If the error function is equal to the negative log-likelihood function, the information criteria AIC and BIC will be calculated. Furthermore the usage of confidence.interval is meaningfull.

Details

The globally convergent algorithm is based on the resilient backpropagation without weight backtracking and additionally modifies one learning rate, either the learningrate associated with the smallest absolute gradient (sag) or the smallest learningrate (slr) itself. The learning rates in the grprop algorithm are limited to the boundaries defined in learningrate.limit.

Value

neuralnet returns an object of class nn. An object of class nn is a list containing at most the following components:

call	the matched call.
response	extracted from the data argument.
covariate	the variables extracted from the data argument.
model.list	a list containing the covariates and the response variables extracted from the formula argument.
err.fct	the error function.
act.fct	the activation function.
data	the data argument.
net.result	a list containing the overall result of the neural network for every repetition.
weights	a list containing the fitted weights of the neural network for every repetition.
generalized.weights	a list containing the generalized weights of the neural network for every repetition.
result.matrix	a matrix containing the reached threshold, needed steps, error, AIC and BIC (if computed) and weights for every repetition. Each column represents one repetition.
startweights	a list containing the startweights of the neural network for every repetition.

Author(s)

Stefan Fritsch, Frauke Guenther, Marvin N. Wright

References

Riedmiller M. (1994) Rprop - Description and Implementation Details. Technical Report. University of Karlsruhe.

Riedmiller M. and Braun H. (1993) A direct adaptive method for faster backpropagation learning: The RPROP algorithm. Proceedings of the IEEE International Conference on Neural Networks (ICNN), pages 586-591. San Francisco.

Anastasiadis A. et. al. (2005) New globally convergent training scheme based on the resilient propagation algorithm. Neurocomputing 64, pages 253-270.

Intrator O. and Intrator N. (1993) Using Neural Nets for Interpretation of Nonlinear Models. Proceedings of the Statistical Computing Section, 244-249 San Francisco: American Statistical Society (eds).

See Also

plot.nn for plotting the neural network.

gwplot for plotting the generalized weights.

predict.nn for computation of a given neural network for given covariate vectors (formerly compute).

confidence.interval for calculation of confidence intervals of the weights.

prediction for a summary of the output of the neural network.

Examples

library(neuralnet)

# Binary classification
nn <- neuralnet(Species == "setosa" ~ Petal.Length + Petal.Width, iris, linear.output = FALSE)
## Not run: print(nn)
## Not run: plot(nn)

# Multiclass classification
nn <- neuralnet(Species ~ Petal.Length + Petal.Width, iris, linear.output = FALSE)
## Not run: print(nn)
## Not run: plot(nn)

# Custom activation function
softplus <- function(x) log(1 + exp(x))
nn <- neuralnet((Species == "setosa") ~ Petal.Length + Petal.Width, iris, 
                linear.output = FALSE, hidden = c(3, 2), act.fct = softplus)
## Not run: print(nn)
## Not run: plot(nn)

---

Plot method for neural networks

Description

plot.nn, a method for the plot generic. It is designed for an inspection of the weights for objects of class nn, typically produced by neuralnet.

Usage

## S3 method for class 'nn'
plot(x, rep = NULL, x.entry = NULL, x.out = NULL,
  radius = 0.15, arrow.length = 0.2, intercept = TRUE,
  intercept.factor = 0.4, information = TRUE, information.pos = 0.1,
  col.entry.synapse = "black", col.entry = "black",
  col.hidden = "black", col.hidden.synapse = "black",
  col.out = "black", col.out.synapse = "black",
  col.intercept = "blue", fontsize = 12, dimension = 6,
  show.weights = TRUE, file = NULL, ...)

Arguments

x	an object of class nn
rep	repetition of the neural network. If rep="best", the repetition with the smallest error will be plotted. If not stated all repetitions will be plotted, each in a separate window.
x.entry	x-coordinate of the entry layer. Depends on the arrow.length in default.
x.out	x-coordinate of the output layer.
radius	radius of the neurons.
arrow.length	length of the entry and out arrows.
intercept	a logical value indicating whether to plot the intercept.
intercept.factor	x-position factor of the intercept. The closer the factor is to 0, the closer the intercept is to its left neuron.
information	a logical value indicating whether to add the error and steps to the plot.
information.pos	y-position of the information.
col.entry.synapse	color of the synapses leading to the input neurons.
col.entry	color of the input neurons.
col.hidden	color of the neurons in the hidden layer.
col.hidden.synapse	color of the weighted synapses.
col.out	color of the output neurons.
col.out.synapse	color of the synapses leading away from the output neurons.
col.intercept	color of the intercept.
fontsize	fontsize of the text.
dimension	size of the plot in inches.
show.weights	a logical value indicating whether to print the calculated weights above the synapses.
file	a character string naming the plot to write to. If not stated, the plot will not be saved.
...	arguments to be passed to methods, such as graphical parameters (see par).

Author(s)

Stefan Fritsch, Frauke Guenther [email protected]

See Also

neuralnet

Examples

XOR <- c(0,1,1,0)
xor.data <- data.frame(expand.grid(c(0,1), c(0,1)), XOR)
print(net.xor <- neuralnet( XOR~Var1+Var2, xor.data, hidden=2, rep=5))
plot(net.xor, rep="best")

---

Neural network prediction

Description

Prediction of artificial neural network of class nn, produced by neuralnet().

Usage

## S3 method for class 'nn'
predict(object, newdata, rep = 1, all.units = FALSE, ...)

Arguments

object	Neural network of class nn.
newdata	New data of class data.frame or matrix.
rep	Integer indicating the neural network's repetition which should be used.
all.units	Return output for all units instead of final output only.
...	further arguments passed to or from other methods.

Value

Matrix of predictions. Each column represents one output unit. If all.units=TRUE, a list of matrices with output for each unit.

Author(s)

Marvin N. Wright

Examples

library(neuralnet)

# Split data
train_idx <- sample(nrow(iris), 2/3 * nrow(iris))
iris_train <- iris[train_idx, ]
iris_test <- iris[-train_idx, ]

# Binary classification
nn <- neuralnet(Species == "setosa" ~ Petal.Length + Petal.Width, iris_train, linear.output = FALSE)
pred <- predict(nn, iris_test)
table(iris_test$Species == "setosa", pred[, 1] > 0.5)

# Multiclass classification
nn <- neuralnet((Species == "setosa") + (Species == "versicolor") + (Species == "virginica")
                 ~ Petal.Length + Petal.Width, iris_train, linear.output = FALSE)
pred <- predict(nn, iris_test)
table(iris_test$Species, apply(pred, 1, which.max))

---

Summarizes the output of the neural network, the data and the fitted values of glm objects (if available)

Description

prediction, a method for objects of class nn, typically produced by neuralnet. In a first step, the dataframe will be amended by a mean response, the mean of all responses corresponding to the same covariate-vector. The calculated data.error is the error function between the original response and the new mean response. In a second step, all duplicate rows will be erased to get a quick overview of the data. To obtain an overview of the results of the neural network and the glm objects, the covariate matrix will be bound to the output of the neural network and the fitted values of the glm object(if available) and will be reduced by all duplicate rows.

Usage

prediction(x, list.glm = NULL)

Arguments

x	neural network
list.glm	an optional list of glm objects

Value

a list of the summaries of the repetitions of the neural networks, the data and the glm objects (if available).

Author(s)

Stefan Fritsch, Frauke Guenther [email protected]

See Also

neuralnet

Examples

Var1 <- rpois(100,0.5)
Var2 <- rbinom(100,2,0.6)
Var3 <- rbinom(100,1,0.5)
SUM <- as.integer(abs(Var1+Var2+Var3+(rnorm(100))))
sum.data <- data.frame(Var1,Var2,Var3, SUM)
print(net.sum <- neuralnet( SUM~Var1+Var2+Var3,  sum.data, hidden=1, 
                 act.fct="tanh"))
main <- glm(SUM~Var1+Var2+Var3, sum.data, family=poisson())
full <- glm(SUM~Var1*Var2*Var3, sum.data, family=poisson())
prediction(net.sum, list.glm=list(main=main, full=full))

---