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I would like to construct a confidence interval around prediction from a neural network, without resorting to bootstrapping - given the computational cost. Can I use the Hessian returned in this way to produce a 95% CI?

1) Can you take the negative inverse of the Hessian as the var/covar matrix? I read here that this depends on what is being maximized or minimized. Is this true and how can you know for certain?

2) Is this an accepted routine for producing a confidence interval around a prediction and if so, how would you do it?

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  • $\begingroup$ Do I understand correctly that you want the confidence interval for the prediction of one case? $\endgroup$ – cbeleites Dec 14 '12 at 1:01
  • $\begingroup$ I am thinking about a confidence interval around the prediction of a new case.....Is there a notion of a confidence interval and a prediction interval for neural nets? If so, I would like to understand both. $\endgroup$ – B_Miner Dec 20 '12 at 1:55
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Try the nnetpredint package. https://cran.r-project.org/web/packages/nnetpredint/

I’ve met the same problem and I also want to construct a prediction confidence interval to the neural networks. So I tried to develop the nnetpredint (R package), using the method from these related papers, which use the Jacobian matrix (first order derivative of the training datasets with gradient function) to estimate model errors instead of the Hessian matrix. The manual is here and the method has the function interface to the models trained by nnet, neuralnet and RSNNS packages:

The example for nnet package is here. The method nnetPredInt takes the model weights, nodes number, training datasets, etc. as input and compute the prediction interval for the new datasets.

    install.packages("nnetpredint")

    # Example: Using the nnet object trained by nnet package
    library(nnet)
    xTrain <- rbind(cbind(runif(150,min = 0, max = 0.5),runif(150,min = 0, max = 0.5)) ,
    cbind(runif(150,min = 0.5, max = 1),runif(150,min = 0.5, max = 1))
    )
    nObs <- dim(xTrain)[1]
    yTrain <- 0.5 + 0.4 * sin(2* pi * xTrain %*% c(0.4,0.6)) +rnorm(nObs,mean = 0, sd = 0.05)
    plot(xTrain %*% c(0.4,0.6),yTrain)

    # Training nnet models
    net <- nnet(yTrain ~ xTrain,size = 3, rang = 0.1,decay = 5e-4, maxit = 500)
    yFit <- c(net$fitted.values)
    nodeNum <- c(2,3,1)
    wts <- net$wts

    # New data for prediction intervals
    library(nnetpredint)
    newData <- cbind(seq(0,1,0.05),seq(0,1,0.05))
    yTest <- 0.5 + 0.4 * sin(2* pi * newData %*% c(0.4,0.6))+rnorm(dim(newData)[1],mean = 0, sd = 0.05)

    # S3 generic method: Object of nnet
    yPredInt <- nnetPredInt(net, xTrain, yTrain, newData, alpha = 0.05) # 95% confidence interval
    print(yPredInt[1:20,])

    # S3 default method for user defined input
    yPredInt2 <- nnetPredInt(object = NULL, xTrain, yTrain, yFit, node = nodeNum, wts = wts, newData, alpha = 0.05, funName = 'sigmoid')

    plot(newData %*% c(0.4,0.6),yTest,type = 'b')
    lines(newData %*% c(0.4,0.6),yPredInt$yPredValue,type = 'b',col='blue')
    lines(newData %*% c(0.4,0.6),yPredInt$lowerBound,type = 'b',col='red') # lower bound
    lines(newData %*% c(0.4,0.6),yPredInt$upperBound,type = 'b',col='red') # upper bound

The keys to the estimation methods:

Use the first order Taylor expansion to expand the f(x) at each weight parameters. And calculate the gradient vector/ Jacobian matrix from the training datasets.

References:

De Veaux R. D., Schumi J., Schweinsberg J., Ungar L. H., 1998, "Prediction intervals for neural networks via nonlinear regression", Technometrics 40(4): 273-282.

Chryssolouris G., Lee M., Ramsey A., "Confidence interval prediction for neural networks models",IEEE Trans. Neural Networks, 7 (1), 1996, pp. 229-232

And also check out this paper for detailed maths. http://cdn.intechopen.com/pdfs-wm/14915.pdf Confidence Intervals for Neural Networks and Applications to Modeling Engineering Materials

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  • $\begingroup$ could kindly help me understand the last article you cited? Does equation (23) relate to model output gradient or model's loss function gradient? The question is about both Jacobian matrix and gradient vector. $\endgroup$ – Alexey Burnakov Nov 14 '17 at 15:04
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I'd not recommend deriving prediction intervals from anything else than independent test cases for ANN. In general, the more prone to overfitting a certain approach is, the more important are independent test cases.

If you can show that your models are stable, you may be able to construct prediction intervals from "within" the model/fitting process. That is commonly done for univariate linear regression, but these are rather restricted models (low complexity, few degrees of freedom).

And anyways, I'd recommend to check the stability of the models by iterated/repeated cross validation. But of course the computational effort for that is just like bootstrapping.

How long does one model take to fit that 3 weeks are not enough for resampling validation?

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