The prediction uncertainty of a neural network is considered from a classical system identification point of view. To know this uncertainty is extremely important when using a network in decision and feedback applications. The asymptotic covariance of the internal parameters in the network due to noise in the observed dependent variables (output) and model class mismatch, i.e., the true system cannot be exactly described by the model class, is first surveyed. This is then applied to the prediction step of the network to get a closed form expression for the asymptotic, in training data information, prediction variance. Another interpretation of this expression is as the non-asymptotic Cramér-Rao Lower Bound. To approximate this expression, only the gradients and residuals, already computed in the gradient descent algorithms commonly used to train neural networks, are needed. Using a toy example, it is illustrated how the uncertainty in the output of a neural network can be estimated.

The uncertainty in the prediction calculated using the delta method for an over-parameterized (parametric) black-box model is shown to be larger or equal to the uncertainty in the prediction of a canonical (minimal) model. Equality holds if the additional parameters of the overparameterized model do not add flexibility to the model. As a conclusion, for an overparameterized black-box model, the calculated uncertainty in the prediction by the delta method is not underestimated. The results are shown analytically and are validated in a simulation experiment where the relationship between the normalized traction force and the wheel slip of a car is modelled using e.g., a neural network.

Despite the great success of neural networks (NN) in many application areas, it is still not obvious how to integrate an NN in a sensor fusion framework. The reason is that the computation of the for fusion required variance of NN is still a rather immature area. Here, we apply a methodology from system identification where uncertainty of the parameters in the NN are first estimated in the training phase, and then this uncertainty is propagated to the output in the prediction phase. This local approach is based on linearization, and it implicitly assumes a good signal-to-noise ratio and persistency of excitation. We illustrate the proposed method on a fundamental problem in advanced driver assistance systems (ADAS), namely to estimate the tire-road friction. This is a single input single output static nonlinear relation that is simple enough to provide insight and it enables comparisons with other parametric approaches. We compare both to existing methods for how to assess uncertainty in NN and standard methods for this problem, and evaluate on real data. The goal is not to improve on simpler methods for this particular application, but rather to validate that our method is on par with simpler model structures, where output variance is immediately provided.

lassifiers based on neural networks (nn) often lack a measure of uncertainty in the predicted class. We propose a method to estimate the probability mass function (pmf) of the different classes, as well as the covariance of the estimated pmf. First, a local linear approach is used during the training phase to recursively compute the covariance of the parameters in the nn. Secondly, in the classification phase, another local linear approach is used to propagate the covariance of the learned nn parameters to the uncertainty in the output of the last layer of the nn. This allows for an efficient Monte Carlo (mc) approach for; (i) estimating the pmf; (ii) calculating the covariance of the estimated pmf; and (iii) proper risk assessment and fusion of multiple classifiers. Two classical image classification tasks, i.e., mnist, and cfar10, are used to demonstrate the efficiency of the proposed method.

Neural Networks (NNS) can solve very hard classification and estimation tasks but are less well suited to solve complex sensor fusion challenges, such as end-to-end control of autonomous vehicles. Nevertheless, NN can still be a powerful tool for particular sub-problems in sensor fusion. This would require a reliable and quantifiable measure of the stochastic uncertainty in the predictions that can be compared to classical sensor measurements. However, current NN'S output some figure of merit, that is only a relative model fit and not a stochastic uncertainty. We propose to embed the NN'S in a proper stochastic system identification framework. In the training phase, the stochastic uncertainty of the parameters in the (last layers of the) NN is quantified. We show that this can be done recursively with very few extra computations. In the classification phase, Monte-Carlo (MC) samples are used to generate a set of classifier outputs. From this set, a distribution of the classifier output is obtained, which represents a proper description of the stochastic uncertainty of the predictions. We also show how to use the calculated uncertainty for outlier detection by including an artificial outlier class. In this way, the NN fits a sensor fusion framework much better. We evaluate the approach on images of handwritten digits. The proposed method is shown to be on par with MC dropout, while having lower computational complexity, and the outlier detection almost completely eliminates false classifications.

This letter considers the problem of detecting and tracking objects in a sequence of images. The problem is formulated in a filtering framework, using the output of object-detection algorithms as measurements. An extension to the filtering formulation is proposed that incorporates class information from the previous frame to robustify the classification. Further, the properties of the object-detection algorithm are exploited to quantify the uncertainty of the bounding box detection in each frame. The complete filtering method is evaluated on camera trap images of the four large Swedish carnivores, bear, lynx, wolf, and wolverine. The experiments show that the class tracking formulation leads to a more robust classification.