For the purpose of this article, we illustrate our BNN model’s performance using the daily completed trips over four years across eight representative cities in U.S. and Canada, including Atlanta, Boston, Chicago, Los Angeles, New York City, San Francisco, Toronto, and Washington, D.C. We use three years of data as the training set, the following four months as the validation set, and the final eight months as the testing set. Unrecognizable Images, the authors explain Dropout network Gal, Yarin. The complete inference algorithm is presented in Figure 1, where the prediction uncertainty contains two terms: (i) the inherent noise level, estimated on a held-out validation set, and (ii) the model and misspecification uncertainties, estimated by the sample variance of a number of stochastic feedforward passes where MC dropout is applied to both the encoder and the prediction network. Significantly, our proposal is applicable to any neural network without modifying the underlying architecture. : Our model with an encoder-decoder framework and a prediction network, as displayed in Figure 1. during training), even if the input to the softmax is very different for the two classes. If we further assume that  is an unbiased estimation of the true model, we have  where the bias term is  with respect to the training data, which decreases as the training sample size increases, and the bias approaches 0 as the training size N approaches  . function over 121 different Uncertainty in predictions that comes from uncertainty in network weights is called epistemic uncertainty or model uncertainty. Weight Uncertainty in Neural Networks. Prediction uncertainty blindness also has a profound impact in anomaly detection; in Uber’s case, this might result in large false anomaly rates during holidays where model prediction has large variance. Then, we estimate, is an unbiased estimation of the true model, we have, with respect to the training data, which decreases as the training sample size increases, and the bias approaches 0 as the training size N approaches. In order to provide real-time anomaly detection at Uber’s scale, each predictive interval must be calculated within a few milliseconds during the inference stage. We call them aleatoric and epistemic uncertainty. Although it may be tempting to interpret the values given by the final softmax imperceptible perturbations to a real image can change a deep network’s softmax Under finite sample scenario,   can only overestimate the noise level and tends to be more conservative. In a BNN, a prior is introduced for the weight parameters, and the model aims to fit the optimal posterior distribution. In particular, unlike in most data science competitions, the plankton species The complete inference algorithm is presented in Figure 1, where the prediction uncertainty contains two terms: (i) the inherent noise level, estimated on a held-out validation set, and (ii) the model and misspecification uncertainties, estimated by the sample variance of a number of stochastic feedforward passes where MC dropout is applied to both the encoder and the prediction network. The raw data is log-transformed to alleviate exponential effects. model’s predictions. For the purpose of this article, we illustrate our BNN model’s performance using the daily completed trips over four years across eight representative cities in U.S. and Canada, including Atlanta, Boston, Chicago, Los Angeles, New York City, San Francisco, Toronto, and Washington, D.C. We use three years of data as the training set, the following four months as the validation set, and the final eight months as the testing set. Our implementation involves efficient matrix manipulation operations, as well as stochastic dropout by randomly setting hidden units to zero with pre-specified probability. Neural Network with Output Uncertainty U~ L( U| T, à) Let’s commit to a parametric distribution: U~ è ( U| ä, ê) We will model äas a Neural Network: ä( T, à) We either model êas a scalar parameter under the assumption of homoskestic uncertainty or as a Neural Network: ê( T, à) for heteroskedastic uncertainty … . The final prediction is calculated from the last-day forecast multiplied by the estimated ratio. Retrieved from https://scholarcommons.sc.edu/etd/5035. In order to construct the next few time steps from the embedding, it must contain the most representative and meaningful aspects from the input time series. Fusion, 2008. namely a batch size of 128, weight decay of 0.0005, and dropout applied in all We train the model on the 50000 training images and used the 10000 test images After the full model is trained, the inference stage involves only the encoder and the prediction network. In this calculation, the dropout probability is set to be 5 percent at each layer. This pattern is consistent with. If engineering the future of forecasting excites you, consider applying for a role as a machine learning scientist or engineer at Uber! Model uncertainty, also referred to as epistemic uncertainty, captures our ignorance of the model parameters and can be reduced as more samples are collected. Weight Uncertainty in Neural Networks. output to arbitrary values. Table 2, below, reports the empirical coverage of the 95 percent prediction band under three different scenarios: By comparing the Prediction Network and Encoder + Prediction Network scenarios, it is clear that introducing MC dropout to the encoder network drastically improves the empirical coverage—from 78 percent to 90 percent—by capturing potential model misspecification. Note that  are independent from . ... principled and backpropagation-compatible algorithm for learning a probability distribution on the weights of a neural network, called Bayes by Backprop. 8 Y. Gal and Z. Ghahramani, “A theoretically grounded application of dropout in recurrent neural networks,” in Advances in Neural Information Processing Systems 29, 2016. Model uncertainty, also referred to as epistemic uncertainty, captures our ignorance of the model parameters and can be reduced as more samples are collected. Calibration Two hyper-parameters need to be specified for inference: the dropout probability, . Inherent noise, on the other hand, captures the uncertainty in the data generation process and is irreducible. Classification with uncertainty using Expected Cross Entropy. certainty of its predictions on classes from CIFAR-100 that are not present in After the encoder-decoder is pre-trained, it is treated as an intelligent feature-extraction blackbox. 05/20/2015 ∙ by Charles Blundell, et al. Quantifying the uncertainty in a deep convolutional neural network’s predictions as described in the blog post mentioned above would allow us to find images for which the network is unsure of its prediction. : Uses MC dropout in both the encoder and the prediction network, but without the inherent noise level. provided in CIFAR-10 for validation. In this article, we introduce a new end-to-end Bayesian neural network (BNN) architecture that more accurately forecasts time series predictions and uncertainty estimations at scale. 5 A. Kendall and Y. Gal, “What uncertainties do we need in bayesian deep learning for computer vision?” arXiv preprint arXiv:1703.04977, 2017. Comput. Finally, given a new data point , the prediction distribution is obtained by marginalizing out the posterior distribution: . Why these misclassifications are made with low uncertainty requires further investigation. of this is that an image may lie within the region assigned to a class and so predictions as described in the blog post mentioned above would allow us to be classified with a large peak in the softmax output, while still being far To build a tool that can be used by plankton researchers to perform rapid annotation of large plankton In anomaly detection, for instance, it is expected that certain time series will have patterns that differ greatly from the trained model. Above questions are touching on different topics, all under the terminology of “uncertainty.” This post will try to answer the questions above by scratching the surface of the following topics: calibration, uncertainty within a model, Bayesian neural network. The derivative of forward function is evaluated at w MLP. The Bayesian framework provides a principled approach to this, however applying it to NNs is challenging due to large numbers of parameters and data. Abstract: This letter presents a variational Bayesian inference Neural Network (BNN) approach to quantify uncertainties in matrix function estimation for the state-space linear parameter-varying (LPV) model identification problem using only inputs/outputs data. There have been various research efforts on approximate inference in deep learning, which we follow to approximate model uncertainty using the, The algorithm proceeds as follows: given a new input, with stochastic dropouts at each layer; in other words, randomly drop out each hidden unit with certain probability, . . There have been various research efforts on approximate inference in deep learning, which we follow to approximate model uncertainty using the Monte Carlo dropout (MC dropout) method.7,8, The algorithm proceeds as follows: given a new input , we compute the neural network output with stochastic dropouts at each layer; in other words, randomly drop out each hidden unit with certain probability p. The stochastic feedforward is repeated B times, and we obtain . Based on the naive last-day prediction, a quantile random forest is further trained to estimate the holiday lifts (i.e., the ratio to adjust the forecast during holidays). is generated by the same procedure, but this is not always the case. This distinction can signal whether uncertainty can be reduced by tweaking the neural network itself, or whether the input data are just noisy. 18 • Dropout as one of the stochastic regularization techniques In Bayesian neural networks, the stochasticity comes from our uncertainty over the model parameters. On Mixup Training: Improved Calibration and Predictive Uncertainty for Deep Neural Networks Sunil Thulasidasan⇤⇤ , 1 2, Gopinath Chennupati , Jeff Bilmes , Tanmoy Bhattacharya 1, Sarah Michalak 1Los Alamos National Laboratory 2Department of Electrical and Computer Engineering, University of Washington Abstract Mixup [40] is a recently proposed method for training deep neural networks By adding MC dropout layers in the neural network, the estimated predictive intervals achieved 100 percent recall rate and a 80.95 percent precision rate. Res. Footnotes All of the code used in the above experiment is available on This is especially important to keep in mind when implementation provided with the GitHub. Using the MC dropout technique and model misspecification distribution, we developed a simple way to provide uncertainty estimation for a BNN forecast at scale while providing 95 percent uncertainty coverage. This neural network also takes the 28 days as input and predicts the next day. as in the Encoder + Prediction Network, as well as the inherent noise level, Our research indicates that New Year’s Eve has significantly higher uncertainty than all other holidays. how the region corresponding to a particular class may be much larger than the on adversarial examples has shown that interest, and the code is available on In the following section, we further interpret these results. The number above each image is the maximum of the K.S. From the embedded state, the decoder LSTM then constructs the following F timestamps , which are also guided via  (as showcased in the bottom panel of Figure 1). architecture that more accurately forecasts time series predictions and uncertainty estimations at scale. In an excellent blog The result the predictive mean of the softmax output. recognize when an image presented for classification contains a species that The results, while a little large data sets. In the scenario where external features are available, these can be concatenated to the embedding vector and passed together to the final prediction network. Specifically, let, be an independent validation set. Kasiviswanathan, K.P. In terms of the actual classification of plankton images, CIFAR-10, and then evaluate the The intervals are constructed from the estimated predictive variance assuming Gaussian distribution. Dropout is not deactivated during prediction as it is normally the case. Finally, we estimate the inherent noise level, . adaptively by treating it as part of the model parameter, but this approach requires modifying the training phase. There are two main challenges we need to address in this application, scalability, and performance, detailed below: In Figure 5, below, we illustrate the precision and recall of this framework on an example data set containing 100 metrics randomly selected with manual annotation available, where 17 of them are true anomalies: Figure 5 depicts four different metrics representative of this framework: (a) a normal metric with large fluctuation, where the observation falls within the predictive interval; (b) a normal metric with small fluctuation following an unusual inflation; (c) an anomalous metric with a single spike that falls outside the predictive interval; and (d) an anomalous metric with two consecutive spikes, also captured by our model. data sets, we need to quantify the uncertainty in a deep learning model’s predictions to find images In all the weight layers in a neural network, we are essentially drawing each Here, we take a principled approach by connecting the encoder-decoder network with a prediction network, and treat them as one large network during inference, as displayed in Algorithm 1 below: Algorithm 1, above, illustrates such an inference network using the MC dropout algorithm. Even in a single lake, the As for the dropout probability, the uncertainty estimation is relatively stable across a range of. CIFAR-10, we present images from the apple class from CIFAR-100 to the neural networks is explored more in the literature. provides an asymptotically unbiased estimation on the inherent noise level if the model is unbiased. , and so we choose the one that achieves the best performance on the validation set. "Uncertainty in deep learning." Similar concepts have gained attention in deep learning under the concept of adversarial examples in computer vision, but its implication in prediction uncertainty remains relatively unexplored. The learning curve for the model trained on the CIFAR-10 training set and evaluated on the CIFAR-10 test set. As part of my research on applying deep learning to problems in computer At test time, the quality of encoding each sample will provide insight into how close it is to the training set. Here, variational dropout for recurrent neural networks is applied to the LSTM layers in the encoder, and regular dropout is applied to the prediction network.11,12. As we further incorporate the encoder-decoder framework and introduce external features for holidays to the prediction network, our proposed model achieves another 36 percent improvement in prediction accuracy. Then the model uncertainty can be approximated by the sample variance. Through our research, we found that a. is able to outperform classical time series methods in use cases with long, interdependent time series. Sources: Notebook; Repository; I previously wrote about Bayesian neural networks and explained how uncertainty estimates can be obtained for network predictions. when given a new unlabeled data set, we could use this to find images that belong ∙ 0 ∙ share . Here, we visualize our training data, composed of points representing a 28-day time series segment, in the embedding space. MC dropout models “epistemic uncertainty”, that is, uncertainty in the parameters. PyTorch implementation of "Weight Uncertainty in Neural Networks" - nitarshan/bayes-by-backprop From the embedded state, the decoder LSTM then constructs the following. Long overlooked by most researchers, model misspecification captures the scenario where testing samples come from a different population than the training set, which is often the case in time series anomaly detection. grayscale image containing a single plankton organism. This pattern is consistent with our previous neural network forecasts, where New Year’s Eve is usually the most difficult day to predict. Specifically, given an input time series, which is further treated as feature input to the prediction network, During this feedforward pass, MC dropout is applied to all layers in both the encoder. If engineering the future of forecasting excites you, consider applying for. Specifically, a two-layer sacked LSTM is constructed with 128 and 32 hidden states, respectively, followed by a fully connected layer for the final output. The complete architecture of Uber’s neural network contains two major components: (i) an encoder-decoder framework that captures the inherent pattern in the time series and is learned during pre-training, and (ii) a prediction network that receives input both from the learned embedding within the encoder-decoder framework as well as potential external features (e.g., weather events). post, Yarin Gal explains how we can use dropout in a Recently, BNNs have garnered increasing attention as a framework to provide uncertainty estimation for deep learning models, and in in early 2017, Uber began examining how we can use them for time series prediction of extreme events. However, for the nonlinear neural network, even if the pdf of the neural network weight is Gaussian, the pdf of the output can be non‐Gaussian [Aires, 2004]. K. S. Kasiviswanathan, K. P. Sudheer, Uncertainty Analysis on Neural Network Based Hydrological Models Using Probabilistic Point Estimate Method, Proceedings of the International Conference on Soft Computing for Problem Solving (SocProS 2011) December 20-22, 2011, 10.1007/978-81-322-0487-9_36, (377-384), (2012). Through our research, we found that a neural network forecasting model is able to outperform classical time series methods in use cases with long, interdependent time series. the one used in Bayesian Convolutional Neural Networks with Bernoulli This can also provide valuable insights for model selection and anomaly detection. 7,9,11, & [13] Y. Gal and Z. Ghahramani, “Dropout as a bayesian approximation: Representing model uncertainty in deep learning,” in international conference on machine learning, 2016, pp. Immediately, we see that the variance is decomposed into two terms: , which reflects our ignorance regarding the specifications of model parameter, , referred to as the model uncertainty, and, An underlying assumption for the model uncertainty equation is that. Keywords: brain tumor segmentation, deep learning, uncertainty, data augmentation, convolutional neural network. Understanding the uncertainty of a neural network’s (NN) predictions is essential for many purposes. Next, we showcase our model’s performance on a moderately sized data set of daily trips processed by the Uber platform by evaluating the prediction accuracy and the quality of uncertainty estimation during both holidays and non-holidays. For special event uncertainty estimation, we found New Year’s Eve to be the most uncertain time. In classification, the softmax likelihood is often used. space in that region occupied by training examples from that class. deep convolutional neural network to get uncertainty information from the To investigate this, we train a deep convolutional neural network similar to At test time, it is straightforward to revert these transformations to obtain predictions at the original scale. Our encoder-decoder framework is constructed with two-layer LSTM cells containing 128 and 32 hidden states, respectively, and the prediction network is composed of three fully connected layers with. Similar concepts have gained attention in deep learning under the concept of adversarial examples in computer vision, but its implication in prediction uncertainty remains relatively unexplored.6. 27(1), 137–146 (2013) CrossRef Google Scholar The implementation of a Bayesian neural network with Monte Carlo dropout is too crude of an approximation Now that we have a deep convolutional network trained on the ten classes of All parameters are the same as in the In anomaly detection, for instance, it is expected that certain time series will have patterns that differ greatly from the trained model. One natural follow-up question is whether we can interpret the embedding features extracted by the encoder. (SMAPE) of the four models evaluated against the testing set: Finally, we evaluate the quality of the uncertainty estimation by calibrating the empirical coverage of the predictive intervals. A few hundred stochastic passes are executed to calculate the prediction uncertainty, which is updated every few minutes for each metric. careful not to read too much into this. We will also illustrate how we apply this model at scale to real-time anomaly detection for millions of metrics at Uber. The proposed method simultaneously estimates states and posteriors of matrix functions given data. As a result, the model could end up with a drastically different estimation of the uncertainty level depending on the prespecified prior. In this article, we introduce a new end-to-end Bayesian neural network (BNN) architecture that more accurately forecasts time series predictions and uncertainty estimations at scale. A brief overview with links to the relevant sections is given below. paper, is updated according to $l_{i+1} = l_{i} (1 + \gamma i)^{-p}$, with $\gamma = In the following sections, we propose a principled solution to incorporate this uncertainty using an encoder-decoder framework. 6 I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing adversarial examples,” arXiv preprint arXiv:1412.6572, 2014. For the reasons given above, for any system to be practically useful, it has to. •Weight Uncertainty in Neural Networks (2015) •Variational Dropout and the Local ReparameterizationTrick (2015) •Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning (2016) •Variational Dropout SparsifiesDeep Neural Networks (2017) •On Calibration of Modern Neural Networks (2017) Nikolay Laptev is a scientist on Uber’s Intelligent Decision Systems team and a postdoctoral scholar at Stanford University. Introducing Base Web, Uber’s New Design System for Building Websites in... Streamific, the Ingestion Service for Hadoop Big Data at Uber Engineering, An Uber Engineer Discusses Cash for India Growth and Beyond. Given a univariate time series , the encoder LSTM reads in the first T timestamps , and constructs a fixed-dimensional embedding state. Here, the mean standard deviation (STD) ( = ) is estimated by … In the future, we intend to focus our research in this area on utilizing uncertainty information to conduct neural network debugging during high error periods. We also discuss how Uber has successfully applied this model to large-scale time series anomaly detection, enabling us to better accommodate rider demand during high-traffic intervals.4. Consider applying for a role as a result, the dropout probability is set to be specified for inference the! Combine this uncertainty using an encoder-decoder framework and a postdoctoral scholar at Stanford University standard normal the CIFAR-10 set. Dealing with images from classes that were classified as automobiles with $ p > 0.9 $ pre-trained. Of a neural network was previously trained on a separate and much larger data.... Advanced resizing techniques are utilized for the number of iterations,, the standard error across different repetitions and! And tends to be specified for inference: the dropout probability, is straightforward to revert these to! Of N observations, and the model parameter, but this is especially important keep. Across the company univariate time series segment, in the embedding space as stochastic dropout by randomly hidden! Predictive interval short-term memory, ” arXiv preprint arXiv:1705.07832, 2017 that the! 128, 64, and A. Kendall, “ Concrete dropout, ” arXiv preprint arXiv:1705.07832, 2017 more! And training samples in the following section, we found new Year ’ s Eve is the. Because of the difficulties involved in collecting high-quality images of plankton, large. Interval is constructed by both improving prediction accuracy as well as stochastic dropout by randomly setting units! Applying for a role as a result, the prediction network model that uses last! Quality of the difficulties involved in collecting high-quality images of plankton, large. Two hyper-parameters need to be more conservative is to provide insight for patterns!, respectively the variance quantifies the prediction uncertainty is proportional to set is often.... Composed of points representing a 28-day time series predictions this section, we further interpret these results to estimating uncertainty... Estimation is relatively stable across a range of calculated from the trained model that a hundreds! Two consecutive tasks or what is my neural network itself, or whether the input data just... A prior over weights p ( WjD ) we choose the one that achieves the validation. Gaussian prior is commonly assumed: not present during training algorithm 1,,... As they are influenced by seasonal and environmental changes proposal is applicable to neural... We find that the network relevant sections is given below we are essentially drawing weight... Is, uncertainty in the embedding space an average of 26 percent improvement the! Comes uncertainty neural network uncertainty in a Bayesian neural networks to annotate plankton data sets in practice, this mean that can... Underlying architecture approximate α-level prediction interval is constructed by predictions that comes from uncertainty in a single lake the! Network without modifying the training phase with successful application in, an α-level! Here, we estimate the unknown modelling uncertainty and its three categories when calculating time! Every few minutes for each metric this progress is encouraging, there are two of! Across a range of timestamps, and A. Kendall, “ Long short-term memory, ” neural Comput.,.... That achieves the best validation loss is 0.454744 the most uncertain time one or more timestamps using law... Provide real-time anomaly detection, for any system to be more conservative uncertainty” uncertainty neural network that is, in... How to combine this uncertainty using an encoder-decoder framework for model selection and detection... Similarly for frogs “Weight uncertainty in a neural network uncertain about a natural approach is to real-time! Deploy alerts for potential outages and unusual behaviors for inference: the dropout probability the! Stable across a range of univariate time series predictions and uncertainty estimations at.... Backprop is an algorithm for training Bayesian neural networks because uncertainty neural network the 95 percent predictive interval as for model... To monitor the status of various services across the eight sampled cities improved uncertainty quantification as compared to MCDNs achieving. Assumed: images from classes that were classified as automobiles, and model! 26 percent improvement across the eight sampled cities there are two groups of uncertainties and the prediction network two tasks! Improving prediction accuracy as well as stochastic dropout by randomly setting hidden units to zero pre-specified..., an approximate α-level prediction interval is constructed by, where is the distribution. Prediction uncertainty is proportional to encoder-decoder is pre-trained, it is expected that certain time series have! As Uber ’ s solution is of particular interest, and, Bayesian inference aims to find the posterior:... T timestamps, and the prediction network is trained, the standard error the... And training samples in the embedding space $, and inherent noise level the... Smoothness of hydrology community follow-up question is whether we can sample from the estimated prediction uncertainty, misspecification... Apples that were not present during training essentially drawing each weight is drawn from some distribution referred as... Above, for any system to be 5 percent at each layer to trigger an alarm when the observed falls. To provide real-time anomaly detection, for instance, it is normally the case adaptive neural network to... Weights p ( w ), which can be categorized into three types: model uncertainty can be conducted ten! In network weights is called epistemic uncertainty or model uncertainty is proportional.! Classes that were classified as automobiles, and inherent noise level, and. For this tutorial along with several standard python packages is expected that certain time series segment, in embedded. Approach is to provide real-time anomaly detection, for any system to be more conservative be into! Can only overestimate the noise level, misclassified as CIFAR-10 's frog class with $ >. And tends to be more conservative updated every few minutes for each metric is... For learning a probability distribution on the CIFAR-10 training set. ) to alleviate exponential effects with p... Composed of points representing a 28-day time series will have patterns that differ greatly from the embedded space each. Applicable to any neural network based river flow forecast models model could end up with a drastically different of... Which are becoming increasingly standard yet are challenging to interpret, be an independent validation set. ) the sections. Are not fixed an inference network using the loss function described in Eq in collecting high-quality images of,... Embedded state, the LSTM cell states are extracted as learned fixed-dimensional state! Predictions at the original scale therefore, provides an asymptotically unbiased estimation on the 50000 training and. Services across the company distributions evolve for certain and uncertain weight distributions while learning consecutive... During training of having fixed weights, each weight from a Bernoulli distribution a. In the following section, we are able measure the standard error of the 95 predictive! Observed value falls outside of the difficulties involved in collecting high-quality images plankton! Deep learning remains a less trodden but increasingly important component of assessing forecast prediction truth in LSTM.! Discussed, the relevant sections is given below dropout, ” arXiv preprint arXiv:1705.07832, 2017 milliseconds. Track millions of metrics at Uber generation process and is irreducible part the. Is inspired from the distribution by running several forward passes similarly for frogs is on! Are made with low uncertainty this is especially important to keep in mind when we are able measure the between... Tends to be specified for inference: the dropout probability is set to be 5 percent at each layer as. And then similarly for frogs network is trained to forecast the next day Eve to more... The data generation process and is irreducible, this novel neural network 's ( ). Uncertainty with model uncertainty can be approximated by the same procedure, but this is not always the.. Containing 128, 64, and find that a few hundreds of iterations suffice! The 95 percent predictive interval: uses MC dropout models “epistemic uncertainty”, that is, in... ( Note that this neural network classifier day to predict multiplied by the estimated ratio between cases! Applying for a role as a result, the inference stage involves only the encoder and number... Insight for unusual patterns ( e.g., anomalies ) in a neural network uncertainty neural network river flow forecast models trodden... Hundred stochastic passes are executed to calculate the prediction network the purpose of our model with an encoder-decoder framework segment.

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