Deep Quantile Regression for Uncertainty Estimation in Unsupervised and Supervised Lesion Detection.

Despite impressive state-of-the-art performance on a wide variety of machine learning tasks in multiple applications, deep learning methods can produce over-confident predictions, particularly with limited training data. Therefore, quantifying uncertainty is particularly important in critical applications such as lesion detection and clinical diagnosis, where a realistic assessment of uncertainty is essential in determining surgical margins, disease status and appropriate treatment. In this work, we propose a novel approach that uses quantile regression for quantifying aleatoric uncertainty in both supervised and unsupervised lesion detection problems. The resulting confidence intervals can be used for lesion detection and segmentation. In the unsupervised setting, we combine quantile regression with the Variational AutoEncoder (VAE). The VAE is trained on lesion-free data, so when presented with an image with a lesion, it tends to reconstruct a lesion-free version of the image. To detect the lesion, we then compare the input (lesion) and output (lesion-free) images. Here we address the problem of quantifying uncertainty in the images that are reconstructed by the VAE as the basis for principled outlier or lesion detection. The VAE models the output as a conditionally independent Gaussian characterized by its mean and variance. Unfortunately, joint optimization of both mean and variance in the VAE leads to the well-known problem of shrinkage or underestimation of variance. Here we describe an alternative Quantile-Regression VAE (QR-VAE) that avoids this variance shrinkage problem by directly estimating conditional quantiles for the input image. Using the estimated quantiles, we compute the conditional mean and variance for the input image from which we then detect outliers by thresholding at a false-discovery-rate corrected p-value. In the supervised setting, we develop binary quantile regression (BQR) for the supervised lesion segmentation task. We show how BQR can be used to capture uncertainty in lesion boundaries in a manner that characterizes expert disagreement.

A robust variational autoencoder using beta divergence.

The presence of outliers can severely degrade learned representations and performance of deep learning methods and hence disproportionately affect the training process, leading to incorrect conclusions about the data. For example, anomaly detection using deep generative models is typically only possible when similar anomalies (or outliers) are not present in the training data. Here we focus on variational autoencoders (VAEs). While the VAE is a popular framework for anomaly detection tasks, we observe that the VAE is unable to detect outliers when the training data contains anomalies that have the same distribution as those in test data. In this paper we focus on robustness to outliers in training data in VAE settings using concepts from robust statistics. We propose a variational lower bound that leads to a robust VAE model that has the same computational complexity as the standard VAE and contains a single automatically-adjusted tuning parameter to control the degree of robustness. We present mathematical formulations for robust variational autoencoders (RVAEs) for Bernoulli, Gaussian and categorical variables. The RVAE model is based on beta-divergence rather than the standard Kullback-Leibler (KL) divergence. We demonstrate the performance of our proposed ß-divergence-based autoencoder for a variety of image and categorical datasets showing improved robustness to outliers both qualitatively and quantitatively. We also illustrate the use of our robust VAE for detection of lesions in brain images, formulated as an anomaly detection task. Finally, we suggest a method to tune the hyperparameter of RVAE which makes our model completely unsupervised.

Quantile Regression for Uncertainty Estimation in VAEs with Applications to Brain Lesion Detection.

The Variational AutoEncoder (VAE) has become one of the most popular models for anomaly detection in applications such as lesion detection in medical images. The VAE is a generative graphical model that is used to learn the data distribution from samples and then generate new samples from this distribution. By training on normal samples, the VAE can be used to detect inputs that deviate from this learned distribution. The VAE models the output as a conditionally independent Gaussian characterized by means and variances for each output dimension. VAEs can therefore use reconstruction probability instead of reconstruction error for anomaly detection. Unfortunately, joint optimization of both mean and variance in the VAE leads to the well-known problem of shrinkage or underestimation of variance. We describe an alternative VAE model, Quantile-Regression VAE (QR-VAE), that avoids this variance shrinkage problem by estimating conditional quantiles for the given input image. Using the estimated quantiles, we compute the conditional mean and variance for input images under the Gaussian model. We then compute reconstruction probability using this model as a principled approach to outlier or anomaly detection. We also show how our approach can be used for heterogeneous thresholding of images for detecting lesions in brain images.

BRAIN LESION DETECTION USING A ROBUST VARIATIONAL AUTOENCODER AND TRANSFER LEARNING.

Automated brain lesion detection from multi-spectral MR images can assist clinicians by improving sensitivity as well as specificity. Supervised machine learning methods have been successful in lesion detection. However, these methods usually rely on a large number of manually delineated images for specific imaging protocols and parameters and often do not generalize well to other imaging parameters and demographics. Most recently, unsupervised models such as autoencoders have become attractive for lesion detection since they do not need access to manually delineated lesions. Despite the success of unsupervised models, using pre-trained models on an unseen dataset is still a challenge. This difficulty is because the new dataset may use different imaging parameters, demographics, and different pre-processing techniques. Additionally, using a clinical dataset that has anomalies and outliers can make unsupervised learning challenging since the outliers can unduly affect the performance of the learned models. These two difficulties make unsupervised lesion detection a particularly challenging task. The method proposed in this work addresses these issues using a two-prong strategy: (1) we use a robust variational autoencoder model that is based on robust statistics, specifically the ß-divergence that can be trained with data that has outliers; (2) we use a transfer-learning method for learning models across datasets with different characteristics. Our results on MRI datasets demonstrate that we can improve the accuracy of lesion detection by adapting robust statistical models and transfer learning for a variational autoencoder model.

A note on the phase locking value and its properties.

We investigate the properties of the Phase Locking Value (PLV) and the Phase Lag Index (PLI) as metrics for quantifying interactions in bivariate local field potential (LFP), electroencephalography (EEG) and magnetoencephalography (MEG) data. In particular we describe the relationship between nonparametric estimates of PLV and PLI and the parameters of two distributions that can both be used to model phase interactions. The first of these is the von Mises distribution, for which the sample PLV is a maximum likelihood estimator. The second is the relative phase distribution associated with bivariate circularly symmetric complex Gaussian data. We derive an explicit expression for the PLV for this distribution and show that it is a function of the cross-correlation between the two signals. We compare the bias and variance of the sample PLV and the PLV computed from the cross-correlation. We also show that both the von Mises and Gaussian models are suitable for representing relative phase in application to LFP data from a visually-cued motor study in macaque. We then compare results using the two different PLV estimators and conclude that, for this data, the sample PLV provides equivalent information to the cross-correlation of the two complex time series.

On temporal connectivity of PFC via Gauss-Markov modeling of fNIRS signals.

Functional near-infrared spectroscopy (fNIRS) is an optical imaging method, which monitors the brain activation by measuring the successive changes in the concentration of oxy- and deoxyhemoglobin in real time. In this study, we present a method to investigate the functional connectivity of prefrontal cortex (PFC) Sby applying a Gauss-Markov model to fNIRS signals. The hemodynamic changes on PFC during the performance of cognitive paradigm are measured by fNIRS for 17 healthy adults. The color-word matching Stroop task is performed to activate 16 different regions of PFC. There are three different types of stimuli in this task, which can be listed as incongruent stimulus (IS), congruent stimulus (CS), and neutral stimulus (NS), respectively. We introduce a new measure, called "information transfer metric" (ITM) for each time sample. The behavior of ITMs during IS are significantly different from the ITMs during CS and NS, which is consistent with the outcome of the previous research, which concentrated on fNIRS signal analysis via color-word matching Stroop task. Our analysis shows that the functional connectivity of PFC is highly relevant with the cognitive load, i.e., functional connectivity increases with the increasing cognitive load.

Classification of respiratory signals by linear analysis.

The aim of this study is the classification of wheeze and non-wheeze epochs within respiratory sound signals acquired from patients with asthma and COPD. Since a wheeze signal, having a sinusoidal waveform, has a different behavior in time and frequency domains from that of a non-wheeze signal, the features selected for classification are kurtosis, Renyi entropy, f(50)/ f(90) ratio and mean-crossing irregularity. Upon calculation of these features for each wheeze and non-wheeze portion, the whole data scattered as two classes in four dimensional feature space is projected using Fisher Discriminant Analysis (FDA) onto the single dimensional space that separates the two classes best. Observing that the two classes are visually well separated in this new space, Neyman-Pearson hypothesis testing is applied. Finally, the correct classification rate is %95.1 for the training set, and leave-one-out approach pursuing the above methodology yields a success rate of %93.5 for the test set.