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Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes
The application of Stochastic Differential Equations (SDEs) to the analysis
of temporal data has attracted increasing attention, due to their ability to
describe complex dynamics with physically interpretable equations. In this
paper, we introduce a non-parametric method for estimating the drift and
diffusion terms of SDEs from a densely observed discrete time series. The use
of Gaussian processes as priors permits working directly in a function-space
view and thus the inference takes place directly in this space. To cope with
the computational complexity that requires the use of Gaussian processes, a
sparse Gaussian process approximation is provided. This approximation permits
the efficient computation of predictions for the drift and diffusion terms by
using a distribution over a small subset of pseudo-samples. The proposed method
has been validated using both simulated data and real data from economy and
paleoclimatology. The application of the method to real data demonstrates its
ability to capture the behaviour of complex systems
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Deep learning networks find unique mammographic differences in previous negative mammograms between interval and screen-detected cancers: a case-case study.
BackgroundTo determine if mammographic features from deep learning networks can be applied in breast cancer to identify groups at interval invasive cancer risk due to masking beyond using traditional breast density measures.MethodsFull-field digital screening mammograms acquired in our clinics between 2006 and 2015 were reviewed. Transfer learning of a deep learning network with weights initialized from ImageNet was performed to classify mammograms that were followed by an invasive interval or screen-detected cancer within 12 months of the mammogram. Hyperparameter optimization was performed and the network was visualized through saliency maps. Prediction loss and accuracy were calculated using this deep learning network. Receiver operating characteristic (ROC) curves and area under the curve (AUC) values were generated with the outcome of interval cancer using the deep learning network and compared to predictions from conditional logistic regression with errors quantified through contingency tables.ResultsPre-cancer mammograms of 182 interval and 173 screen-detected cancers were split into training/test cases at an 80/20 ratio. Using Breast Imaging-Reporting and Data System (BI-RADS) density alone, the ability to correctly classify interval cancers was moderate (AUC = 0.65). The optimized deep learning model achieved an AUC of 0.82. Contingency table analysis showed the network was correctly classifying 75.2% of the mammograms and that incorrect classifications were slightly more common for the interval cancer mammograms. Saliency maps of each cancer case found that local information could highly drive classification of cases more than global image information.ConclusionsPre-cancerous mammograms contain imaging information beyond breast density that can be identified with deep learning networks to predict the probability of breast cancer detection
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