Time series are difficult to monitor, summarize and predict. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). For scalability, we require fast linear time algorithms. The popular piecewise linear model can determine where the data goes up or down and at what rate. Unfortunately, when the data does not follow a linear model, the computation of the local slope creates overfitting. We propose an adaptive time series model where the polynomial degree of each interval vary (constant, linear and so on). Given a number of regressors, the cost of each interval is its polynomial degree: constant intervals cost 1 regressor, linear intervals cost 2 regressors, and so on. Our goal is to minimize the Euclidean (l_2) error for a given model complexity. Experimentally, we investigate the model where intervals can be either constant or linear. Over synthetic random walks, historical stock market prices, and electrocardiograms, the adaptive model provides a more accurate segmentation than the piecewise linear model without increasing the cross-validation error or the running time, while providing a richer vocabulary to applications. Implementation issues, such as numerical stability and real-world performance, are discussed.
While we focus on segmentation, there are many methods available for fitting models to continuous variables, such as a regression, regression decision trees, Neural Networks @cite_23 , Wavelets @cite_39 , Adaptive Multivariate Splines @cite_46 , Free-Knot Splines @cite_26 , Hybrid Adaptive Splines @cite_19 , etc.
@cite_26 @cite_39 @cite_19 @cite_23 @cite_46
2068120653 2145316937 2049228615 1801768344
We describe a Bayesian method, for fitting curves to data drawn from an exponential family, that uses splines for which the number and locations of knots are free parameters. The method uses reversible jump Markov chain Monte Carlo to change the knot configurations and a locality heuristic to speed up mixing. For nonnormal models, we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the Bayesian information criterion, BIC, under priors that make this approximation accurate. Our technique is based on a marginalised chain on the knot number and locations, but we provide methods for inference about the regression coefficients, and functions of them, in both normal and nonnormal models. Simulation results suggest that the method performs well, and we illustrate the method in two neuroscience applications. An increasing number of computer vision and pattern recognition problems require structured regression techniques. Problems like human pose estimation, unsegmented action recognition, emotion prediction and facial landmark detection have temporal or spatial output dependencies that regular regression techniques do not capture. In this paper we present continuous conditional neural fields (CCNF) – a novel structured regression model that can learn non-linear input-output dependencies, and model temporal and spatial output relationships of varying length sequences. We propose two instances of our CCNF framework: Chain-CCNF for time series modelling, and Grid-CCNF for spatial relationship modelling. We evaluate our model on five public datasets spanning three different regression problems: facial landmark detection in the wild, emotion prediction in music and facial action unit recognition. Our CCNF model demonstrates state-of-the-art performance on all of the datasets used. The selection of variables in regression problems has occupied the minds of many statisticians. Several Bayesian variable selection methods have been developed, and we concentrate on the following methods: Kuo & Mallick, Gibbs Variable Selection (GVS), Stochastic Search Variable Selection (SSVS), adaptive shrinkage with Jefireys' prior or a Laplacian prior, and reversible jump MCMC. We review these methods, in the context of their difierent properties. We then implement the methods in BUGS, using both real and simulated data as examples, and investigate how the difierent methods perform in practice. Our results suggest that SSVS, reversible jump MCMC and adaptive shrinkage methods can all work well, but the choice of which method is better will depend on the priors that are used, and also on how they are implemented. Time series are difficult to monitor, summarize and predict. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). For scalability, we require fast linear time algorithms. The popular piecewise linear model can determine where the data goes up or down and at what rate. Unfortunately, when the data does not follow a linear model, the computation of the local slope creates overfitting. We propose an adaptive time series model where the polynomial degree of each interval vary (constant, linear and so on). Given a number of regressors, the cost of each interval is its polynomial degree: constant intervals cost 1 regressor, linear intervals cost 2 regressors, and so on. Our goal is to minimize the Euclidean (l_2) error for a given model complexity. Experimentally, we investigate the model where intervals can be either constant or linear. Over synthetic random walks, historical stock market prices, and electrocardiograms, the adaptive model provides a more accurate segmentation than the piecewise linear model without increasing the cross-validation error or the running time, while providing a richer vocabulary to applications. Implementation issues, such as numerical stability and real-world performance, are discussed.