Similarity-based approaches represent a promising direction for time series analysis. However, many such methods rely on parameter tuning, and some have shortcomings if the time series are multivariate (MTS), due to dependencies between attributes, or the time series contain missing data. In this paper, we address these challenges within the powerful context of kernel methods by proposing the robust (TCK). The approach taken leverages the missing data handling properties of Gaussian mixture models (GMM) augmented with informative prior distributions. An ensemble learning approach is exploited to ensure robustness to parameters by combining the clustering results of many GMM to form the final kernel. We evaluate the TCK on synthetic and real data and compare to other state-of-the-art techniques. The experimental results demonstrate that the TCK is robust to parameter choices, provides competitive results for MTS without missing data and outstanding results for missing data.
In several cases, the assumption of a single parametric model underlying all the data may be too strong. Additionally, finding the most suitable parametric model is a crucial and often difficult task, which must be repeated every time a new dataset is processed. This issue is addressed by the autoregressive kernel @cite_27 , which evaluates the similarity of two time series on the corresponding likelihood profiles of a vector autoregressive model of a given order, across all possible parameter settings, controlled by a prior. The kernel is then evaluated as the dot product in the parameter space of such profiles, used as sequence representations. The reservoir based kernels @cite_80 , map the time series into a high dimensional, dynamical feature space, where a linear readout is trained to discriminate each signal. These kernels fit reservoir models sharing the same fixed reservoir topology to all time series. Since the reservoir provides a rich pool of dynamical features, it is considered to be generic'' and, contrarily to kernels based on a single parametric model, it is able to represent a wide variety of dynamics for different datasets.
@cite_80 @cite_27
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Pictorial structure (PS) models are extensively used for part-based recognition of scenes, people, animals and multi-part objects. To achieve tractability, the structure and parameterization of the model is often restricted, for example, by assuming tree dependency structure and unimodal, data-independent pairwise interactions. These expressivity restrictions fail to capture important patterns in the data. On the other hand, local methods such as nearest-neighbor classification and kernel density estimation provide non-parametric flexibility but require large amounts of data to generalize well. We propose a simple semi-parametric approach that combines the tractability of pictorial structure inference with the flexibility of non-parametric methods by expressing a subset of model parameters as kernel regression estimates from a learned sparse set of exemplars. This yields query-specific, image-dependent pose priors. We develop an effective shape-based kernel for upper-body pose similarity and propose a leave-one-out loss function for learning a sparse subset of exemplars for kernel regression. We apply our techniques to two challenging datasets of human figure parsing and advance the state-of-the-art (from 80 to 86 on the Buffy dataset [8]), while using only 15 of the training data as exemplars. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practi cal limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n 2 ). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p 2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such method s, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performancethan existing algorithms, for any given problem instance, and not only for worst-case situations. Time series data mining has received much greater interest along with the increase in temporal data sets from different domains such as medicine, finance, multimedia, etc. Representations are important to reduce dimensionality and generate useful similarity measures. High-level representations such as Fourier transforms, wavelets, piecewise polynomial models, etc., were considered previously. Recently, autoregressive kernels were introduced to reflect the similarity of the time series. We introduce a novel approach to model the dependency structure in time series that generalizes the concept of autoregression to local autopatterns. Our approach generates a pattern-based representation along with a similarity measure called learned pattern similarity (LPS). A tree-based ensemble-learning strategy that is fast and insensitive to parameter settings is the basis for the approach. Then, a robust similarity measure based on the learned patterns is presented. This unsupervised approach to represent and measure the similarity between time series generally applies to a number of data mining tasks (e.g., clustering, anomaly detection, classification). Furthermore, an embedded learning of the representation avoids pre-defined features and an extraction step which is common in some feature-based approaches. The method generalizes in a straightforward manner to multivariate time series. The effectiveness of LPS is evaluated on time series classification problems from various domains. We compare LPS to eleven well-known similarity measures. Our experimental results show that LPS provides fast and competitive results on benchmark datasets from several domains. Furthermore, LPS provides a research direction and template approach that breaks from the linear dependency models to potentially foster other promising nonlinear approaches.