Journal ArticleDOI
Applying a kernel function on time-dependent data to provide supervised-learning guarantees
TLDR
A kernel function is applied to reconstruct time-dependent open-ended sequences of observations, also referred to as data streams in the context of Machine Learning, into multidimensional spaces in attempt to hold the data independency assumption.Abstract:
We employ a Monte-Carlo approach to find the best phase space for a given data stream.We propose kFTCV, a novel approach to validate data stream classification.Results show Taken's theorem can transform data streams into independent states.Therefore, we can rely on SLT framework to ensure learning when dealing with data streams. The Statistical Learning Theory (SLT) defines five assumptions to ensure learning for supervised algorithms. Data independency is one of those assumptions, once the SLT relies on the Law of Large Numbers to ensure learning bounds. As a consequence, this assumption imposes a strong limitation to guarantee learning on time-dependent scenarios. In order to tackle this issue, some researchers relax this assumption with the detriment of invalidating all theoretical results provided by the SLT. In this paper we apply a kernel function, more precisely the Takens' immersion theorem, to reconstruct time-dependent open-ended sequences of observations, also referred to as data streams in the context of Machine Learning, into multidimensional spaces (a.k.a. phase spaces) in attempt to hold the data independency assumption. At first, we study the best immersion parameterization for our kernel function using the Distance-Weighted Nearest Neighbors (DWNN). Next, we use this best immersion to recursively forecast next observations based on the prediction horizon, estimated using the Lyapunov exponent. Afterwards, predicted observations are compared against the expected ones using the Mean Distance from the Diagonal Line (MDDL). Theoretical and experimental results based on a cross-validation strategy provide stronger evidences of generalization, what allows us to conclude that one can learn from time-dependent data after using our approach. This opens up a very important possibility for ensuring supervised learning when it comes to time-dependent data, being useful to tackle applications such as in the climate, animal tracking, biology and other domains.read more
Citations
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The Laws of Large Numbers
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Decomposing time series into deterministic and stochastic influences: A survey
Felipe Simões Lage Gomes Duarte,Ricardo Araújo Rios,Eduardo R. Hruschka,Eduardo R. Hruschka,Rodrigo Fernandes de Mello +4 more
TL;DR: Each decomposition strategy is better devoted to particular scenarios, however, without any previous knowledge on data, GHKSS confirmed to work as a fair and general baseline besides its time complexity.
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Semi-supervised time series classification on positive and unlabeled problems using cross-recurrence quantification analysis
TL;DR: The use of the Maximum Diagonal Line of the Cross-Recurrence Quantification Analysis (MDL-CRQA), applied on the time series phase space, as similarity measurement, improves classification results for PU time series when compared against the mostly used time-domain similarity measurements.
References
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Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond
TL;DR: Learning with Kernels provides an introduction to SVMs and related kernel methods that provide all of the concepts necessary to enable a reader equipped with some basic mathematical knowledge to enter the world of machine learning using theoretically well-founded yet easy-to-use kernel algorithms.
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Tapas Kanungo,David M. Mount,Nathan S. Netanyahu,Christine D. Piatko,Ruth Silverman,Angela Y. Wu +5 more
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Nonlinear time series analysis
Holger Kantz,Thomas Schreiber +1 more
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