144 research outputs found

    Auto-adaptive multi-scale Laplacian Pyramids for modeling non-uniform data

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    Kernel-based techniques have become a common way for describing the local and global relationships of data samples that are generated in real-world processes. In this research, we focus on a multi-scale kernel based technique named Auto-adaptive Laplacian Pyramids (ALP). This method can be useful for function approximation and interpolation. ALP is an extension of the standard Laplacian Pyramids model that incorporates a modified Leave-One-Out Cross Validation procedure, which makes the method stable and automatic in terms of parameters selection without extra cost. This paper introduces a new algorithm that extends ALP to fit datasets that are non-uniformly distributed. In particular, the optimal stopping criterion will be point-dependent with respect to the local noise level and the sample rate. Experimental results over real datasets highlight the advantages of the proposed multi-scale technique for modeling and learning complex, high dimensional dataThey wish to thank Prof. Ronald R. Coifman for helpful remarks. They 525 also gratefully acknowledge the use of the facilities of Centro de Computación Científica (CCC) at Universidad Autónoma de Madrid. Funding: This work was supported by Spanish grants of the Ministerio de Ciencia, Innovación y Universidades [grant numbers: TIN2013-42351-P, TIN2015-70308-REDT, TIN2016-76406-P]; project CASI-CAM-CM supported by Madri+d 530 [grant number: S2013/ICE-2845]; project FACIL supported by Fundación BBVA (2016); and the UAM–ADIC Chair for Data Science and Machine Learnin

    Diffusion maps and local models for wind power prediction

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    The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-33266-1_70Proceedings of 22nd International Conference on Artificial Neural Networks, Lausanne, Switzerland, September 11-14, 2012In this work we will apply Diffusion Maps (DM), a recent technique for dimensionality reduction and clustering, to build local models for wind energy forecasting. We will compare ridge regression models for K–means clusters obtained over DM features, against the models obtained for clusters constructed over the original meteorological data or principal components, and also against a global model. We will see that a combination of the DM model for the low wind power region and the global model elsewhere outperforms other options.With partial support from grant TIN2010-21575-C02-01 of Spain’s Ministerio de Economía y Competitividad and the UAM–ADIC Chair for Machine Learning in Modelling and Prediction. The first author is also supported by an FPI-UAM grant and kindly thanks the Applied Mathematics Department of Yale University for receiving her during a visit. The second author is supported by the FPU-MEC grant AP2008-00167. We also thank Red Eléctrica de España, Spain’s TSO, for providing historic wind energy dat

    Diffusion, methods and applications

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    Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Escuela Politécnica Superior, Departamento de Ingeniería Informática. Fecha de lectura: junio de 2014Big Data, an important problem nowadays, can be understood in terms of a very large number of patterns, a very large pattern dimension or, often, both. In this thesis, we will concentrate on the high dimensionality issue, applying manifold learning techniques for visualizing and analyzing such patterns. The core technique will be Di usion Maps (DM) and its Anisotropic Di usion (AD) version, introduced by Ronald R. Coifman and his school at Yale University, and of which we will give a complete, systematic, compact and self-contained treatment. This will be done after a brief survey of previous manifold learning methods. The algorithmic contributions of the thesis will be centered in two computational challenges of di usion methods: the potential high cost of the similarity matrix eigenanalysis that is needed to define the di usion embedding coordinates, and the di culty of computing this embedding over new patterns not available for the initial eigenanalysis. With respect to the first issue, we will show how the AD set up can be used to skip it when looking for local models. In this case, local patterns will be selected through a k-Nearest Neighbors search using a properly defined local Mahalanobis distance, that enables neighbors to be found over the latent variable space underlying the AD model while we can work directly with the observable patterns and, thus, avoiding the potentially costly similarity matrix eigenanalysis. The second proposed algorithm, that we will call Auto-adaptative Laplacian Pyramids (ALP), focuses in the out-of-sample embedding extension and consists in a modification of the classical Laplacian Pyramids (LP) method. In this new algorithm the LP iterations will be combined with an estimate of the Leave One Out CV error, something that makes possible to directly define during training a criterion to estimate the optimal stopping point of this iterative algorithm. This thesis will also present several application contributions to important problems in renewable energy and medical imaging. More precisely, we will show how DM is a good method for dimensionality reduction of meteorological weather predictions, providing tools to visualize and describe these data, as well as to cluster them in order to define local models. In turn, we will apply our AD-based localized search method first to find the location in the human body of CT scan images and then to predict wind energy ramps on both individual farms and over the whole of Spain. We will see that, in both cases, our results improve on the current state of the art methods. Finally, we will compare our ALP proposal with the well-known Nyström method as well as with LP on two large dimensional problems, the time compression of meteorological data and the analysis of meteorological variables relevant in daily radiation forecasts. In both cases we will show that ALP compares favorably with the other approaches for out-of-sample extension problemsBig Data es un problema importante hoy en día, que puede ser entendido en términos de un amplio número de patrones, una alta dimensión o, como sucede normalmente, de ambos. Esta tesis se va a centrar en problemas de alta dimensión, aplicando técnicas de aprendizaje de subvariedades para visualizar y analizar dichos patrones. La técnica central será Di usion Maps (DM) y su versión anisotrópica, Anisotropic Di usion (AD), introducida por Ronald R. Coifman y su escuela en la Universidad de Yale, la cual va a ser tratada de manera completa, sistemática, compacta y auto-contenida. Esto se llevará a cabo tras un breve repaso de métodos previos de aprendizaje de subvariedades. Las contribuciones algorítmicas de esta tesis estarán centradas en dos de los grandes retos en métodos de difusión: el potencial alto coste que tiene el análisis de autovalores de la matriz de similitud, necesaria para definir las coordenadas embebidas; y la dificultad para calcular este mismo embedding sobre nuevos datos que no eran accesibles cuando se realizó el análisis de autovalores inicial. Respecto al primer tema, se mostrará cómo la aproximación AD se puede utilizar para evitar el cálculo del embedding cuando estamos interesados en definir modelos locales. En este caso, se seleccionarán patrones cercanos por medio de una búsqueda de vecinos próximos (k-NN), usando como distancia una medida de Mahalanobis local que permita encontrar vecinos sobre las variables latentes existentes bajo el modelo de AD. Todo esto se llevará a cabo trabajando directamente sobre los patrones observables y, por tanto, evitando el costoso cálculo que supone el cálculo de autovalores de la matriz de similitud. El segundo algoritmo propuesto, que llamaremos Auto-adaptative Laplacian Pyramids (ALP), se centra en la extensión del embedding para datos fuera de la muestra, y se trata de una modificación del método denominado Laplacian Pyramids (LP). En este nuevo algoritmo, las iteraciones de LP se combinarán con una estimación del error de Leave One Out CV, permitiendo definir directamente durante el periodo de entrenamiento, un criterio para estimar el criterio de parada óptimo para este método iterativo. En esta tesis se presentarán también una serie de contribuciones de aplicación de estas técnicas a importantes problemas en energías renovables e imágenes médicas. Más concretamente, se muestra como DM es un buen método para reducir la dimensión de predicciones del tiempo meteorológico, sirviendo por tanto de herramienta de visualización y descripción, así como de clasificación de los datos con vistas a definir modelos locales sobre cada grupo descrito. Posteriormente, se aplicará nuestro método de búsqueda localizada basado en AD tanto a la búsqueda de la correspondiente posición de tomografías en el cuerpo humano, como para la detección de rampas de energía eólica en parques individuales o de manera global en España. En ambos casos se verá como los resultados obtenidos mejoran los métodos del estado del arte actual. Finalmente se comparará el algoritmo de ALP propuesto frente al conocido método de Nyström y al método de LP, en dos problemas de alta dimensión: el problema de compresión temporal de datos meteorológicos y el análisis de variables meteorológicas relevantes para la predicción de la radiación diaria. En ambos casos se mostrará cómo ALP es comparativamente mejor que otras aproximaciones existentes para resolver el problema de extensión del embedding a puntos fuera de la muestr

    Positive semi-definite embedding for dimensionality reduction and out-of-sample extensions

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    In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space Rd\mathbb{R}^d. This paper introduces a dimensionality reduction method where the embedding coordinates are the eigenvectors of a positive semi-definite kernel obtained as the solution of an infinite dimensional analogue of a semi-definite program. This embedding is adaptive and non-linear. A main feature of our approach is the existence of a non-linear out-of-sample extension formula of the embedding coordinates, called a projected Nystr\"om approximation. This extrapolation formula yields an extension of the kernel matrix to a data-dependent Mercer kernel function. Our empirical results indicate that this embedding method is more robust with respect to the influence of outliers, compared with a spectral embedding method.Comment: 16 pages, 5 figures. Improved presentatio
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