Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping.
The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree...
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Format: | info:eu-repo/semantics/doctoralThesis |
Language: | eng |
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Servicio de Publicaciones. Universidad de Navarra
2023
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Online Access: | https://hdl.handle.net/10171/67243 |
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author | Martínez Lopez, I. (Iñigo) Viles-Díez, E. (Elisabeth) García Olaizola, I. (Igor) |
author_facet | Martínez Lopez, I. (Iñigo) Viles-Díez, E. (Elisabeth) García Olaizola, I. (Igor) |
author_sort | Martínez Lopez, I. (Iñigo) |
collection | DSpace |
description | The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data.
Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers.
This thesis proposes novel elastic alignment methods that use parametric & diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable & invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence.
Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include:
(a) an enhanced temporal transformer network for time series alignment and averaging,
(b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy,
(c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally,
(d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers.
Taken together, these advancements demonstrate the versatility and potential of closed-form diffeomorphic transformations for a range of time series applications.
In summary, this thesis aims to enhance time-series tasks such as alignment, averaging, classification and clustering by leveraging the power of fast, efficient, parametric \& diffeomorphic warping methods. |
format | info:eu-repo/semantics/doctoralThesis |
id | oai:dadun.unav.edu:10171-67243 |
institution | Universidad de Navarra |
language | eng |
publishDate | 2023 |
publisher | Servicio de Publicaciones. Universidad de Navarra |
record_format | dspace |
spelling | oai:dadun.unav.edu:10171-672432023-09-18T05:16:40Z Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. Martínez Lopez, I. (Iñigo) Viles-Díez, E. (Elisabeth) García Olaizola, I. (Igor) Time series Nonlinear time warping Similarity distance Diffeomorphic transformations Machine learning Deep learning The proliferation and ubiquity of temporal data across many disciplines has sparked interest for similarity, classification and clustering methods specifically designed to handle time series data. A core issue when dealing with time series is determining their pairwise similarity, i.e., the degree to which a given time series resembles another. Traditional distance measures such as the Euclidean are not well-suited due to the time-dependent nature of the data. Elastic metrics such as dynamic time warping (DTW) offer a promising approach, but are limited by their computational complexity, non-differentiability and sensitivity to noise and outliers. This thesis proposes novel elastic alignment methods that use parametric & diffeomorphic warping transformations as a means of overcoming the shortcomings of DTW-based metrics. The proposed method is differentiable & invertible, well-suited for deep learning architectures, robust to noise and outliers, computationally efficient, and is expressive and flexible enough to capture complex patterns. Furthermore, a closed-form solution was developed for the gradient of these diffeomorphic transformations, which allows an efficient search in the parameter space, leading to better solutions at convergence. Leveraging the benefits of these closed-form diffeomorphic transformations, this thesis proposes a suite of advancements that include: (a) an enhanced temporal transformer network for time series alignment and averaging, (b) a deep-learning based time series classification model to simultaneously align and classify signals with high accuracy, (c) an incremental time series clustering algorithm that is warping-invariant, scalable and can operate under limited computational and time resources, and finally, (d) a normalizing flow model that enhances the flexibility of affine transformations in coupling and autoregressive layers. Taken together, these advancements demonstrate the versatility and potential of closed-form diffeomorphic transformations for a range of time series applications. In summary, this thesis aims to enhance time-series tasks such as alignment, averaging, classification and clustering by leveraging the power of fast, efficient, parametric \& diffeomorphic warping methods. La proliferación y ubicuidad de los datos temporales en diversas disciplinas ha despertado un gran interés por los métodos de similitud, clasificación y agrupación diseñados específicamente para manejar series temporales. Uno de los principales desafíos al tratar con este tipo de datos es determinar su similitud. Las medidas de distancia tradicionales, como la euclídea, resultan inadecuadas debido a la naturaleza temporal de los datos. Métricas elásticas como la deformación temporal dinámica (DTW) ofrecen un enfoque correcto, pero están limitadas por su complejidad computacional, no diferenciabilidad y sensibilidad al ruido. Esta tesis propone nuevos métodos de alineación elástica que emplean transformaciones de deformación paramétricas y difeomórficas para superar las limitaciones de las métricas basadas en DTW. El método propuesto es diferenciable e invertible, lo que lo hace adecuado para arquitecturas de aprendizaje profundo. Además, es robusto frente al ruido, computacionalmente eficiente y lo suficientemente expresivo y flexible como para capturar patrones complejos. Asimismo, se ha desarrollado una solución cerrada para el cálculo del gradiente de estas transformaciones difeomórficas, lo que permite una búsqueda eficiente en el espacio de parámetros y conduce a mejores soluciones en convergencia. Aprovechando los beneficios de estas transformaciones difeomórficas de forma cerrada, esta tesis propone un conjunto de avances que incluyen: (a) un transformador temporal vitaminado para la alineación y promediado de series temporales, (b) un modelo de clasificación de series temporales basado en aprendizaje profundo para alinear y clasificar simultáneamente señales con alta precisión, (c) un algoritmo incremental de agrupación de series temporales que es invariable a la deformación, escalable y puede funcionar con recursos computacionales limitados, y finalmente, (d) un modelo de flujo normalizador que mejora la flexibilidad de las transformaciones afines en las capas de acoplamiento y autorregresivas. En conjunto, estos avances demuestran la versatilidad y el potencial de las transformaciones difeomórficas en forma cerrada para diversas aplicaciones relacionadas con las series temporales. En resumen, esta tesis pretende mejorar tareas como la alineación, el promediado, la clasificación y la agrupación de series temporales aprovechando la potencia y eficiencia de los métodos paramétricos de deformación difeomórfica. 2023-09-15T10:34:11Z 2023-09-15T10:34:11Z 2023-09-11 2023-07-17 info:eu-repo/semantics/doctoralThesis https://hdl.handle.net/10171/67243 eng info:eu-repo/semantics/openAccess application/pdf Servicio de Publicaciones. Universidad de Navarra |
spellingShingle | Time series Nonlinear time warping Similarity distance Diffeomorphic transformations Machine learning Deep learning Martínez Lopez, I. (Iñigo) Viles-Díez, E. (Elisabeth) García Olaizola, I. (Igor) Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title | Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title_full | Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title_fullStr | Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title_full_unstemmed | Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title_short | Diffeomorphic transformations for time series analysis: An efficient approach to nonlinear warping. |
title_sort | diffeomorphic transformations for time series analysis: an efficient approach to nonlinear warping. |
topic | Time series Nonlinear time warping Similarity distance Diffeomorphic transformations Machine learning Deep learning |
url | https://hdl.handle.net/10171/67243 |
work_keys_str_mv | AT martinezlopeziinigo diffeomorphictransformationsfortimeseriesanalysisanefficientapproachtononlinearwarping AT vilesdiezeelisabeth diffeomorphictransformationsfortimeseriesanalysisanefficientapproachtononlinearwarping AT garciaolaizolaiigor diffeomorphictransformationsfortimeseriesanalysisanefficientapproachtononlinearwarping |