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Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/220091
La signatura de un camino y una aplicación práctica
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En este trabajo de final de grado se definirá el concepto de la signatura para caminos de clase $C^{1}$ a trozos y se estudiarán y demostrarán algunas de sus propiedades más importantes, que nos indican que existe una estrecha relación entre el camino y su signatura, cosa que nos permite utilizar esta última para describirlo. También, motivados por las propiedades comentadas previamente, se analizará el papel de la signatura en el
campo del machine learning, una de sus aplicaciones más recientes y activas, estudiando como se puede utilizar para construir atributos que sean capaces de mostrar y exponer las propiedades geométricas de las secuencias de datos consideradas. Además, pondremos en práctica esta última parte, aplicando la signatura junto con técnicas de machine learning al problema clásico de clasificación de dígitos manuscritos. Finalmente, propondremos dos nuevas generalizaciones de la signatura, motivadas por el cálculo fraccionario, de las
que estudiaremos también algunas de sus propiedades y aplicaciones, comparando sus resultados con los de la signatura original en el problema de reconocimiento de dígitos manuscritos, donde se observan mejoras significativas en las tasas de acierto.
In this undergraduate thesis we will define the concept of the signature for piecewise class $C^{1}$ paths and we will study and show some of its most important properties, which indicate that there is a close relationship between the path and its signature, allowing us to use the latter to describe it. Furthermore, motivated by the forementioned properties, we will analyse the role of the signature in the field of machine learning, one of its most recent and active applications, and we will study how it can be used to build attributes that are able to show and reveal the geometric properties of the considered data sequences. In addition, we will put this last part into practice by applying the signature together with machine learning techniques to the well-known problem of handwritten digit classification. Finally, we will propose two novel generalisations of the signature, motivated by fractional calculus, of which we will also study some of their properties and applications, comparing their results with those of the original signature in the handwritten digit recognition problem, where significant improvements in accuracy rates are observed.
In this undergraduate thesis we will define the concept of the signature for piecewise class $C^{1}$ paths and we will study and show some of its most important properties, which indicate that there is a close relationship between the path and its signature, allowing us to use the latter to describe it. Furthermore, motivated by the forementioned properties, we will analyse the role of the signature in the field of machine learning, one of its most recent and active applications, and we will study how it can be used to build attributes that are able to show and reveal the geometric properties of the considered data sequences. In addition, we will put this last part into practice by applying the signature together with machine learning techniques to the well-known problem of handwritten digit classification. Finally, we will propose two novel generalisations of the signature, motivated by fractional calculus, of which we will also study some of their properties and applications, comparing their results with those of the original signature in the handwritten digit recognition problem, where significant improvements in accuracy rates are observed.
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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2024, Director: José Manuel Corcuera Valverde
Matèries (anglès)
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JIMÉNEZ LUMBRERAS, Rubén. La signatura de un camino y una aplicación práctica. [consulta: 20 de gener de 2026]. [Disponible a: https://hdl.handle.net/2445/220091]