Introduction to the analysis of dependent data.
Authorship
N.D.D.
Bachelor of Mathematics
N.D.D.
Bachelor of Mathematics
Defense date
09.15.2025 11:30
09.15.2025 11:30
Summary
In this work, the study of time series is addressed through the analysis of real datasets that present a dependence over time. This dependence makes traditional statistical methods, based on the assumption of independence between observations, inadequate. For this reason, it is necessary to use specific techniques that take into account the temporary structure of this data. The main objective of this work is to apply several tools about time series analysis in order to model, interpret and predict the behaviour of a variable measured over time. To this end, an exploratory description of the available series will be carried out, the presence of patterns such as trends or seasonality will be identified, and appropriate models will be adjusted, such as autoregressive moving average models (ARMA) or autoregressive integrated moving average models (ARIMA), which allow us to predict at future time points. Likewise, Statistical Inference procedures and the validation techniques of the considered models will also be presented. Finally, all the methodology presented will be put into practice using the statistical software R, in order to highlight the usefulness of the presented techniques in the analysis of real data with temporal dependence.
In this work, the study of time series is addressed through the analysis of real datasets that present a dependence over time. This dependence makes traditional statistical methods, based on the assumption of independence between observations, inadequate. For this reason, it is necessary to use specific techniques that take into account the temporary structure of this data. The main objective of this work is to apply several tools about time series analysis in order to model, interpret and predict the behaviour of a variable measured over time. To this end, an exploratory description of the available series will be carried out, the presence of patterns such as trends or seasonality will be identified, and appropriate models will be adjusted, such as autoregressive moving average models (ARMA) or autoregressive integrated moving average models (ARIMA), which allow us to predict at future time points. Likewise, Statistical Inference procedures and the validation techniques of the considered models will also be presented. Finally, all the methodology presented will be put into practice using the statistical software R, in order to highlight the usefulness of the presented techniques in the analysis of real data with temporal dependence.
Direction
CONDE AMBOAGE, MERCEDES (Tutorships)
CONDE AMBOAGE, MERCEDES (Tutorships)
Court
CONDE AMBOAGE, MERCEDES (Student’s tutor)
CONDE AMBOAGE, MERCEDES (Student’s tutor)
Clustering methods based on high-density region estimation
Authorship
A.G.L.
Bachelor of Mathematics
A.G.L.
Bachelor of Mathematics
Defense date
09.15.2025 13:00
09.15.2025 13:00
Summary
Density-based clustering offers a flexible alternative to classical clustering techniques, as it allows the detection of clusters with arbitrary shapes without requiring the number of groups to be specified in advance. Moreover, it adapts the shape of the clusters to the actual distribution of the data, without assuming any prior structure. After reviewing the main clustering approaches, the project focuses on non-parametric density estimation and analyzes two strategies to identify high-density regions: Delaunay triangulation and line segments between pairs of points. These techniques are applied to a real dataset on breast tumors and are compared to traditional algorithms such as $k$-means, Gaussian mixture models, and hierarchical clustering. The results allow for an assessment of the strengths and limitations of each clustering method on this dataset, as well as the potential of these models as support tools for medical diagnosis.
Density-based clustering offers a flexible alternative to classical clustering techniques, as it allows the detection of clusters with arbitrary shapes without requiring the number of groups to be specified in advance. Moreover, it adapts the shape of the clusters to the actual distribution of the data, without assuming any prior structure. After reviewing the main clustering approaches, the project focuses on non-parametric density estimation and analyzes two strategies to identify high-density regions: Delaunay triangulation and line segments between pairs of points. These techniques are applied to a real dataset on breast tumors and are compared to traditional algorithms such as $k$-means, Gaussian mixture models, and hierarchical clustering. The results allow for an assessment of the strengths and limitations of each clustering method on this dataset, as well as the potential of these models as support tools for medical diagnosis.
Direction
SAAVEDRA NIEVES, PAULA (Tutorships)
SAAVEDRA NIEVES, PAULA (Tutorships)
Court
SAAVEDRA NIEVES, PAULA (Student’s tutor)
SAAVEDRA NIEVES, PAULA (Student’s tutor)
Difference equations: fundamentals, convergence and applications
Authorship
B.G.V.
Bachelor of Mathematics
B.G.V.
Bachelor of Mathematics
Defense date
09.15.2025 12:15
09.15.2025 12:15
Summary
This Bachelor's Thesis focuses on the study of difference equations, which describe dynamic processes evolving in discrete time. Building on foundational concepts such as sequences and recurrence relations, the work explores various types of equations, including higher-order cases and multidimensional systems. Special attention is given to the convergence of solutions and the stability of equilibrium points, with the Copson Theorem presented as a generalization of the classical monotone convergence theorem. Through theoretical examples and practical applications, this work highlights the relevance of these tools in both mathematical and scientific contexts.
This Bachelor's Thesis focuses on the study of difference equations, which describe dynamic processes evolving in discrete time. Building on foundational concepts such as sequences and recurrence relations, the work explores various types of equations, including higher-order cases and multidimensional systems. Special attention is given to the convergence of solutions and the stability of equilibrium points, with the Copson Theorem presented as a generalization of the classical monotone convergence theorem. Through theoretical examples and practical applications, this work highlights the relevance of these tools in both mathematical and scientific contexts.
Direction
Rodríguez López, Rosana (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
Rodríguez López, Rosana (Tutorships)
BUEDO FERNANDEZ, SEBASTIAN (Co-tutorships)
Court
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
Rodríguez López, Rosana (Student’s tutor)
BUEDO FERNANDEZ, SEBASTIAN (Student’s tutor)
Rodríguez López, Rosana (Student’s tutor)
Introduction to time series analysis
Authorship
I.G.A.
Bachelor of Mathematics
I.G.A.
Bachelor of Mathematics
Defense date
09.15.2025 16:30
09.15.2025 16:30
Summary
A time series is not anything other than a data set collected in regular time lapses (hours, days, weeks, years...) This type of data naturally appears in a variety of fields such as economy (monthly employement rate), industry (daily production of a factory), demography (region population throughout some decades), enviroment (seasonal maximum daily temperature)... In this project, we will present a brief introduction to time series analysis, both from a descriptive perspective and from a more formal viewpoint based on the concept of a stochastic process. The latter will allow us to introduce and define the main model families according to the Box-Jenkins methodology. In addition to that, we will provide real-life examples to illustrate the application of these models using the statistic software R as our main tool.
A time series is not anything other than a data set collected in regular time lapses (hours, days, weeks, years...) This type of data naturally appears in a variety of fields such as economy (monthly employement rate), industry (daily production of a factory), demography (region population throughout some decades), enviroment (seasonal maximum daily temperature)... In this project, we will present a brief introduction to time series analysis, both from a descriptive perspective and from a more formal viewpoint based on the concept of a stochastic process. The latter will allow us to introduce and define the main model families according to the Box-Jenkins methodology. In addition to that, we will provide real-life examples to illustrate the application of these models using the statistic software R as our main tool.
Direction
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
BORRAJO GARCIA, MARIA ISABEL (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Survival analysis in multistate models
Authorship
L.G.D.
Bachelor of Mathematics
L.G.D.
Bachelor of Mathematics
Defense date
09.15.2025 11:00
09.15.2025 11:00
Summary
This work addresses Survival Analysis using multi-state models. These are a key statistical tool for describing processes that evolve over time through different phases, such as the progression of chronic diseases. A theoretical foundation based on Markov chains and Poisson processes is presented, and methods are explored to estimate transition rates via maximum likelihood from intermittently observed data. The use of covariates is also incorporated to model the effect of biomedical factors. The theoretical development is accompanied by examples that illustrate practical applications of the theory, using both simulated and real data. A more detailed simulation study made it possible to analyze the properties of the estimators depending on the number of individuals observed and the observation frequency. The aim of this work is to provide a comprehensive overview, from theory to practical application, of the potential of these models in the context of clinical studies.
This work addresses Survival Analysis using multi-state models. These are a key statistical tool for describing processes that evolve over time through different phases, such as the progression of chronic diseases. A theoretical foundation based on Markov chains and Poisson processes is presented, and methods are explored to estimate transition rates via maximum likelihood from intermittently observed data. The use of covariates is also incorporated to model the effect of biomedical factors. The theoretical development is accompanied by examples that illustrate practical applications of the theory, using both simulated and real data. A more detailed simulation study made it possible to analyze the properties of the estimators depending on the number of individuals observed and the observation frequency. The aim of this work is to provide a comprehensive overview, from theory to practical application, of the potential of these models in the context of clinical studies.
Direction
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
SANCHEZ SELLERO, CESAR ANDRES (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Linear regression with measurement errors in data
Authorship
P.L.L.
Bachelor of Mathematics
P.L.L.
Bachelor of Mathematics
Defense date
09.15.2025 17:15
09.15.2025 17:15
Summary
Study of linear regression models with measurement errors in the sample: introduction of measurement errors, analysis of their effect on the inference of the main model parameters, and explanation of different techniques for correcting them whenever possible.Theoretical study of common regression models considering measurement errors, followed by the practical application of the results in real-world examples.
Study of linear regression models with measurement errors in the sample: introduction of measurement errors, analysis of their effect on the inference of the main model parameters, and explanation of different techniques for correcting them whenever possible.Theoretical study of common regression models considering measurement errors, followed by the practical application of the results in real-world examples.
Direction
AMEIJEIRAS ALONSO, JOSE (Tutorships)
ALONSO PENA, MARIA (Co-tutorships)
AMEIJEIRAS ALONSO, JOSE (Tutorships)
ALONSO PENA, MARIA (Co-tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Dynamic systems in economics
Authorship
D.M.G.
Bachelor of Mathematics
D.M.G.
Bachelor of Mathematics
Defense date
09.15.2025 11:45
09.15.2025 11:45
Summary
Dynamical systems are a fundamental tool for explaining real-world phenomena, including those in economics. This work presents a qualitative analysis of various economic models formulated as autonomous differential equations, exploring their behavior using different analytical techniques. Three models of particular importance in economic dynamics are studied in detail: the Solow Swan growth model, the Goodwin business cycle model, and the Kaldor cycle model. Their equilibrium points, stability properties, and possible oscillatory dynamics are examined. In addition, bifurcation theory, especially Hopf bifurcations, is employed to explain the emergence of limit cycles and endogenous fluctuations in nonlinear systems. This approach highlights how small changes in parameters can lead to qualitative shifts in economic behavior, offering a rigorous and visual understanding of the dynamics of economic systems.
Dynamical systems are a fundamental tool for explaining real-world phenomena, including those in economics. This work presents a qualitative analysis of various economic models formulated as autonomous differential equations, exploring their behavior using different analytical techniques. Three models of particular importance in economic dynamics are studied in detail: the Solow Swan growth model, the Goodwin business cycle model, and the Kaldor cycle model. Their equilibrium points, stability properties, and possible oscillatory dynamics are examined. In addition, bifurcation theory, especially Hopf bifurcations, is employed to explain the emergence of limit cycles and endogenous fluctuations in nonlinear systems. This approach highlights how small changes in parameters can lead to qualitative shifts in economic behavior, offering a rigorous and visual understanding of the dynamics of economic systems.
Direction
OTERO ESPINAR, MARIA VICTORIA (Tutorships)
Diz Pita, Érika (Co-tutorships)
OTERO ESPINAR, MARIA VICTORIA (Tutorships)
Diz Pita, Érika (Co-tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Fixed-point index theory and applications to differential equations
Authorship
V.L.M.A.
Bachelor of Mathematics
V.L.M.A.
Bachelor of Mathematics
Defense date
09.15.2025 12:30
09.15.2025 12:30
Summary
The fixed point index theory is a valuable tool for its usefulness in many branches of mathematics and for its numerous applications to economics, game theory or analysis. In this Bachelor's Final Project, we collect the main concepts related to this theory from a mainly analytical approach, without forgetting its close relation with topology and differential geometry.First we will give a definition of the topological degree or Brouwer's degree for continuous functions in finite dimensional spaces, which will allow us to develop interesting properties of this theory, and in particular, to prove Brouwer's Fixed Point Theorem. Subsequently, we will extend these concepts to infinite dimensional spaces with the Leray-Schauder Theorem. We will seek to give practical sense to the results presented, analyzing some of their applications in the study of the solutions of differential equations. Finally, we will study an extension of the degree to operators defined on cones, and we will present a concrete example of application: the analysis of a tubular reactor.
The fixed point index theory is a valuable tool for its usefulness in many branches of mathematics and for its numerous applications to economics, game theory or analysis. In this Bachelor's Final Project, we collect the main concepts related to this theory from a mainly analytical approach, without forgetting its close relation with topology and differential geometry.First we will give a definition of the topological degree or Brouwer's degree for continuous functions in finite dimensional spaces, which will allow us to develop interesting properties of this theory, and in particular, to prove Brouwer's Fixed Point Theorem. Subsequently, we will extend these concepts to infinite dimensional spaces with the Leray-Schauder Theorem. We will seek to give practical sense to the results presented, analyzing some of their applications in the study of the solutions of differential equations. Finally, we will study an extension of the degree to operators defined on cones, and we will present a concrete example of application: the analysis of a tubular reactor.
Direction
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
MAJADAS MOURE, ALEJANDRO OMAR (Co-tutorships)
FERNANDEZ TOJO, FERNANDO ADRIAN (Tutorships)
MAJADAS MOURE, ALEJANDRO OMAR (Co-tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Introduction to bifurcation theory
Authorship
P.P.B.
Bachelor of Mathematics
P.P.B.
Bachelor of Mathematics
Defense date
09.15.2025 13:15
09.15.2025 13:15
Summary
What happens when a minimal variation in a parameter leads to a drastic change in the behavior of a system? This question is what motivates the study of bifurcation theory. This work presents an accessible overview of this theory, focused on local bifurcations in continuous dynamical systems represented by ordinary differential equations (ODEs). Firstly, basic concepts of dynamical systems and differential equations are introduced, which are necessary to understand the phenomenon of bifurcation. Subsequently, using a motivating example, it is shown how small variations in parameters can lead to qualitatively different behaviors, highlighting the need to develop an appropriate theoretical framework. On this basis, different types of bifurcations (such as fold, transcritical, pitchfork, and Hopf) are analyzed both theoretically and graphically. Finally, the initial example is revisited to apply the developed theory. The work aims to combine the formal development of the theory with its application in order to understand real-world phenomena.
What happens when a minimal variation in a parameter leads to a drastic change in the behavior of a system? This question is what motivates the study of bifurcation theory. This work presents an accessible overview of this theory, focused on local bifurcations in continuous dynamical systems represented by ordinary differential equations (ODEs). Firstly, basic concepts of dynamical systems and differential equations are introduced, which are necessary to understand the phenomenon of bifurcation. Subsequently, using a motivating example, it is shown how small variations in parameters can lead to qualitatively different behaviors, highlighting the need to develop an appropriate theoretical framework. On this basis, different types of bifurcations (such as fold, transcritical, pitchfork, and Hopf) are analyzed both theoretically and graphically. Finally, the initial example is revisited to apply the developed theory. The work aims to combine the formal development of the theory with its application in order to understand real-world phenomena.
Direction
OTERO ESPINAR, MARIA VICTORIA (Tutorships)
OTERO ESPINAR, MARIA VICTORIA (Tutorships)
Court
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
CABADA FERNANDEZ, ALBERTO (Chairman)
BORRAJO GARCIA, MARIA ISABEL (Secretary)
MUÑOZ SOLA, RAFAEL (Member)
Modeling of Macroeconomic Time Series
Authorship
A.P.S.
Bachelor of Mathematics
A.P.S.
Bachelor of Mathematics
Defense date
09.15.2025 18:00
09.15.2025 18:00
Summary
This project is about Time Series, a useful tool in statistics to study the evolution in time of some data. The approach given here will be of macroeconomic time series, produced with mensual or greater periodicity, because of the ability of these series to predict future values from past values observed from the data. The aim of this study is to model a particular time series, in this case Total air traffic passengers in Galicia, to predict future values of this variable. First of all, the concept of time series will be introduced, together with the necessary tools and techniques for the analysis. A useful method to analyze time series is presented here, it is Box-Jenkins methodology, and it is classified in three stages (detection, estimation and model checking) using ARIMA models in order to model the time series. ARIMA models (Autoregressive Integrated Moving Average models) are the most used models in time series, and they are described in this section. The last part of this process consists of obtaining future forecasts of time series, in this case, the total number of air traffic passengers.
This project is about Time Series, a useful tool in statistics to study the evolution in time of some data. The approach given here will be of macroeconomic time series, produced with mensual or greater periodicity, because of the ability of these series to predict future values from past values observed from the data. The aim of this study is to model a particular time series, in this case Total air traffic passengers in Galicia, to predict future values of this variable. First of all, the concept of time series will be introduced, together with the necessary tools and techniques for the analysis. A useful method to analyze time series is presented here, it is Box-Jenkins methodology, and it is classified in three stages (detection, estimation and model checking) using ARIMA models in order to model the time series. ARIMA models (Autoregressive Integrated Moving Average models) are the most used models in time series, and they are described in this section. The last part of this process consists of obtaining future forecasts of time series, in this case, the total number of air traffic passengers.
Direction
FEBRERO BANDE, MANUEL (Tutorships)
FEBRERO BANDE, MANUEL (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
A review of finite difference algorithms in parabolic models
Authorship
A.R.P.
Bachelor of Mathematics
A.R.P.
Bachelor of Mathematics
Defense date
09.15.2025 13:00
09.15.2025 13:00
Summary
In this work we will study the main finite difference methods, more specifically their application to the numerical resolution of parabolic partial differential equations (PDEs). At first, the basic structure of finite difference methods is introduced and some fundamental concepts for the analysis of numerical methods are defined. Then, we will make the theoretical analysis of each of the finite difference methods that we will address. These are the explicit and implicit methods, the Crank-Nicolson method and the method of lines. We will also make use of academic tests to analyze the behavior of these methods in response to changes in the parameters of each PDE. The results will be compared with the ones obtained using own functions of MATLAB. Finally, we will present the application to a heat transfer problem.
In this work we will study the main finite difference methods, more specifically their application to the numerical resolution of parabolic partial differential equations (PDEs). At first, the basic structure of finite difference methods is introduced and some fundamental concepts for the analysis of numerical methods are defined. Then, we will make the theoretical analysis of each of the finite difference methods that we will address. These are the explicit and implicit methods, the Crank-Nicolson method and the method of lines. We will also make use of academic tests to analyze the behavior of these methods in response to changes in the parameters of each PDE. The results will be compared with the ones obtained using own functions of MATLAB. Finally, we will present the application to a heat transfer problem.
Direction
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
QUINTELA ESTEVEZ, PEREGRINA (Tutorships)
Court
QUINTELA ESTEVEZ, PEREGRINA (Student’s tutor)
QUINTELA ESTEVEZ, PEREGRINA (Student’s tutor)
Fermat's Last Theorem for regular primes
Authorship
M.R.C.
Bachelor of Mathematics
M.R.C.
Bachelor of Mathematics
Defense date
09.15.2025 18:45
09.15.2025 18:45
Summary
This thesis presents an exposition of Fermat’s Last Theorem in the case where the exponent is a regular prime, as well as its proof when n= 4. The theorem, formulated in the 17th century, states that there are no non-zero integer solutions of the equation xn + yn = zn when n greater than 2. In the mid-19th century, Kummer proved the result for regular primes, introducing innovative tools such as cyclotomic integers, ideal factorization, and the concept of regularity of a prime. This thesis reviews these fundamental ideas and also presents the proof for the case n= 4, which can be approached with more elementary methods. The aim is to provide a clear introduction to Kummer’s algebraic approach and to show how his work laid the foundations of modern number theory, paving the way for Wiles’ general proof in the 20th century.
This thesis presents an exposition of Fermat’s Last Theorem in the case where the exponent is a regular prime, as well as its proof when n= 4. The theorem, formulated in the 17th century, states that there are no non-zero integer solutions of the equation xn + yn = zn when n greater than 2. In the mid-19th century, Kummer proved the result for regular primes, introducing innovative tools such as cyclotomic integers, ideal factorization, and the concept of regularity of a prime. This thesis reviews these fundamental ideas and also presents the proof for the case n= 4, which can be approached with more elementary methods. The aim is to provide a clear introduction to Kummer’s algebraic approach and to show how his work laid the foundations of modern number theory, paving the way for Wiles’ general proof in the 20th century.
Direction
Jeremías López, Ana (Tutorships)
Jeremías López, Ana (Tutorships)
Court
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
RODRIGUEZ CASAL, ALBERTO (Chairman)
ALONSO TARRIO, LEOVIGILDO (Secretary)
SALGADO SECO, MODESTO RAMON (Member)
Inequalities on Olympic Problems
Authorship
J.V.Z.
Bachelor of Mathematics
J.V.Z.
Bachelor of Mathematics
Defense date
09.16.2025 11:00
09.16.2025 11:00
Summary
The main goal of this work is to present the different inequalities used to solve many of the problems proposed in the different Mathematical Olympiad competitions, from the regional level to the international -and the relatively new EGMO, at a female level. In adition, we present a selection of solved problems using one or more of these inequalities, noting the power they have to aproximate or solve each problem.
The main goal of this work is to present the different inequalities used to solve many of the problems proposed in the different Mathematical Olympiad competitions, from the regional level to the international -and the relatively new EGMO, at a female level. In adition, we present a selection of solved problems using one or more of these inequalities, noting the power they have to aproximate or solve each problem.
Direction
GAGO COUSO, FELIPE (Tutorships)
GAGO COUSO, FELIPE (Tutorships)
Court
GAGO COUSO, FELIPE (Student’s tutor)
GAGO COUSO, FELIPE (Student’s tutor)
Review and Comparison of Clustering Methodologies
Authorship
M.V.B.
Bachelor of Mathematics
M.V.B.
Bachelor of Mathematics
Defense date
09.16.2025 11:30
09.16.2025 11:30
Summary
With the growing volume of available data, understanding its internal structure without relying on prior information has become a crucial task in data analysis. This work explores clustering methodologies, or unsupervised grouping, which allow the discovery of natural groupings within the data without prior knowledge of their categories. The main objective is to analyze and compare different families of clustering algorithms. We examine partitional methods such as k-means and k-medoids, which are effective in scenarios where the clusters have simple and well-defined shapes; hierarchical methods, which build progressive groupings; model-based approaches, which provide a probabilistic and more flexible representation of the data; and density-based algorithms such as DBSCAN, capable of identifying clusters with irregular shapes as well as isolated observations. Throughout the work, we describe the mathematical foundations, core algorithms, and the main advantages and limitations of each method. We also analyze factors such as the choice of parameters and the types of groups each method is able to detect. Finally, a dataset is presented as a case study in which these methodologies are applied, with the aim of visualizing and comparing the results in a practical manner, thus facilitating the understanding of the different approaches.
With the growing volume of available data, understanding its internal structure without relying on prior information has become a crucial task in data analysis. This work explores clustering methodologies, or unsupervised grouping, which allow the discovery of natural groupings within the data without prior knowledge of their categories. The main objective is to analyze and compare different families of clustering algorithms. We examine partitional methods such as k-means and k-medoids, which are effective in scenarios where the clusters have simple and well-defined shapes; hierarchical methods, which build progressive groupings; model-based approaches, which provide a probabilistic and more flexible representation of the data; and density-based algorithms such as DBSCAN, capable of identifying clusters with irregular shapes as well as isolated observations. Throughout the work, we describe the mathematical foundations, core algorithms, and the main advantages and limitations of each method. We also analyze factors such as the choice of parameters and the types of groups each method is able to detect. Finally, a dataset is presented as a case study in which these methodologies are applied, with the aim of visualizing and comparing the results in a practical manner, thus facilitating the understanding of the different approaches.
Direction
PATEIRO LOPEZ, BEATRIZ (Tutorships)
PATEIRO LOPEZ, BEATRIZ (Tutorships)
Court
PATEIRO LOPEZ, BEATRIZ (Student’s tutor)
PATEIRO LOPEZ, BEATRIZ (Student’s tutor)