This thesis aims to give insight into a novel approach for quantifying car traffic in developing cities. This is necessary to improve efficiency in resource allocation for improvements in infrastructure. The project took form of a case study of neighborhoods in the city of Nairobi, Kenya. The approach consists of a method which relies on topics from the field of Topological Data Analysis, together with the use of large data sources from taxi services in the city. With this, both qualitative and quantitative insight can be given about the traffic. The method was proven useful for understanding how traffic spreads, and to differentiate between levels of congestion: quantifying it. However, it failed to detect the effect of previous improvements of infrastructure. / Målet med rapporten är att ge insikt i en innovativ ansats för att kvantifiera biltrafik i utvecklingsstäder. Detta kommer som en nödvändighet för att kunna förbättra resursfördelning i utvecklandet av infrastruktur. Projektet utspelade sig som en fallstudie där stadsdelar i Nairobi, Kenya studerades. Ansatsen innefattar en metod som bygger på tekniker från topologisk dataanalys (eng. \textit{Topological Data Analysis}), tillsammans med stora datakällor från taxitjänster i staden. Detta hoppas ge både kvalitativ och kvantitativ information om trafiken i staden. Metoden visade sig vara användbar för att förstå hur trafik sprider sig och att differentiera mellan nivåer av trafik, alltså att kvantifiera den. Tyvärr så misslyckades metoden visa sig användbar för att mäta förbättringar i infrastruktur.

This thesis concerns the topological features of plane geometric shapes and financial transaction data. Topological properties of the data such as homology groups and their stable ranks are analysed. It is investigated how to mathematically describe differences between data sets and it is found that stable ranks can be used to capture these differences. Sub sampling is introduced as a way to apply stochastic methods to geometric structures. It is found that the average stable rank can be used to differentiate data sets. Furthermore, the sensitivity of average stable ranks to random noise is explored and it is studied how a single point changes the average stable ranks of geometric shapes and financial transaction data. A method to incorporate categorical data within the analysis is introduced. The theory is applied to financial transaction data with the objective to understand if there are topological differences between fraudulent and legit transactions which can be used to classify them. / I denna uppsats analyseras finansiell transaktionsdata samt plana geometriska objekt med hjälp av verktyg inom Topologisk Dataanalys. Topologiska egenskaper såsom homologi samt stabil rang analyseras och det undersöks hur en matematiskt kan beskriva skillnaden mellan geometriska objekt. Det visar sig att simplistiska komplex och dess motsvarande stabila rang kan användas för att beskriva dessa skillnader. Det undersöks även hur stokastiska metoder kan appliceras på geometrisk data och begreppet genomsnittlig stabil rang introduceras. Känsligheten för brus hos den genomsnittliga stabila rangen undersöks för plana objekt och det undersöks hur den genomsnittliga stabila rangen av en datamängd ändras om en datapunkt läggs till. En metod för att beskriva avstånd på kategorisk data introduceras eftersom analysen av stabil rang kräver ett definierat avstånd mellan datapunkter. Det undersöks huruvida det finns topologiska skillnader mellan bedrägliga och icke-bedrägliga transaktioner, samt om det finns skillnader mellan olika typer av bedrägliga transaktioner.

Mudanças de padrão são variações nos dados da série temporal. Tais mudanças podem representar transições que ocorrem entre estados. A análise de dados topológicos (TDA) permite uma caracterização de dados de séries temporais obtidos a partir de sistemas dinâmicos complexos. Neste trabalho, apresentamos uma técnica de detecção de mudança de padrão baseada em TDA. Especificamente, a partir de uma determinada série temporal, dividimos o sinal em janelas deslizantes sem sobreposição e para cada janela calculamos a homologia persistente, ou seja, o barcode associado. A partir desse barcode, o intervalo médio e a entropia persistente são calculados e plotados em relação à duração do sinal. Resultados experimentais em conjuntos de dados reais e artificiais mostram bons resultados do método proposto: 1) Detecta mudança de padrões identificando a mudança no intervalo médio e calculando a entropia persistente para os barcodes gerados pelo conjunto de dados de entrada. 2) Mostra qualitativamente quão sensível é a escolha do método de filtragem para evidenciar características topológicas do espaço original sob exame. Isto é conseguido usando duas filtragens: uma filtragem métrica e uma do tipo lower-star. 3) Variando o tamanho da janela, o método pode caracterizar a presença de estruturas locais do conjunto de dados, como o período de convulsão nos sinais EEG. 4) O método proposto é capaz de caracterizar a complexidade pela medida de entropia persistente dos barcodes, uma medida de entropia baseada na definição de entropia de Shannon. Além disso, neste trabalho, mostramos a evidência de mudanças de complexidade associadas a um período de convulsão de um sinal de EEG / Pattern changings are variations in time series data. Such changes may represent transitions that occur between states. Topological data analysis (TDA) allows characterization of time-series data obtained from complex dynamical systems. In this work, we present a pattern changing detection technique based on TDA. Specifically, starting from a given time series, we divide the signal in slicing windows with no overlapping and for each window we calculate the persistent homology, i.e., the associated barcode. From the barcode the average interval size and persistent entropy are calculated and plotted against the signal duration. Experimental results on artificial and real data sets show good results of the proposed method: 1) It detects pattern changing by identifying the change in the average interval size and calculated persistent entropy for the barcodes generated by the input data set. 2) It shows qualitatively how sensible the choice of filtration method is to evidence topological features of the original space under examination. This is accomplished by using two filtrations: a metric and a lower-star filtration. 3) By varying the slice window size, the method can characterize the presence of local structures of the data set such as the seizure period in EEG signals. 4) The proposed method can characterize complexity by the measure persistent entropy for barcodes, an entropy measure based on Shannon´s entropy definition. Moreover, in this work, we show the evidence of complexity changes associated with a seizure period of an EEG signal

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No description available.

Parkinson’s Disease (PD) is the fastest growing neurodegenerative disease, currently affecting two to three percent of the population over 65. Studying functional connectivity (FC) in PD patients may provide new insights into how the disease alters brain organization in different subjects. We explored persistent homology (PH) as a method for studying FC based on the functional magnetic resonance imaging (fMRI) recordings of 63 subjects, of which 56 were diagnosed with PD.  We used PH to translate each set of fMRI recordings into a stable rank. Stable ranks are homological invariants that are amenable for statistical analysis. The pipeline has multiple parameters, and we explored the effect of these parameters on the shape of the stable ranks. Moreover, we fitted functions to reduce the stable ranks to points in two or three dimensions. We clustered the stable ranks based on the fitted parameter values and based on the integral distance between them. For some of the parameter combinations, not all clusters were located in the space covered by controls. These clusters correspond to patients with a topologically distinct connectivity structure, which may be clinically relevant. However, we found no relation between the clusters and the medication status or cognitive ability of the patients. It should be noted that this study was an exploration of applying persistent homology to PD data, and that statistical testing was not performed. Consequently, the presented results should be considered with care. Furthermore, we did not explore the full parameter space, as time was limited and the data set was small. In a follow-up study, a measurable desired outcome of the pipeline should be defined and the data set should be expanded to allow for optimizing over the full parameter space. / Parkinsons sjukdom är den snabbast växande neurodegenerativa sjukdomen och drabbar för närvarande två till tre procent av befolkningen över 65 år. Att studera funktionell konnektivitet (FC) hos patienter med Parkinson kan ge nya insikter om hur sjukdomen förändrar hjärnans uppsättning i olika områden. Vi använde oss av persistent homologi (PH) som en metod för att studera FC baserat på inspelningar av funktionell magnetresonanstomografi (fMRI) av 63 försökspersoner varav 56 hade diagnosen PD. Vi använde oss av persistent homologi (PH) som en metod för att studera FC baserat på inspelningar av funktionell magnetresonanstomografi (fMRI) av 63 försökspersoner varav 56 hade diagnosen PD. Vi använde PH för att översätta varje uppsättning fMRI-prov vardera till en stable rank. Stable ranks är homologiska invarianter som är lämpliga för statistisk analys. Pipelinen har flera parametrar och vi undersökte effekten av dessa parametrar på formen av dessa stable ranks. Vi anpassade funktioner för att reducera alla stable ranks till punkter i två eller tre dimensioner. Vi grupperade alla stable ranks utifrån de anpassade parametervärdena och utifrån det integrala avståndet mellan dem. För vissa parameterkombinationer kunde inte alla kluster inom det område som täcks av kontrollerna bli funna. Dessa kluster motsvarar patienter med en topologiskt distinkt konnektivitetsstruktur, vilket kan vara kliniskt relevant. Vi fann dock inget samband mellan klustren och patienternas läkemedelsstatus eller kognitiva förmåga.  Det bör noteras att den här studien var en undersökning på tillämpningen av persistent homologi på PD-data och att statistiska tester inte utfördes. Följaktligen bör de presenterade resultaten betraktas med försiktighet. Dessutom undersökte vi inte hela parameterutrymmet eftersom tiden var begränsad och datamängden liten. I en uppföljningsstudie bör man definiera ett mätbart önskat resultat av pipelinen och datamängden bör utökas för att möjliggöra optimering av hela parameterutrymmet.

This thesis explores the possibility of using session-based customers data from Svenska Handelsbanken AB to detect fraudulent sessions. Tools within Topological Data Analysis are employed to analyse customers behavior and examine topological properties such as homology and stable rank at the individual level. Furthermore, a RNN-LSTM model is, on a general behaviour level, trained to predict the customers next event and investigate its potential to detect anomalous behavior. The results indicate that simplicial complexes and their corresponding stable rank can be utilized to describe differences between genuine and fraudulent sessions on individual level. The use of a neural network suggests that there are deviant behaviors on general level concerning the difference between fraudulent and genuine sessions. The fact that this project was done without internal bank knowledge of fraudulent behaviour or historical knowledge of general suspicious activity and solely by data handling and anomaly detection shows great potential in session-based detection. Thus, this study concludes that the use of Topological Data Analysis and Neural Networks for detecting fraud and anomalous events provide valuable insight and opens the door for future research in the field. Further analysis must be done to see how effectively one could detect fraud mid-session. / I följande uppsats undersöks möjligheten att använda sessionbaserad kunddata från Svenska Handelsbanken AB för att detektera bedrägliga sessioner. Verktyg inom Topologisk Dataanalys används för att analysera kunders beteende och undersöka topologiska egenskaper såsom homologi och stabil rang på individnivå. Dessutom tränas en RNN-LSTM modell på en generell beteende nivå för att förutsäga kundens nästa händelse och undersöka dess potential att upptäcka avvikande beteende. Resultaten visar att simpliciella komplex och deras motsvarande stabil rang kan användas för att beskriva skillnader mellan genuina och bedrägliga sessioner på individnivå. Användningen av ett neuralt nätverk antyder att det finns avvikande beteenden på en generell nivå avseende skillnaden mellan bedrägliga och genuina sessioner. Det faktum att detta projekt genomfördes utan intern bankkännedom om bedrägerier eller historisk kunskap om allmäna misstänksamma aktiviteter och enbart genom datahantering och anomalidetektion visar stor potential för sessionbaserad detektion. Därmed drar denna studie slutsatsen att användningen av topologisk dataanalys och neurala nätverk för att upptäcka bedrägerier och avvikande händelser ger värdefulla insikter och öppnar dörren för framtida fortsätta studier inom området. Vidare analyser måste göras för att se hur effektivt man kan upptäcka bedrägerier mitt i sessioner.

La quantité de données disponibles n'a jamais été aussi grande. Se poser les bonnes questions, c'est-à-dire des questions qui soient à la fois pertinentes et dont la réponse est accessible est difficile. L'analyse topologique de données tente de contourner le problème en ne posant pas une question trop précise mais en recherchant une structure sous-jacente aux données. Une telle structure est intéressante en soi mais elle peut également guider le questionnement de l'analyste et le diriger vers des questions pertinentes. Un des outils les plus utilisés dans ce domaine est l'homologie persistante. Analysant les données à toutes les échelles simultanément, la persistance permet d'éviter le choix d'une échelle particulière. De plus, ses propriétés de stabilité fournissent une manière naturelle pour passer de données discrètes à des objets continus. Cependant, l'homologie persistante se heurte à deux obstacles. Sa construction se heurte généralement à une trop large taille des structures de données pour le travail en grandes dimensions et sa robustesse ne s'étend pas au bruit aberrant, c'est-à-dire à la présence de points non corrélés avec la structure sous-jacente.Dans cette thèse, je pars de ces deux constatations et m'applique tout d'abord à rendre le calcul de l'homologie persistante robuste au bruit aberrant par l'utilisation de la distance à la mesure. Utilisant une approximation du calcul de l'homologie persistante pour la distance à la mesure, je fournis un algorithme complet permettant d'utiliser l'homologie persistante pour l'analyse topologique de données de petite dimension intrinsèque mais pouvant être plongées dans des espaces de grande dimension. Précédemment, l'homologie persistante a également été utilisée pour analyser des champs scalaires. Ici encore, le problème du bruit aberrant limitait son utilisation et je propose une méthode dérivée de l'utilisation de la distance à la mesure afin d'obtenir une robustesse au bruit aberrant. Cela passe par l'introduction de nouvelles conditions de bruit et l'utilisation d'un nouvel opérateur de régression. Ces deux objets font l'objet d'une étude spécifique. Le travail réalisé au cours de cette thèse permet maintenant d'utiliser l'homologie persistante dans des cas d'applications réelles en grandes dimensions, que ce soit pour l'inférence topologique ou l'analyse de champs scalaires. / Massive amounts of data are now available for study. Asking questions that are both relevant and possible to answer is a difficult task. One can look for something different than the answer to a precise question. Topological data analysis looks for structure in point cloud data, which can be informative by itself but can also provide directions for further questioning. A common challenge faced in this area is the choice of the right scale at which to process the data.One widely used tool in this domain is persistent homology. By processing the data at all scales, it does not rely on a particular choice of scale. Moreover, its stability properties provide a natural way to go from discrete data to an underlying continuous structure. Finally, it can be combined with other tools, like the distance to a measure, which allows to handle noise that are unbounded. The main caveat of this approach is its high complexity.In this thesis, we will introduce topological data analysis and persistent homology, then show how to use approximation to reduce the computational complexity. We provide an approximation scheme to the distance to a measure and a sparsifying method of weighted Vietoris-Rips complexes in order to approximate persistence diagrams with practical complexity. We detail the specific properties of these constructions.Persistent homology was previously shown to be of use for scalar field analysis. We provide a way to combine it with the distance to a measure in order to handle a wider class of noise, especially data with unbounded errors. Finally, we discuss interesting opportunities opened by these results to study data where parts are missing or erroneous.

With the arrival of 5G networks, telecommunication systems are becoming more intelligent, integrated, and broadly used. This thesis focuses on predicting the upcoming traffic to efficiently promote resource allocation, guarantee stability and reliability of the network. Since networks modeled as graphs potentially capture more information than tabular data, the construction of the graph and choice of the model are key to achieve a good prediction. In this thesis traffic prediction is based on a time-evolving graph, whose node and edges encode the structure and activity of the system. Edges are created by dynamic time-warping (DTW), geographical distance, and $k$-nearest neighbors. The node features contain different temporal information together with spatial information computed by methods from topological data analysis (TDA). To capture the temporal and spatial dependency of the graph several dynamic graph methods are compared. Throughout experiments, we could observe that the most successful model GConvGRU performs best for edges created by DTW and node features that include temporal information across multiple time steps. / Med ankomsten av 5G nätverk blir telekommunikationssystemen alltmer intelligenta, integrerade, och bredare använda. Denna uppsats fokuserar på att förutse den kommande nättrafiken, för att effektivt hantera resursallokering, garantera stabilitet och pålitlighet av nätverken. Eftersom nätverk som modelleras som grafer har potential att innehålla mer information än tabulär data, är skapandet av grafen och valet av metod viktigt för att uppnå en bra förutsägelse. I denna uppsats är trafikförutsägelsen baserad på grafer som ändras över tid, vars noder och länkar fångar strukturen och aktiviteten av systemet. Länkarna skapas genom dynamisk time warping (DTW), geografisk distans, och $k$-närmaste grannarna. Egenskaperna för noderna består av dynamisk och rumslig information som beräknats av metoder från topologisk dataanalys (TDA). För att inkludera såväl det dynamiska som det rumsliga beroendet av grafen, jämförs flera dynamiska grafmetoder. Genom experiment, kunde vi observera att den mest framgångsrika modellen GConvGRU presterade bäst för länkar skapade genom DTW och noder som innehåller dynamisk information över flera tidssteg.

<p dir="ltr">Deep Neural Networks (DNNs) successfully perform object detection tasks, and the Con- volutional Neural Network (CNN) backbone is a commonly used feature extractor before secondary tasks such as detection, classification, or segmentation. In a DNN model, the relationship between the features learned by the model from the training data and the features leveraged by the model during test and deployment has motivated the area of feature interpretability studies. The work presented here applies equally to white-box and black-box models and to any DNN architecture. The metrics developed do not require any information beyond the feature vector generated by the feature extraction backbone. These methods are therefore the first methods capable of estimating black-box model robustness in terms of latent space complexity and the first methods capable of examining feature representations in the latent space of black box models.</p><p dir="ltr">This work contributes the following four novel methodologies and results. First, a method for quantifying the invariance and/or equivariance of a model using the training data shows that the representation of a feature in the model impacts model performance. Second, a method for quantifying an observed domain gap in a dataset using the latent feature vectors of an object detection model is paired with pixel-level augmentation techniques to close the gap between real and synthetic data. This results in an improvement in the model’s F1 score on a test set of outliers from 0.5 to 0.9. Third, a method for visualizing and quantifying similarities of the latent manifolds of two black-box models is used to correlate similar feature representation with increase success in the transferability of gradient-based attacks. Finally, a method for examining the global complexity of decision boundaries in black-box models is presented, where more complex decision boundaries are shown to correlate with increased model robustness to gradient-based and random attacks.</p>