The clustering of events can have a large impact on society. The extremal index $\theta$ tells how much extreme events cluster. We will compare different types of estimators in this project. First, we review the extremes of different sequences which have different values of $\theta$. We have found significant differences between the extremes. Then, 2 different types of estimators are introduced which both use different ways to divide the data, using disjoint blocks and using sliding blocks. The optimal block lengths are simulated for all those estimators. Using those block lengths, $\theta$ is simulated with all the estimators. From the simulations we conclude that the estimators using sliding blocks perform better. The best-performed estimator that we found from the simulations is used to estimate $\theta$ on data from the KNMI, comparing wind gusts and precipitation at weather stations De Bilt and Vlissingen.

This paper is a research on the segmentation of customers. The clustering of customers is done based on the variables recency, frequency and monetary value. Such a clustering is called an RFM-model. The clustering is done using the K-means clustering method. To find the optimal number of clusters the following performance metrics are used: Elbow method, Silhouette Analysis, and Davies-Bouldin Index. The RFM-model is extended by introducing the loyalty variable. This model is called an RFML-model. Lastly, further clustering is done within one of the clusters.

The primary goal of this report is to provide a general overview of offline change-point literature as it is known today. Change-point methods are important statistical problems, where we are interested in determining whenever a certain data-set changes in structure. Furthermore, the term "off-line" is meant to indicate that the data itself is already known, whereas on the other hand we have "on-line" methods which deal with situations where new data is yet being received during localisation of the change-points. In this report we mainly consider off-line methods, as we feel off-line methods provide a more friendly introduction into change-point analysis and on-line methods are in principle just an extension of their off-line counterparts. First off, these off-line change-point methods are considered under different assumptions (parametric, non-parametric). In each case, we treat a solution to the change-point problem under different models(normal and gamma model, mean or varianche change etc.). Eventually we shall also treat some widely used algorithms, meant to extend the problem into the localisation of multiple change-points.indent Aside from theoretical considerations, an equally important part of this report will be focused on empirical results. Both for the statistics as algorithms will there be a performance study where the different methods will be empirically assessed and compared under different models, namely the robustness against "outliers"(extreme data-values) will be investigated. So to complement the primary goal, we will also focus on the following two subgoals:
1) Assessment and comparison of different change-point models
2) Evaluating and improving robustness against outliers

The most common way to evaluate traffic safety is investigating the occurrence and severity of crashes using historical data. This approach however has a number of limitations, the most important of which is probably its reactive nature. An alternative method using non-crash events has gained a lot of attention recently, especially thanks to the rapid improvement of sensing technologies. By gathering trajectory data and calculating various Surrogate Measures of Safety it has become possible to analyse safety without waiting for accidents to happen. Using these indicators combined with Extreme Value Theory (EVT) one can estimate the probability of crashes as extreme (unobserved) events. The primary goal of this thesis is to contribute to the research that has been done so far on the application of Extreme Value Theory to Surrogate Measures for traffic safety analysis. Research questions seek for answers to what we can learn from applying univariate EVT using indicators describing collision course and crossing course interactions, and how we can predict nearness to collision and severity using bivariate EVT models.

In this thesis we are going to study outlier detection methods and propose a new method. Classical outlier detection is typically based on the assumption that the data is from a Gaussian/normal distribution. When the underlying distribution of a random sample is heavy tailed, so not normal , it is likely to have some extreme observations which would be identified as outlier using the classical procedure. This paper aims to address this issue by proposing a procedure
to identify real ‘outliers’ for heavy tailed data set. We first dive in the some existing methods and see how they work, try to understand them, simulate them and see their shortcomings in the case of a heavy tailed distribution. Then we study Extreme Value Theory (EVT) which we shall use to set up our proposed method of detecting outliers. Once we have constructed the proposed method, we are going to simulate and compare it with the existing methods. The goal in the case of normality is that the new method is not worse than the existing ones, at least not extremely, and in the case of a heavy tailed function to work better.

In this thesis the theory of depth functions is researched. Depth functions are functions that measure data depth and order multivariate observations. Two depth functions are discussed: the halfspace and simplicial depth function. The halfspace depth of a point is defined as the smallest probability for which a closed halfspace contains that point. The simplicial depth of a point is defined as the probability of that point being contained in a simplex for which its vertices are independent and identically distributed. Contours allow us to visualize these depth functions. This theory is applied to simulations with multivariate distributions and to weather statistics.

Time series analysis is used to predict future behaviour of processes and is widely used in the finance sector. In this paper we will analyse the modelling of multivariate time series of financial data using vector autoregressive processes. The goal is that the reader will understand the presented models and could theoretically perform time series analysis by himself. Two specific models will be explained: the Vector Autoregressive model (VAR model) and the Vector Error Correction Model (VECM). We will describe various methods to analyse multivariate time series using these models, such as forecasting the process, variance decomposition of the forecast error, causality analysis and impulse response analysis. Examples of these models and analysis methods will be presented and investigated. Finally, we will perform a time series analysis with these models on Dutch indices and stock data. We conclude that real-world data often does not fit the VAR model and VECM requirements and that further improved models should be considered as well.

In dit onderzoek is een Monte Carlo algoritme beschreven voor het schatten van de dominante eigenwaarde en bijbehorende eigenvector van een niet-negatieve, irreducibele matrix. Naast de werking van het Monte Carlo algoritme is ook de convergentiesnelheid onderzocht. Tevens zijn een aantal simulaties in de praktijk uitgevoerd voor zowel het Monte Carlo algoritme als de power methode en aan de hand hiervan bekeken of het Monte Carlo algoritme een goede concurrent is van de power methode.

This thesis gathers, develops and evaluates several characterizations of multivariate tail dependence. It is established that the stable tail dependence function (STDF) is a suitable copula-based dependence function that fully captures the multivariate extremal dependence structure in all dimensions d≥2 and can be used to visualize the tail dependence structure for bivariate and trivariate problems. Based on the STDF, we propose a multivariate tail dependence coefficient (TDC) as an extension of the well-known bivariate TDC. Importantly, we show that the proposed measure can identify tail independence in all dimensions d≥2, similar to its bivariate variant. The performance of nonparametric estimators for the STDF and, inherently, the multivariate TDC, is assessed with an extensive simulation study, including smoothed and bias-corrected versions of the empirical STDF. Based on the estimators for the STDF and the multivariate TDC, test statistics under the null hypothesis of tail independence are developed and evaluated in another simulation study. The STDF-based estimation and testing procedures are applied to foreign exchange (FX) data to characterize the tail dependence structure between three European FX rates and five worldwide FX rates.

Three interest rate models are researched: Displaced Exponential-Vasicek, Hull-White one factor and Hull-White two factors with time-dependent volatility parameters. The motivation for this is two-fold: firstly, we would like to understand how the capital calculations would be impacted when yield curves are modelled under the three different models. This is done by looking at both magnitude and stability of the risk profiles and scalar risk-measures for three counterparties, which are highly representative for the bank. Secondly, we investigate the benefits and drawbacks of using one model and its corresponding calibration method over the others, with a special attention to the impact on yields correlations.

The first model, calibrated to historical data, is used as a nine-factors model for forward rates and is currently being used within the bank for PFE profiles and CVA regulatory capital. Historical backtest has proven the current model to perform reasonably well on real data and therefore it is used as a benchmark against which the other two models are tested. The two Hull-White models, used as short rate models, are calibrated to the risk-neutral measure (namely, to European swap- tions). Precisely, a two-steps calibration procedure suited for piece-wise constant volatility functions is implemented for both. The stability analysis reveals that the variation of Exposure at Defaults is significant, which might be undesired. On the other side, the two short rate models retain the correlation structure of interest rates better than the current model. This in turn translates into higher capital impact.

Precipitation has high spatial and temporal uncertainty, which makes it challenging to predict. We focus specifically on extreme amounts of precipitation. The Royal Dutch Meteorological Institute (KNMI) uses a numerical model, approximating the solutions to partial differential equations, to forecast precipitation and other metrics about the weather. These forecasts have systematic errors, due to the model’s high sensitivity to input parameters. These errors can be corrected with statistical methods, by looking at the relation between the predicted and actual precipitation. We use a non-parametric regression set-up to estimate the conditional expectation of the weather given the forecasts of the numerical weather prediction model of the KNMI. Specifically, we focus on predicting the maximum precipitation in a three by three kilometers area in the Netherlands. There are several existing methods for solving non-parametric regression problems; in this thesis we will focus on k-nearest neighbors and random forests. A simulation study shows, however, that both these methods are not capable of dealing with more complex regression problems, such as forecasting extreme precipitation. Therefore, we are proposing a newly developed method, called k-nearest forest neighbors, which is a generalization of the random forests approach. This new method performs significantly better on the simulated data, compared to k-nearest neighbors and random forests. When applying the methods on a precipitation data set obtained from the KNMI, it also turns out that the method we developed has more predictive power than the numerical weather model and the existing non-parametric regression approaches.

Over the past three decades, the singular value decomposition has been increasingly used for various big data applications. As it allows for rank reduction of the input data matrix, it is not only able to compress the information contained, but can even reveal underlying patterns in the data through feature identification. This thesis explores algorithms for large-scale SVD calculation, and uses these to demonstrate how the SVD can be applied to a variety of fields, including information retrieval, recommender systems and image processing.

The algorithms discussed are Golub-Kahan-Lanczos bidiagonalization, randomized SVD and block power SVD. Each algorithm is implemented in Matlab and both error and time taken by each algorithm are compared. We find that the block power SVD is very effective, especially when only the truncated SVD is required. Due to its simplicity, speed and relatively small error for low-rank matrix approximation, it is an ideal method for the applications discussed in this thesis.

We show how the SVD can be used for information retrieval, through Latent Semantic Indexing. The method is tested on the Time collection and we find that the SVD removes much of the noise present in the data and solves the issues of synonymy and polysemy. Then, SVD-based algorithms for recommender systems are presented. We implement a basic SVD algorithm called Average Rating Filling, and a (biased) stochastic gradient descent algorithm, which was developed for the Netflix recommender-system prize. These are tested on the Movielens 100k dataset, resulting in the best performance by biased stochastic gradient descent. Finally, the SVD is used for image compression and we find that, while not very useful for face recognition, the SVD could provide a time- and space-efficient method for searching through an image database for similar images.

This thesis project developed an alternative PM2.5 concentration prediction model and early warning system of extreme air pollution based on the long short-term memory (LSTM) and achieved satisfying performance. To research more deeply, we divided the task into two parts. The first task was predicting the PM2.5 concentration of next 24 hours and another one was building early warning system of extreme air pollution of next 12 hours.
To solve the first task, we started from the 1-hour prediction problem, that was predicting PM2.5 of next hour based on the last hours’ data. We did parameter optimization to derive the best network architecture and we got a RMSE of 19.7863. We then successfully built 24-hour prediction model that was predicting PM2.5 concentration of next 24 hours according to the optimal 1-hour prediction model. The proposed 24-hour prediction model exhibited satisfactory performance, including the 13-24 h prediction task which is predicting the mean PM2.5 concentration among next 13-24 hours (RMSE=49.41).
Although we got a satisfying RMSE for the PM2.5 prediction problem, we didn’t get accurate prediction for extreme conditions and that’s why we continued to focus on the second task. We regarded the highest PM2.5 value among 12 hours as the extreme air pollution of this period and we divided the warning level into 4 parts. Then we built the early warning system based on the LSTM to predict the warning level of highest PM2.5 value of next 12 hours. As indicated by the ACC and AUC, our LSTM model achieved sound performance (ACC=86.7%, AUC=0.837).
To improve the prediction performance, we focused on several model optimization techniques for the 1-hour prediction model and each technique has effectively improved the accuracy. Moreover, we combined these optimization methods together, which leaded to the lowest RMSE of 14.1937. The combined optimization method performed better than any single optimization method, which suggested that we can use some effective optimization methods together to improve the prediction accuracy of LSTM model. In addition, we also compared our model with the random forest (RF) model and the comparison result proved that LSTM network worked better for both tasks.

In deze scriptie is onderzocht of regressie, een postprocessing methode, een goede toevoeging is aan het model dat het Europees Centrum voor Weersverwachtingen op Middellange Termijn (ECMWF) ontwikkeld heeft en waarvan het KNMI de uitvoer gebruikt, specifiek om te bepalen of er minstens 5 mm neerslag op een dagdeel (00-12 UTC en 12-00 UTC) zal vallen. Hiervoor is gebruik gemaakt van logistische regressie. Voor het verifiëren van de resultaten is er gekeken naar de Brier-Scores en de reliability diagrammen van de regressiemodellen. Deze zijn vervolgens vergeleken met de resultaten van het ECMWF-model. De resultaten zijn vrijwel gelijk, maar de Brier-Scores van het originele model zijn net iets beter dan die van de regressiemodellen. De conclusie die na het onderzoek wordt getrokken is dat het gemaakte regressiemodel geen toevoeging heeft op het ECMWF-model.

Intraday liquidity risk is a subject that applies to all banks, and arises whenever there is a timing mismatch between incoming and outgoing payments within a business day. In case such a mismatch occurs, the bank is exposed to the risk that it is unable to meet its payment obligations at the time expected. A liquidity buffer could help to mitigate this risk.

This thesis presents a framework for intraday liquidity risk management within ABN AMRO Bank, while taking different priorities of transactions into account. We examine the use of extreme value theory (EVT) and propose two metrics to capture the risk: the univariate and multivariate risk metric. The univariate risk metric represents the size of the liquidity buffer for each priority group separately and provides granular view. Making use of a Monte Carlo simulation algorithm in combination with univariate EVT, we
are able to estimate the size of the liquidity buffer for a specified time interval within a business day. We forecast the buffer size 30 days out-of-sample and test the violations against the conditional coverage (CC) hypothesis. Satisfactory results are obtained for the
groups with high and moderate priority when the highest confidence levels are considered: $\alpha$ = 0.1 and 0.05. For the group with low priority, the risk metric performs well for the lowest confidence levels: $\alpha$ = 0.025 and 0.01. The multivariate risk metrics aggregates
the size of the liquidity buffer, while taking the diversification of the priority groups into account. We define a failure set and investigate the use of multivariate EVT.