# Probability and Statistics Seminar

## Past sessions

### The max-semistable laws: characterization, estimation and testing

In this talk we present the class of max-semistable distribution functions that appear as the limit, in distribution, of the maximum, suitably centered and normalized, of $k_n$ independent and identically distributed random variables, where $k_n$ is an integer-valued geometric sequence with ratio $r$ (larger or equal to $1$). This class of distributions includes all the max-stable distributions but also multimodal distributions and discrete distributions. We will characterize the max-semistable laws, discuss the estimation of the parameters and the fractal component and propose a test that allow us to distinguish between max-stable and max-semistable laws.

Join work with Luísa Canto e Castro and Maria da Graça Temido.

### Comparison of joint schemes for multivariate normal i.i.d. output

The performance of a product frequently relies on more than one quality characteristic. In such a setting, joint control schemes are used to determine whether or not we are in the presence of unfavorable disruptions in the location and spread of a vector of quality characteristics. A common joint scheme for multivariate output comprises two constituent control charts: one for the mean vector based on a weighted Mahalanobis distance between the vector of sample means and the target mean vector; another one for the covariance matrix depending on the ratio between the determinants of the sample covariance matrix and the target covariance matrix. Since we are well aware that there are plenty of quality control practitioners who are still reluctant to use sophisticated control statistics, this paper tackles Shewhart-type charts for the location and spread based on a few pairs of control statistics that depend on the nominal mean vector and covariance matrix. We recall or derive the joint probability density functions of these pairs of control statistics in order to investigate the impact on the ability of the associated joint schemes to detect shifts in the process mean vector or covariance matrix for various out-of-control scenarios.

Joint work with Wolfgang Schmid, Patrícia Ferreira Ramos, Taras Lazariv, António Pacheco.

### Modelling extremal temporal dependence in stationary time series

Extreme value theory concerns the statistical study of the extremal properties of random processes. The most common problems treated by extreme value methods involve modeling the tail of an unknown distribution function from a set of observed data with the purpose of quantifying the frequency and severity of events more extreme than any that have been observed previously. A fundamental issue in applied multivariate extreme value (MEV) analysis is modelling dependence within joint tail regions. In this seminar we suggest modelling joint tails of the distribution of two consecutive pairs $(X_i;X_{i+1})$ of a first-order stationary Markov chain by a dependence model described in Ramos and Ledford (2009). Applications of this modelling approach to real data are then considered.

Ramos and Ledford (2009). A new class of models for bivariate joint tails. J. R. Statist. Soc., B. 71. p. 219-241.

### Binary autoregressive geometric modelling in a DNA context

Symbolic sequences occur in many contexts and can be characterized e.g. by integer-valued intersymbol distances or binary-valued indicator sequences. The analysis of these numerical sequences often sheds light on the properties of the original symbolic sequences. This talk introduces new statistical tools to explore the autocorrelation structure in indicator sequences and to evaluate its impact on the probability distribution of intersymbol distances. The methods are illustrated with data extracted from mitochondrial DNA sequences.

This is a joint work with Manuel Scotto (IST, Lisbon, Portugal), Christian Weiss (Helmut Schmidt University, Hamburg, Germany) and Paulo Ferreira (DETI, IEETA, Aveiro, Portugal).

### On the peaks-over-threshold method in extreme value theory

The origin, the development and the use of the peaks-over-threshold method (in particular in higher-dimensional spaces) will be discussed as well as some issues that need clarification.

### Spatial and Spatio-Temporal Nonlinear Time Series

In this talk we present a new spatial model that incorporates heteroscedastic variance depending on neighboring locations. The proposed process is regarded as the spatial equivalent to the temporal autoregressive conditional heteroscedasticity (ARCH) model. We show additionally how the introduced spatial ARCH model can be used in spatiotemporal settings. In contrast to the temporal ARCH model, in which the distribution is known given the full information set of the prior periods, the distribution is not straightforward in the spatial and spatiotemporal setting. However, it is possible to estimate the parameters of the model using the maximum-likelihood approach. Via Monte Carlo simulations, we demonstrate the performance of the estimator for a specific spatial weighting matrix. Moreover, we combine the known spatial autoregressive model with the spatial ARCH model assuming heteroscedastic errors. Eventually, the proposed autoregressive process is illustrated using an empirical example. Specifically, we model lung cancer mortality in 3108 U.S. counties and compare the introduced model with two benchmark approaches.

(joint work with Robert Gartho and Philipp Otto)

### Distributed and robust network localization

Signal processing over networks has been a broad and hot topic in the last few years. In most applications networks of agents typically rely on known node positions, even if the main goal of the network is not localization. Also, mobile agents need localization for, e.g., motion planning, or formation control, where GPS might not be an option. Also, real-world conditions imply noisy environments, and the network real-time operation calls for fast and reliable estimation of the agents’ locations. So, galvanized by the compelling applications researchers have dedicated a great amount of work to finding the nodes in networks. With the growing network sizes of devices constrained in energy expenditure and computation power, the need for simple, fast, and distributed algorithms for network localization spurred this work. Here, we approach the problem starting from minimal data collection, aggregating only range measurements and a few landmark positions. We explore tailored solutions recurring to the optimization and probability tools that can leverage performance under noisy and unstructured environments. Thus, the contributions are, mainly:
• Distributed localization algorithms characterized for their simplicity but also strong guarantees;
• Analyses of convergence, iteration complexity, and optimality bounds for the designed procedures;
• Novel majorization approaches which are tailored to the specific problem structure.

### An ARL-unbiased $np-$chart

We usually assume that counts of nonconforming items have a binomial distribution with parameters $(n,p)$, where $n$ and $p$ represent the sample size and the fraction nonconforming, respectively.

The non-negative, discrete and usually skewed character and the target mean $(np_0)$ of this distribution may prevent the quality control engineer to deal with a chart to monitor $p$ with: a pre-specified in-control average run length (ARL), say $1/\alpha$; a positive lower control limit; the ability to control not only increases but also decreases in $p$ in a expedient fashion. Furthermore, as far as we have investigated, the $np-$ and $p-$charts proposed in the Statistical Process Control literature are ARL-biased, in the sense that they take longer, in average, to detect some shifts in the fraction nonconforming than to trigger a false alarm.

Having all this in mind, this paper explores the notions of uniformly most powerful unbiased tests with randomization probabilities to eliminate the bias of the ARL function of the $np-$chart and to bring its in-control ARL exactly to $1/\alpha$.

### The Block Maxima and POT methods and, an extension of POT to integrated stochastic processes

We shall review the classical maximum domain of attraction condition underlying BM and POT, two fundamental methods in Extreme Value Theory. A theoretical comparison between the methods will be presented.

Afterwards, the maximum domain of attraction condition to spatial context will be discussed. Then a POT-type result for the integral of a stochastic process verifying the maximum domain of attraction condition will be obtained.

### On Eigenvalues of the Transition Matrix of some Count Data Markov Chains

A stationary Markov chain is uniquely determined by its transition matrix, the eigenvalues of which play an important role for characterizing the stochastic properties of a Markov chain. Here, we consider the case where the monitored observations are counts, i.e., having values in either the full set of non-negative integers, or in a finite set of the form ${0,...,n}$ with a prespecified upper bound $n$. Examples of count data time series as well as a brief survey of some basic count data time series models is provided.

Then we analyze the eigenstructure of count data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models. Our method relies on the computation of appropriate conditional (factorial) moments.

### From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification

The softmax transformation is a key component of several statistical learning models, encompassing multinomial logistic regression, action selection in reinforcement learning, and neural networks for multi-class classification. Recently, it has also been used to design attention mechanisms in neural networks, with important achievements in machine translation, image caption generation, speech recognition, and various tasks in natural language understanding and computation learning. In this talk, I will describe sparsemax, a new activation function similar to the traditional softmax, but able to output sparse probabilities. After deriving its properties, I will show how its Jacobian can be efficiently computed, enabling its use in a neural network trained with backpropagation. Then, I will propose a new smooth and convex loss function which is the sparsemax analogue of the logistic loss. An unexpected connection between this new loss and the Huber classification loss will be revealed. We obtained promising empirical results in multi-label classification problems and in attention-based neural networks for natural language inference. For the latter, we achieved a similar performance as the traditional softmax, but with a selective, more compact, attention focus.

### Geostatistical History Matching with Ensemble Updating

In this work, a new history matching methodology is proposed, coupling within the same framework the advantages of using geostatistical sequential simulation and the principles of ensemble Kalman filters: history matching based on ensemble updating.  The main idea of this procedure is to use simultaneously the relationship between the petrophysical properties of interest and the dynamical results to update the static properties at each iteration, and to define areas of influence for each well. This relation is established through the experimental non-stationary covariances, computed from the ensemble of realizations. A set of petrophysical properties of interest is generated through stochastic sequential simulation. For each simulated model, we obtain its dynamic responses at the wells locations by running a fluid flow simulator over each single model. Considering the normalized absolute deviation between the dynamic responses and the real dynamic response in each well as state variables, we compute the correlation coefficients of the deviations with each grid cell through the ensemble of realizations. Areas of high correlation coefficients are those where the permeability is more likely to play a key role for the production of that given well. Using a local estimation of the response of the deviations, through a simple kriging process, we update the subsurface property of interest at a given localization.

### Statistical Modeling of Integer-valued Time Series: An Introduction

Modeling and predicting the temporal dependence and evolution of low integer-valued time series have attracted a lot of attention over the last years. This is partially due to the increasing availability of relevant high-quality data sets in various fields of applications ranging from finance and economy to medicine and ecology. It is important to stress, however, that there is no a unifying approach applicable to modeling all integer-valued time series and, consequently, the analysis of such time series has to be restricted to special classes of integer-valued models. A useful division of these models can be made as being either observation-driven or parameter-driven models.  A suitable class of observation-driven models is the one including models based on thinning operators. Models belonging to this class are obtained by replacing the multiplication in the conventional time series models by an appropriate thinning operator, along with considering a discrete distribution for the sequence of innovations in order to preserve the discreteness of the counts.

This talk aims at providing an overview of recent developments in thinning-based time series models paying particular attention to models obtained as discrete counterparts of conventional univariate and multivariate autoregressive moving average models, with either finite or infinite support. Finally, we also outline and discuss likely directions of future research.

### Robust heritability and predictive accuracy estimation in plant breeding

Genomic prediction is used in plant breeding to help find the best genotypes for selection. Here, the  accurate estimation of  predictive accuracy (PA) and heritability (H) is essential for genomic selection (GS). As in other applications,  field data are analyzed via regression models, which are known to lead to biased estimation when the normality premise is violated, biases that may translate into inaccurate H and PA estimates and negatively impact GS. Therefore, a robust analogue of a method from the literature used for H and PA estimation is presented. Both techniques are then compared through simulation.

(Joint work with Hans-Peter Piepho & Joseph O. Ogutu, Bioinformatics Unit, Institute of Crop Science, University of Hohenheim, Stuttgart, Germany)

### On stochastic ordering and control charts for the traffic intensity

The traffic intensity is a crucial parameter of a queueing system since it is a measure of the average occupancy of a server. Expectedly, an increase in the traffic intensity must be detected quickly so that appropriate corrective actions are taken.

In this talk, we:

• briefly review existing procedures used to monitor the traffic intensity of M/G/1, GI/M/1 and a few other queues;
• focus on control charts to detect increases in traffic intensity whose control statistics are integer-valued and/or can be (approximately) modelled by discrete time Markov chains;
• investigate the stochastic monotonicity properties of the associated probability transition matrices;
• explore the implications of these properties to provide insights on the performance of such control charts.

(Joint work with António Pacheco)

### Behind the myth of Option Trading

Starting from the basics of derivative products, you will learn more about financial markets, how they are organized, what products are traded, by who and why. But we won’t stop there, and you will also have to properly put yourself into traders’ shoes and realize why derivatives lay where mathematics meet finance. We will go through some notions (Derivatives, tangible assets, financial contracts, Call/Put options, European/American options, option pricing, maxima…) focusing in the pratical application of pricing models like B&S formula limit & reality, new statistical models studied, etc.

### Symbolic Covariance Matrices and Principal Component Analysis for Interval Data

Recent years witnessed a huge breakthrough of technology which enables the storage of a massive amounts of information. Additionally, the nature of the information collected is also changing. Besides the traditional format of recording single values for each observation, we have now the possibility to record lists, intervals, histograms or even distributions to characterize an observation. However, conventional data analysis is not prepared for neither of these challenges, and does not have the necessary or appropriate means to treat extremely large databases or data with a more complex structure. As an answer to these challenges Symbolic Data Analysis, introduced in the late 1980s by Edwin Diday, extends classical data analysis to deal with more complex data by taking into account inner data variability and structure.

Principal component analysis is one of the most popular statistical methods to analyse real data. Therefore, there have been several proposals to extend this methodology to the symbolic data analysis framework, in particular to interval-valued data.

In this talk, we discuss the concepts and properties of symbolic variance and covariance of an interval-valued variable. Based on these, we develop population formulations for four symbolic principal component estimation methods. This formulation introduces simplifications, additional insight and unification of the discussed methods. Additionally, an explicit and straightforward formula that defines the scores of the symbolic principal components, equivalent to the representation by Maximum Covering Area Rectangle, is also presented.

Joint work with António Pacheco (CEMAT and DM-IST, Univ. de Lisboa), Paulo Salvador (IT, Univ. Aveiro),  Rui Valadas (IT, Univ. de Lisboa), and Margarida Vilela (CEMAT).

### On ARL-unbiased c-charts for i.i.d. and INAR(1) Poisson counts

In Statistical Process Control (SPC) it is usual to assume that counts have a Poisson distribution. The non-negative, discrete and asymmetrical character of a control statistic with such distribution and the value of its target mean may prevent the quality control practitioner to deal with a c-chart with:

1. a positive lower control limit and the ability to control not only increases but also decreases in the mean of those counts in a timely fashion;
2. a pre-specified in-control average run length (ARL).

Furthermore, as far as we have investigated, the c-charts proposed in the SPC literature tend not to be ARL-unbiased. (The term ARL-unbiased is used here to coin any control chart for which all out-of-control ARL values are smaller than  the in-control ARL.)

In this talk, we explore the notions of unbiased, randomized and uniformly most powerful unbiased tests (resp. randomization of the emission of a signal and a nested secant rule search procedure) to:

1. eliminate the bias of the ARL function of the c-chart for the mean of i.i.d. (resp. first-order integer-valued autoregressive, INAR(1)) Poisson counts;
2. bring the in-control ARL exactly to a pre-specified and desired value.

We use the R statistical software to provide striking illustrations of the resulting ARL-unbiased c-charts.

Joint work with Sofia Paulino and Sven Knoth

### Transmission and Power Generation Investment under Uncertainty

The challenge of deregulated electricity markets and ambitious renewable energy targets have contributed to an increased need of understanding how market participants will respond to a transmission planner’s investment decision. We study the optimal transmission investment decision of a transmission system operator (TSO) that anticipates a power company’s (PC) potential capacity expansion. The proposed model captures both the investment decisions of a TSO and PC and accounts for the conflicting objectives and game-theoretic interactions of the distinct agents. Taking a real options approach allows to study the effect of uncertainty on the investment decisions and taking into account timing as well as sizing flexibility.

We find that disregarding the power company’s optimal investment decision can have a large negative impact on social welfare for a TSO. The corresponding welfare loss increases with uncertainty. The TSO in most cases wants to invest in a higher capacity than is optimal for the power company. The exception is in case the TSO has no timing flexibility and faces a relatively low demand level at investment. This implies that the TSO would overinvest if it would disregard the PC’s optimal capacity decision. On the contrary, we find that if the TSO only considers the power companies sizing flexibility, it risks installing a too small capacity. We furthermore conclude that a linear subsidy in the power company's investment cost could increase its optimal capacity and therewith, could serve as an incentive for power companies to invest in larger capacities.

Joint work with Nora S. Midttun, Afzal S. Siddiqui, and Jannicke S. Sletten.

### Optional-Contingent-Product Pricing in Marketing Channels

This paper studies the pricing strategies of firms belonging to a vertical channel structure where a base and an optional contingent products are sold. Optional contingent products are characterized by unilateral demand interdependencies. That is, the base product can be used independently of a contingent product. On the other hand, the contingent product’s purchase is conditional on the possession of the base product.

We find that the retailer decreases the price of the base product to stimulate demand on the contingent-product market. Even a loss-leader strategy could be optimal, which happens when reducing the base product’s price has a large positive effect on its demand, and thus on the number of potential consumers of the contingent product. The price reduction of the base product either mitigates the double-marginalization problem, or leads to an opposite inefficiency in the form of a too low price compared to the price maximizing vertically integrated channel profits. The latter happens when the marginal impact of both products’ demands on the base product’s price is low, and almost equal in absolute terms.

Joint work with Sihem Taboubi and Georges Zaccour.

Immediately followed by another seminar session.

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