Stan Tendijck
Statistician and Operational Researcher
Who am I?

I am Stan Tendijck, a Dutch statistician and operational researcher.
At the moment, I am a Ph.D. student at the STOR-i Centre for Doctoral Training, Lancaster University.

• Age:

24

• Residence:

Lancaster, UK

Fun Facts
3 Bronze Medals at the IMC
2,321 Reputation on mse
17 JaneStreet puzzles solved
78 Project Euler problems solved

Since I was young I always liked to solve difficult puzzles. That is also why I participated in all sorts of different mathematics competitions from a local level to an international level.

Hence, I made some brain teasers for you! Click on the link below to visit the puzzles section.

Click here to go to the puzzles section.

What is the STOR-i?

The STOR-i Doctoral Training Centre based in Lancaster is a place where students are completing their PhD in topics that are covered in statistics and operational research.

The program at the STOR-i consists of two phases: the MRes year, which is a one year program devoted to acquiring a high level understanding of statistics and operational research in general, and the main phase, which takes approximately 3 years and this is where the students complete their PhD.

The centre has links with academic partners but it is unique in its links with industrial partners. Leading industries, such as BT, Shell, Janssen, and Rolls Royce, collaborate with STOR-i to produce research with impact.

For more information about the institute, visit the website for more detailed information or watch the two videos below.

# STOR i - MRes Experience

Resume
Experience
2019 - present
Teaching Assistant
Lancaster University

Assisted students to pass mathematically related courses and assessed them.

2017
Intern
Shell Global Solutions International B.V.

Applied extreme value theory to model severe storms at sea.

2014 - 2018
Teaching Assistant
Delft University of Technology

Assisted students to pass mathematically related courses and assessed them.

Education
2019 - present
Lancaster University
Lancaster

PhD student at the STOR-i doctoral training centre

2018 - 2019
Lancaster University
Lancaster

MRes student at the STOR-i doctoral training centre

2016 - 2018
Delft University of Technology
Delft

MSc in Applied and Industrial Mathematics with a specialisation in statistics

2004 - 2005
Delft University of Technology
Delft

BSc in Applied and Industrial Mathematics

My Skills
Coding
• MATLAB
• Python
• R
• C
Posters, Publications, and Talks

### Publications:

A statistical model for the directional evolution of severe ocean storms with David Randell, Emma Ross and Philip Jonathan, Environmetrics e2541, 2019

Modelling the extremes of bivariate mixture distributions with application to oceanographic data with Emma Eastoe, Jonathan Tawn, David Randell, and Philip Jonathan, under review, 2021

Description of my Ph.D.

### Introduction

In the design of old or new offshore facilities, e.g. oil platforms or vessels, it is very important - both for safety and reliability reasons they can survive the most extreme storms. In particular, companies that build and/or maintain offshore facilities are interested in failure probabilities, e.g. what is the chance it breaks down in the next year? Mathematically, we define a variable $$X$$ denoting the response variable. This might be the force applied to a structure over some time due to high waves, the maximum heave movement of a vessel, or anything else. We make the assumption that a failure does not happen in a period of time if and only if the maximum observed response $$X$$ over this period denoted by $$X_{\max}$$ does not exceed a threshold $$x_f$$. Hence, we are interested in modelling the extremes of the response $$X$$.

It is not uncommon that there exist physical models that can calculate or estimate a response given a state $$\boldsymbol{e}$$ of an environment $$\boldsymbol{E}$$, i.e., the density $$f_{X|\boldsymbol{E}}$$ of the response $$X$$ given the environment $$\boldsymbol{E}$$ is mostly understood in a physical sense. This includes both extreme and not extreme $$X$$ given $$\boldsymbol{E}$$. As a motivating example, we look at the three responses, maximum offset, maximum heave, and base shear. These responses are calculated using functions of the environment based on physical models. In the three bottom plots of Figure 1, the evolution over time of these responses are shown for certain parts where they are relatively large. The top four plots of Figure 1 show the corresponding evolution over time of some oceanographic variables. From these figures, it can be seen that there exists some relation between significant wave height $$H_S$$ and wind speed $$W_S$$, and most of the responses. We note that also wave peak period $$T_p$$ plays some role in the value of the responses that are not explained by $$H_S$$ and wind speed. Moreover, we note that $$H_S$$ and $$W_S$$ are highly correlated.

In our application, the environment $$\boldsymbol{E}$$ would thus be the ocean environment, which contains all variables that describe the ocean. It should be noted that, in theory, $$\boldsymbol{E}$$ is very high dimensional, and can contain hundreds of variables. Hence, we give an incomplete list of examples of components of $$\boldsymbol{E}$$: (1) significant wave height $$H_S$$; (2) wind speed $$W_S$$; (3) wave peak period $$T_p$$; (4) current speed; (5) directional covariates; and (6) seasonal covariates. We note that all the random variables that form the ocean environment $$\boldsymbol{E}$$ are summary statistics or averages which try to describe the main features of the water basin for a fixed amount of time, usually three hours.

It follows that if the density $$f_{\boldsymbol{E}}$$ of the environment $$\boldsymbol{E}$$ is known, then the marginal probabilty density function of the response $$X$$ can be calculated according to $f_X(x) = \int f_{X|\boldsymbol{E}}(x|\boldsymbol{e}) f_{\boldsymbol{E}}(\boldsymbol{e})\,\mathrm{d} \boldsymbol{e}.$ Hence, there is a need to model the distribution of $$\boldsymbol{E}$$ well, and, in particular, to model the part of the distribution of $$\boldsymbol{E}$$ that causes high responses. It will thus be the focus of this research to develop models to compute $$f_{\boldsymbol{E}}(\boldsymbol{e})$$ for the $$\boldsymbol{e}$$ that are considered extreme environments, i.e., the $$\boldsymbol{e}$$ that are associated with high responses. For completeness reasons, also the process how $$X|\boldsymbol{E}$$ is modelled will be discussed.

A company active in the offshore area usually has more than one facility in operation. It might be the case that each offshore facility satisfies a set of safety guidelines, however, it is not true that failure events of different facilities are independent. Indeed, it can even be true that given one fails, the other facilities in the vicinity are almost guaranteed to fail which yields an interest in both the distribution of $$\boldsymbol{E}$$ given $$X_1>x_{f1}$$, and $$\boldsymbol{X}_{-1}|\boldsymbol{E},X_1>x_{f1}$$. Moreover, even if the density $$f_X(x)$$ is well understood, there is a need to calculate $$f_{\boldsymbol{X}}(\boldsymbol{x})$$ where $$\boldsymbol{X}=(X_1,\dots,X_m)$$ (and $$\boldsymbol{x}=(x_1,\dots,x_m)$$) is the joint vector of response variables that a company is interested in, for some $$m\geq1$$. This might be the same response at different locations as well as a different response at the same location. In particular, engineers are interested in probabilities of the following type $$\mathbb{P}(\boldsymbol{X}>\boldsymbol{x}_{f}) = \int\mathbb{P}(\boldsymbol{X}>\boldsymbol{x}_{f}\mid\boldsymbol{E}=\boldsymbol{e}) f_{\boldsymbol{E}}(\boldsymbol{e})\,\mathrm{d} \boldsymbol{e},$$

It can also be the case that we are interested in large values of a function $$\phi(\boldsymbol{E})$$ of the environment. This might be because $$\phi(\boldsymbol{E})$$ can be considered a driver of a response variable. In that setting, we note that the same line of reasoning in the above applies. We get $\mathbb{P}(\phi(\boldsymbol{E}) > e_f) = \int 1\{\phi(\boldsymbol{E})>e_f\} f_\boldsymbol{E}(\boldsymbol{e})\,\mathrm{d}\boldsymbol{e}.$ In the computation of integrals of these types, it is noted that if a component of $$\boldsymbol{E}$$ is independent of the responses under investigation, then this component integrates out, and it would not be necessary to accurately model this component to yield useful results. Moreover, it is not necessary to model $$f_{\boldsymbol{E}}(\boldsymbol{e})$$ well in the region $$\boldsymbol{e}\in\{\boldsymbol{e}'\in\boldsymbol{E}:\ \mathbb{P}(\boldsymbol{X} \le \boldsymbol{x}_f|\boldsymbol{e}') \approx 1\}$$, where the definition of $$p\approx 1$$ depends on the problem.

In my Ph.D, we will try to model $$\boldsymbol{E}$$. This comes with many interesting research questions. For example, waves can be classified into two different categories: swell waves and wind waves. This means that the distribution of the wave height is a mixture distribution. Little analysis has been performed before on how to deal with mixtures in extremes.

Moreover, there is a need to develop models for the evolution of $$\boldsymbol{E}$$ over time. In a recent paper by Tendijck et al (2019), a model for the joint evolution of $$H_S$$ and dominant wave direction over time was developed. In this Ph.D. this model will be extended to also include other oceanographic variables, such as wind speed and/or wave peak period.

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