# Optimisation in Car-Sharing Systems

For many people the costs of owning, running and maintaining a car are simply too big to justify having one. Having to pay hundreds, if not thousands of pounds to cover insurance, road tax and fuel costs for only a few short journeys a week to work or to the shops doesn’t make financial sense. One alternative to this that has proven popular is ‘car-sharing‘ or ‘car-clubs‘. These schemes allow members to access vehicles owned by the company by reserving them in advance so they can use them for the short trips they might make in a day. This could be anything from picking up the kids from school, popping down to the shops or going out for a few hours at the weekend.

There are several other benefits to using a car-sharing schemes in addition to any potential financial savings. By using a single vehicle to do several different people’s short trips means that less cars are on the road in total reducing traffic congestion and vehicle emissions. Even something as simple as being able to guarantee a parking space when you reach your destination can be attractive, particularly in busy inner cities.

# Optimal Search Problems

One active area of research within STOR-i is the problem of how to optimally search for something or someone. This is a problem of practical importance with it being used for many real world applications. Bomb disposal squads need to search for unexploded ordinance, search parties want to look for survivors of disasters and the police want to find people on the run from the law.

Like most things the best way to build intuition to a problem is to start by constructing smaller toy models of the system. We can represent the places where an object might be hidden as arcs in a network which are connected to each other with nodes. This type of structure is deliberately similar to that of a road system, the arcs represent streets and the nodes are the street corners. The object that we want to find will be hidden along one of the streets and the job of those searching is to find it as quickly as possible.

Example Network for Optimal Searches

# Why Lawyers Need Statistics

I previously talked about Bayes’ theorem and its often misunderstood applications. Normally these mistakes aren’t particular costly or harmful in the world of statistics, but if they are used to make decisions that impact on the real world then getting things wrong can be extremely costly.

One place where statistics can be called upon to influence important matters is the court. Throughout the last 50 years there has been an increase in the use of statistics in court matters and it is important that everybody involved understands them and their use. If any or all of the prosecution, defence or jury misinterpret the information given to them then the chances of a miscarriage of justice will greatly increase.

$\text{Prob of matching a description} \neq \text{Prob of matching a description and being guilty}$

The classical mistake made in the past by many prosecuting teams is that of the ‘prosecutors fallacy‘. This is when the prosecution or defence have presented the jury with some statistic such as a probability that has been calculated incorrectly, yet manage to convince the jury to accept its truth.

Scales of justice; source

# Expected Goals in Football

Previously I talked about one way to measure the success of a football team over a year through `Pythagorean Expectation‘. Whilst this is a pretty good metric for predicting success, it can only be applied over a certain number of games and can’t tell us anything about a particular match. Since being able to determine how well a team performed in a particular match is the ultimate goal of analysing games, many ideas have been developed to try and do this with increasing accuracy.

#### A Short List of (Increasingly Better) Football Metrics

1. Goals Scored & Goals Conceded
2. Shots
3. Shots on Target (SoT)
4. Total Shots Ratio (TSR)
5. Expected Goals (xG)
6. Expected Goals with tracking data

At the most basic level that you see in the overall league table is the goals scored and conceded for each team. Teams that tend to score lots of goals and concede few goals win more matches and hence finish higher up in the table at the end of league.

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# New STOR-i Space & Posters

We recently had the grand opening of the new STOR-i building on campus so we are no longer borrowing space from other departments, but finally have our own STOR-i exclusive space.

We also displayed the recent posters we made from the previous simulation masterclass given by Barry Nelson in the new MRes baseroom. The one I worked on with Emily, Anna and Rob is below and is titled “Simulation Confidence Intervals for Input Uncertainty”.

My fellow MRes student Harjit, looking at the posters. Harjit’s blog.

# What is Bayes’ Theorem?

One of the most famous theorems in statistics and probability is that of Bayes’ Theorem, which first appeared around 250 years ago. It allows us to calculate reverse probabilities and use new evidence to update our beliefs. For example the probability of a hypothesis given a set of evidence can be found from the probability of that evidence given a hypothesis.

To understand Bayes’ Theorem it is important to have a basic understanding of conditional probability. This is the probability of something happening given that some event has already happened. Some examples of conditional probabilities are given below,

• Given that Watford scored a goal, what was the probability that Odion Ighalo scored?
• Given that it rained yesterday, what is the probability that it will remain tomorrow?
• Given a sports centre has a swimming pool, what is the probability it also has a gym?