Consider an art gallery with several rooms. Some of these rooms are connected directly by doorways but for some pairs of rooms it may be necessary to pass through one or more intermediary rooms in order to travel between them. Each room in the gallery contains various valuable pieces of artwork. At night, when the gallery is closed, a single guard must patrol the area to prevent thievery or vandalism from instituters (attackers). The Patrol Problem is to find a patrol route that minimizes the expected cost of any damage caused by attackers.

To approach this problem we must first create a model and make some modelling assumptions.

We can use the ideas from my post on The seven bridges of Königsberg to represent the rooms of the gallery as nodes on a graph as shown in the example below:

We assume that the total value of the artwork in each room is known to both the patroller and any potential attacker. We also assume that the length of time taken to carry out an attack in any given room is random but is sampled from a known distribution.

Our patrol model assumes that the attackers arrive according to a Poisson process with a known rate and then decide which room to attack in one of the two follow ways:

1. The target of the attack is chosen at random with known probabilities.
2. The target of each attack is chosen strategically with the presence of a patroller in mind and the aim to maximize the total expected cost of the attacks.

The patroller is assumed to move between rooms in discrete time-steps. If the patroller interrupts an attack in progress, we assume that no damage is caused.

We need a way to tell the patroller which is the best route to take.

If the attackers choose where to attack using the randomised method we have the following:

While visiting a location the patroller either determines that no attacks are underway or apprehends the attacker. Thus, we know that immediately after a visit to a location, no attackers are present. It therefore makes sense to characterize the system by a vector containing the number of time-steps since patroller visited each room. We call this the state of the model.

If we assume that the time it takes to carry out an attack has some maximum, we can ensure the number of states is finite. This is because once we have neglected a room long enough, increasing the time since the last visit will not change the probability that an attack is ongoing.

The current room can be determined from the state as it will correspond to the entry with the lowest value. A patrol policy then tells the patroller what to do in any given state: either stay where you are or move to an adjacent room.

Since there are a finite number or states and a finite number of rooms we have a finite number of policies. An optimal policy can be found using linear programming.

If the attackers choose where to attack strategically we can create a two-person zero sum game as discussed by in my post on game theory.

In either case the optimal solution is very computationally expensive to calculate and so approximate methods are often preferred.

This year STOR-i hosted its ninth annual conference, with talks from a wide variety of speakers from the UK — including some of its own PhD students and alumni — and from overseas. We listened to 12 presentations, so for the sake of brevity I will mention all of them but only go into more detail for a few.

We were welcomed to the conference by Prof. Kevin Glazebrook who spoke a bit about STOR-i’s new round of funding and introduced the first speaker: from the .

Jacquillat spoke about analytics in air transportation. In particular he discussed how we can use air traffic flow management to absorb delays upstream by holding planes on the runway so they don’t get held up in the air, expending extra fuel as they wait for their turn to land. He also spoke about the benefits of adjusting existing integer programs for scheduling so that they are optimised to minimise passenger delays. This would give greater priority to larger flights and ensuring connecting flights arrive on time.

The second talk was by third year STOR-i PhD student, Henry Moss, who introduced us to a Bayesian optimisation method called MUMBO, which he has been developing for his thesis.

Next up was from the who talked about her work with the Mallows ranking model as well as some applications and recent advances in the area.

After lunch we were given two more talks. The first by , from the , and the other by Georgia Souli, a third year STOR-i PhD student, before another break for refreshments.

We came back to a presentation titled “The Use of Shape Constraints for Modelling Time Series of Counts” by from Columbia University.

, a STOR-i alumni then talked to us about his work at using machine learning to detect fraud. One of the major problems here is that machine leaning requires data to help it learn but as the nature of fraud changes the algorithm must be able to adapt. The difficulty here is that since all the obvious fraud attempts are blocked future iterations will have no experience of them and so will have difficulties detecting them. Flowerdew suggested that allowing suspected fraudulent transactions be completed with some small probability and then proportionally increasing the weight  of these outcomes in the learning stage would allow the algorithm to learn more effectively and therefore prevent more fraud in the long run.

The final presentation of the day was: “Making random things better: Optimisation of Stochastic Systems” by from the .

We reconvened in the evening to look at posters made by the PhD students about each of their projects. This was a really good opportunity for them to develop their presentation skills by explaining their findings to knowledgeable academics in closely related fields. It was also an opportunity for us MRes students to learn a bit more about the research going on at the university and the sort of projects we might be interested in.

The following day we kicked off with a presentation by from . This focused on using the network of transactions between small and mediums sized businesses to improve credit risk models. Since transaction network data is difficult to get hold of, she also spoke about what approaches one can use without access to this data.

Next up was from who spoke about the balance between accuracy and interpretability for data science models and how this can be achieved.

Another STOR-i alumni, Ciara Pike-Burke, then talked about her recent work with multi-armed bandits. A multi-armed bandit can be thought of as a slot machine where pulling each arm will give a reward from some unknown distribution.  The usual problem is balancing exploration to learn more about the different reward distributions for each arm while also trying to maximise the total rewards by exploiting the arm that is performing best. The reward distributions are usually constant but Pike-Burke considered the case where the rewards are dependent on the previous actions of the player. For example a company can suggest different products to a customer on their website and the reward is dependent upon whether the customer follows that link. If the customer has just bought a bed they are probably less likely to buy another bed. However, that same customer might be more likely to buy new pillowcases.

Finally from presented his talk on “Model Based Clustering with Sparse Covariance Matrices.”