Genius Mathematics Consultants

Interested in applying machine learning to your business, but not sure where to start?

Having heard about the many spectacular achievements of machine learning and it’s growing adoption, many businesses are keen to use it to gain competitive advantage. And, worried about being left behind if they don’t!

However, despite grand ambitions, the rapid growth of the field means there is a lack of people with the expertise to implement actual working solutions. Many businesses are in the position of having a lot of data, but no idea how they can use it!

Machine learning has been applied to such a diverse set of applications that one can only sample them:

• Predicting which units are faulty early in a production line rather than late, to reduce wastage
• Automatic fruit grading, sorting and shelf-life estimation based on attributes like weight, shape, and colour
• Predicting which machine parts are likely to fail and when, to optimize when machine parts are replaced or scheduled for maintenance. A solution must be found which optimizes how limited maintenance and replacement budget is spent, while also minimizing failure and lost profit due to downtime
• Optimizing the structure of a network to minimize the probability of failure for a given network cost
• Automation of human roles in industry and manufacturing
• Optimizing system configurations based on input like weather data, time of year, order numbers etc.

We’re interested in developing mutually beneficial collaborations with industry. To that end, we’d be interested in helping you explore if and how machine learning can increase profits, reduce wastage, and optimize the efficiency of your business.

Please explore the site to learn more about our algorithm consulting services (we offer general math consulting as well). Then, we’d love to get this conversation started, so please contact us to express your interest!

Remote sensing systems (such as LIDAR) allow us to measure a plethora of variables on the earth’s surface and in the air. This includes temperature, wind, vegetation, clouds, ice and many more. However, the relationship between the variable to be measured and the signal received can be complex, as radiation passing through the atmosphere is scattered and absorbed by clouds and molecules in the atmosphere. Attempting to determine this relationship using theory alone may be arduous. Machine learning is a natural alternative.

Let \(v\) denote some variable we wish to measure (perhaps on the earth’s surface), and \(s\) the signal received by the sensor (possibly located on a satellite). We wish to find a relationship

\[v = f(s),\]

that will allow us to convert the properties of the signal, which may contain data from several frequency bands, to the properties we actually wish to measure. Suppose we have gathered a large number of data pairs \((s_i,v_i)\). We can do this be recording the signal received \(s_i\) when the sensor is pointed at a location whose properties \(v_i\) are already known. We then use these data points to train or teach a neural network.

Ideally, the data used for fitting is only a subset of the total amount of data collected, so that the neural network can be cross-validated against data that was not used in the fitting process.

Machine learning has been applied to remote sensing in many practical applications including:

• Measuring the chlorophyll concentration in the ocean
• Classifying vegetation
• Classifying cloud types
• Measuring precipitation
• Identifying snow cover
• Forecasting

Of course, remote sensing is only one of an unlimited number of applications of machine learning. Whenever you wish to determine a relationship present in your data that may be too complex to determine using scientific theory, machine learning is an exciting alternative.

Network reliability optimization problems first appeared for telecommunication and transport systems. These days, they also have applications in computer networks, and electric and gas networks. In the modern world, the consequences of a network failing can be catastrophic in terms of cost and even lives lost. Since many non-mathematical professionals need to design networks of various kinds, network optimization is a great example of how math consulting can deliver value for industry.

The goal of reliability optimization is to minimize the cost of the network, while ensuring it still meets some minimum standard of “reliability”, represented by the probability of the network failing.  Conceptually, a network can be represented by an undirected graph \(G = (N,E)\) with nodes \(N\) and edges \(E\):

Let \(x_{ij}\) represent which nodes we build connections between, i.e.\(x_{ij}=1\) if there is an edge between node \(i\) and node \(j\) and \(x_{ij}=0\) otherwise. Let \(c_{ij}\) represent the cost of building a connection between node \(i\) and node \(j\). Then the cost of building the network is

\[C(x) = \sum_{i=1}^{n-1} \sum_{j=i+1}^{n}c_{ij}x_{ij}.\]

Let \(p\) represent the probability that a node will be operational. There are two commonly used standards for when a network is considered to be operational, depending on the application:

1. Two specified nodes are connected
2. Every node is connected to every other node

It is then possible to assign to every network \(x\) a reliability \(R(x)\), which is the probability that the network will be operational. Suppose our tolerance for network failure is \(R_{min}\). Then we wish to minimize the cost function \(C(x)\) subject to the constraint \(R(x) \geq R_{min}\).

Mathematically speaking, network reliability problems are examples of integer programming problems, and can be solved in a variety of ways, such as using genetic algorithms.

A common choice of distribution for operational risk modelling is lognormal. This distribution has a fairly heavy tail and is mathematically and conceptually simple. Often a separate lognormal will be used to model the body and the tail of each risk cell, so that they can be parameterised separately. The body distribution can be chosen to closely match the common losses that occur in the internal data for that risk cell, which the tail can chosen to match the expectation of extreme losses. The necessity of this is questionable, since the high quantiles computed in op risk modelling are overwhelmingly driven by the tail. In most cases, if model simplicity is desired, a single distribution using the tail parameters is sufficient.

Once the loss severity distribution has been estimated for each risk cell, along with a corresponding frequency distribution (typically Poisson, although op risk models are not sensitive to this choice), then Monte Carlo simulation is used to sum them and produce the bank’s total or aggregate loss distribution. The modeller may then be surprised to discover that the median aggregate loss is far higher than the bank’s typical annual loss! After all, the median loss from the body distribution matches that in the internal data for each risk cell, so shouldn’t the  same be true of the aggregate distribution?

The explanation here lies in the heavy tailed nature of the lognormal distribution. In particular, the lognormal has the property that for large \(N\), the sum of \(N\) randomly generated losses approaches the maximum loss, i.e.

\[\sum_{i=1}^N x_i \to \max(\{x_i\}).\]

This means that, surprisingly, even a typical loss in the aggregate distribution (say, the median) is primarily constituted by a small number of tail losses from the individual distributions, and not losses near the medians! Thus, the medians of the individual distributions end up being pretty much irrelevant.

Of course, op risk models are typically constructed to estimate the 99.9% quantile, not the median (which banks have data for anyway). However, it is instructive to note that it is very difficult to model op risk in a way that accurately reproduces both the extreme losses, and the more common losses, due to the heavy tailed nature of the distributions.

This illustration shows how even relatively simple financial modelling problems can result in technical complexities that are very difficult for finance profressionals to understand and resolve. This is a great example of how our quantitative analysis consulting services can deliver great value and impress the regulators!

Anyone who has worked in financial risk modelling can testify to the fact that Monte Carlo simulations can take a long time to run – sometimes an entire day or longer! The problem this causes is that model testing – particularly sensitivity testing around the inputs, parameters and methodologies – ends up taking weeks and months. The loss in productivity (and the sanity of the modeler) is huge.

This is a really simple technique that can reduce run time by as much as half.

Often, risk modelers are interested in simulating from an aggregate distribution that is the sum of multiple other distributions. Usually it is the very high quantiles of this aggregate distribution that they want to estimate, such as the 99.9% quantile or 1 in 1000 event.  They would therefore like to generate a lot of losses from the tail of the distribution, to get sufficient resolution for estimating the high quantiles. Unfortunately, by definition most of the losses generated during a Monte Carlo simulation will not fall in the tail of the constituent distributions, and 50% will even fall below the median. for heavy-tailed distributions, these small losses usually don’t contribute much to the large losses in the aggregate distribution. This means that a tremendous amount of computational power is used to generate high density losses in the body of the distribution, just to get a handful of losses in the tail.

The solution: re-use those body losses! Put a cap on the number of body losses you wish to generate, say 10,000 losses below the 90% quantile. Before beginning the Monte Carlo simulation, generate these 10,000 losses and store them in a vector. Now, whenever the simulation calls for a loss below the 90% quantile, simply locate the nearest corresponding loss in this vector. In other words, if the random number generates the number 0.44, we would calculate 0.44*10,000 and round to the nearest whole number and use the loss with the corresponding index. This procedure is much faster than the calculations required to invert a lognormal cdf. However, losses above the 90% quantile should continue to be uniquely generated to preserve accuracy in the tail.

While there are more sophisticated approaches to improving Monte Carlo performance, a simple approach like this is quick to implement and less prone to errors.

Another tip that’s handy to know is that with certain distributions, there may be a shortcut that doesn’t require you to generate all the losses again when you change a parameter. For a lognormal distribution, for example, the losses generated are proportional to \(\exp(\mu)\). This means that if wish to change the \(\mu\) parameter from \(\mu_1\) to \(\mu_2\), all you have to do is scale all the losses by \(\exp(\mu_2-\mu_1)\)! This can save a tremendous amount of time when conducting sensitivity testing.

Does your bank have Monte Carlo models which take all day to run? Why not drop us an email to find out how our financial modelling consulting services can work for you.