How many cheeses?

I looked into the program of Mathematica User Cónference 2009 and when reading Wine-and-Cheese Reception I thought: "how many cheeses will people know in different territories in average?" In some, probably 7? I think, I know approx. 100, preferably French, Italian, English, Spanish, Swiss .... with very local ones like Ami-du-Chambertin, Castelmagno, Oxford Blue, Cabrales, Sbrinz, ..

Then I typed into Wolfram Alpha "cheese", knowing this is asking for static info and not computational knowledge. But yes, I got a list of more general cheese types (40?). Not bad, I thought and then "Roquefort", "Limburger", "Gruyere", ... returning useful info, like nutrition facts, nutrients compared to other food, calories, vitamins, ... Wow.

Beware Wind!

A little more finance. Again from Derman's Blog. In the following paper it is described that studies have found evidence that wind speed has strong influence on mood and consequently effect in stock returns. All extracted from data of wind speed and daily stock market returns across 18 European countries from 1994 to 2004. The authors claim: our findings contradict the rational asset-pricing hypothesis ...

This is a joke, or weather forecasters will become immensely rich and will only publish fake forecasts in the future to avoid sharing.

I really like data-driven methods, but this example shows, how important it is to analyse the full range of factors that might influence and apply multi-method analysis and how dangerous a naive approach can be.

Humans WANT to see patterns and machine learning shall help us to see the right ones. It often requires comprehensive cross-model checks, sub domain analysis, .... This is why we have passionately developed multiple methods and task builders for mlf .

I Saw It With My Eyes

Our machine learning framework is a multi-method-multi-strategy system. One of the methods: self organizing maps (SOM). A SOM is a type of artificial neural network that is trained using unsupervised learning to produce a, say, two-dimensional map from a usually high-dimensional input space of the training samples. They use a neighbourhood function to preserve the topological properties of the input space. This makes SOM useful for visualizing low-dimensional views of high-dimensional parameter spaces.

In the above maps you see the intensity of missing dots on printed paper (right side) and the values of a special parameter of the printing process (left). One sees that there is a region of correlation in the left/down corner but the many-missing-dots region in the center has no correspondence at this parameter (it might at another process parameter). One could imagine to have a monitoring board where all process parameters and quality parameters are represented as SOM und you control your process by staying in the high-quality regions of your quality parameters.

Surrogate Models

Remember the HPC post? Simulating a portfolio of financial instruments across scenarios might take days on single processors. What you have to do is solving many millions of complex differential or integro-differential equations. Mathematica helps us to exploit multi-core environments, but to get results in real-time, you need more. In certain cases (VaR), we gain enormous speed-up by principle component application, or, if Montecarlo simulation is required, we put valuations on GPU co-processors.

But we are also studying the possibility to use Surrogate Models (SuM). SuMs are easy-to-compute compact models that mimic the complex behaviour of the models in special ranges used in a simulation. Surrogate models are constructed using data-driven approaches as we know from machine learning in our mlf . Simplified: run the simulation with the "original" models, trying to cover the whole working-space, sample the results and extract the SUMs from them.

The finding process is time consuming, because you need to do massive model and cross-model testing.

In Mathematica's declarative environment it is easy to define higher level tasks, as in mlf, where one can define such machine learning tasks, which then run automatically.

In UnRisk this is an ongoing project, but if we think of principle component analysis, GPU co-processing, SuM and grid computing, we might gain gigantic speed-ups. With the indispensable support of Mathematica .

Asymptotic Mathematics

is not a common name. IMO, it suggests that striving for exact solutions, one could decompose a domain so that exact solutions are possible in the sub-domains, when impossible in the domain.

Above you see the famous Black-Scholes-Equation for the pricing of European Call options (with an end condition you can solve it exactly). But its assumptions: constant volatility, no early exercise, discrete dividend payments, ... are unrealistic.

In UnRisk we use integration techniques for the numerical treatment of deal types within a generalized Black-Scholes world. We decompose (time, underlying,..) in such a way that we can calculate so called Green-Functions which we integrate. The total solution comes from an asymptotic recomposition.

It is a proprietary method, which we call Adaptive Integration.

Mathematica offers comprehensive numerical and integrated symbolic computation and it often uses symbolic techniques behind the scenes to optimize numerical computations for time and accuracy. Again, it is the AND effect that matters.