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Science, models and machine learning

The members of the Azimuth Project have been working on both predicting and understanding the El Niño phenomenon, along withexpository articles such as this . So far we've mostly talked about the physics and data of the El Niño, along with looking at one method of actually trying to predict El Niño events. Since there's going to more data exploration using methods more typical of machine learning, it's an opportune time to briefly describe the mindset and highlight some of differences between different kinds of predictive models.

statistics It ML should same also be pointed out that there's not a fundamental distinction betweenmachine learning and statistics. There are certainly differences in culture, background, terminology and typical real-world tasks considered, but in terms of the actual algorithms and mathematics used there's a great commonality. Throughout the rest of the article we'll use the talk about machine learning, models but could equally have used statistical models.

For our purposes here, a model is any systematic procedure for taking some input data and providing a prediction of some output data. There's a spectrum of models, ranging from physically based models at one end to purely data based models at the other. As a very simple example, suppose you have a commute by car from your place of work to your home and you want to leave work in order to arrive home at 6.30pm. You can tackle this by building a model which takes as input the day of the week and gives you back a time to leave.

  • There's the data driven approach, where you try various leaving times on various days and record whether or not you get home by 6.30pm. You might find that the traffic is lighter on weekend days so you can leave at 6.10pm while on weekdays you have to leave at 5.45pm, except on Wednesdays when you have to leave at 5.30pm. Since you've just crunched on the data you have no idea why it works, but it's a very reliable rule when you use it to predict when you need to leave.

  • There's the physical model approach, where you figure out how many people are doing on any given day and then figure out what that implies for the traffic levels and thence what time you need to leave. In this case you find out that there's a mid-week sports game on Wednesday evenings which leads to even higher traffic. By proceeding from first principles you've got a more detailed framework which is equally predictive but at the cost of having to investigate a lot of complications.

The one of those American medication adverts which presents a big message about "Using a purely data driven techniques is wonderful" while the voiceover lays out all sorts of small print. The remainder of this post will try to cover some of the simplest parts of the

How much data? data do we have?

A lot of problems vanish with unlimited data usage

data usage and burning data

So suppose we're trying to model a system where the some aspect has a true probability distribution $ P()$. We can't directly observe that, but we have some samples $ S$ obtained from observation of the system and hence come from $ P()$. Clearly there are problems if we generate this sample in a way that will bias the area of the distribution we sample from: it wouldn't be a good idea to try to get training data featuring heights in the American population by only handing out surveys in the locker rooms of basketball facilities. But if we take care to avoid (as much as possible) any bias, then we can make various kinds of estimates of the distribution that we think $ S$ comes from; lets call the estimate implied for $ S$ by some particular procedure $ P'()$. It would be nice if $ P = P'$ wouldn't it? And indeed many good estimators have the property that as the size of $ S$ tends to infinity $ P'$ will tend to $ P$. However, for finite sizes of $ S$, and especially for small sizes, $ P'$ may have some spurious detail that's not present in $ P$.

As a simple illustration of this, my computer has an a pseudo-random number generator which generates essentially uniformly distributed random numbers between 0 and 32767. I just asked for 8 numbers and got

  • 2928, 6552, 23979, 1672, 23440, 28451, 3937, 18910.

Note that we've got 3 values (23440, 23979 and 28451) an interval of length 5012 between 23440 and 28451. For this uniform distribution the expected value of the number within that range is 1.2ish. Readers will be familiar with how the expectation of a random quantity for a small sample will have a large amount of variation around its value that only reduces as the sample size increases, so this isn't a surprise. However, it does highlight that even with completely unbiased sampling from the full distribution will typically give rise to extra "structure" within the distribution implied by the samples.

Generalisation

Now almost all modelling techniques, while not necessarily estimating a full model of the probability distributions from the training samples, can be seen as building functions that are related to the probability distribution: for example, a thresholding classifier for dividing input into two output classes will place the threshold at the optimal point for the distribution implied by the samples. As a consequence, one important aim in building machine learning models is to try to estimate the features that are present in the full probability distribution while not learning such fine details that they are likely to be spurious features due to the small sampling. If you think about this, it's a bit counter-intuitive: you deliberately don't want to perfectly reflect every single the pattern in the training data. Indeed, specialising a model too closely to the training is given the name over-fitting.

This brings us to generalisation. Strictly speaking generalization is the ability of a model to work well upon unseen instances of the problem (which may be difficult for a variety of reasons). In practice however one tries hard to get representative training data so that the main issue in generalization is in preventing overfitting.

sparsity One factor that's often related to generalization issparsity . This refers to the degree to which a model has empty elements, typically represented as 0 coefficients. There are various reasons for wanting sparse models: the obvious one is speed of model evaluation, although that is much less significant with modern computing power. It's often possible to incorporate aprior into the modelling procedure which will encourage the model to be sparse. (Recall that in Bayesian modelling the prior represents our initial ideas of how likely various different parameter values are.) There are some cases where we have various detailed Bayesian priors about sparsity for problem specific reasons. However the more common case is having a "general modelling" belief, based upon general experience in modelling, that sparser models have a better generalisation performance.

<!-- <div align = "center"> <a href = "http://www.shrimpnews.com/FreeReportsFolder/WeatherFolder/ElNino.html#Monster"> <img width = "445" src = "http://math.ucr.edu/home/baez/ecological/el_nino/ElNinoMap1998.jpg"> </a> </div> -->

There are a couple of other reasons for wanting sparse models. The obvious one is speed of model evaluation, although that is much less significant with modern computing power. A less obvious reason is that one can often only use a sparse model, eg, if you're attempting to see how the input factors should be physically modified in order to affect the real system in a particular way. In this case one might want a good sparse model rather than an excellent dense model.

Inference

<!-- <div align = "center"> <a href = "http://www.shrimpnews.com/FreeReportsFolder/WeatherFolder/ElNino.html#Monster"> <img width = "445" src = "http://math.ucr.edu/home/baez/ecological/el_nino/ElNinoMap1998.jpg"> </a> </div> -->

Preprocessing and feature selection

without feature selection often won't converge.

often embodies some physical knowledge.

Inferring a physical model from a ML model

It is certainly possible to take a predictive model obtained by machine learning and use it to figure out a physically based model; this is one way of performing what's known a data mining. However in practice there are a couple of reasons why it's necessary to exhibit some care when doing this:

  • The variables in the training set may be related by some non-observed latent variables which may be difficult to reconstruct without knowledge of the physical laws that are in play.

  • x Machine learning models have a maddening ability to find variables that are predictive due to the way the training data was gathered. For example, in an early vision system aimed at finding tanks all the images of tanks were taken during one day on a military base when there was accidentally a speck of grime on the camera lens, while all the images of things that weren't tanks were taken on other days. The neural net cunningly learned that to decide if it was being shown a tank it should see if the shadow from the grime was in place or not.

  • It's common to have very highly correlated input variables. In that case a model will generally learn a function which utilises an arbitrary combination of the correlated variables and an equally good model would result from using any other combination. Certain sparsity encouraging priors have the useful property of encouraging the model to select only one representative from a group of correlated variables. However, even in that case it's still important not to assign too much importance to the particular division of model parameters in groups correlated variables.

  • y One can often come up with good machine learning models even when physically important variables haven't been collected in the training data. A related issue is that if all the training data is collected from a particular subspace factors that aren't important there won't be found. For example, if in a collision system to be modelled all data is collected at low speeds the machine learning model won't learn about relativistic effects that only have a big effect at a substantial fraction of the speed of light.

Conclusions

All of the ideas discussed above are really just ways of making sure that work on statistical/machine learning models are producing meaningful results in situations where the training data is scarce. As Bob Dylan (almost) sang , "to work outside the physical law, you must be honest, honest; I know you always say that you agree".

category: blog