# The 10-minute marathon challenge

Not content with having recently won the London marathon for the fourth time in a record time of 2:02:37, the phenomenal Kenyan athlete Eliud Kipchoge has announced a new bid to run the marathon distance in under two hours. The time Kipchoge set in the London marathon was already the second fastest in history and Kipchoge also holds the record for the fastest ever marathon, at 2:01:39, made in Berlin in 2018. But the sub- 2 hour marathon remains an elusive goal.

In 2016 Nike sponsored an attempt to break the 2-hour target. Three elite runners, including Kipchoge, trained privately to run a marathon-length distance in circuits around the Monza racetrack in Italy. Kipchoge won the race, but in a time of 2:00:25, therefore failing by 25 seconds to hit the 2-hour target. The specialised conditions for this attempt, including the use of relay teams of pace setters, meant that the race fell outside of IAAF standards, and therefore the 2:00:25 is not valid as a world record. Kipchoge’s planned attempt in London will also be made under non-standard conditions, so whatever time he achieves will also not be considered as valid in respect of IAAF rules. Regardless of this, beating the 2-hour barrier would represent a remarkable feat of human achievement, and this is Kipchoge’s goal.

But this begs the question: is a sub- 2 hour marathon under approved IAAF standards plausible? The following graphic shows how the marathon record has improved in the last 100 years or so, from Johnny Hayes’ record of 2:55:18 in 1908, right up to Kipchoge’s Berlin record.

Clearly there’s a law of diminishing returns in operation: the very substantial improvements in the first half of the graph are replaced by much smaller incremental improvements in the second half. This is perfectly natural: simple changes in training regimes and running equipment initially enabled substantial advances; more recent changes are much more subtle, and result in only in marginal improvements. So, the shape of the graph is no surprise. But if you were extrapolating forward to what might happen in the next 10, 100 or 1000 years, would your curve go below the 2-hour threshold or not?

Actually, it’s straightforward to take a set of data, like those contained in the graphic above, and fit a nice smooth curve that does a reasonable job at describing the overall pattern of the graph. And we could then extrapolate that curve into the future and see whether it goes below 2 hours or not. And if it does, we will even have a prediction of when it does.

But there’s a difficulty – the question of whether the solution crosses the 2-hour threshold or not is likely to depend very much on the type of curve we use to do the smoothing. For example, we might decide that the above graphic is best broken down into sections where the pattern has stayed fairly similar. In particular, the most recent section from around 1998 to 2018 looks reasonably linear, so we might extrapolate forward on that basis, in which case the 2-hour threshold is bound to be broken, and pretty soon too. On the other hand we might decide that the whole period of data is best described by a kind of ‘ell’-shaped curve which decreases to a lower horizontal limit. And then the question will be whether that limit is above or below 2 hours. In both cases the data will determine the details of the curve – the gradient of the straight line, for example, or the limit of the ‘ell’-shaped curve – but the form of the graph – linear, ‘ell’-shaped or something else – is likely to be made on more subjective grounds. And yet that choice will possibly determine whether the 2-hour threshold is predicted to be beaten or not.

There’s no way round this difficulty, though statistical techniques have been used to try to tackle it more rigorously. As I mentioned in a previous post, since athletics times are fastest times – whether it’s the fastest time in a race, or the fastest time ever when setting a record – it’s natural to base analyses on so-called extreme value models, which are mathematically suited to this type of process. But this still doesn’t resolve the problem of how to choose the curve which best represents the trend seen in the above picture. And the results aren’t terribly reliable. For example, in an academic paper ‘Records in Athletics Through Extreme-Value Theory‘ written in 2008 the authors John Einmahl and Jan Magnus predicted the absolute threshold times or distances (in case of field events) for a number of athletic events. At the time of writing their paper the world record for the marathon was 2:04:26, and they predicted a best possible time of 2:04:06.

History, of course, proved this to be completely wrong. To be fair to the authors though, they gave a standard error on their estimate of 57 minutes. Without going into detail, the standard error is a measure of how accurate the authors think their best answer is likely to be, and one rule of thumb interpretation of the standard error is that if you give a best answer and a standard error, then you’re almost certain the true value lies within 2 standard errors of your best answer. So, in this case, the authors were giving a best estimate of 2:04:06, but – rather unhelpfully – saying the answer could be as much as 114 minutes faster than that, taking us down to a marathon race time of 0:10:06.

So, come on Kipchoge, never mind the 2-hour marathon, let’s see if you’re up to the 10-minute marathon challenge.

Footnote: don’t trust everything you read in statistical publications. (Except in this blog, of course 😉).

Had a bad weekend? Maybe your team faded against relegated-months-ago Huddersfield Town, consigning your flickering hopes of a Champions League qualification spot to the wastebin. Or maybe you support Arsenal.

Anyway, Smartodds loves Statistics is here to help you put things in perspective: ‘We are in trouble‘. But not trouble in the sense of having to play Europa League qualifiers on a Thursday night. Trouble in the sense that…

Human society is under urgent threat from loss of Earth’s natural life

Yes, deep shit trouble.

This is according to a Global Assessment report by the United Nations, based on work by hundreds of scientists who compiled as many as 15,000 academic studies. Here are some of the headline statistics:

• Nature is being destroyed at a rate of tens to hundreds of times greater than the average over the last 10 million years;
• The biomass of wild mammals has fallen by 82%;
• Natural ecosystems have lost around half of their area;
• A million species are at risk of extinction;
• Pollinator loss has put up to £440 billion of crop output at risk;

The report goes on to say:

The knock-on impacts on humankind, including freshwater shortages and climate instability, are already “ominous” and will worsen without drastic remedial action.

But if only we could work out what the cause of all this is. Oh, hang on, the report says it’s…

… all largely as a result of human actions.

For example, actions like these:

• Land degradation has reduced the productivity of 23% of global land;
• Wetlands have drained by 83% since 1700;
• In the years 2000-2013 the area of intact forest fell by 7% – an area the size of France and the UK combined;
• More than 80% of wastewater, as well as 300-400m tons of industrial waste, is pumped back into natural water reserves without treatment;
• Plastic waste is a factor of tens greater than in 1980, affecting 86% of marine turtles, 44% of seabirds and 43% of marine animals.
• Fertiliser run-off has created 400 dead zones – an area the size of the UK.

You probably don’t need to be a bioscientist and certainly not a statistician to realise none of this is particularly good news. However, the report goes on to list various strategies that agencies, governments and countries need to adopt in order to mitigate against the damage that has already been done and minimise the further damage that will unavoidably be done under current regimes.  But none of it’s easy, and evidence so far is not in favour of collective human will to accept the responsibilities involved.

Josef Settele of the Helmholtz Centre for Environmental Research in Germany said

People shouldn’t panic, but they should begin drastic change. Business as usual with small adjustments won’t be enough.

So, yes, cry all you like about Liverpool’s crumbling hopes for a miracle against Barcelona tonight, but keep it in perspective and maybe even contribute to the wider task of saving humanity from itself.

<End of rant. Enjoy tonight’s game.>

Correction: *Bareclona’s* crumbling hopes

# More or Less

In a recent post I included a link to an article that showed how Statistics can be used to disseminate bullshit. That article was written by Tim Harford, who describes himself as ‘The Undercover Economist’, which is also the title of his blog. Besides the blog, Tim has written several books, one of which is also called ‘The Undercover Economist‘.

As you can probably guess from all of this, Tim is an economist who, through his writing and broadcasting, aims to bring the issues of economics to as wide an audience as possible. But there’s often a very thin boundary between what’s economics and what’s Statistics, and a lot of Tim’s work can equally be viewed from a statistical perspective.

The reason I mention all this is that Tim is also the presenter of a Radio 4 programme ‘More or Less’, whose aim is to…

…try to make sense of the statistics which surround us.

‘More or Less’ is a weekly half-hour show, which covers 3 or 4 topics each week. You can find a list of, and link to, recent episodes here.

As an example, at the time of writing this post the latest episode includes the following items:

• An investigation of a claim in a recent research paper that claimed floods had worsened by a factor of 15  since 2005;
• An investigation into a claim by the Labour Party that a recent resurgence in the number of cases of Victorian diseases is due to government  austerity policy;
• An interview with Matt Parker, who was referenced in this blog here, about his new book ‘Humble Pi’;
• An investigation into a claim in The Sunday Times that drinking a bottle of wine per week is equivalent to a losing ￡2,400 per year in terms of reduction in happiness.

Ok, now, admittedly, the whole tone of the programme is about as ‘Radio 4’ as you could possibly get. But still, as a means for learning more about the way Statistics is used – and more often than not, mis-used – by politicians, salespeople, journalists and so on, it’s a great listen and I highly recommend it.

If Smartodds loves Statistics was a radio show, this is what it would be like (but less posh).

# Taking things to extremes

One of the themes I’ve tried to develop in this blog is the connectedness of Statistics. Many things which seem unrelated, turn out to be strongly related at some fundamental level.

Last week I posted the solution to a probability puzzle that I’d posted previously. Several respondents to the puzzle, including Olga.Turetskaya@smartodds.co.uk, included the logic they’d used to get to their answer when writing to me. Like the others, Olga explained that she’d basically halved the number of coins in each round, till getting down to (roughly) a single coin. As I explained in last week’s post, this strategy leads to an answer that is very close to the true answer.

Anyway, Olga followed up her reply with a question: if we repeated the coin tossing puzzle many, many times, and plotted a histogram of the results – a graph which shows the frequencies of the numbers of rounds needed in each repetition – would the result be the typical ‘bell-shaped’ graph that we often find in Statistics, with the true average sitting somewhere in the middle?

Now, just to be precise, the bell-shaped curve that Olga was referring to is the so-called Normal distribution curve, that is indeed often found to be appropriate in statistical analyses, and which I discussed in another previous post. To answer Olga, I did a quick simulation of the problem, starting with both 10 and 100 coins. These are the histograms of the results.

So, as you’d expect, the average values (4.726 and 7.983 respectively) do indeed sit nicely inside the respective distributions. But, the distributions don’t look at all bell-shaped – they are heavily skewed to the right. And this means that the averages are closer to the lower end than the top end. But what is it about this example that leads to the distributions not having the usual bell-shape?

Well, the normal distribution often arises when you take averages of something. For example, if we took samples of people and measured their average height, a histogram of the results is likely to have the bell-shaped form. But in my solution to the coin tossing problem, I explained that one way to think about this puzzle is that the number of rounds needed till all coins are removed is the maximum of the number of rounds required by each of the individual coins. For example, if we started with 3 coins, and the number of rounds for each coin to show heads for the first time was 1, 4 and 3 respectively, then I’d have had to play the game for 4 rounds before all of the coins had shown a Head. And it turns out that the shape of distributions you get by taking maxima is different from what you get by taking averages. In particular, it’s not bell-shaped.

But is this ever useful in practice? Well, the Normal bell-shaped curve is somehow the centrepiece of Statistics, because averaging, in one way or another, is fundamental in many physical processes and also in many statistical operations. And in general circumstances, averaging will lead to the Normal bell-shaped curve.

Consider this though. Suppose you have to design a coastal wall to offer protection against sea levels. Do you care what the average sea level will be? Or you have to design a building to withstand the effects of wind. Again, do you care about average winds? Almost certainly not. What you really care about in each case will be extremely large values of the process: high sea-levels in one case; strong winds in the other. So you’ll be looking through your data to find the maximum values – perhaps the maximum per year – and designing your structures to withstand what you think the most likely extreme values of that process will be.

This takes us into an area of statistics called extreme value theory. And just as the Normal distribution is used as a template because it’s mathematically proven to approximate the behaviour of averages, so there are equivalent distributions that apply as templates for maxima. And what we’re seeing in the above graphs – precisely because the data are derived as maxima – are examples of this type. So, we don’t see the Normal bell-shaped curve, but we do see shapes that resemble the templates that are used for modelling things like extreme sea levels or wind speeds.

So, our discussion of techniques for solving a simple probability puzzle with coins, leads us into the field of extreme value statistics and its application to problems of environmental engineering.

But has this got anything to do with sports modelling? Well, the argument about taking the maximum of some process applies equally well if you take the minimum. And, for example, the winner of an athletics race will be the competitor with the fastest – or minimum – race time. Therefore the models that derive from extreme value theory are suitable templates for modelling athletic race times.

So, we moved from coin tossing to statistics for extreme weather conditions to the modelling of race times in athletics, all in a blog post of less than 1000 words.

Everything’s connected and Statistics is a very small world really.

# One touch football

Remember that thing about statistical diagrams of data telling their own story? Well, here’s a graphic showing the location of every touch of the ball by Alexis Sánchez in the 12 minutes he played as a substitute in the Manchester derby for United against City:

I recently posted a problem that had been shown to me by Benoit.Jottreau@smartodds.co.uk. Basically, you have a bunch of coins. You toss them and remove the ones that come up heads. You then repeat this process over and over till all the coins have been removed. The question was, if you start with respectively 10 or 100 coins, how many rounds of this game does it take on average till all the coins have been removed?

I’m really grateful to all of you who considered the problem and sent me a reply. The answers you sent me are summarised in the following graphs.

The graph on the left shows the counts of the guesses for the number of rounds needed when starting with 10 coins; the one on the right is the counts  but starting with 100 coins. The main features are as follows:

• Starting with 10 coins, the most popular answer was 4 rounds; with 100 coins the most popular answer was either 7 or 8 rounds.
• Almost everyone gave whole numbers as their answers. This wasn’t necessary. Even though the result of every experiment has to be a whole number, the average doesn’t. In a similar way, the average number of goals in a football match is around 2.5.
• The shape of the distribution of answers for the two experiments is much the same: heavily skewed to the right. This makes sense given the nature of the experiment: we can be pretty sure a minimum number of rounds will be needed, but less sure about the maximum. This is reflected in your collective answers.
• Obviously, with more coins, there’s more uncertainty about the answer, so the spread of values is much greater when starting with 100 coins.

Anyway, I thought the replies were great, and much better than I would have come up with myself if I’d just gone with intuition instead of solving the problem mathematically.

A few people also kindly sent me the logic they used to get to these answers. And it goes like this…

Each coin will come up heads or tails with equal probability. So, the average number of coins that survive each round is half the number of coins that enter that round. This is perfectly correct. So, for example, when starting with 10 coins, the average number of coins in the second round is 5. By the same logic, the average number of coins in the second round is 2.5. And the average number of coins in the third round is 1.25. And in the fourth round it’s 0.625. So, the first time the average number of coins goes below 1 is on the fourth round, and it’s therefore reasonable to assume 4 is the average number of rounds for all the coins to be removed.

Applying the same logic but starting with 100 coins, it takes 7 rounds for the average number of coins to fall below 1.

With a slight modification to the logic, to always round to whole numbers, you might get slightly different answers: say 5 and 8 instead of 4 and 7. And looking at the answers I received, I guess most respondents applied an argument of this type.

This approach is really great, since it shows a good understanding of the main features of the process: 50% of coins dropping out, on average, at each round of the game. And it leads to answers that are actually very informative: knowing that I need 4 rounds before the average number of coins drops below 1 is both useful and very precise in terms of explaining the typical behaviour of this process.

However… you don’t quite get the exact answers for the average number of rounds, which are 4.726 when you start with 10 coins, and 7.983 when you start with 100 coins. But where do these numbers come from, and why doesn’t the simple approach of dividing by 2 until you get below 1 work exactly?

Well, as I wrote above, starting with 10 coins you need 4 rounds before the average number of coins falls below 1. But this is a statement about the average number of coins. The question I actually asked was about the average number of rounds. Now, I don’t want to detract from the quality of the answers you gave me. The logic of successively dividing by 2 till you get below one coin is great, and as I wrote above, it will give an answer that is meaningful in its own right, and likely to be close to the true answer. But, strictly speaking, it’s focussing on the wrong aspect of the problem: the number of coins instead of the number of rounds.

The solution is quite technical. Not exactly rocket science, but still more intricate than is appropriate for this blog. But you might still find it interesting to see the strategy which leads to a solution.

So, start by considering just one of the coins. Its pattern of results (writing H for Heads and T for Tails) will be something like

• T, T, H; or
• T, T, T, H; or
• H

That’s to say, a sequence of T’s followed by H (or just H if we get Heads on the first throw).

But we’ve seen something like this before. Remember the post Get out of jail? We kept rolling a dice until we got the first 6, and then stopped. Well, this is the same sort of experiment, but with a coin. We keep tossing the coin till we get the first Head. Because of the similarity between these experiments, we can apply the same technique to calculate the probabilities for the number of rounds needed to get the first Head for this coin. One round will have probability 1/2, two rounds 1/4, three rounds 1/8 and so on.

Now, looking at the experiment as a whole, we have 10 (or 100) coins, each behaving the same way. And we repeat the experiment until all of the coins have shown heads for the first time. What this means is that the total number of rounds needed is the maximum of the number of rounds for each of the individual coins. It turns out that this gives a simple method for deriving a formula that gives the probabilities of the number of rounds needed for all of the coins to be removed, based on the probabilities for a single coin already calculated above.

So, we now have a formula for the probabilities for the numbers of rounds needed. And there’s a standard formula for converting this formula into the  average. It’s not immediately obvious when you see it, but with a little algebraic simplification it turns out that you can get the answer in fairly simple mathematical form. Starting with n coins – we had n=10 and n=100 –  the average number of rounds needed turns out to be

$1+\sum_{k=1}^n(-1)^{k-1}{n \choose k} (2^k-1)^{-1}$

The $\sum$ bit means do a sum, and the $~{n \choose k}~$ term is the number of unique combinations of k objects chosen from n. But don’t worry at all about this detail; I’ve simply included the answer to show that there is a formula which gives the answer.

With 10 coins you can plug n=10 into this expression to get the answer 4.726. With 100 coins there are some difficulties, since the calculation of $~{n \choose k}~$ with n=100 is numerically unstable for many values of k. But accurate approximations to the solution are available, which don’t suffer the same numerical stability problems, and we get the answer 7.983.

So, in summary, with a fair bit of mathematics you can get exact answers to the problem I set. But much more importantly, with either good intuition or sensible reasoning you can get answers that are very similar. This latter skill is much more useful in Statistics generally, and it’s fantastic that the set of replies I received showed collective strength in this respect.

# Do I feel lucky?

Ok, I’m going call it…

This is, by some distance:

The Best Application of Statistics in Cinematic History‘:

It has everything: the importance of good quality data; inference; hypothesis testing; prediction; decision-making; model-checking. And Clint Eastwood firing rounds off a 44 Magnum while eating a sandwich.

But, on this subject, do you feel lucky? (Punk)

Richard Wiseman is Professor in Public Understanding of Psychology at the University of Hertfordshire. His work touches on many areas of human psychology, and one aspect he has studied in detail is the role of luck. A summary of his work in this area is contained in his book The Luck Factor.

This is from the book’s Amazon description:

Why do some people lead happy successful lives whilst other face repeated failure and sadness? Why do some find their perfect partner whilst others stagger from one broken relationship to the next? What enables some people to have successful careers whilst others find themselves trapped in jobs they detest? And can unlucky people do anything to improve their luck – and lives?

Richard’s work in this field is based over many years of research involving a study group of 400 people. In summary, what he finds, perhaps unsurprisingly, is that people aren’t born lucky or unlucky, even if their perception is that they are. Rather, our attitude to life generally determines how the lucky and unlucky events we experience determine the way our lives pan out. In other words, we really do make our own luck.

He goes on to identify four principles we can adopt in order to make the best out of the opportunities (and difficulties) life bestows upon us:

1. Create and notice chance opportunities;
3. Create self-fulfilling prophesies via positive expectations;

In summary: if you have a positive outlook on life, you’re likely to make the best of the good luck that you have, while mitigating as well as is possible  against the bad luck.

But would those same four principles work well for a sports modelling company? They could probably adopt 1, 3 and 4 as they are, perhaps reinterpreted as:

1. Seek out positive value trading opportunities wherever possible.

3. Build on success. Keep a record of what works well, both in trading and in the company generally, and do more of it.

Principle 2 is a bit more problematic: much better to stress the need to avoid the trap of following instinct, when models and data suggest a different course of action. However, I think the difficulty is more to do with the way this Principle has been written, rather than what’s intended. For example, I found this description in a review of the book:

Lucky people actively boost their intuitive abilities by, for example… learning to dowse.

Learning to dowse!

But this isn’t what Wiseman meant at all. Indeed, he writes:

Superstition doesn’t work because it is based on outdated and incorrect thinking. It comes from a time when people thought that luck was a strange force that could only be controlled by magical rituals and bizarre behaviors.

So, I don’t think he’s suggesting you start wandering around with bits of wood in a search for underground sources of water. Rather, I think he’s suggesting that you be aware of the luck in the events around you, and be prepared to act on them. But in the context of a sports modelling company, it would make sense to completely replace reference to intuition with data and research. So…

2. Invest in data and research and develop your trading strategy accordingly.

And putting everything together:

1. Seek out positive value trading opportunities wherever possible.
2. Invest in data and research and develop your trading strategy accordingly.
3. Build on success. Keep a record of what works well, both in trading and in the company generally, and do more of it.

And finally, what’s that you say?  “Go ahead, make my day.” Ok then…

# “I don’t like your mum”

VAR, eh?

So, does video-assisted refereeing (VAR) improve the quality of decision-making in football matches?

Of course, that’s not the only question about VAR: assuming there is an improvement, one has to ask whether it’s worth either the expense or the impact it has on the flow of games when an action is reviewed. But these are subjective questions, whereas the issue about improvements in decision-making is more objective, at least in principle. With this in mind, IFAB, the body responsible for determining the laws of football, have sponsored statistical research into the extent to which VAR improves the accuracy of refereeing decisions.

But before looking at that, it’s worth summarising how the VAR process works. VAR is limited to an evaluation of decisions made in respect of four types of events:

• Goals
• Penalties
• Straight red cards
• Mistaken identity in the award of cards

And there are two modes of operation of VAR:

• Check mode
• Review mode

The check mode runs in the background throughout the whole game, without initiation by the referee. All incidents of the above type are viewed and considered  by the VAR, and those where a potential error are checked, with the assistance of replays if necessary. Such checks are used to identify situations where the referee is judged to have made a ‘clear and obvious error’ or there has been a ‘serious missed incident’.  Mistakes for other types of incidents – e.g. the possible award of a free kick – or mistakes that are not judged to be obvious errors should be discarded during the check process.

When a check by VAR does reveal a possible mistake of the above type, the referee is notified, who is then at liberty to carry out a review of the incident. The review can consist solely of a description of the event from the VAR to the referee, or it can comprise a video review of the incident by the referee using a screen at the side of the pitch. The referee is not obliged to undertake a review of an incident, even if flagged by the VAR following a check. On the other hand, the referee may choose to carry out a review of an incident, even if it has not been flagged by the VAR.

Hope that’s all clear.

Anyway, the IFAB report analysed more than 800 competitive games in which VAR was used, and includes the following statistics:

• 56.9% of checks were for penalties and goals; almost all of the others were for red card incidents;
• On average there were fewer than 5 checks per match;
• The median check time of the VAR was 20 seconds
• The accuracy of reviewable decisions before VAR was applied was 93%.
• 68.8% of matches had no review
• On average, there is one clear and obvious error every 3 matches
• The decision accuracy after VAR is applied is 98.9%.
• The median duration of a review is 60 seconds
• The average playing time lost due to VAR is less than 1% of the total playing time.
• In 24% of matches, VAR led to a change in a referee’s decision; in 8% of matches this change led to a decisive change in the match outcome.
• A clear and obvious error was not corrected by VAR in around  5% of matches.

This all seems very impressive. A great use of Statistics to check the implementation of the process and to validate its ongoing use. And maybe that’s the right conclusion. Maybe. It’s just that, as a statistician, I’m still left with a lot of questions. Including:

1. What was the process for checking events, both before and after VAR? Who decided if a decision, either with or without VAR, was correct or not?
2. It would be fairest if the analysis of incidents in this experiment were done ‘blind’. That’s to say, when an event is reviewed, the analyst should be unaware of what the eventual decision of the referee was. This would avoid the possibility of the experimenter – perhaps unintentionally – being drawn towards incorrect agreement with the VAR process decision.
3. It’s obviously the case when watching football, that even with the benefit of slow-motion replays, many decisions are marginal. They could genuinely go either way, without being regarded as wrong decisions. As such, the impressive-looking 93% and 98.9% correct decision rates are probably more fairly described as rates of not incorrect decisions.
4. There’s the possibility that incidents are missed by the referee, missed by VAR and missed by whoever is doing this analysis. As such, there’s a category of errors that are completely ignored here.
5. Similarly, maybe there’s an average of only 5 checks per match because many relevant incidents are being missed by VAR.
6. The use of the median to give average check and review times could be disguising the fact that some of these controls take a very long time indeed. It would be a very safe bet that the mean times are much bigger than the medians, and would give a somewhat different picture of the extent to which the process interrupts games when applied.

So, I remain sceptical. The headline statistics are encouraging, but there are aspects about the design of this experiment and the presentation of results that I find questionable. And that’s before we assess value in terms of cost and impact on the flow of games.

On the other hand, there’s at least some evidence that VAR is having incidental effects that aren’t picked up by the above experiment. It was reported that in Italy Serie A,  the number of red cards given for dissent during the first season of VAR was one, compared with eleven in the previous season. The implication being that VAR is not just correcting mistakes, but also leading to players moderating their behaviour on the pitch. Not that this improvement is being universally adopted by all players in all leagues of course. But anyway, this fact that VAR might actually be improving the game in terms of the way it’s played, above and beyond any potential improvements to the refereeing process, is an interesting aspect, potentially in VAR’s favour, which falls completely outside the scope of the IFAB study discussed above.

But in terms of VAR’s impact on refereeing decisions, I can’t help feeling that the IFAB study was designed, executed and presented in a way that shines the best possible light on VAR’s performance.

Incidentally, if you’re puzzled by the title of this post, you need to open the link I gave above, and exercise your fluency in Spanish vernacular.

# Picture this

You can’t help but be amazed at the recent release of the first ever genuine image of a black hole. The picture itself, and the knowledge of what it represents, are extraordinary enough, but the sheer feat of human endeavour that led to this image is equally breathtaking.

Now, as far as I can see from the list of collaborators that are credited with the image, actual designated statisticians didn’t really contribute. But, from what I’ve read about the process of the image’s creation, Statistics is central to the underlying methodology. I don’t understand the details, but the outline is something like this…

Although black holes are extremely big, they’re also a long way away. This one, for example, has a diameter that’s bigger than our entire solar system. But it’s also at the heart of the Messier 87 galaxy, some 55 million light years away from Earth. Which means that when looking towards it from Earth, it occupies a very small part of space. The analogy that’s been given is that capturing the black hole’s image in space would be equivalent to trying to photograph a piece of fruit on the surface of the moon. And the laws of optics imply this would require a telescope the size of our whole planet.

To get round this limitation, the Event Horizon Telescope (EHT) program uses simultaneous signals collected from a network of eight powerful telescopes stationed around the Earth. However, the result, naturally, is a sparse grid of signals rather than a complete image. The rotation of the earth means that with repeat measurements this grid gets filled-out a little. But still, there’s a lot of blank space that needs to be filled-in to complete the image. So, how is that done?

In principle, the idea is simple enough. This video was made some years ago by Katie Bouman, who’s now got worldwide fame for leading the EHT program to produce the black hole image:

The point of the video is that to recognise the song, you don’t need the whole keyboard to be functioning. You just need a few of the keys to be working – and they don’t even have to be 100% precise – to be able to identify the whole song. I have to admit that the efficacy of this video was offset for me by the fact that I got the song wrong, but in the YouTube description of the video, Katie explains this is a common mistake, and uses the point to illustrate that with insufficient data you might get the wrong answer. (I got the wrong answer with complete data though!)

In the case of the music video, it’s our brain that fills in the gaps to give us the whole tune. In the case of the black hole data, it’s sophisticated and clever picture imaging techniques, that rely on the known physics of light transmission and a library of the patterns found in images of many different types. From this combination of physics and library of image templates, it’s possible to extrapolate from the observed data to build proposal images, and for each one find a score of how plausible that image is. The final image is then the one that has the greatest plausibility score. Engineers call this image reconstruction; but the algorithm is fundamentally statistical.

At least, that’s how I understood things. But here’s Katie again giving a much  better explanation in a Ted talk:

Ok, so much for black holes. Now, think of:

1. Telescopes as football matches;
2. Image data as match results;
3. The black hole as a picture that contains information about how good football teams really are;
4. Astrophysics as the rules by which football matches are played;
5. The templates that describe how an image changes from one pixel to the next as a rule for saying how team performances might change from one game to the next.

And you can maybe see that in a very general sense, the problem of reconstructing an image of a black hole has the same elements as that of estimating the abilities of football teams. Admittedly, our football models are rather less sophisticated, and we don’t need to wait for the end of the Antarctic winter to ship half a tonne of hard drives containing data back to the lab for processing. But the principles of Statistics are generally the same in all applications, from black hole imaging to sports modelling, and everything in between.

# Out of the woods

I’ve made the point before that sometimes a graphical representation of data tells its own story, without any need for words or statistical analysis…