Coronavirus maths

As a growing number of people seek to educate themselves on coronavirus COVID-19, while confined to their homes, a better understanding can be gained by taking a look at how to model an epidemic.

Researchers have created highly complex models of the spread of infections. For example, BlueDot’s disease-tracking model, described in this podcast, monitors the Internet with AI language translators and evaluates the network effects of transmission based on air travel itineraries. However, a surprising amount of insight can be gained from a very simple approach called the SIR model.

SIR model

The SIR model divides the population into three classes. The susceptible class (S) includes everyone who can catch the infection. In the case of a novel virus like corona, it seems that the entire global population was initially susceptible. The infected class (I) includes all those currently infected and able to transmit the virus to susceptible people. The removed class (R) includes everyone who has recovered from the virus or, unfortunately, died. In the model, these people no longer transmit the disease nor are they susceptible. The idea is that people move from the susceptible class to the infected class to the removed class.

Although there is much focus in the media on the exponential rise of the total number of cases of coronavirus, this figure includes recoveries and deaths. In one sense this is a huge underestimate, because the figures only includes people who have taken a test and returned a positive result. As explained by Tomas Pueyo, many people do not display symptoms until around 5 days after infection and for over 90% these symptoms are mild, so there could be ten times more people infected than the official figures suggest. In another sense, the figures are a huge exaggeration, because people who have recovered are unlikely to be infectious, because their immune systems have fought off the virus.

The SIR model measures the number of infectious people. On the worldometers site these are called “active cases”. The critical insight of the SIR model, shown in the diagram above, is that the class of infected people grows if the daily number of new cases exceeds the number of closed cases.

Closed cases – removal rate

In a real epidemic, experts don’t really know how many people are infected, but they can keep track of those who have died or recovered. So it is best to start by considering the rate of transfer from infected to removed. After some digging around, it appears that the average duration of an infection is about 2 weeks. So in a steady state situation, about on person in 14 or 7% of those infected would recover every day. This percentage would be a bit less than this if the epidemic is spreading fast, because there would be more people who have recently acquired the virus, so let’s call it 6%. For the death rate, we need the number of deaths divided by the (unknown) total number of people infected. This is likely to be lower than the “case fatality rate” reported on worldometers because that divides the number of deaths only be the number of positive tests. The death rate is estimated to be 2-3%. If we add 3% to the 6% of those recovering, the removal rate (call it “a”) is estimated to be 9%.

In the absence of a cure or treatment for the virus, it is unlikely that the duration of infectiousness can be reduced. As long as hospitals are not overwhelmed, those who might otherwise have died may be saved. However, there is not much that governments or populations can do to speed up the daily rate of “closed cases”. The only levers available are those to reduce the number of “new cases” below 9%. This appeared to occur as a result of the draconian actions taken in China in the second half of February, but the sharp increase in new cases that became apparent over the weekend of 6/7 March spooked the financial markets.

New cases – infection rate

In the SIR model, the number of new infections depends on three factors: the number of infectious people, the number of susceptible people and the something called the infection rate (r), which measures the probability that an infected person passes on the virus to a susceptible person either through direct contact or indirectly, for example, by contaminating a surface, such as a door handle.

Governments can attempt to reduce the number of new infections by controlling each of the three driving factors. Clearly, hand washing and avoiding physical contact can reduce the infection rate. Similarly infected people are encouraged to isolate themselves in order to reduce the proportion of susceptible people exposed to direct or indirect contact. Guidance to UK general practitioners is to advise patients suffering from mild symptoms to stay at home and call 111, while those with serious symptoms should call 999.

When will it end?

As more people become infected, the number of susceptible people naturally falls. Until there is a vaccine, there is nothing governments can do to speed up this decline. Eventually, when enough people will have caught the current strain of the virus, so-called “herd immunity” will prevail. It is not necessary for everyone to have come into contact with the virus, rather it is sufficient for the number of susceptible people to be smaller than a critical value. Beyond this point infected people, on average, recover before they encounter a susceptible person. This is how the epidemic will finally start to die down.

When people refer to the transmission rate or reproduction rate of a virus, they mean the number of secondary infections produced by a primary infection across a susceptible population. This is equal to the number of susceptible people times the infection rate divided by the removal rate. This determines the threshold number of susceptible people below which the number of infections falls. The critical value is equal to removal rate relative to the infection rate (a/r) . When the number of susceptible people falls to this critical value, the number of infected people will reach a peak and subsequently decline. More susceptible people will still continue to be infected, but at a decreasing rate, until the infection dies out completely, by which time a significant part of the population will have been infected.

This looks very scary indeed

Running the figures through the SIR model produces some extremely scary predictions. At the time of writing, new cases of infection were rising at a rate of about 15%. At this rate, the virus could spread to 2/3 of the population before it dies out. If three percent of those infected die, the virus would kill 2 percent of the population. Based on results so far these would be largely elderly people or those suffering from complications, so it is extremely important that they are protected from infection. If the virus continues to run out of control, the number of deaths could run into the millions before the epidemic ends.

It is absolutely essential to reduce the infection rate and keep it low, particularly among the elderly and vulnerable groups

We should watch China carefully for new cases. If none arise, it suggests that a large proportion of the population gained immunity through infection, even though lower numbers of infections were reported. However, if the imposition of constraints temporary reduced the infection rate, leaving a large susceptible pool still vulnerable, the epidemic could re-emerge once the constraints are relaxed.

What to do?

The imposition of governments restrictions on travel and large gatherings are forcing the people to rethink their options. Where possible, office workers, university students, schoolchildren and sportsmen may find themselves congregating online in virtual environments rather than in the messy and dangerous real world.

Among cyclists, this ought to be good news for Zwift and other online platforms. Zwift seems to be particularly well-positioned, due to its strong social aspect, which allows riders to meet and race against each other in virtual races. It also has the potential to allow world tour teams to compete in virtual races.

In fact the ability to meet friends and do things together virtually would have applications across all walks of life. Sports fans need something between the stadium and the television. Businesses need a medium that fills the gap between a physical office and a conference call. Schools and universities require better ways to ensure that students can learn, while classrooms and lecture theatres are closed. These innovations may turn out to be attractive long after the coronavirus scare has subsided.

While the prospect of going to the pub or a crowded nightclub loses its appeal, cycling offers the average person a very attractive alternative way to meet friends while avoiding close proximity with large groups of people. As the weather improves, the chance to enjoying some exercise in the fresh air looks ever more enticing.

End notes

SIR model based on: Mathematical Biology J.D. Murray, chapter 19

Modelling Strava Fitness and Freshness

Since my blog about Strava Fitness and Freshness has been very popular, I thought it would be interesting to demonstrate a simple model that can help you use these metrics to improve your cycling performance.

As a quick reminder, Strava’s Fitness measure is an exponentially weighted average of your daily Training Load, over the last six weeks or so. Assuming you are using a power meter, it is important to use a correctly calibrated estimate of your Functional Threshold Power (FTP) to obtain an accurate value for the Training Load of each ride. This ensures that a maximal-effort one hour ride gives a value of 100. The exponential weighting means that the benefit of a training ride decays over time, so a hard ride last week has less impact on today’s Fitness than a hard ride yesterday. In fact, if you do nothing, Fitness decays rate is about 2.5% per day.

Although Fitness is a time-weighted average, a simple rule of thumb is that your Fitness Score equates to your average daily Training Load over the last month or so. For example, a Fitness level of 50 is consistent with an average daily Training Load (including rest days) of 50. It may be easier to think of this in terms of a total Training Load of 350 per week, which might include a longer ride of 150, a medium ride of 100 and a couple of shorter rides with a Training Load of 50.

How to get fitter

The way to get fitter is to increase your Training Load. This can be achieved by riding at a higher intensity, increasing the duration of rides or including extra rides. But this needs to be done in a structured way in order be effective. Periodisation is an approach that has been tried and tested over the years. A four-week cycle would typically include three weekly blocks of higher training load, followed by an easier week of recovery. Strava’s Fitness score provides a measure of your progress.

Modelling Fitness and Fatigue

An exponentially weighted moving average is very easy to model, because it evolves like a Markov Process, having the following property, relating to yesterday’s value and today’s Training Load.
$F_{t} = \lambda * F_{t-1}+\left ( 1-\lambda \right )*TrainingLoad_{t}$
where
$F_{t}$ is Fitness or Fatigue on day t and
$\lambda = exp(-1/42) \approx 0.976$ for Fitness or
$\lambda = exp(-1/7) \approx 0.867$ for Fatigue

This is why your Fitness falls by about 2.5% and your Fatigue eases by about 13.5% after a rest day. The formula makes it straightforward to predict the impact of a training plan stretching out into the future. It is also possible to determine what Training Load is required to achieve a target level of Fitness improvement of a specific time period.

Ramping up your Fitness

The change in Fitness over the next seven days is called a weekly “ramp”. Aiming for a weekly ramp of 5 would be very ambitious. It turns out that you would need to increase your daily Training Load by 33. That is a substantial extra Training Load of 231 over the next week, particularly because Training Load automatically takes account of a rider’s FTP.

Interestingly, this increase in Training Load is the same, regardless of your starting Fitness. However, stepping up an average Training Load from 30 to 63 per day would require a doubling of work done over the next week, whereas for someone starting at 60, moving up to 93 per day would require a 54% increase in effort for the week.

In both cases, a cyclist would typically require two additional hard training rides, resulting in an accumulation of fatigue, which is picked up by Strava’s Fatigue score. This is a much shorter term moving average of your recent Training Load, over the last week or so. If we assume that you start with a Fatigue score equal to your Fitness score, an increase of 33 in daily Training Load would cause your Fatigue to rise by 21 over the week. If you managed to sustain this over the week, your Form (Fitness minus Fatigue) would fall from zero to -16. Here’s a summary of all the numbers mentioned so far.

Whilst it might be possible to do this for a week, the regime would be very hard to sustain over a three-week block, particularly because you would be going into the second week with significant accumulated fatigue. Training sessions and race performance tend to be compromised when Form drops below -20. Furthermore, if you have increased your Fitness by 5 over a week, you will need to increase Training Load by another 231 for the following week to continue the same upward trajectory, then increase again for the third week. So we conclude that a weekly ramp of 5 is not sustainable over three weeks. Something of the order of 2 or 3 may be more reasonable.

A steady increase in Fitness

Consider a rider with a Fitness level of 30, who would have a weekly Training Load of around 210 (7 times 30). This might be five weekly commutes and a longer ride on the weekend. A periodised monthly plan could include a ramp of 2, steadily increasing Training Load for three weeks followed by a recovery week of -1, as follows.

This gives a net increase in Fitness of 5 over the month. Fatigue has also risen by 5, but since the rider is fitter, Form ends the month at zero, ready to start the next block of training.

To simplify the calculations, I assumed the same Training Load every day in each week. This is unrealistic in practice, because all athletes need a rest day and training needs to mix up the duration and intensity of individual rides. The fine tuning of weekly rides is a subject for another blog.

A tougher training block

A rider engaging in a higher level of training, with a Fitness score of 60, may be able to manage weekly ramps of 3, before the recovery week. The following Training Plan would raise Fitness to 67, with sufficient recovery to bring Form back to positive at the end of the month.

A general plan

The interesting thing about this analysis is that the outcomes of the plans are independent of a rider’s starting Fitness. This is a consequence of the Markov property. So if we describe the ambitious plan as [3,3,3,-2], a rider will see a Fitness improvement of 7, from whatever initial value prevailed: starting at 30, Fitness would go to 37, while the rider starting at 60 would rise to 67.

Similarly, if Form begins at zero, i.e. the starting values of Fitness and Fatigue are equal, then the [3,3,3,-2] plan will always result in a in a net change of 6 in Fatigue over the four weeks.

In the same way, (assuming initial Form of zero) the moderate plan of [2,2,2,-1] would give any rider a net increase of Fitness and Fatigue of 5.

Use this spreadsheet to experiment.

Predicting the World Champion

A couple of years ago I built a model to evaluate how Froome and Dumoulin would have matched up, if they had not avoided racing against each other over the 2017 season. As we approach the 2019 World Championships Road Race in Yorkshire, I have adopted a more sophisticated approach to try to predict the winner of the men’s race. The smart money could be going on Sam Bennett.

Deep learning

With only two races outstanding, most of this year’s UCI world tour results are available. I decided to broaden the data set with 2.HC classification European Tour races, such as the OVO Energy Tour of Britain. In order to help with prediction, I included each rider’s weight and height, as well as some meta-data about each race, such as date, distance, average speed, parcours and type (stage, one-day, GC, etc.).

The key question was what exactly are you trying to predict? The UCI allocates points for race results, using a non-linear scale. For example, Mathieu Van Der Poel was awarded 500 points for winning Amstel Gold, while Simon Clarke won 400 for coming second and Jakob Fuglsang picked up 325 for third place, continuing down to 3 points for coming 60th. I created a target variable called PosX, defined as a negative exponential of the rider’s position in any race, equating to 1.000 for a win, 0.834 for second, 0.695 for third, decaying down to 0.032 for 20th. This has a similar profile to the points scheme, emphasising the top positions, and handles races with different numbers of riders.

A random forest would be a typical choice of model for this kind of data set, which included a mixture of continuous and categorical variables. However, I opted for a neural network, using embeddings to encode the categorical variables, with two hidden layers of 200 and 100 activations. This was very straightforward using the fast.ai library. Training was completed in a handful of seconds on my MacBook Pro, without needing a GPU.

After some experimentation on a subset of the data, it was clear that the model was coming up with good predictions on the validation set and the out-of-sample test set. With a bit more coding, I set up a procedure to load a start list and the meta-data for a future race, in order to predict the result.

Predictions

With the final start list for the World Championships Road Race looking reasonably complete, I was able to generate the predicted top 10. The parcours obviously has an important bearing on who wins a race. With around 3600m of climbing, the course was clearly hilly, though not mountainous. Although the finish was slightly uphill, it was not ridiculously steep, so I decided to classify the parcours as rolling with a flat finish

It was encouraging to see that the model produced a highly credible list of potential top 10 riders, agreeing with the bookies in rating Mathieu Van Der Poel as the most likely winner. Sagan was ranked slightly below Kristoff and Bennett, who are seen as outsiders by the pundits. The popular choice of Philippe Gilbert did not appear in my top 10 and Alaphilippe was only 9th, in spite of their recent strong performances in the Vuelta and the Tour, respectively. Riders in positions 5 to 10 would all be expected to perform well in the cycling classics, which tend to be long and arduous, like the Yorkshire course.

For me, 25/1 odds on Sam Bennett are attractive. He has a strong group of teammates, in Dan Martin, Eddie Dunbar, Connor Dunne, Ryan Mullen and Rory Townsend, who will work hard to keep him with the lead group in the hillier early part of the race. Then he will then face an extremely strong Belgian team that is likely to play the same game that Deceuninck-QuickStep successfully pulled off in stage 17 of the Vuelta, won by Gilbert. But Bennett was born in Belgium and he was clearly the best sprinter out in Spain. He should be able to handle the rises near the finish.

A similar case can be made for Kristoff, while Matthews and Van Avermaet both had recent wins in Canada. Nevertheless it is hard to look past the three-times winner Peter Sagan, though if Van Der Poel launches one of his explosive finishes, there is no one to stop him pulling on the rainbow jersey.

Appendix

After the race, I checked the predicted position of the eventual winner, Mads Pedersen. He was expected to come 74th. Clearly the bad weather played a role in the result, favouring the larger riders, who were able to keep warmer. The Dane clearly proved to be the strongest rider on the day.

References

Code used for this project

Cycling Physique

It is easy to assume that successful professional cyclists are all skinny little guys, but if you look at the data, it turns out that they have an average height of 1.80m and an average weight of around 68kg. If we are to believe the figures posted on ProCyclingStats, hardly any professional cyclists would be considered underweight. In fact, they would struggle to perform at the required level if they did not maintain a healthy weight.

Taller than you might think

According to a study published in 2013 and updated in 2019, the global average height of adult males born in 1996 was 1.71m, but there is considerable regional variation. The vast majority of professional cyclists come from Europe, North America, Russia and the Antipodes where men tend to be taller than those from Asia, Africa and South America. For the 41 Colombians averaging 1.73m, there are 85 Dutch riders with a mean height of 1.84m. See chart below.

Furthermore, road cycling involves a range of disciplines, including sprinting and time trialling, where size and raw power provide an advantage. The peloton includes larger sprinters alongside smaller climbers.

Not as light as expected

While 68kg for a 1.80m male is certainly slim, it equates to a body mass index of 21 (BMI = weight / (height)²), which is towards the middle of the recommended healthy range. BMI is not a sophisticated measure, as it does not distinguish between fat and muscle. Since muscle is more dense than fat and cyclists tend to have it a higher percentage of lean body mass, they will look slimmer than a lay person of equivalent height and weight. Nevertheless doctors use BMI as a guide and become concerned when it falls below 18.5.

The chart includes over 1,100 professional cyclists, but very few pros would be considered underweight. The majority of riders have a BMI of between 20 and 22. Although Colombian riders (red) tend to be smaller, specialising in climbing, their average BMI of 20.8 is not that different from larger Dutch riders (orange) with a mean BMI of 21.2. The taller Colombians include the sprinters Hodeg, Gaviria and Molano.

Types of rider

This chart shows the names of a sample of top riders. All-out sprinters tend to have a BMI of around 24, even if they are small like Caleb Ewan. Sprints at the end of more rolling courses are likely to be won by riders with a BMI of 22, such as Greipel, van Avermaet, Sagan, Gaviria, Groenewegen, Bennet and Kwiatkowski. Time trial specialists like Dennis and Thomas have similar physiques, though Dumoulin and Froome are significantly lighter and remarkably similar to each other.

GC contenders Roglic, Kruiswijk and Gorka Izagirre are near the centre of the distribution with a BMI around 21, close to Viviani, who is unusually light for a sprinter. Pinot, Valverde, Dan Martin, the Yates brothers and Pozzovivo appear to be light for their heights. Interestingly climbers such as Quintana, Uran, Alaphilippe, Carapaz and Richie Porte all have a BMI of around 21, whereas Lopez is a bit heavier.

If the figures reported on ProCyclingStats are accurate, George Bennet and Emanuel Buchmann are significantly underweight. Weighting 58kg for a height of 1.80m does not seem to be conducive to strong performance, unless they are extraordinary physical specimens.

Conclusions

Professional cyclists are lean, but they would not be able to achieve the performance required if they were underweight. It is possible that the weights of individual riders might vary over time by a couple of kilos, moving them a small amount vertically on the chart, but scientific approaches are increasingly employed by expert nutritionists to avoid significant weight loss over longer stage races. The Jumbo Foodcoach app was developed alongside the Jumbo-Visma team and, working with Team Sky, James Morton strove to ensure that athletes fuel for the work required. Excessive weight loss can lead to a range of problems for health and performance.

Betting on the Tour

On the eve of the Tour de France, the pundits have made their predictions, but when the race is over, they will be long forgotten. One way of checking your own forecasts is to take a look at the odds offered on the betting markets. These are interesting, because they reflect the actions of people who have actually put money behind their views. In an efficient and liquid market, the latest prices ought to reflect all information available. This blog takes a look at the current odds, without wishing to encourage gambling in any way.

The website oddchecker.com collates the odds from a number of bookmakers across a large range of bets. It is helpful to convert the odds into predicted probabilities. Focussing on the overall winner,  Egan Bernal is the favourite at 5/2 (equating to a 29% probability taking the yellow jersey), followed by Geraint Thomas at 7/2 (22%) and Jakob Fuglsang at 6/1 (14%). This gives a 51% chance of a winner being one of the two Team Ineos riders. The three three leading contenders are some distance ahead of Adam Yates, Richie Porte, Thibaut Pinot and Nairo Quintana. Less fancied riders include Roman Bardet, Steven Kruijswijk, Rigoberto Uran, Mikel Landa, Enric Mas and Vincenzo Nibali. Anyone else is seen as an outsider.

Ups and downs

The odds change over time, as the markets evaluate the performance and changing fortunes of the riders. In the following chart shows the fluctuations in the average daily implied winning chances of the three current favourites since the beginning of the year, according to betfair.com.

The implied probability that Geraint Thomas would repeat last year’s win has hovered between 20% and 30%, spiking up a bit during the Tour of Romandie. Unfortunately, Chris Froome’s odds are no longer available, as he was most likely the favourite earlier this year. However, his crash on 11 June instantaneously improved the odds for other riders, particularly Thomas and Bernal, though expectations for the Welshman declined after he crashed out of the Tour de Suisse on 18 June.

The betting on Fuglsang spiked up sharply during the Tirreno Adriatico, where he won a stage and came 3rd on GC, and the Tour of the Basque country, where he finished strongly. Apparently, his three podium results in the the Ardenne Classics had no effect on his chances of a yellow jersey, whereas his victory in the Critérium Dauphiné had a significant positive impact.

Egan Bernal, appeared from the shadows. At the beginning of the year, he was seen as a third string in Team Ineos. His victory in Paris Nice hardly registered on his odds for the Tour. But since Froome’s crash and Thomas’s departure from the Tour de Suisse, he became the bookies’ favourite.

With 65% of the money on the three main contenders, there are some pretty good odds available on other riders. A couple of crashes, an off day or a bit of bad luck could turn the race on its head. Clearly the Ineos and Astana teams are capable of protecting their GC contenders, but so too are Movistar, EF Education First, Michelton Scott, Groupama-FDJ, Bahrain Merida and others.

References

Code I used can be found here

Strava – Tour de Richmond Park Clockwise

Following my recent update on the Tour de Richmond Park leaderboard, a friend asked about the ideal weather conditions for a reverse lap, clockwise around the park. This is a less popular direction, because it involves turning right at each mini-roundabout, including Cancellara corner, where the great Swiss rouleur crashed in the 2012 London Olympics, costing him a chance of a medal.

An earlier analysis suggested that apart from choosing a warm day and avoiding traffic, the optimal wind direction for a conventional anticlockwise lap was a moderate easterly, offering a tailwind up Sawyers Hill. It does not immediately follow that a westerly wind would be best for a clockwise lap, because trees, buildings and the profile of the course affect the extent to which the wind helps or hinders a rider.

Currently there are over 280,000 clockwise laps recorded by nearly 35,000 riders, compared with more than a million anticlockwise laps by almost 55,000 riders. As before, I downloaded the top 1,000 entries from the leaderboard and then looked up the wind conditions when each time was set on a clockwise lap.

In the previous analysis, I took account of the prevailing wind direction in London. If wind had no impact, we would expect the distribution of wind directions for leaderboard entries to match the average distribution of winds over the year. I defined the wind direction advantage to be the difference between these two distributions and checked if it was statistically significant. These are the results for the clockwise lap.

The wind direction advantage was significant (at p=1.3%). Two directions stand out. A westerly provides a tailwind on the more exposed section of the park between Richmond Gate and Roehampton, which seems to be a help, even though it is largely downhill. A wind blowing from the NNW would be beneficial between Roehampton and Robin Hood Gate, but apparently does not provide much hindrance on the drag from Kingston Gate up to Richmond, perhaps because this section of the park is more sheltered. The prevailing southwesterly wind was generally unfavourable to riders setting PBs on a clockwise lap.

The excellent mywindsock web site provides very good analysis for avid wind dopers. This confirms that the wind was blowing predominantly from the west for the top ten riders on the leaderboard, including the KOM, though the wind strength was generally light.

The interesting thing about this exercise is that it demonstrates a convergence between our online and our offline lives, as increasing volumes of data are uploaded from mobile sensors. A detailed analysis of each section of the million laps riders have recorded for Richmond Park could reveal many subtleties about how the wind flows across the terrain, depending on strength and direction. This could be extended across the country or globally, potentially identifying local areas where funnelling effects might make a wind turbine economically viable.

References

Jupyter notebook for calculations

Creating artistic images from Strava rides

When you upload a ride, Strava draws a map using the longitude and latitude coordinates recorded by your GPS device. This article explores ways in which these numbers, along with other metrics, can be used to create interesting images that might have some artistic merit.

The idea was motivated by the huge advances made in the field of Deep Learning, particularly applications for image recognition. However, since datasets come in all shapes and forms, researchers have explored ways of converting different types of data into images.  In a paper published in 2015, the authors achieved success in identifying standard time series by converting them into images.

GPS bike computers typically record snapshots of information every second. What kind of images could these time series generate? It turns out that there are several ways to convert a time series into an image.

Spectrogram

Creating a spectrogram is a standard approach from signal processing that is particularly useful for analysing acoustic files. The spectrogram is a heat map that shows how the underlying frequencies contributing to the signal change over time. Technically, it is derived by calculating the discrete Fourier transform of a window that slides across the time series. I applied this to my regular Saturday morning club ride of four laps around Richmond Park. The image changes a bit once the ride gets going after about 1200 seconds (20 minutes), but, frankly, the result was not particularly illuminating. There is no obvious reason to consider cycling power data as a superposition of frequencies.

Ah! Now we are getting somewhere

The authors of the referenced paper took a different approach to produce things called Gramian Angular Summation Field (GASF), Gramian Angular Difference Field (GADF), and Markov Transition Field (MTF). Read the paper if want to know the details. I created these and something call a Recurrence Plot. All of these methods generate a matrix, by combining every element in the time series with every other element. The underling observations occurring at times $t_{1}$ and $t_{2}$ determine the colour of the pixel at position ($t_{1}$, $t_{2}$). Images are symmetric along the lower-left to upper-right diagonal, apart from GADF, which is antisymmetric.

Let’s see how do they look for on four laps of Richmond Park. We have six time series, with corresponding sets of images below. The segmentation of the images is due to periodicity of the data. This is particularly clear in the geographic data (longitude, latitude and altitude). The higher intensity of the main part of the ride is most obvious in the heart rate data. The MTF plots are quite interesting. Scroll down through the images to the next section

From cycle ride to art

It is one thing to create an image of each item, but how can we combine these to summarise a ride in a single image. I considered two methods of combining time series into a single image: a) create a new image where the vertical and horizontal axes represent different series and b) create a new image by simply adding the corresponding values from two underlying images.

One problem is that some cyclists don’t have gadgets like heart rate monitors and power meters, so I initially restricted myself to just the longitude, latitude and altitude data. Nevertheless, as noted in an earlier blog, it is possible to work out speed, because the time interval is one second between each reading. Furthermore, one can estimate power, from the speed and changes in elevation.

Another problem is that rides differ in length. For this I split the ride into, say, 128 intervals and took the last observation in each interval. So for a 3 hour ride, I’d be sampling about once every 84 seconds.

The chart at the top of this blog was created by first normalising each series to a standard range (-1, +1). Method a) was used to create two images: longitude was added to latitude and altitude was multiplied by speed. These were added using method b). Using these measures will produce pretty much the same chart each time the ride is done. In contrast, an image that is totally unique to the ride can be produced using data relating to the individual rider. The image below uses the same recipe to combine speed, heart rate, power and cadence. If this had been a particularly special ride, the image would be a nice personal memento.

For anyone interested in the underlying code, I have posted a Jupyter notebook here.

References

Encoding Time Series as Images for Visual Inspection and Classification Using Tiled Convolutional Neural Networks, Wang Z Oates T, https://www.aaai.org/ocs/index.php/WS/AAAIW15/paper/viewFile/10179/10251