What time did you go to bed? A simple Bayesian model to improve user experience in HRV4Training

Blog post by Marco Altini

In this post I'll cover the data science and modeling behind the next version of the bedtime detection algorithms in HRV4Training. As current users know already, we hadn't put much effort in detecting bedtime until now. In this post I'll cover a few methods I've been working on in he past few days and highlight how combining noisy data sources using a simple Bayesian model can provide good results in detecting bedtime. The new model will be part of HRV4Training approximately around mid April. 

How can we detect bedtime? 

​Since HRV4Training is used first thing in the morning, detecting wakeup time is straightforwards. That's simply a few minutes before you open the app. But what about bedtime? To keep things simple we started by setting bedtime to 8 hours before wakeup time, and let the user manually change it as part of the tags that are filled in each morning.

Last week I started looking at this problem more closely and developed a series of algorithms that I will now describe with the aim of understanding limitations and possible improvements that led to the current implementation based on a simple Bayesian model. Here are the models we will be comparing:

• 8 hours before wake up time
• Pedometer output
• Moving average of your historical data
• Bayesian model combining Pedometer output and historical data

Pedometer output

A few years back, for another app, I wrote an algorithm that was trying to detect bedtime based on Pedometer output. All iPhones starting with the 5 include a dedicated processor to count steps (that's why, for example, you can see your daily steps in the Health app without bothering about battery drain). My thought was, since the pedometer can detect movement, it can also detect lack of movement, and therefore we might be able to detect when people are sleeping. Once we go that way, we also make some assumptions, e.g. users pick up the phone first thing in the morning, and most importantly users bring the phone to their bedroom before going to bed. The drawing below shows the rationale behind this approach:

However, I was never happy about the output of this algorithm, as it could fail often. The pedometer is not perfect and sometimes it doesn't detect steps. Some other times I would simply leave the phone next to my bed hours before actually going to sleep. Especially when you live in a small place (see for example our current living situation), and don't walk that much to get to your bedroom, the algorithm can easily fail as no steps are detected.

As a result, it was accurate only sometimes and I decided not to deploy these algorithms in HRV4Training. However, as I've beed automating more and more of the daily tags in HRV4Training, for example your workouts data using Strava's APIs, I went back to the bedtime problem and looked at a few alternatives. ​

Moving average of your historical data​

A simple alternative is to use historical data. For example, a moving average of your last week. As you annotate your bedtime for a few days, the app could learn from your input and start providing your recent average bedtime as the default one to edit. We are creatures of habit, and as such, our bedtime might be similar across days. 

This approach is simple, makes sense, and will most likely get close to your actual bedtime. If the goal is to "limit the amount of manual editing", it might be good enough. However, unless we always go to bed at the exact same time, we will never get the estimate exactly right.

Bayesian model combining Pedometer output and historical data

Both our previous models suffer from limitations:

• Pedometer output works great sometimes. Meaning that when it works, it detects exactly when you went to bed. When it doesn't, it's typically off by hours, and provides a terrible estimate. 
• Historical data works well on average, gets close enough to the estimate, but can hardly be correct. It will always require some adjusting and it does not rely on any actual data from your previous night behavior. 

If the pedometer output was reliable, that would probably be my preferred way to detect bedtime, as we could deterministically find the exact moment in time we went to bed. However, we live in a noisy world, and data is hardly perfect or reliable.

Bayes to the rescue

The current problem suits well a Bayesian framework. We have noisy data, historical annotations, and new evidence we gather daily based on Pedometer output. We can define a simple Bayesian model with normal conjugate priors to determine bedtime. In a Bayesian framework, we have three entities of interest; the prior, the likelihood, and the posterior:

• The prior represents our knowledge of the parameters before we observe any data. For example, what time do you normally go to bed? 
• The likelihood tells us what is the probability of a certain hypothesis, given that we have observed some data. For example, what is the probability that we went to bed at a certain hour, given our pedometer output? Here we fix the data, as it was observed using the phone Pedometer, and determine the likelihood of different bedtime hypothesis. Is it more likely that we went to bed at 11pm or midnight? Since this model is rather simple, we will be assuming a normal bedtime distribution and use a conjugate prior, i.e. another normal distribution, so that we can easily compute the posterior (that will be yet another normal distribution).
• The posterior is what we really care about, i.e. given our prior knowledge and observed data, what time do we think we went to bed? Obtaining the posterior is as simple as multiplying the likelihood by our prior, thanks to Bayes rule. Since we used a normal likelihood and conjugate prior, the math of this procedure turns out to be extremely simple, and the posterior is also a normal distribution. 

Personal data

Let's try to make this a bit more practical by looking at my recent data. We will assume we have one week of historical data with manual bedtime annotations. Using this historical data we can determine the prior parameters. The parameters of a normal distribution are the mean and variance, that we can simply derive from our initial week of recordings.

Another parameter we need in order to proceed, is the variance of our likelihood. The variance of our likelihood basically represents how much we trust our Pedometer output. I will postpone the discussion on the choice of this parameter to later, as I think looking at some data first can help in better understanding what we are doing here:

In the figure above, we can see the posterior, and therefore our model estimate, for three different hypothesis. We are answering the question: what would this model predict if the Pedometer output showed that we went to bed at 11pm? Similarly, what would this model predict if the Pedometer output showed that we went to bed at midnight or 1 am?

The x axis shows time of the day (hour) with respect to midnight, so -1 is 11 pm. On the y axis we have the density of the distributions.

My data shows a prior quite peaked just before midnight, meaning that I typically go to bed around that time, as our prior comes from my historical data. The likelihood (second row), is centered around our data point, so 11pm, midnight or 1am, and shows high variance. The distribution is quite spread out because we don't trust much our Pedometer output. Coming back to the variance of the likelihood, this parameter was estimated as the variance of the difference between the actual bedtime (manually annotated) and the Pedometer output estimated bedtime. If the accelerometer Pedometer was always correct (or almost always correct), we would have a much lower variance and trust more our likelihood, as I will show in another example later.

As a result, the posterior is always very close to the prior. The distribution on the third row changes only slightly, moving to the right, as we see evidence of going to bed later (11 pm to 1am). This is exactly what we want, as we trust more historical data than Pedometer output, given my particular case (see later to understand how this naturally generalizes to each individual).

Simulated data: sticking to our habits

As a second example, I want to show what would happen if we were really good at having a consistent lifestyle and going to bed every day pretty much at the same time. In this case, the Pedometer output would basically be completely ignored because we know we can rely much more on historical data. The likelihood is even more spread out, and the prior even more peaked than before:

Simulated data: trusting the Pedometer

Finally, as a third case, let's hypothesize we can actually trust our Pedometer. There might be individual cases in which you always carry your phone to bed, and maybe have to climb a flight of stairs on your way to the bedroom. This seems a case in which your phone might be good at determining bedtime, or at least better than your historical data. Since we trust the Pedometer more, the variance on the likelihood is much reduced, and our posterior follows the likelihood more than the prior, showing that the estimate will be much closer to the Pedometer output than the historical data:

We can see from the posterior plots above how for the first time we definitely move farther away from midnight as the data also moves away (11pm or 1am cases).

How good is this Bayesian model?

Now that I have explained the different models, and how the Bayesian model relies on both historical data and new evidence gathered from the phone Pedometer, we need to look at what all of this means for a user that simply wants to track sleep time without having to spend too much time annotating or editing tags in the morning. 

Using my last week of data, I computed absolute errors for the following models:

• Constant: 8 hours before wake up time
• Historical: Moving average of your historical data
• Pedometer: Pedometer output
• Bayesian: Bayesian model combining Pedometer output and historical data

​
​Root mean square errors between the actual manually annotated time and the model estimated time, were as follows:

• Constant: 43 minutes
• Historical: 19 minutes
• Pedometer: 55 minutes
• Bayesian: 10 minutes

Interestingly, even though the Pedometer output got the time right 3 times (out of 7), it shows the biggest average error, as it fails completely sometimes. 

​The Historical data approach provides also decent results, showing that I tend to go to bed around the same time every day.

​Finally, the Bayesian model further reduces error and provides the best estimates, just 10 minutes off the actual bedtime on average.

Summary

In this post I covered a few different methods to detect bedtime, proposing a Bayesian model able to combine noisy data sources for optimal results.

One of the best aspects of this model, is that it adapts to your own case. As I've shown above, confidence in historical or Pedometer data will change based on your own annotations and data. Thus, the algorithm will tune based on your behavior.

We'll be rolling out this feature soon, and hopefully make your life easier for the ones interested in tracking sleep time and bedtime.

If you are an iPad user or have an iphone 4S, as these devices do not have Pedometers, we will be implementing our second best model, i.e. the one using historical data.

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