Reporting delays etc

Made this GIF overnight. It's showing how my forecast evolves as more data is added to the system. Initially it doesn't assume any control will be imposed, and then at the appropriate day (23rd March) it assumes a change in R (value to be estimated) which feeds through into deaths after a couple of weeks according to the model dynamics.

But something about it bugged me. Why did the model fail in mid-late March? If it's supposed to be a decent forecasting system, it should be predicting where the future data will lie. The late March data were not affected by the lockdown, it's just that the model is overestimating the pace of growth from early data. I played around with a few ideas and really didn't manage to fix it. I did, however, notice that the data jumped sharply from the 13th to 14th of March, from 1-2 deaths per day, to 10+ deaths per day, without a single day in between. This is actually pretty implausible from a statistical point of view if the underlying death rate is growing steadily in an exponential way, as theory and practice expects.

So I went and had a more careful look at the data. 

These data are actually not, as people might assume, the number of people who have died on a given day. They are actually the number of reports collected in a given 24h period, which may represent deaths on any previous time. I already knew this and also knew that it shouldn't affect the growth rate estimate, so long as the delays are fairly consistent over time. This can be checked with an alternative data set in which deaths are actually collated by true date of death rather than date of report, and this is what the next plot does:

Oops I didn't label the axes properly. Deaths vs dates. The red/pink circles are the same as the previous data, that is to say, the number in the daily report that features on the news each day. The blue/cyan triangles, on the other hand, are the deaths that are known to have actually occurred on on each day. The first thing to note is that the blue/cyan points are for the most part higher, and that's despite these numbers only relating to England, so the UK as a whole will be another 10-20% higher still. There is a drop-off towards the present day where the totals are probably not yet complete and some to-be-added deaths will turn out to have occurred on these days. This is specifically warned about in the data. Ignoring these points, the slopes of the two data set are strikingly similar just as theory expects (which is good, as I was relying on this for my analyses to make sense).These are the dotted pink and cyan lines, with the extent of the lines both showing the points I used to derive them. So far, so good.

So now look at the initial segment where I have drawn a couple of bolder lines in red and dark blue. These are linear fits to the darker blue and red data points, and their slopes are quite different. The blue one agrees withe the cyan - I also extended this with the thin solid cyan line and it's coincidentally (confusingly) identical to the dotted one. The red one, on the other hand, extends as the solid pink line and clearly misses the future data. Just as my slightly more complex model fit did. You can also see that the blue dots show no fewer than 5 days actually had 3-9 deaths inclusive, despite there being none of these in the red data. My fit using the full model is not actually just a linear regression but it's rather similar, and creates the same effect.

My conclusion is that the reason my prediction struggles here is the red data were just particularly poor at this point, and this wasn't due either to bad luck with the randomness of death, or that the model doesn't represent the underlying dynamics well (the blue data are perfectly linear), but instead almost certainly due to some weird reporting errors being significantly larger than I had allowed for in my estimation. Because the chance of not getting a single day in that range of intermediate values is extremely low in a world where we had roughly a whole week with the expected death rate in that range. I don't know if they actually changed the system around that time or not - might do a bit more digging.