A couple of weeks ago, I spent some time exploring Firefox crash data so that I can come up with a model to catch "explosive crashes" in a timely fashion. Over the last couple of weeks, I have evaluated and identified a model that is "good enough". Currently, the code is sitting in a pull request, waiting to be landed in Socorro.

This post will mainly focus on many of the models I have tried and did not include in my final implementation.

At the end of the last post, I identified some trends in the data. We looked at crash counts and the possibility of using a predictive model and measure the deviation to figure out if a crash is explosive.

The first thing I tried is a sinusoidal model as we intuitively knew that our data has a seasonal pattern and the ACF plot shows that this is the case.

The sinusoidal model uses the following formula for prediction:

$$\hat{y}_t = \mu + \alpha \sin(\frac{2\pi}{\omega} t + \phi)$$

We need to fit for \( \alpha \), \( \omega \), \( \mu \) and \( \phi \) with this formula. For each signature, we take the last 2 weeks (or any other periods) of data find the values. Then, we simply plug in tomorrow's \( t \) to get a predicted value.

We can lock \( \omega \) to be 7 as we know that our period is 7 due to weekly data fluctuations.

This formula is not exactly ideal as it does not account for an overall upward or downward trend in the crashes, which we see in crash data.

This is easily accounted for if we replace \( \mu \) with a linear function so our model becomes:

$$\hat{y}_t = \beta_0 + \beta_1 t + \alpha \sin(\frac{2\pi}{\omega} t + \phi)$$

This model worked reasonably well. In fact, it is one of the better models from my arsenal that I've tried (average error is only 5%, with a catch). Here is an attempt at predicting tomorrow's global crash volume:

The residuals graphs also look pretty good as it seems fairly random.

Residual histogram looks fairly normal. Lagged scatter plot does not show significant correlations. Residual vs fits has an average of approximately zero.

However, The model required the time series to be a sinusoidal curve. This is to say that the number of crashes must show a pattern over the week. Otherwise it performs poorly as shown below:

A problem with the sinusoidal model is that there are hundreds of thousands of time series and I cannot feasibly look through all of them. While the sinusoidal model is good for certain types of time series, I cannot say that all crash signatures at all times exhibits the same patterns, which they don't.

It turns out, this is also the problem with many other models as well. Specifically, exponential weighted moving average and decompositions models worked terribly as they just do not fit the data like they do in textbooks. With these models, 20% to 30% errors are not uncommon.

So at this point I realized that I'm generalizing too much. Although many crash signatures exhibits roughly the same behaviour, I cannot just come up with a model by looking at a small subset of the signatures and say that this model fits for all signatures. It just doesn't. In order for me to get good prediction results, I (or an algorithm) would have to look at every single crash signatures and decide on a prediction model for each signature. It is not feasible for me to look through the entire data set and an algorithm would perhaps be even more challenging.

Another important thing that I realized at this point is that I do not actually know how well these prediction models will perform when it comes to classifying explosive crashes. There are no training examples other than the Olympic doodle crash. Without a proper test set, all I can do is cross my fingers and hope that the model works.

In summary, there are several problems that I faced with the time series based approach:

- Technical limitations which would mean that I have to implement algorithms that are difficult to implement and usually not implemented as we rely on tools such as scipy.
- Time series approach assumes that all crash signatures can be modeled by the same type of time series.
- We lacked test set examples so we cannot evaluate how well the model performed after it is developed.

Since my internship is coming rapidly to a close, it was decided, after some discussions, that we should first try a much simpler model than the ones proposed thus far. I can use this model to first test for additional explosive examples so that I can use it to verify whatever models I can come up with. It so turned out that the method used to find these examples, despite being very simple, is also very good. I'll detail this in a follow up blog post.