Statistics

Covid-19 Coronavirus

So this virus outbreak thing is interesting isn't it?

As a mathematician-in-training, what I'm finding more interesting is the lack of good statistics on what's happening. The UK government were initially publishing new infections within England and in the whole of the UK. In addition, they were publishing the total UK infected, and total UK deaths. They stopped reporting anything on March 5th.

Though as well as keeping track of government announcements, I've also been keeping record of daily updates that the World Health Organisation (WHO) have been making.

Here's a link to their European map.

Here's a link to their global map.

On a daily basis, I've been recording UK totals from their site. You can see them graphed below:

You can't see it, but the values for the first 9 days are just two people.

Though see how there's a dip on March 16th? How could there possibly be a dip in cases for one day by over 100 people? It seems even the pros can get their numbers wrong. Bad work, WHO. 🙁

Also, you'll notice from around March 5th to March 8th, it goes flat. These are days on which I didn't check the figures.

From these total cases per day, I've calculated new cases per day.

See that negative number of new cases on March 16th? Insert eyeroll emoji.

There's also a spike on March 9th, this is just total new cases from March 5th to March 9th. -catching up from when I didn't check on the totals on these days.

But as you can see, as predicted, this initial growth is basically exponential. Fairly typical for an epidemic apparently.

So the big question is... what next?! -and here's where my mathematics and statistics study comes in handy... I've recently finished a section on epidemics! What better time to apply some of my learning!

First off, it's worth mentioning that all the epidemic modelling I've learned about assumes homogeneous mixing. This is the kind of mixing that occurs in a family home. ie: there's more or less an equal chance of me having contact with one person than there is anyone else. In real life this, of course, isn't true. Living in London, I'm less likely to be in contact with someone in Edinburgh than I am someone I travel with on the London Underground every day. Also important point: homogeneous mixing means no quarantining. So all of the below is essentially average worst-case.

So with the proviso that these results will probably (more than likely) be wildly inaccurate, let's get started! We need a few important numbers, some of which we've already got:

  1. We need the starting number of infected (y0), in this case 2.
  2. We need the starting number of non-infected (x0), that's the total population of the UK minus 2. I decided to estimate the current population at 67.44 million.
  3. We need the epidemic number, ρ. This is calculated in the following way:

\rho =\frac{n\gamma}{\beta}

Where:

  • n is the total population (67.44 mil).
  • γ is the parameter in the exponential distribution that describes the mean recovery time of the virus. (It's random).
  • β is the parameter in the Poisson distribution that describes the mean contact rate (per day). (Also random).

Which is fine, but how do we know what γ and β are? Well there's a number called R0 ("R-naught") that I've NOT been taught about that represents the contagiousness of a virus. Interestingly, I've found two completely opposed descriptions of this number:

  1. https://en.wikipedia.org/wiki/Basic_reproduction_number
  2. https://towardsdatascience.com/social-distancing-to-slow-the-coronavirus-768292f04296

Though we all know how reliable wikipedia is, and I've just found this Stanford paper which supports the towardsdatascience description:

R_{0}=\frac{\beta}{\gamma}

Therefore:

\rho =\frac{n}{R_{0}}

Hooray. Imperial College London seems to estimate the R0 of COVID-19 to be 2.4, so let's run with that.

\rho =\frac{67440000}{2.4}=28,100,000

Max Number Of Infectives At One Time

Once we have all this, we can work out the maximum number of infectives at any one time. ie: the peak of the infection in the population, ymax:

y_{\text{max}} = y_{0} + x_{0} -\rho -\rho\:\text{log}\left(\frac{x_{0}}{\rho}\right)

Now we can plug all the numbers in to find ymax! So:

y_{\text{max}} = 2 + 67439998 -28100000 -28100000\times\:\text{log}\left(\frac{67439998 }{28100000 }\right)

Hence:

y_{\text{max}} = 28,656,065

Which is 42.5% of the population infected at one time! Ouch! At least you'll know that if we (in our theoretical UK) hit 28 million with no quarantining, we'd be at the peak of the outbreak.

Other sources state that COVID-19 actually has a range of R0 values, 1.4-3.8-ish, so the band of possible outcomes without quarantining is actually quite broad. But from this it's possible to work out a best-case/worse-case comparison:

An R0 of 1.4 would mean a max of 12.2 million (18% of the population), and an R0 of 3.8 would mean a max of 39.4 million (58% of the population) at one time.

 

Number Of People Not Affected

The following assumes that the whole debacle is over. Everyone that has caught the virus from it has now recovered. How many people were not affected?

This is found using the following iteration formula:

x_{\infty,j+1}=x_{0}\:\text{exp}\:\left(\frac{x_{\infty,j}-\left(x_{0}+y_{0}\right)}{\rho}\right),\:\:\: j=0,1,2,\:\ldots

Initially x_{inf,j} is zero, and you use your result x_{inf,j+1} to calculate x_{inf,j+2}, and so on. This eventually settles down to the number of people not affected!

So using the power of spreadsheets, and not taking up the space here with columns and columns of numbers:

  • R0 of 1.4 would mean 32.98 million people are not affected.
  • R0 of 2.4 would mean 8.29 million people are not affected.
  • R0 of 3.8 would mean 1.66 million people are not affected.

You can imagine that an increase in quarantining means a lower R0. Seems that could have a big effect.

#ImNotAStatisticianButItsStillFunLookingAtNumbers

Critique, Statistics

Statistics Assignment 2

Two thirds of the way through my assignments!

Again, fairly happy with the mark I received for this, but there were some aspects of this assignment I found challenging, and some where I thought I might've done quite well on, but slipped up in some way.

Let's cover some areas here:

Finding A Real-World Process

For these questions I had to find a real-world process that could be modelled with the given mathematical objects/processes. Kind of the opposite of a mathematical modelling problem.

I found these tasks really difficult. What I found to be the worst aspect about getting this kind of question wrong is that it's not necessarily my understanding of the mathematical process that's flawed. I feel in each of these cases, I did my best to find a real world example, knowing that the example I gave, itself, was slightly flawed. So despite the fact that I can perfectly explain each mathematical process, I couldn't explain how each could be applied to a real world process so lost marks.

The two models were the Galton-Watson branching process, and the simple random walk (specifically, a particle executing a simple random walk on the line with two absorbing barriers).

The two typical examples that are referred to in my texts are genetics and mutations for the Galton-Watson branching process:

"A mutation is a spontaneous transformation of a particular gene into a different form, and this can occur by chance at any time... The mutant gene becomes the ancestor of a branching process, and geneticists are particularly interested in the probability that the mutation will eventually die out."

For the simple random walk, the example of the "gambler's ruin" was given. Imagine two people with £10 each, each of them betting on an event. If one of the two loses the bet, they give £1 to the other (the random walk on the line). If one of them runs out of money, then they lose (one of the "absorbing barriers" are hit).

In coming up with answers, I could've used Google, but that would've been cheating. However, now I've completed the assignment and received my grade, Google is my best friend in finding suitable answers here...

Seems you can use the Galton-Watson branching process to determine the extinction of a family name, and I found a good example of a random walk with absorbing barriers in this MIT paper, featuring a little flea called Stencil. It discusses the probability of him falling over the Cliff of Doom in front of him, or the Pit of Disaster behind him.

 

Concluding An Answer

A couple of my answers here and there were classed as being incomplete. Generalising each case:

1)
Upon finding that an answer resembles a certain construction (a probability distribution function, cumulative distribution function or generating function), as well as saying which distribution the function belongs to, you should also explicitly state the variables that appear in it. Even to anyone non-mathematical, it would be obvious to see that the variables in the general case are associated with the specific answer you arrived at. Though for assignments (and exams, presumably) this is not enough. If a general function has variables explicitly state what each one is in your answer.

eg: the p.g.f. of the modified geometric distribution is

\frac{a\:-\:bs}{c\:-\:ds}

If your answer resembles this, say what a, b, c and d are.

In addition, if your answer is a probability (or set thereof), include a statement describing them.

Note that one whole mark can be deducted for an insufficient conclusion (apparently).

2)
Don't forget your definitions.

Specifically:

To calculate the variance of the position of a particle (along the random walk line) after n steps, you can just sum the variances of each step. However this only works because each individual step is independent of the last (one of the properties of the random walk). Due to the fact that I didn't mention this definition of the variance of a particle in a random walk, I lost half a mark. Not massive, but where you can mention a definition, mention it.

 

Different Routes In A Markov Chain

I struggled with this, and although I arrived at the correct answer, the method I had used was entirely wrong (and also a little inelegant).

In this question, I covered all routes separately and so had a small handful of different probability calculations. Though when considering potential routes in a Markov chain, you can consider all routes simultaneously by taking advantage of something called an absolute probability (of the Markov chain being in a particular state at a particular time), given an initial distribution. (for my own reference this is covered in Book3, Subsection 11.2, p.87. And the handbook, p.23 item 17).

 

Arbitrary Constants

Does it matter if an arbitrary constant is positive or negative? (my ref: Q6a). I previously thought not. In this instance my constant in an integral calculation absorbed the negative sign that was in front if it. After all, a negative general constant is still a general constant, right? Well I lost half a mark here because of the absorption, and it's not currently clear why. I've asked my tutor, and I'll update it on here once I hear back from him.

Critique, Statistics

Statistics Assignment 1

Very happy with the high mark I achieved for this first assignment. Though as usual, there's a decent about to be improving. Let's start by looking at some of the more major things my tutor pointed out.

Variance

First thing is variance. How do you calculate it? Well it turns out there are a couple of ways. I just decided to use the most cumbersome way...

Calculating the mean \mu (expectation E(X)) is easy. Multiply each number with its probability and sum them all:

\mu = E(x) = \sum x\: p(x)

The variance can then be calculated in one of two different ways:

\sigma^{2}=E[(X-\mu)^{2}]=\sum (x-\mu)^{2}\: p(x)
or
\sigma^{2}=E(X^{2})-(E(X))^{2}=\left(\sum x^{2}\: p(x)\right)-\mu^{2}

When they're written out like this, it's fairly obvious to see which method is more like the method used to calculate the mean and as  such would be far less hassle. (x is an integer and p(x) and \mu can be reals/rationals).

 

Multivariate Poisson Process

This was the question in which I lost the most marks:

Customers arrive at a shoe shop according to a Poisson process with a rate of 20 per hour.
15% of customers buy men's shoes.
60% buy women's shoes.
25% buy children's shoes.
Calculate the probability that exactly eight customers arrive in half an hour, exactly three of whom wish to purchase children's shoes.

This is such a typical mistake for me to make in statistics. I'm sure I've made this kind of mistake before...

What I ended up doing was working out the probability of the number of customers being 8 using the Poisson distribution's probability function. This was fine.

Then I used the same probability function to find the probability of 3 people wanting to buy children's shoes and multiplied them together. Wrong. At this point I needed to find the conditional probability that of the 8 customers, 3 bought children's shoes. Hence, here I shouldn't have used the Poisson distribution, I should've used the Binomial distribution instead. ie: from 8, choose 25%.

Reflecting back on the question, the correct answer seems slightly more obvious now. Especially given the "...exactly three of whom..." part of the question. I struggle to be mindful of stuff like this in the moment of answering a stats question. I suppose this part of the "translating English into maths" issue comes with more practise...

Index Of Dispersion

Again, my issue here was to not observe subtleties in the question. Given information about the associated distributions, I was initially meant to calculate the mean and variance of the total number of books bought in 9 hours. I managed to get this first part right, but the second part of the question asked me to calculate the index of dispersion for "this process". It turns out that "this process" refers to the process in the main question generally and not the process of books being bought within 9 hours. In this instance, ignoring the total number of books bought in 9 hours (kind of) simplifies the answer too.

Other Issues

In this first assignment, I lost a half a mark here and there for incorrect arithmetic. (GASP!). Upon completing my draft submission, instead of just reading through it, I should sit down and verify all my working. It will take more time, but if it scrapes 2 marks back, it could be worth it.

Other issue that occurred more than once was a lack of units when talking about rates of things happening. So there's a requirement to state "\lambda=20 per hour" instead of just "\lambda=20".

 

Calculus

Integration Revision - Integration By Substitution

Now, looking at the other integral from Oct 13th...

\int^{x}_{0}4xe^{-2x^{2}}\:dx=1-e^{-2x^{2}}

In my opinion, this was scarier than the first, but looking at it again, what makes it scary is the various powers. e to the power of x to the power of 2 and such.

But why should this be scary? There's practically only one tool that one could use to solve this: the only tool that includes functions of functions, integration by substitution:

\int f(g(x))\:g^{\prime}(x)\:dx=\int f(u)\: du

In our case, the inner function:
g(x)=-2x^{2}=u
so:
g^{\prime}(x)=-4x

and the outer function:
f(x)=e^{x}

Plugging all these into our "integration by substitution" tool gives:

\int -4xe^{-2x^{2}}\:dx

But this is slightly different to the original integral due to that minus sign. Of course this is trivial to deal with as:

\int^{x}_{0}4xe^{-2x^{2}}\:dx=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx

Great! So now we have the structure we require to apply integration by substitution! We can substitute all of that with f(u), and all the scary bits go away. So:

\int^{x}_{0}4xe^{-2x^{2}}\:dx

=-\int^{x}_{0} -4xe^{-2x^{2}}\:dx

=-\int^{u}_{0} e^{u}\:du

=-[ e^{u}]^{u}_{0}

=-(e^{u}-1)

=1-e^{u}

Then substituting u (=g(x)) back in:

=1-e^{-2x^{2}}

Calculus

Integration Revision - Integration By Parts, Twice

So looking at the first one from a few days ago (Oct 13th):

\int^{\infty}_{0}x^{2}\:\lambda\: e^{-\lambda x}\:dx=\frac{2}{\lambda^{2}}

I was totally lost with this. But running through some old tricks made this seem a lot more approachable. First off, moving the constant out brings a bit more clarity to the integrand,

\int^{\infty}_{0}x^{2}\lambda e^{-\lambda x}\:dx=\lambda\int^{\infty}_{0}x^{2} e^{-\lambda x}\:dx

But given the x^{2} and the exponential, we'll also probably need to use integration by parts:

\int^{b}_{a} f(x)g^{\prime}(x)dx=[f(x)g(x)]^{b}_{a}-\int^{b}_{a} f^{\prime}(x)g(x)dx

So, as above, letting:
f(x)=x^{2}
g^{\prime}(x)=e^{-\lambda x},

we have:
f^{\prime}(x)=2x
g(x)=-\frac{1}{\lambda}e^{-\lambda x}

Then plugging f, f^{\prime}, g and g^{\prime} into our lovely integration by parts tool above gives:

\lambda\int^{\infty}_{0}x^{2}\: e^{-\lambda x}\:dx

=\lambda\left(\left[-x^{2}\: \frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}-\int^{\infty}_{0}-2x\:\frac{1}{\lambda}\: e^{-\lambda x}\:dx\right)

=\lambda\left(0-0+2\frac{1}{\lambda}\int^{\infty}_{0}x\: e^{-\lambda x}\:dx\right)

=2\int^{\infty}_{0}x\: e^{-\lambda x}\:dx

Which looks very familiar with what we've started with, except x^{2} is now just x! If we can reduce x by another power, we'll end up with just 1 which will surely give us a much easier integral to solve.

So applying integration by parts again, but letting:
f(x)=x
g^{\prime}(x)=e^{-\lambda x},

we have:
f^{\prime}(x)=1
g(x)=-\frac{1}{\lambda}e^{-\lambda x}

Then:

2\int^{\infty}_{0}x\: e^{-\lambda x}\:dx

=2\left(\left[-x\: \frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}-\int^{\infty}_{0}-\frac{1}{\lambda}\: e^{-\lambda x}\:dx\right)

=2\left(0-0+\frac{1}{\lambda}\int^{\infty}_{0} e^{-\lambda x}\:dx\right)

=\frac{2}{\lambda}\int^{\infty}_{0} e^{-\lambda x}\:dx

=\frac{2}{\lambda}\left[-\frac{1}{\lambda}\:e^{-\lambda x}\right]^{\infty}_{0}

=\frac{2}{\lambda}\left(0-\left(-\frac{1}{\lambda}\right)\right)

=\frac{2}{\lambda}\:\frac{1}{\lambda}

=\frac{2}{\lambda^{2}}

Statistics

End of Book 1 - Review of Understanding

A quick look at the mind map I produced for the whole of Book 1, very much a foundation for the rest of the materials on the module.

The complexity has increased necessarily, but I'd like to think the clutter has been reduced to a minimum. I was rearranging nodes as I was adding them in the hope of reducing clutter. You'll notice there are two white floating boxes not connected to anything externally. This was just to avoid clutter.

Some things to note about it that help me refer back to it:

It's split roughly into thirds, vertically. Each third is more or less a sub topic. More like themes, perhaps. Far left is foundational principles. Middle relates to basics of distributions. Far right is concepts surrounding the C.D.F. and P.D.F. (the definitions of which are in the white box in the middle of the far-right section).

Colouring helps a lot when needing to refer back to it, I found. Axioms are pink, properties are green, and definitions are blue. I found that in referring back to it, I needed some kind of differentiation between the definition of a certain distribution, and a normal definition. So all the yellow nodes you see are definitions of distributions.

In summary, to assist my understanding, I now have a graph that leads me from the most basics concept (like what an "event" is), to the definition of the standard normal distribution. I'll be referring back to this as I go!

Calculus, Statistics

Integration Revision

Putting these in a safe place for later. Found these to be a really good revision questions for integration.

\int^{\infty}_{0}x^{2}\lambda e^{-\lambda x}\:dx=\frac{2}{\lambda^{2}}

and

\int^{x}_{0}4xe^{-2x^{2}}\:dx=1-e^{-2x^{2}}

For context, the first one formed part of a question that required me to find the variance of a random variable, and the second one involved having to find the c.d.f. from a p.d.f.

(for my own reference, this was Book 1, Activity 5.2, p. 54 and Activity 6.1, p.62)

General, Maths, Statistics

Current Application of Mind Maps

I've finally completed the second section of the first book, and it's served as a good revision/introduction session. I'm glad I went to the trouble of creating a mind map for the main concepts that were covered. Here's how my mind map grew as I added to it:

\/

\/

I've used this new mind map to refer back to several times during the exercises I've been set, and even the first assignment question. Admittedly it takes longer to get through all the learning material, as you have to take time out to contribute to the mind map as you go. Though I would say the time spent is worth it.

I also felt that this last image was the maximum size a mind map could take without becoming too cluttered. For the next section of the book, I'll be starting a brand new mind map. I'll try to keep going with this method for as long as I can, but it seems it's very useful.

Statistics

Module Number 7!

Summer is over!

Autumn is here!

That can only mean the new academic year is about to begin! (and the first module of the third and final stage of my degree!)

Up next? Applications of Probability (M343)!

I'm excited about starting this. When I started on this degree, I saw that there weren't many options for stats modules in the final stage unless you were specifically aiming toward a BSc in Statistics. I've read a number of articles discussing how there isn't enough stats in education at the moment and I'm inclined to agree. I'm hoping this will round my skills out nicely.

For all the info on this next module, check out the official page here.

The module website for students is due to open tomorrow, so I'll get access to all my materials and be allocated a tutor. What makes this module a little bit different is that it's a single credit module. The last two modules I've had (mathematical modelling and pure mathematics) were double-credit. So I'm wondering whether the workload will feel really light or not. I'm hoping it will be lighter, so I can contribute on here more frequently than for the last two!