Cheat Sheets: The New Programmer’s Friend

Cheat sheets are brilliant: whether you’re learning to program for the first time or you’re picking up a new language. Most data scientists are probably programming regularly in multiple languages at any given time: cheat sheets are a handy reference guide that saves you from googling how to “do that thing you know I did it in python yesterday but how does it go in stata?”

This post is an ongoing curation of cheat sheets in the languages I use. In other words, it’s a cheat sheet for cheat sheets. Because a blog post is more efficient than googling “that cheatsheet, with the orange bit and the boxes.” You can find my list of the tutorials and how-to guides I enjoyed here.

R cheat sheets + tutorials

Python cheat sheets

Stata cheat sheets

  • There is a whole list of them here, organised by category.
  • Stata cheat sheet, I could have used this five years ago. Also very useful when it’s been awhile since you last played in the stata sandpit.
  • This isn’t a cheat sheet, but it’s an exhaustive list of commands that makes it easy to find what you want to do- as long as you already have a good idea.

SPSS cheat sheets

  • “For Dummies” has one for SPSS too.
  • This isn’t so much a cheat sheet but a very basic click-by-click guide to trying out SPSS for the first time. If you’re new to this, it’s a good start. Since SPSS is often the gateway program for many people, it’s a useful resource.

General cheat sheets + discusions

  • Comparisons between R, Stata, SPSS, SAS.
  • This post from KD Nuggets has lots of cheat sheets for R, Python, SQL and a bunch of others.

I’ll add to this list as I find things.

Law of Large Numbers vs the Central Limit Theorem: in GIFs

I’ve spoken about these two fundamentals of asymptotics previously here and here. But sometimes, you need a .gif to really drive the point home. I feel this is one of those times.

Firstly, I simulated a population of 100 000 observations from the random uniform distribution. This population looks nothing like a normal distribution and you can see that below.

histogram of uniform distribution

Next, I took 500 samples from the data with varying sample sizes. I used n=5, 10, 20, 50, 100 and 500. I calculated the sample mean (x-bar) and the z score for each and I plotted their kernel densities using ggplot in R.

Here’s a .gif of what happens to the z score as the sample size increases: we can see that the distribution is pretty normal looking, even when the sample size is quite low. Notice that the distribution is centred on zero.

z score gif

Here’s a .gif of what happens to the sample mean as n increases: we can see that the distribution collapses on the population mean (in this case µ=0.5).

sample mean gif

For scale, here is a .gif of both frequencies as n gets large sitting on the same set of axes: the activity is quite different.

Sample mean vs z score

 If you want to try this yourself, the script is here. Feel free to play around with different distributions and sample sizes, see what turns up.

The Law of Large Numbers: It’s Not the Central Limit Theorem

I’ve spoken about asymptotics before. It’s the lego of the modelling world, in my view. Interesting, hard and you can lose years of your life looking for just the right piece that fits into the model you’re trying to build.

The Law of Large Numbers (LLN) is another simple theorem that’s widely misunderstood. Most often it’s conflated with the central limit theorem (CLT), which deals with the studentised sample mean or z-score. The LLN pertains to the sample mean itself.

Like the CLT, the LLN is actually a collection of theorems, strong and weak. I’ll confine myself to the simplest version here, Khinchine’s weak law of large numbers. It states that for a random, independent and identically distributed sample of n observations from any distribution with a finite mean (µ) and variance: then the sample mean has a probability limit equal to the population mean, µ. That is, the sample mean is a consistent estimator of the population mean under these conditions.

Put simply, as n gets very big, the sample mean is equal to the population mean.

Notice there is nothing about normal distributions as n gets large. That’s the key difference between the LLN and the CLT. One deals with the sample mean alone, the other with the studentised version. On its own, the distribution of the sample mean collapses onto a single point as n gets large: µ. This is the implication of the LLN.

Appropriately scaled, centred and at the correct rate, the studentised sample mean has a normal distribution in the limit as N gets large: that’s the CLT.

As usual, here’s an infographic to go: put side by side the two theorems have different results but are dealing with something quite similar.

CLT vs LLN infographic

Three things every new data scientist should know

Anyone who has spent any time in the online data science community knows that this kind of post is a genre all on its own. “N things you should know/do/be/learn/never do” is something that pops up in my twitter feed several times a day. These posts range from useful ways to improve your own practice to clickbait listing reams of accomplishments that make Miss Bingley’s “accomplished young ladies” speech in Pride and Prejudice appear positively unambitious.

Miss Bingley’s pronouncement could be easily be applied to data scientists everywhere:

“Oh! certainly,” cried his faithful assistant, “no [woman] can be really esteemed accomplished who does not greatly surpass what is usually met with. A woman must have a thorough knowledge of music, singing, drawing, dancing, and the modern languages, to deserve the word; and besides all this, she must possess a certain something in her air and manner of walking, the tone of her voice, her address and expressions, or the word will be but half-deserved.”

Swap out the references to women with “data scientist”, throw in a different skill set and there we have it:

“Oh! certainly,” cried his faithful assistant, “no data scientist can be really esteemed accomplished who does not greatly surpass what is usually met with. A data scientist must have a thorough knowledge of programming in every conceivable language that was, is or shall be, linear algebra, business acumen, obscure models only ever applied in obscure places, and whatever is “hot” this year, to deserve the title; and besides all this, she must possess a certain something in her air and manner of tweeting, the tone of her blogging, her linkedin profile and be a snappy dresser, or the title will be but half-deserved.”

Put like that, you’d be forgiven for not allowing the Miss Bingleys of the world to define you.

If I had a list of things to say to new data scientists, they wouldn’t have much to do with data science at all:

  1. You define yourself and your own practice. Not twitter, not an online community, not blogs from people who may or may not know your work. Data science is an incredibly broad array of people, ideas and tools. Maybe you’re in the middle of it, maybe you’re on the edge. That’s OK, it’s all valuable.
  2. You’re more than a bot. This is an industry that is increasing automation every day. You add value to your organisation in ways that a bot never can. What is the value you add? Cultivate and grow it.
  3. The online community is a wonderful place full of people who want to help you grow your practice and potential. Dive in and explore: but remember that the advice and pronouncements are just that. They don’t always apply to you all the time. Take what’s useful today and put the rest aside until it’s useful later.

It’s a short list!

It got wet

NSW got wet this weekend. In our own particular case we lost a large amount of our driveway and several paddocks spontaneously attained lake status. So there was nothing else to do but to poke around and see what I could turn up in the historic record (find yours here).

Some locals recorded up to 250mm in 24 hours this weekend. I thought that was an extraordinary amount until I checked the data (only available up until April this year so far, alas).

It turns out that sometime in the late sixties the local rainfall station recorded an extraordinary 392mm in 24 hours. Now that’s an outlier…!

I’ll invest in a new pair of gumboots just in case.

Smooth scatter plot rainfall

If you’re into this sort of thing, the plot was done using the “smoothScatter” function in R. It’s a change from the usual time series line chart. I think I’m a convert.

Continuous, Censored and Truncated Data: what are the differences and do you need to care?

Whenever I work with someone whose statistical or econometric experience has been more practical than theoretical, two things happen. The first is that the poor person inexplicably develops a twitch whenever I launch into an enthusiastic tangent that requires a sheet of graph paper and extensive hand waving.

The other thing that inevitably happens is that the digression comes to an end and the question is asked “but does that matter in practice?”

When it comes to model section, the difference between data types really does matter. You may make choices one way or another, but understanding the differences (both obvious and subtle) lets you make those choices understanding that you do have them.

This post is a cliff-notes version of the issue. Maybe you’ve heard of these differences in data types and just need a memory jog. Maybe you’ve not heard of them at all and want somewhere simple to start.

Continuous data is pretty simple: it’s data that can lie anywhere on the real line with a positive probability. That is, it can be anywhere from very large negative numbers to very large positive numbers. The normal distribution is an example of continuous data.

Truncated data, on the other hand, is data which is continuous but has the added complication of only being observed above or below a certain point. The classic example suggested by Greene is income [1]. One example would be if we only surveyed the income of those earning above the tax-free threshold: then we would have truncated data.

Censored data is similar. It’s an issue not of observation but in the way the data is sampled. Some parts of the distribution are obscured, but not ignored. The survey may, for example, interview all income levels, but only record those above the tax free threshold and describe the rest as “under the tax threshold” rather than giving the income in dollar terms. In this case all parts of the distribution are reported on, but the level of information differs above or below a threshold.

Most people are aware of issues modelling categorical data using techniques designed for continuous data. However, censored and truncated data also need special treatment. A lot of the data we deal with has a natural truncation point: distance isn’t negative, prices are not (well, hardly ever) negative. Recognising that you may be dealing with truncated or censored data is an important part of initial data analysis. For a thorough discussion, see W.H. Green’s chapter on the subject here.

In practice, continuous data methodologies may work quite well for these types of data as long as there isn’t a large amount of data sitting at or near the truncation or censoring point (which is often zero).

Test scores are something I’ve worked a lot with. In my experience, once the proportion of test scores began to approach around 20% zeros, I needed to switch over to models designed for the issue. In the 10%-20% range I will often try a few of different models to see which is most appropriate. That’s just a general rule of thumb- your mileage may vary.

Hand waving and furious graph-paper drawing aside: yes in this case knowing the differences does matter in practice.

Notes:

[1] W. H. Green, Econometric Analysis, is a classic text and here I’m looking at p. 756 in the fifth edition. There are three copies of this book living in my house. Definitely worth the investment if you are looking for either a classic text covering everything econometrics or a useful TV stand. What can I say? We were young and poor and a matched set of texts made up for deficits in our furniture budget. I’ve owned this book for nearly twenty years and I still use it- even long after we can afford furniture.

The Central Limit Theorem: Misunderstood

Asymptotics are the building blocks of many models. They’re basically lego: sturdy, functional and capable of allowing the user to exercise great creativity. They also hurt like hell when you don’t know where they are and you step on them accidentally. I’m pushing it on the last, I’ll admit. But I have gotten very sweary over recalcitrant limiting distributions in the past (though I may be in a small group there).

One of the fundamentals of the asymptotic toolkit is the Central Limit Theorem, or CLT for short. If you didn’t study eight semesters of econometrics or statistics, then it’s something you (might have) sat through a single lecture on and walked away with the hot take “more data is better”.

The CLT is actually a collection of theorems, but the basic entry-level version is the Lindberg-Levy CLT. It states that for any sample of n random, independent observations drawn from any distribution with finite mean (μ) and standard deviation (σ), if we calculate the sample mean x-bar then,

central limit theorem

In my time both in industry and in teaching, I’ve come across a number of interpretations of this result: many of them very wrong from very smart people. I’ve found it useful to clarify what this result does and does not mean, as well as when it matters.

Not all distributions become normal as n gets large. In fact, most things don’t “tend to normality” as N gets large. Often, they just get really big or really small. Some distributions are asymptotically equivalent to normality, but most “things”- estimators and distributions alike- are not.

The sample mean by itself does not become normal as n gets large. What would happen if you added up a huge series of numbers? You’d get a big number. What would happen if you divided your big number by your huge number? Go on, whack some experimental numbers into your calculator!

Whatever you put into your calculator, it’s not a “normal distribution” you get when you’re done. The sample mean alone does not tend to a normal distribution as N gets large.

The studentised sample mean has a distribution which is normal in the limit. There are some adjustments we need to make before the sample mean has a stable limiting distribution – this is the quantity often known as the z-score. It’s this quantity that tends to normality as n gets large.

How large does n need to be? This theorem works for any distribution with a finite mean and standard deviation, e.g. as long as x comes from a distribution with these features. Generally, statistics texts quote the figure of n=30 as a “rule of thumb”. This works reasonably well for simple estimators and models like the sample mean in a lot of situations.

This isn’t to say, however, that if you have “big data” your problems are gone. You just got a whole different set, I’m sorry. That’s a different post, though.

So that’s a brief run down on the simplest of central limit theorems: it’s not a complex or difficult concept, but it is a subtle one. It’s the building block upon which models such as regression, logistic regression and their known properties have been based.

The infographic below is the same information, but for some reason my students find information in that format easier to digest. When it comes to asymptotic theory, I am disinclined to argue with them: I just try to communicate in whatever way works. On that note, if this post was too complex or boring, here is the CLT presented with bunnies and dragons.** What’s not to love?CLT infographic

** I can’t help myself: The reason why the average bunny weights distribution gets narrower as the sample size gets larger is because this is the sample mean tending towards the true population mean. For a discussion of this behaviour vs the CLT see here.

It’s my only criticism of what was an otherwise a delightful video. Said video being in every way superior to my own version done late one night for a class with my dog assisting and my kid’s drawing book. No bunnies or dragons, but it’s here.

Q&A vs the Leaders’ Debate: is everyone singing from the same song sheet?

The election campaign is in full swing here in Australia and earlier this week the leaders of the two main parties, Malcolm Turnbull and Bill Shorten, faced off in a heavily scripted debate in which few questions were answered and the talking points were well practiced. An encounter described as “diabolical” and “boring“, fewer Australians tuned in compared to recent years. Possibly this was because they expected to hear what they had already heard before.

Since the song sheet was well rehearsed, this seemed like the perfect opportunity for another auspol word cloud. The transcript of the debate was made available on Malcolm Turnbull’s website and it was an easy enough matter of poking around and seeing what could be found. Chris Ullmann, who moderator, was added to the stop words list as he was a prominent feature in earlier versions of the cloud.

debate word cloud

The song sheet was mild: the future tense “will” was in the middle with Shorten, labor, plan, people and Turnbull. Also featured were tax, economic, growth, change and other economic nouns like billion, (per)cent, economy, budget, superannuation. There was mention of climate, (people) smugglers, fair and action, but these were relatively isolated as topics.

In summary, this word cloud is not that different to that generated from the carefully strategised twitter feeds of Turnbull and Shorten I looked at last week.

The ABC’s program Q and A could be a better opportunity for politicians to depart from the song sheet and offer less scripted insight: why not see what the word cloud throws up?

This week’s program aired the day after the leader’s debate and featured Steve Ciobo (Liberal: minister for trade), Terri Butler (Labor: shadow parliamentary secretary for child safety and prevention of family violence), Richard di Natale (Greens, leader, his twitter word cloud is here), Nick Xenophon (independent senator) and Jacqui Lambie (independent senator).  Tony Jones hosted the program and suffered the same fate as Chris Uhlmann.

QandA word cloud

The word cloud picked up on the discursive format of the show: names of panellists feature prominently. Interestingly, Richard di Natale appears in the centre. Also prominent are election related words such as Australia, government, country, question, debate.

Looking at other topics thrown up by the word cloud, there is a broad range: penalty rates, coal, senate, economy, businesses, greens, policy, money, Queensland, medicare, politician, commission.

Two different formats, two different panels and two different sets of topics. Personally, I prefer it when the song sheet has a few more pages.

Social Networks: The Aeneid Again

Applying social network analysis techniques to the Aeneid provides an opportunity to visualise literary concepts that Virgil envisaged for the text. It occurred to me that this was a great idea when I saw this social network analysis of Game of Thrones. If there is a group of literary figures more blood thirsty, charming and messed around by cruel fate than the denizens of Westeros, it would be those in the golden age of Roman literature.

Aeneid social network

This is a representation of the network of characters in the Aeneid. Aeneas and Turnus, both prominent figures in the wordcloud I created for the Aeneid are also prominent in the network. Connected to Aeneas is his wife Lavinia, his father Anchises, the king of the Latins (Latinus) and Pallas, the young man placed into Aeneas’ care.

Turnus is connected to Aeneas directly along with his sister Juturna, Evander (father of Pallas. Cliff notes version: the babysitting did not go well) and Allecto, a divine figure of rage.

Between Aeneas and Turnus is the “Trojan contingent”. Virgil deliberately created parallels between the stories surrounding the fall of Troy and Aeneas’ story. Achilles, the tragic hero, is connected to Turnus directly, while Aeneas is connected to Priam (king of Troy) and Hector (the great defender of Troy). Andromache is Hector’s widow whom Aeneas meets early in the epic.

Also of note is the divine grouping: major players in directing the action of the epic. Jupiter, king of the gods and Apollo the sun god are directly connected to our hero. Venus, Neptune, Minerva and Cupid are all present. In a slightly different grouping, Juno, Queen of the Gods and Aeneas’ enemy is connected to Dido, Aeneas’ lover. Suffice it to say, the relationship was not a “happily ever after”.

I used this list of the characters in the Aeneid as a starting point and later removed all characters who were peripheral to the social network. If you’re interested in trying this yourself, I posted the program I used here. Once again, the text used is the translation by J.W. Mackail and you can download it from Project Gutenberg here.

There were a number of resources I found useful for this project:

  • This tutorial from R DataMining provided a substantive amount of the code required for the social network analysis
  • While this tutorial from the same place was very helpful for creating a text document matrix. I’ve used it previously a number of times.
  • This article from R Bloggers on using igraph was also very useful
  • There were a number of other useful links and I’ve documented those in the R script.

Whilst text mining is typically applied to modern issues, the opportunity to visualise an ancient text is an interesting one. I was interested in how the technique grouped the characters together. These groupings were by and large consistent not only with the surface interpretation of the text, but also deeper levels of political and moral meaning within the epic.