Multiple plots in R: lesson zero

Today, in one of my more productive days, I managed to create a sleek R script that plotted several histograms in a lattice, allowing for easy identification of the underlying trend. Although the majority of the time taken consisted of collecting the data and making various adjustments, it took a not inconsiderable amount of work to write the code.

As I was cursing the apply function – not for the last time I am sure – I suddenly realised the insane level of productivity that I have come to see as "par for the course". The level of computational analysis that can be conducted in a few hours with nothing more than a desktop PC,  a broadband connection, and copious amounts of caffeine is phenomenal. No longer can a lack of computing power or software be blamed for a lack of productivity growth: rate-limiting factors are now exclusively human.

Here at least is my contribution to the collective intelligence of biologicals. I have noticed that the most common reason that people avoid R is that they cannot rapidly make graphs that meet the high standards of their clients. In the next few editions of this blog we will build up a basic repertoire of plotting techniques, focusing on graphics that are sickeningly impressive.

Of course, Rome was not built in a day, and a thorough knowledge of R plotting cannot be built in one. Instead we will progress one layer at a time, adding additional levels of complexity and functionality. We start with a simple script that allows us to plot several graphs at the same time, each with a different value of a key variable. Here is the output:

Figure 1: Multiple plots using par

The code

# Clear all objects
rm(list=ls())

# Create a data set using random variables
df<-data.frame(x=rnorm(160),y=runif(160),a=sample(c(1,0,-1,10),
replace=TRUE,160))

df$z<-with(df,{3*a*y+x})

# Create a function that plots the value of "z" against the "y" value
plotM<-function(l){
  
  df.temp<-df[df$a==l,]
  plot(df.temp$y,df.temp$z,xlab="Y Value",ylab="Z Value",
  main=paste("Value of key variable: ",toString(l)))
  abline(lm(df.temp$z~df.temp$y),col="red")
  
}

# Create a grid to plot the different values of "a"
par(mfrow=c(2,2))

# Loop through each value of "a" and call the plotM function
for (i in c(1,0,-1,10)){
    plotM(i)
}

  
Walk-through
The first few lines of code create a data frame with 4 variables: x,y,z,a. Three of these variables are randomly generated, with the z variable dependent upon the other 3. Suppose that we are analysing this data set, unaware of the relationship between the x,y and z variables. A preliminary inspection shows that there are only 4 observed values of a. It seems sensible to plot z against y for each of these different a values.

The relevant code to create this plot starts with the function "plotM", standing for plot Multiple. This function takes an argument "l" that determines which value of the variable we are plotting. The line

df.temp<-df[df$a==l,]

filters the data frame to include only those rows where the variable a=l. The next line simply creates a standard R plot of y versus z. Finally, we use the abline function to plot a linear fit to highlight the trend. Too easy.

Now we come to the fun part. Using the par(mfrow=c(2,2)) we create a 2x2 grid in which to place the next four plots. Whatever plots we now create will be placed sequentially into this grid. Hence we can iterate over a vector containing the values of a,  calling plotM each time. This gives us our grid.

Comments
Okay, so the graph does not look like it came from NASA – or to be honest with NASA's ailing reputation maybe it does. But note that once we have created the plotM function, we only have to write 3 lines of code to make 4 separate charts. Moreover, the code would not increase even if we were plotting 100 charts.

Of course, we have not yet even drawn upon any of R's custom plotting packages. In the next edition of this blog we will look at how to use the ggplot2 package to add colour and a wide range of other features to our graphs.

Keep It Linear Stupid: the cost of achieving perfection under pressure

"That gives you 48 hours to develop a methodology – since we haven't done one of these before – prototype the model, build it, calibrate it to client expectations, and write the report. But don't worry, the client is paying a premium for the rush."

Every analyst has been in this sort of situation. The client wants something now, and your boss has agreed to deliver it. The trouble is that before that perfectly crafted 3-line email three is sent, accompanied by a hefty invoice, you actually have to build the thing. What is more, lack of time is no excuse for sloppy work. No mistakes, after all we're professionals.

Working to a deadline

The Mythical Man-Month is the best book about managing projects to a deadline that I have ever read.  Although focused on software engineering, the book has lessons that apply to project management in any technical field. The author, Frederick P. Brooks, provides a mathematical argument for why adding manpower to a technical project that is running late makes it later (Brook's Law).  The problem is that  time spent getting new people up to speed (initiation) and coordinating additional manpower (co-ordination) exceeds the additional productivity. There is a fixed time-cost to incorporate a new worker onto a project. Unless the project has a long way to go, adding people just slows it down.

An obvious corollary to Brook's Law is that there is a minimum time required to complete any project that is directly proportional to its complexity. Moreover, this minimum time corresponds to a unique number of workers: at some point additional manpower has decreasing returns to scale in the form of co-ordination and initiation costs. What does this mean for your 48-hour rush-job? It is a one-man, or at most two-man, show. Looks like you will be staying back in the office on your own.

No mistakes...well sort of

The big consultancy houses and banks have built a powerful image for their industries over time. That image is in every logo, every website, every banner at a conference. The image is one of flawlessness, of crystalline perfection, of exacting standards.

This notion of a clinical, relentless pursuit of perfection has managed to convince the non-technical members of society that for the right price perfection is not only achievable, but the norm. On the first day of my first job, I remember my boss emphasising the many processes that were in place to prevent errors. My boss was, naturally, a manager. Managers do not run the numbers: if no-one finds an error then they assume there were none to find. I rapidly realised that although it was unacceptable for errors to be discovered, it was perfectly acceptable for errors to exist.

A one page document can contain perfect spelling; a single table of data can be copied exactly from another source. But the problem for the analyst is managing complexity. Large documents are more likely to house inconsistencies; models that attempt to capture more variables are more prone to calculation errors. The Law of Large Numbers sucks, doesn't it.

Of course, the notion of perfection is somewhat flexible for the rush-job.  The odd inaccurate assumption is forgivable,  as is a minor inconsistency in the results. But if there is a major mistake, don't dare blame the deadline. So how do you avoid mistakes during a rush-job? The answer is a variation of the "Keep It Simple Stupid" (KISS) rule.

Keep it Linear Stupid

Let's assume for a second that you are young and naive, some might even say a cock-eyed optimist who has gotten caught up in the world of international modelling intrigue. Despite the impending deadline, you nevertheless are determined to capture every aspect of the phenomenon at hand. One crucial variable is clearly quadratic,  and to assume it is linear would be a travesty against all that mathematics stands for. Was it all for nothing, Archimedes? 

You have a choice in front of you: you pick the blue line – the complexity ends, you wake up in your bed and believe whatever you want to believe. You pick the red line – you stay in mathematical wonderland and this model is going to show you just how deep the rabbit-hole goes. You are young, you are fearless, you are stupid – you choose the red line.

"I know what you're asking yourself: why oh why didn't I pick the blue line?"

Down the rabbit-hole

Okay, so its not as bad as waking up in a petri dish to find that you have a massive fire-wire port in the back of your skull. But by 2 am in the morning of the second day, you are going to wish that you could go back and choose that blue line – I guarantee it.

You see the problem with a rush job is that you don't have time to go back. The client expects the results to make sense, and there is always one number that can't be explained by anything other than an error. But when you go to iron it out, that red line comes back to haunt you. Every tiny alteration to the inputs results in a non-linear change to the results. 

Ordinarily, this is fine as you have plenty of time to get it right  Given time you would be able to rework the model, but that is time you don't have. Every time you pin down one set of numbers, another one jumps out of its place. Non-linearity is hard to explain and even harder to defend, especially when you know that there may be errors. If you are lucky, you will be able to explain away the aberration. If not, you have a report that may as well be lit with a neon sign saying "Error: Division by zero".

The price of sanity

Okay, so you learned your lesson. Next time you get a rush job, you restrict your analysis to the one-dimensional case (and even that seems gutsy), tether your key variables to a stake in the numeric al equivalent of Alcatatraz, and assume that everything from the inflation rate to the diffusion of information throughout the economy can be approximated by a linear function. Nice job.

You have done well by your boss, you have done well by yourself. You have even done right be your client, as they have the report they wanted. But what good is it? The purpose of quantitative analysis is to enlighten decision making, to add something that could not have been uncovered by logic alone. Once you have removed all complexity, you have also removed all value. You have created consistency and clarity but at vulgar cost. In truth, you have obscured more than you have explained.

An analyst often has little choice in the matter at the time, but they can make clear how banal the results are that spring from these sorts of exercises. More time may be an option, absolute internal consistency may not be essential. If the alternative is simplification ad absurdum, then it may be better to pass on the job. 

Keep it simple, keep it trivial. Keep it linear? Stupid.

 

Getting a first class education without the luxury price tag: ITunes University

Permit me to paint a picture. You have just finished your undergraduate degree in a business or quantitative discipline (eg, economics, computer science, or mathematics). You finished with first class honours, so you have landed a great job in a medium-sized firm. You are fed up to the teeth with formal study, and are looking to gain some experience in the real world and finally repay your student loans. The time has come for your study to start paying off...or has it?

The problem is that in the modern world there is an arms race for qualifications. As demand for skilled graduates has increased, so have the number of students completing university. But as a result, there are more and more members of the work-force completing advanced degrees. In addition, this has been fueled by the behemoth style corporate factories financing postgraduate education. Masters degrees have become the norm, and no sooner have students left university than they are back, but this time in the evening after work.

I am highly sceptical of these Masters programs, particularly those in my home country of Australia. The entry requirements for these degrees are extremely low, most notably in my own area of applied mathematics. Demand for postgraduate qualifications in mathematical finance is so great that universities have slowly lowered the level of prerequisite knowledge. Many of these students have barely completed first year mathematics, but are placed in classes alongside honours students (who often coach them through the course).

As the quality of candidature drops, so does the quality of the course. Students are so concerned about passing that they have no interest in engaging with the material. Courses are crammed into a single evening, usually a three to four hour block, allowing those who work full-time to attend with minimal inconvenience. Of course, by half way through the second hour no one is paying attention, and everyone is secretly hoping that an impromptu visit from the fire department will put everyone out of their misery.

If you are looking for an extra line on your resume then go right ahead and sign up to one of these programs. But what if you genuinely want to gain further education. I guess you can always consider a PhD. It still represents a real qualification that is going to last you for the duration (we hope!). But if you were squeamish about signing up for a Masters, a PhD is likely to make you throw your rifle into the cornfield and run for the hills. After all, you don't want to become an academic, you just want to ensure that you continue learning.

Until recently the obvious answer was self-education and, for all intents and purposes, it still is. Picking up a few books on your area of expertise (eg, computational economics, operations research, stochastic analysis) and working through them as though you were embarking on a PhD is a reasonable way to simulate the experience. Many of my friends who have stayed at university to complete doctorates have noted that they could do most of their work at home, provided that they were still funded to do so. Of course, the greatest benefits of being enrolled in a formal program stem from interaction with other researchers, and this is hard to replicate on your own. However, there are also advantages to working in industry where you are constantly faced with real world problems that can direct and focus your research.

It is, however, this focus or structure that so often is lacking from self-education. However, with the advent of the internet this too can be achieved without needing to drop everything to go back to university. The solution is now freely available on ITunes.

Truly open universities: ITunesU

The availability of online university courses has exploded in the last year. My first exposure to online courses was through MIT Opencourseware. I watched a series of lectures on Linear Algebra to supplement a course that I was studying at the time. To be honest, the course I was taking was far more in depth and of a far higher quality, but I really benefited from being able to watch a completely different course in its entirety and from the comfort of my own home. What was so revolutionary about MIT Opencourseware was its attempt to allow online users to have the full course experience. Every lecture was posted online, and all class materials were available for download. Its not exactly like being enrolled, but it is about 90% of the way there. The problem was that only a very limited number of courses were available. The 101 course was there but 102 often was not. Darn.

In the last 12 months, however, more and more courses have become available online. Some of the best are offered by Stanford, MIT, and Yale and are all available at the ITunes store in the ITunes University Section (ITunesU). Below is a short list of some of the highlights:

• Convex Optimization (A and B)– Taught by Stephen Boyd – Stanford University
• Financial Markets – Taught by Robert Schiller – Yale University
• Information and Entropy – Taught by Paul Penfield & Seth Lloyd – MIT

Sure, you are not going to be able to replicate a degree course-for-course, but chances are you can cover most of the main subjects and fill in the gaps with self-study by reading through the recommended texts.

Better than the real thing – the multiplier effect

The major advantage of online courses is that they are free and thus available to the financially challenged (or alternatively those who don't want to fork out a hundred grand for a degree). As a consequence of this, these courses can actually be better than the real thing. Pray tell how so, I hear you ask?

I found during my university degree that it was sometimes subjects outside of one's discipline that were the most valuable. In my case, I gained more from a couple of well-chosen courses in computer science than I would have from a year of further mathematics. I like to think of cross-disciplinary study as having a multiplier effect – the military meaning as opposed to the economic term. On the battlefield snipers are often referred to as having a multiplier effect, as the presence of a sniper increases the effectiveness of all other members of the unit. In the same way, a basic training in a neighbouring  discipline can leverage your existing knowledge.

As online courses can be taken without cost, they are custom made for this type of 'field-hopping' (tell me if you come up with a better term). With no program requirements, or restrictions on choices of electives you are free to study what you like. I have found that this improves the learning experience, and provides a truly liberal education.

Of course, you still have to put in the hours, do the tutorial exercises, and gain mastery of the material.  Whatever degree you enroll in, no matter how prestigious, the final responsibility will always rest with you. Sandstone spires and grassy quadrangles do not a genius make. As online courses increase in quality and number, I expect that this simple truth will become evermore evident. Make sure you aren't left behind!