Outliers have more fun

What's an outlier and why should you care?

Years ago I worked for a company that gave me t-shirt that said "Outliers have more fun". I've no idea what it meant, but outliers are interesting, and not in a good way. They'll do horrible things to your data and computing costs if you don't get a handle on them.

Simply put, an outlier is one or more data items that are extremely different from your other data items. Here's a joke that explains the idea:

There's a group of office workers drinking in a bar in Seattle. Bill Gates walks in and suddenly, the entire bar starts celebrating. Why? Because on average, they'd all become multi-millionaires.

Obviously, Bill Gates is the outlier in the data set. In this post, I'm going to explain what outliers do to data and what you can do to protect yourself.

(Jessica  Tam, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons)

Outliers and the mean

Let's start with the explaining the joke to death because everyone enjoys that.

Before Bill Gates walks in, there are 10 people in the bar drinking. Their salaries are: $80,000, $81,000, $82,000, $83,000, $84,000, $85,000, $86,000, $87,000, $88,000, and $89,000 giving a mean of $84,500. Let's assume Bill Gates earns $1,000,000,000 a year. Once Bill Gates walks into the bar, the new mean salary is $90,985,909; which is plainly not representative of the bar as a whole. Bill Gates is a massive outlier who's pulled the average way beyond what's representative.

How susceptible your data is to this kind of outlier effect depends on the type and distribution of your data. If your data is scores out of 10, and a "typical" score is 5, the average isn't going to be pulled too far away by an outlier (because the maximum is 10 and the minimum is zero, which are not hugely different from the typical value of 5). If there's no upper or lower limit (e.g salaries, house prices, amount of debt etc.), then you're vulnerable, and you may be even more vulnerable if your distribution is right skewed (e.g. something like a log normal distribution).

What can you do if this is the case? Use the median instead. The median is the middle value. In our Seattle bar example, the median is $84,500 before Bill Gates walks in and $85,000 afterwards. That's not much of a change and is much more representative of the salaries of everyone. This is the reason why you hear "median salaries" reported in government statistics rather than "mean salaries".

If you do use the median, please be aware that it has different mathematical properties from the mean. It's fine as a measure of the average, but if you're doing calculations based on medians, be careful.

Outliers and the standard deviation

The standard deviation is a representation of the spread of the data. The bigger the number, the wider the spread. In our bar example, before Bill Gates walks in, the standard deviation is $2,872. This seems reasonable as the salaries are pretty close together. After Bill Gates walks in, the standard deviation is $287,455,495 which is even bigger than the new mean. This number suggests all the salaries are quite different, which is not the case, only one is.

The standard deviation is susceptible to outliers in the same way the mean is, but for some reason, people often overlook it. I've seen people be very aware of outliers when they're calculating an average, but forget all about it when they're calculating a standard deviation.

What can you do? The answer here's isn't as clear. A good choice is the interquartile range (IQR), but it's not the same measurement. The IQR represents 75% of the data and not the 67% that the standard deviation does. For the bar, the IQR is $4,500 before Bill Gates walks in and $5,000 afterwards.  If you want a measure of 'dispersion', the IQR is a good choice, if you want a drop in replacement for the standard deviation, you'll have to give it more thought. 

Why the median and IQR are not drop in replacements for the mean and standard deviation

The mean and median are subtly different measures and have different mathematical properties. The same applies to standard deviation and IQR. It's important to understand the trade-offs when you use them.

Combining means is easy, we can do it through formula understood for hundreds of years. But we can't combine medians in the same way; the math doesn't work like that. Here's an example, let's imagine we have two bars, one with 10 drinkers earning a mean of $80,000, the other with 10 drinkers earning a mean of $90,000. The mean across the two bars is $85,000. We can do addition, subtraction, multiplication, division, and other operations with means. But if we know the median of the first bar is $81,000 and the median of the second bar is $89,000, we can't combine them. The same is true of the standard deviation and IQR, there are formula to combine standard deviations, but not IQRs.

In the Seattle bar example, we wanted one number to represent the salaries of the people in the bar. The best average is the median and the best measure of spread is the IQR, the reason being outliers. However, if we wanted an average we could apply across multiple bars, or if we wanted to do some calculations using the average and spread, we'd be better off with the mean and standard deviation.

Of course, it all comes down to knowing what you want and why. Like any job, you've got to know your tools.

The effect of more samples

Sometimes, more data will save you. This is especially true if your data is normally distributed and outliers are very rare. If your data distribution is skewed, it might not help that much. I've worked with some data sets with massive skews and the mean can vary widely depending on how many samples you take. Of course, if you have millions of samples, then you'll mostly be OK.

Outliers and calculation costs

This warning won't apply to everyone. I've built systems where the computing cost depends on the range of the data (maximum - minimum). The bigger the range, the more the cost. Outliers in this case can drive computation costs up, especially if there's a possibility of a "Bill Gates" type effect that can massively distort the data. If this applies to your system, you need to detect outliers and take action.

Final advice 

If you have a small sample size (10 or less): use the median and the IQR.

If your data is highly right skewed: use the median and the IQR.

Remember the median and the IQR are not the same as the mean and the standard deviation and be extremely careful using them in calculations.

If your computation time depends on the range of the data, check for outliers.

Why assuming independence can be very, very bad for business

Independence in probability

Why should I care about independence?

Many models in the finance industry and elsewhere assume events are independent. When this assumptions fails, catastrophic losses can occur as we saw in 2008 and 1992. The problem is, developers and data scientists assume independence because it greatly simplifies problems, but the executive team often don't know this has happened, or even worse, don't understand what it means. As a result, the company ends up being badly caught out when circumstances change and independence no longer applies.

(Sergio Boscaino from Busseto, Italy, CC BY 2.0 , via Wikimedia Commons)

In this post, I'm going to explain what independence is, why people assume it, and how it can go spectacularly wrong. I'll provide a some guidance for managers so they know the right questions to ask to avoid disaster. I've pushed the math to the end, so if math isn't your thing, you can leave early and still get benefit.

What is independence?

Two events are independent if the outcome of one doesn't affect the other in any way. For example, if I throw two dice, the probability of me throwing a six on the second die isn't affected in any way by what I throw on the first die. 

Here are some examples of independent events:

• Throwing a coin and getting a head, throwing a dice and getting a two.
• Drawing a king from a deck of cards, winning the lottery having bought a ticket.
• Stopping at at least one red light on my way to the store, rain falling two months from now.

By contrast, some events are not independent (they're dependent):

• Raining today and raining tomorrow. Rain today increases the chances of rain tomorrow.
• Heavy snow today and a football match being played. Heavy snow will cause the match to be postponed.
• Drawing a king from a deck of cards, then without replacing the card, drawing a king on the second draw.

Why people assume independence

People assume independence because the math is a lot, lot simpler. If two events are dependent, the analyst has to figure out the relationship between them, something that can be very challenging and time consuming to do. Other than knowing there's a relationship, the analyst might have no idea what it is and there may be no literature to guide them.  For example, we know that smoking increases the risk of lung cancer (and therefore a life insurance claim), so how should an actuary price that risk? If they price it too low, the insurance company will pay out too much in claims, if they price it too high, the insurance company will lose business to cheaper competitors. In the early days when the link between smoking and cancer was discovered, how could an actuary know how to model the relationship?

Sometimes, analysts assume independence because they don't know any better. If they're not savvy about probability theory, they may do a simple internet search on combining probabilities that will suggest all they have to do is multiple probabilities, which is misleading at best. I believe people are making this mistake in practice because I've interviewed candidates with MS degrees in statistics who made this kind of blunder.

Money and fear can also drive the choice to assume independence. Imagine you're an analyst. Your manager is on your back to deliver a model as soon as possible. If you assume independence, your model will be available on time and you'll get your bonus, if you don't, you won't hit your deadline and you won't get your bonus. Now imagine the bad consequences of assuming independence won't be visible for a while. What would you do?

Harder examples

Do you think the following are independent?

• Two unrelated people in different towns defaulting on their mortgage at the same time

• Houses in different towns suffering catastrophic damage (e.g. fire, flood, etc.)

Most of the time, these events will be independent. For example, a house burning down because of poor wiring doesn't tell you anything about the risk of a house burning down in a different town (assuming a different electrician!). But there are circumstances when the independence assumption fails:

• A hurricane hits multiple towns at once causing widespread catastrophic damage in different insurance categories (e.g. hurricane Andrew in 1992).

• A recession hits, causing widespread lay-offs and mortgage defaults, especially for sub-prime mortgages (2008).

Why independence fails

Prior to 1992, the insurance industry had relatively simple risk models. They assumed independence; an assumption that worked well for some time. In an average year, they knew roughly how many claims there would be for houses, cars etc. Car insurance claims were independent of house insurance claims that in turn were independent of municipal and corporate insurance claims and so on. 

When hurricane Andrew hit Florida in 1992, it  destroyed houses, cars, schools, hospitals etc. across multiple towns. The assumption of independence just wasn't true in this case. The insurance claims were sky high and bankrupted several companies. 

(Hurricane Andrew, houses destroyed in Dade County, Miami. Image from FEMA. Source: https://commons.wikimedia.org/wiki/File:Hurricane_andrew_fema_2563.jpg)

To put it simply, the insurance computer models didn't adequately model the risk because they had independence baked in.  

Roll forward 15 years and something similar happened in the financial markets. Sub-prime mortgage lending was build on a set of assumptions, including default rates. The assumption was, mortgage defaults were independent of one another. Unfortunately, as the 2008 financial crisis hit, this was no longer valid. As more people were laid-off, the economy went down, so more people were laid-off. This was often called contagion but perhaps a better analogy is the reverse of a well known saying: "a rising tide floats all boats".

(Image credit: Secret London 123, CC BY-SA 2.0, via Wikimedia Commons)

The assumption of independence simplified the analysis of sub-prime mortgages and gave the results that people wanted. The incentives weren't there to price in risk. Imagine your company was making money hand over fist and you stood up and warned people of the risks of assuming independence. Would you put your bonus and your future on the line to do so?

What to do - recommendations

Let's live in the real world and accept that assuming independence gets us to results that are usable by others quickly.

If you're a developer or a data scientist, you must understand the consequences of assuming independence and you must recognize that you're making that assumption.  You must also make it clear what you've done to your management.

If you're a manager, you must be aware that assuming independence can be dangerous but that it gets results quickly. You need to ask your development team about the assumptions they're making and when those assumptions fail. It also means accepting your role as a risk manager; that means investing in development to remove independence.

To get results quickly, it may well be necessary for an analyst to assume independence.  Once they've built the initial model (a proof of concept) and the money is coming in, then the task is to remove the independence assumption piece-by-piece. The mistake is to stop development.

The math

Let's say we have two events, A and B, with probabilities of occurring P(A) and P(B). 

If the events are independent, then the probability of them both occurring is:

\[P(A \ and \ B) = P(A  \cap B) = P(A) P(B)\]

This equation serves as both a definition of independence and test of independence as we'll see next.

Let's take two cases and see if they're independent:

1. Rolling a dice and getting a 1 and a 2
2. Rolling a dice and getting a (1 or 2) and (2, 4, or 6)

For case 1, here are the probabilities:

• \(P(A) = 1/6\)
• \(P(B) = 1/6\)
• \(P(A  \cap B) = 0\), it's not possible to get 1 and 2 at the same time
• \(P(A )P(B) = (1/6) * (1/6)\)

So the equation \(P(A \ and \ B) = P(A  \cap B) = P(A) P(B)\) isn't true, therefore the events are not independent.

For case 2, here are the probabilities:

• \(P(A) = 1/3\)
• \(P(B) = 1/2\)
• \(P(A  \cap B) = 1/6\)
• \(P(A )P(B) = (1/2) * (1/3)\)

So the equation is true, therefore the events are independent.

Dependence uses conditional probability, so we have this kind of relationship:

\[P(A \ and \ B) = P(A  \cap B) = P(A | B) P(B)\]

The expression \(P(A | B)\) means the probability of A given that B has occurred (e.g the probability the game is canceled given that it's snowed). There are a number of ways to approach finding \(P(A | B)\), the most popular over the last few years has been Bayes' Theorem which states:

\[P(A | B) = \frac{P(B | A) P(A)}{P(B)}\]

There's a whole methodology that goes with the Bayesian approach and I'm not going to go into it here, except to say that it's often iterative; we make an initial guess and progressively refine it in the light of new evidence. The bottom line is, this process is much, much harder and much more expensive than assuming independence. 

Using AI (LLM) to generate data science code

What AI offers data science code generation and what it doesn't

Using generative AI for coding support has become increasingly popular for good reason; the productivity gain can be very high. But what are its limits? Can you use code gen for real data science problems?

(I, for one, welcome our new AI overlords. D J Shin, CC BY-SA 3.0 , via Wikimedia Commons)

To investigate, I decided to look at two cases: a 'simple' piece of code generation to build a Streamlit UI, and a technically complicated case that's more typical of data science work. I generated Python code and evaluated it for correctness, structure, and completeness. The results were illuminating, as we'll see, and I think I understand why they came out the way they did.

My setup is pretty standard, I'm using Github copilot in Microsoft Visual Studio and Github Copilot directly from the website. In both cases, I chose the Claude model (more on why later).

Case 1: "commodity" UI code generation

The goal of this experiment was to see if I could automatically generate a good enough complete multi-page Streamlit app. The app was to have multiple dialog boxes on each page and was to be runnable without further modification.

Streamlit provides a simple UI for Python programs. It's several years old and extremely popular (meaning, there are plenty of code examples in Github). I've built apps using Streamlit, so I'm familiar with it and its syntax. 

The specification

The first step was a written English specification. I wrote a one-page Word document detailing what I wanted for every page of the app. I won't reproduce it here for brevity's sake, but here's a brief except:

The second page is called “Load model”. This will allow the user to load an existing model from a file. The page will have some descriptive text on what the page does. There will be a button that allows a user to load a file. The user will only be able to load a single with a file extension “.mdl”. If the user successfully loads a model, the code will load it into a session variable that the other pages can access. The “.mdl” file will be a JSON file and the software will check that the file is valid and follows some rules. The page will tell the user if the file has been successfully loaded or if there’s an error. If there’s an error, the page will tell the user what the error is.

In practice, I had to iterate on the specification a few times to get things right, but it only a took a couple of iterations.

What I got

Code generation was very fast and the results were excellent. I was able to run the application immediately without modification and it did what I wanted it to do.

(A screen shot of part of the generated Streamlit app.)

It produced the necessary Python files, but it also produced:

• a requirements.txt file - which was correct
• a dummy JSON file for my data, inferred from my description
• data validation code
• test code

I didn't ask for any of these things, it just produced them anyway.

There were several downsides though. 

I found the VS Code interface a little awkward to use, for me the Github Copilot web page was a much better experience (except that you have to copy the code).

Slight changes to my specification sometimes caused large changes to the generated code. For example, I added a sentence asking for a new dialog box and the code generation incorrectly dropped a page from my app. 

It seemed to struggle with long "if-then" type paragraphs, for example "If the user has loaded a model ...LONG TEXT... If the user hasn't loaded a model ...LONG TEXT...".

The code was quite old-fashioned in several ways. Code generation created the app pages in a pages folder and prefixed the pages with "1_", "2_" etc. This is how the demos on the Streamlit website are structured, but it's not how I would do it, it's kind of old school and a bit limited. Notably, the code generation didn't use some of the newer features of Streamlit; on the whole it was a year or so behind the curve.

Dependency on engine

I tried this with both Claude 3.5 and GPT 4o. Unequivocally, Claude gave the best answers.

Overall

I'm convinced by code generation here. Yes, it was a little behind the times and a little awkwardly structured, but it worked and it gave me something very close to what I wanted within a few minutes.

I could have written this myself (and I have done before), but I find this kind of coding tedious and time consuming (it would have taken me a day to do what I did using code gen in an hour). 

I will be using code gen for this type of problem in the future.

Case 2: data science code generation

What about a real data science problem, how well does it perform?

I chose to use random variables and quasi-Monte Carlo as something more meaty. The problem was to create two random variables and populate them with samples drawn from a quasi-Monte Carlo "random" number generator with a normal distribution. For each variable, work out the distribution (which we know should be normal). Combine the variables with convolution to create a third variable, and plot the resulting distribution. Finally, calculate the mean and standard deviation of all three variables.

The specification

I won't show it here for brevity, but it was a slightly longer than the description I gave above. Notably, I had to iterate on it several times.

What I got

This was a real mixed bag.

My first pass code generation didn't use quasi Monte Carlo at all. It normalized the distributions before the convolution for no good reason which meant the combined result was wrong. It used a histogram for the distribution which was kind-of OK. It did generate the charts just fine though. Overall, it was the kind of work a junior data scientist might produce.

On my second pass, I told it to use Sobel' sequences and I told it to use kernel density estimation to calculate the distribution. This time it did very well. The code was nicely commented too. Really surprisingly, it used the correct way of generating sequences (using dimensions).

(After some prompting, this was my final chart, which is correct.)

Dependency on engine

I tried this with both Claude 3.5 and GPT 4o. Unequivocally, Claude gave the best answers.

Overall

I had to be much more prescriptive here to get what I wanted, but the results were good, but only because I knew to tell it to use Sobel' and I knew to tell it to use kernel density estimation. 

Again, I'm convinced that code gen works.

Observations

The model

I tried the experiment with both Claude 3.5 and GPT 4o. Claude gave much better results. Other people have reported similar experiences.

Why this works and some fundamental limitations

Github has access to a huge code base, so the LLM is based on the collective wisdom of a vast number of programmers. However, despite appearances, it has no insight; it can't go beyond what others have done. This is why the code it produced for the Streamlit demo was old-fashioned. It's also why I had to be prescriptive for my data science case, for example, it just didn't understand what quasi Monte Carlo meant without additional prompting.

AI is known to hallucinate, and we see see something of that here. You really have to know what you're doing to use AI generated code. If you blindly implement AI generated code, things are going to go badly for you very quickly.

Productivity

Code generation and support is a game changer. It ramps up productivity enormously. I've heard people say, it's like having a (free) senior engineer by your side. I agree. Despite the issues I've come across, code generation works "good enough".

Employment

This has obvious implications for employment. With AI code generation and with AI coding support, you need fewer software engineers/analysts/data scientists. The people you do need are those with more insight and the ability spot where the AI generated code has gone wrong, which is bad news for for more junior people or those entering the workforce. It may well be a serious problem for students seeking internships.

Let me say this plainly: people will lose their jobs because of this technology.

My take on the employment issue and what you can do

There's an old joke that sums things up. "A householder calls in a mechanic because their washing machine had broken down. The mechanic looks at the washing machine and rocks it around a bit. Then the mechanic kicks the machine. It starts working! The mechanic writes a bill for $200. The householder explodes, '$200 to kick a washing machine, this is outrageous!'. The mechanic thinks for a second and says, 'You're quite right. Let me re-write the bill'. The new bill says 'Kicking the washing machine $1, knowing where to kick the washing machine $199'." To put it bluntly, you need to be the kind of mechanic that knows where to kick the machine.

(You've got to know where to kick it. LG전자, CC BY 2.0 , via Wikimedia Commons)

Code generation has no insight. It makes errors. You have to have experience and insight to know when it's gone wrong. Not all human software engineers have that insight.

You should be very concerned if:

• You're junior in your career or you're just entering the workforce.
• You're developing BI-type apps as the main or only thing you do.
• There are many people doing exactly the same software development work as you.

If that applies to you, here's my advice:

• Use code generation and code support. You need to know first hand what it can do and the threat it poses. Remember, it's a productivity boost and the least productive people are the first to go.
• Develop domain knowledge. If your company is in the finance industry, make sure you understand finance, which means knowing the legal framework etc.. If it's a drug discovery, learn the principles of drug discovery. Get some kind of certification (online courses work fine). Apply your knowledge to your work. Make sure your employer knows it.
• Develop specialist skills, e.g. statistics. Use those skills in your work.
• Develop human skills. This means talking to customers, talking to people in other departments.

Some takeaways

• AI generated code is good enough for use, even in more complicated cases.
• It's a substantial productivity boost. You should be using it.
• It's a tool, not a magic wand. It does get things wrong and you need to be skilled enough to spot errors.

Python formatting

Python string formatters

I use Python and I output data for reports, which means I need to format strings precisely. I find the string formatters hard to use and resources to explain them are scattered over the web. So I decided to write up my own guide to using formatters. This is mainly for me to have a 'cheat sheet', but I hope you find some use for it too. Of course, I've liberally copied and pointed to the Python documentation.

(This is a Python blog post! Image source: Wikimedia Commons. License: Creative Commons.)

Overview

Python string formatters have this general form:

{identifier : format specifier}

The term identifier is something I made up for easier reference.

Identifiers

The identifier ties the string format to the code in the format statement. Identifiers can be positional (numbered) or named. Numbered identifiers must start from zero and should increment by 1, but you can re-use identifiers like this:

'Today is {0}-{1}-{2} the year is {0}'.format(2020, 10, 22)

You can also use names as identifiers:

'Today is {year}-{month}-{day} the year is {year}'.format(year=2020, month=10, day=22)

and relatedly, you can use a dict:

'Today is {year}-{month}-{day} the year is {year}'.format(**date)

You can read more clever uses of identifiers here: https://docs.python.org/3.4/library/string.html#format-string-syntax

Format specifiers

There's an entire format specifier mini language: https://docs.python.org/3.4/library/string.html#formatspec

The general form is:

[[fill]align][sign][#][0][width][,][.precision][type]

• fill is the character to use to fill padded spaces
• align is the instruction on how to align the string (left, center, right)
• sign, the + or - sign, only makes sense for numbers
• # indicates an alternate form for conversion
• 0 - used for sign aware zero padding for numbers
• width - the width in characters of the field
• , - use of the thousand separator
• .precision - the number of digits after the decimal place
• type - one of these special types: "b", "c", "d", "e", "E", "f", "F", "g", "G", "n", "o", "s", "x", "X", "%"

Type	Meaning
's'	String format. This is the default type for strings and may be omitted.
None	The same as 's'.

Type	Meaning
'b'	Binary format. Outputs the number in base 2.
'c'	Character. Converts the integer to the corresponding unicode character before printing.
'd'	Decimal Integer. Outputs the number in base 10.
'o'	Octal format. Outputs the number in base 8.
'x'	Hex format. Outputs the number in base 16, using lower- case letters for the digits above 9.
'X'	Hex format. Outputs the number in base 16, using upper- case letters for the digits above 9.
'n'	Number. This is the same as 'd', except that it uses the current locale setting to insert the appropriate number separator characters.
None	The same as 'd'.

Type	Meaning
'e'	Exponent notation. Prints the number in scientific notation using the letter ‘e’ to indicate the exponent. The default precision is 6.
'E'	Exponent notation. Same as 'e' except it uses an upper case ‘E’ as the separator character.
'f'	Fixed point. Displays the number as a fixed-point number. The default precision is 6.
'F'	Fixed point. Same as 'f', but converts nan to NAN and inf to INF.
'g'	General format. For a given precision p >= 1, this rounds the number to p significant digits and then formats the result in either fixed-point format or in scientific notation, depending on its magnitude. The precise rules are as follows: suppose that the result formatted with presentation type 'e' and precision p-1 would have exponent exp. Then if -4 <= exp < p, the number is formatted with presentation type 'f' and precision p-1-exp. Otherwise, the number is formatted with presentation type 'e' and precision p-1. In both cases insignificant trailing zeros are removed from the significand, and the decimal point is also removed if there are no remaining digits following it. Positive and negative infinity, positive and negative zero, and nans, are formatted as inf, -inf, 0, -0 and nan respectively, regardless of the precision. A precision of 0 is treated as equivalent to a precision of 1. The default precision is 6.
'G'	General format. Same as 'g' except switches to 'E' if the number gets too large. The representations of infinity and NaN are uppercased, too.
'n'	Number. This is the same as 'g', except that it uses the current locale setting to insert the appropriate number separator characters.
'%'	Percentage. Multiplies the number by 100 and displays in fixed ('f') format, followed by a percent sign.
None	Similar to 'g', except that fixed-point notation, when used, has at least one digit past the decimal point. The default precision is as high as needed to represent the particular value. The overall effect is to match the output of str() as altered by the other format modifiers.

f-strings

These are a way of simplifying Python string formatting and really should be your preferred way of outputting strings. Very usefully, they allow you to embed expressions. Here are a couple of examples.

number = 10

print(f"The number is {number}")

10

print(f"The expression is {number + 100}")

110

Examples

'{:<30}'.format('left aligned')

'{:*^30}'.format('centered')

"int: {0:d};  hex: {0:x};  oct: {0:o};  bin: {0:b}".format(42)

'{:,}'.format(1234567890)

'Correct answers: {:.2%}'.format(points/total)