I recently saw this chart (below) from PNAS.  It’s one of those popular psych studies that asks “does having more money make people happier?”  They discovered that higher annual income does indeed make people happier, but in a logarithmic relationship.  This reinforces something I wrote in Sensitivity Testing Model Assumptions, namely: know your time series.

In today’s post, I’ll explain about log scales using an example from stock trading, and then circle back to the happiness data.  I’ll wind up with an example from my own experience, contrasting exponential versus polynomial functions.  You may also want to check out my earlier posts on seasonal adjustment, and Bayesian probability.

“When interpreting these results, it bears repeating that well-being rose approximately linearly with log(income), not raw income”

Reported “life satisfaction” and “well-being” increase linearly with log income, not straight income.  That is, the next notch up in happiness requires an order of magnitude more money.  Previous studies had found a plateau around \$75,000, with little or no increase in happiness after that.

So, this is a new finding – or is it?  Take another look at that income scale.  If that were a straight scale, the chart would show diminishing returns from additional income.  To enjoy steadily increasing happiness, you have to earn exponentially increasing income.

## Log Scaling for Stock Charts

Here is a chart of Carvana during 2018, when the stock was rising rapidly.  Comparing the tiny candles in March with the longer ones in June and beyond, you might conclude that the stock had become more volatile.  The average daily trading range increased from one dollar to three over the period.

But a dollar when the stock is at \$20 is not the same as a dollar when the stock is at \$60.  To have the candles represent percentage change, you must set the price scale to logarithmic.  See, in the chart below, how the price intervals get closer together as they proceed up the scale.

Whenever a stock chart covers a wide price range, you’re better off using a logarithmic scale.  You may recall from school that adding logs is the same as multiplying the numbers.  So, a linear scale shows additive change, and a log scale shows relative change.

log ab = log a + log b

Take another look at the first Carvana chart above.  Stock traders call that “going parabolic.”  Parabolic growth, also known as “quadratic,” is another rapid growth trend, easily confused with exponential growth.  I did a quick regression analysis, and both models fit the Carvana data pretty well.

Pro tip: Never use “exponential” to describe something that’s not a time series.  Some people seem to think it just means “big,” as in “last month was exponential!”

The point to “know your time series” is to understand the mechanisms underlying your data.  Exponential growth comes from compounding, like if you increase sales by ten percent, and then you increase the new, higher, base by another ten percent – and you keep doing that.

I’ll provide an example of quadratic growth later, but first let’s finish up the PNAS chart.  I think of this as a time series because I picture someone earning steadily more income over their career (the data is actually different people at different income levels).

When I say “steadily more income,” I mean exponentially.  Note that each tick mark on the PNAS income scale doubles the value.  This is a log scale, like the Carvana price scale, above.

Many real-world metrics are based on log scales, like decibels and the Richter scale.  An earthquake of magnitude 6.0 on the Richter scale is ten times as powerful as a 5.0.

The chart below shows what this blessed career looks like.  If you start making \$15,000 at age 18 and double your salary every six years or so, then you will experience steadily increasing “well-being.”  My red line is the same red line as in the PNAS chart.

I think showing the data as someone’s career is a good way to tell the story.  Income and well-being are shown together, with straight scales, and mediated by the hypothetical age.  On the other hand, the correlation has disappeared.  To show that, we must apply a log scale to the income series:

This is why the authors make clear that, to enjoy steadily increasing (linear) happiness, you must earn steadily increasing (exponential) income.  To put it another way, if you only earn increasing (linear) income, then you will have only increasing (log) happiness.

Many real-world metrics are based on log scales, like decibels and the Richter scale.  An earthquake of magnitude 6.0 on the Richter scale is ten times as powerful as a 5.0.

## Polynomial versus Exponential Functions

Once upon a time, way back when databases had size constraints, I observed that parabolic growth in BMW Financial lease transactions would pose a danger to the database.  I ran a regression analysis, calculated when the database would fail, and sent a memo to my boss.

I also worked out a mitigation strategy, but let’s stick with, “the database will blow up on April 21,” for dramatic effect.  Instantly my office filled up with expensive auditors and consultants.

“No, it’s not a malfunction.”

“No, it’s not growing exponentially.”

Lease transactions were growing quadratically, which is why I chose this example.  If new leases are steady at 1,200 per month, that’s a flat line.  Total rows in the lease table will thus increase by 1,200 per month.  That’s a sloping line.

Now, if each lease generates roughly two transactions per month, then total transactions will be a parabola.  Readers with a little calculus will recognize this as integrating, twice, from the constant rate of new leases to the second-order rate of transactions.

This is the essence of “know your time series.”  The regression analysis showed quadratic growth, unequivocally, and it was also supported by how we expected the data to behave.

The auditors milled around for a while, charged us about a million bucks, and decided I was panicking over nothing because the database wasn’t really going to blow up until April 29.  Not to mention my mitigation strategy.

## Big O Notation

In the example above, I showed that a linear function of a linear function is a parabola, also known as a “quadratic” or second-order polynomial.  Compose another linear function on top of that, and it’s a third-order polynomial.

Excel has a handy feature, shown below, for fitting polynomial functions of different order.  This is a little dangerous, because you can easily find a fit without thinking about it.  It’s not enough to get a good R-Square (fit) value.  You must understand why the data behaves as it does.

This “order” thing is sufficiently important to data analysts that they have a notation for it, called “Big O.”  The quadratic example we just worked through would be O(n2) or “order n-squared.”  I notice that Excel will also fit a Power Law, or “Pareto” series.  That will have to wait for a later post.

## Data Lakes Explained

Last month, I wrote an explainer on AI and it was well-received, so here is one on data lakes.  If you already know the concepts, you may still find this framing helpful in client discussions.  Our audience this time is the CFO, or maybe the CMO, and our motivation is that their analytical needs are not well-served by the transactional database.

## Transactional Processing with a Relational Database

The data that runs your business – most of it, anyway – is probably stored in a relational database like Microsoft’s venerable SQL Server.  Without going into details about the “relational” structure, the key is that this database is optimized for the daily operations of the business.

New policies are booked, premiums collected, and claims paid.  These are transactions that add, change, or delete records.  There are also “read only” operations, like producing invoices, but the database is designed primarily for transaction processing.

A well-designed transactional database will resist anomalies

A well-designed transactional database will resist anomalies, like a line item with no invoice, or two sales of the same item.  The database designer will have used a technique called normalization, breaking the data up into smallish tables with relationships that enforce integrity.

Think of how your chart of accounts is organized.  Everything you need to account for is broken down to the lowest relevant level, and then rolled up for reporting.  Every journal entry hits two accounts, debit and credit, so that they’re kept in balance.  Your meticulously normalized database is kind of like that.

When a customer places an order, a row is added to the Order table.  You don’t need to open the Customer table unless there’s a change to the customer.  Built around these normalized tables is the machinery of indexes, clusters, and triggers, which support speed and integrity.

Pro Tip: Take time to confirm that the transactional database is stable and supporting the business satisfactorily.  You don’t want to start building pipelines and then discover there’s a problem with your data source.

## Analytical Processing with a Data Warehouse

Transaction processing involves adding and changing data, with carefully limited scope.  Analytical processing, by contrast, is mostly reading data – not changing it – and holistic in scope.  To support this, the data must be copied into a separate database and denormalized.

Let’s say you want to know whether Dent protection sells better as a standalone product, or as part of a bundle – corrected for the number of dealers who don’t offer the bundle, and segmented by the vehicle’s make and price range.

You could run this query against the transactional database, but it would be difficult.  The query is complicated enough without having to piece together data from multiple tables.  The normalization which served so well for transaction processing is now an obstacle.

Confession: I am a normalization bigot.  I bought C.J. Date’s textbook, read the original papers in the ACM journal, and even coded Bernstein’s algorithm.  To me, organized data is normalized data, and de-normalizing is like leaving your clothes on the floor.

So, this is a good guide to denormalization.  Everything we learned not to do in relational databases – wide tables, nested data, repeating groups – is useful here.

Analytical data is stored in cubes, stars, snowflakes, hearts, and clovers

Analytical work requires not only a new database design, but a new database system.  Out goes SQL Server and now we have Big Query, Redshift, and Snowflake.  You may hear this buzzword, OLAP, which means “online analytical processing.”  This concept was invented for marketing purposes, to describe the new category of software.

Analytical data is stored in cubes, stars, snowflakes, hearts, and clovers (see sidebar).  Just kidding about the hearts and clovers.  Also, while your transactional database may be running SQL Server “on premise,” the analytical database will almost certainly be on a cloud service from Amazon, Microsoft, or Google.

To be honest, not everyone needs an OLAP database.  As CIO for BMW Financial Services, I did not recommend one because our analytical workload was small, at the time, and could be served adequately without a lot of new gear and expensive consultants.  Since then, I have gone over to the side of the consultants.

## Sidebar: What’s an OLAP Cube?

In the early days of analytical processing, software vendors thought it would be a good idea to use a multidimensional data structure called a hypercube. Think of a typical spreadsheet, with rows representing an income statement and one column for each month. That’s two dimensions. Now, add a stack of spreadsheets, one for each region. That makes three dimensions, like a cube. I put myself through grad school working at Comshare, one of the first OLAP software vendors. It supported seven dimensions. That’s a hypercube. Nowadays, there are better data structures, and this leads to some confusion. Older analysts may assume that if they’re doing OLAP, then they must be using a cube. They may use the term “OLAP cube” to mean any analytical database, even though cubes have largely been replaced by newer structures.

## Pooling Data in a Data Lake

You can think of the data lake as a way station between the transactional database and the data warehouse.  We want to collect all the data into a common repository before loading it into the data warehouse.

Why not simply extract, transform, and load data straight from the transactional database?  Well, we could, but it would be brittle.  Any change on either side would require an update to the pipeline.  The data lake decouples the OLTP and OLAP data stores.

The data lake serves the very important function of storing all the data, in whatever format, whether or not it’s amenable to organization.  The term’s originator, James Dixon, wanted to suggest a large volume of data with no preconceived organization.

The key thing is to collect all the data in one place, and think about organization later.  This calls for an “object data store,” like Google Cloud Storage.  GCP and AWS both use “buckets.”  You get the idea – this is where you leave your clothes on the floor.

Most of your data will indeed be structured data coming from the transactional database, and on its way into the OLAP database – but not all of it.  Here are some real-life examples I have encountered:

• Logs of API traffic. Details of who is using our ecommerce API, including copies of the payload for each request and response.
• Text snippets. A file of the several paragraphs that make our standard Texas contract different from the one in Wisconsin, so that we can produce new contracts automatically.  Same goes for product copy on the web site.
• Telephone metadata. A list of timestamps, durations, phone numbers, and extensions for all calls in the call center, both inbound and outbound.

These examples are better served by special-purpose databases like Hadoop, Bigtable, and Mongo.  It’s best to take stock of all the data your analysts might need, broadly speaking, and start collecting it before you go too far with designing the OLAP database.

## Speculation on Fractal Programming Language

We flew east out of Panama City, and I looked down on the faceted green hills of the Cordillera de San Blas, perhaps for the last time.  In the sky were statistically similar puffs of white cumulus cloud.  Naturally, I was thinking of fractals.

I had spent the past few days coding technical analysis indicators in Python, which reminded me of coding same in C#.  This, in turn, reminded me that although the TA community talks a lot about geometric repetition, we have yet to invent a single fractal indicator, much less a trading strategy.

I write my trading strategies in C# on the MultiCharts platform.  Its procedures for time series data look a lot like the vector-oriented syntax of Python.  Here is how to do Bollinger bands in each:

• StandardDeviationCustom(length, devs)
• df[price].rolling(length).std() * devs

I have to admit not having much intuition about vector operations.  Matrices and summations just look like for loops to me – clearly an obstacle to the proper appreciation of Python.  I have worked with SAS and SYSTAT, though, so Python at the command prompt seems natural.

What I noticed with the Python exercise is that the classic TA indicators were designed with an iterative mindset, reflecting the programming languages of the day – Sapir’s theory, again – and so I imagine that the reason we have no fractal indicators is that our language can’t express them.

Here are some basic things I would expect from a fractal-oriented programming language:

• Create a dataset from a generator function
• Derive fractal metrics, like the Hausdorff dimension
• Compare two datasets for statistical similarity
• Compare a dataset to subsets of itself

Admittedly, I have only a cursory notion of how this would work.  That’s why this piece has “speculation” in the title.  Meanwhile, I will continue plugging away in C# and Python.

## Predictive Selling in F&I

We have all seen how Amazon uses predictive selling, and now this approach is finding its way into our industry.  In this article I compare and contrast different implementations, and discuss how the technique may be better suited to online than to the F&I suite.

If you read Tom Clancy, you might like Lee Childs.  If you bought a circular saw, you might need safety goggles.  To draw these inferences, Amazon scans for products that frequently occur together in the order histories of its customers.  You can imagine that given their volume of business, Amazon can fine-tune the results by timeframe, department, price, and so on.

The effectiveness of predictive selling depends on two things: the strength of your algorithms, and the depth of your database.  Automotive Mastermind claims to use “thousands of data points,” mined from the DMS, social media, and credit bureaus.  An online auto retailer or platform site (see my taxonomy here) will also have data about which web pages the customer viewed.  Your typical F&I menu is lucky if it can read data from the DMS.

The face of predictive selling in F&I is the automated interview.  We all know the standard questions:

• How long do you plan on keeping the car?
• How far do you drive to work?
• Do you park the car in a garage?
• Do you drive on a gravel road?
• Do you transport children or pets?

A system that emulates the behavior of an expert interviewer is called, appropriately, an “expert system.”  I alluded to expert systems for F&I here, in 2015, having proposed one for a client around the same time.  This is where we can begin to make some distinctions.

Rather than a set of canned questions, a proper expert system includes a “rules editor” wherein the administrator can add new questions, and an “inference engine” that collates the results.  Of course, the best questions are those you can answer from deal data, and not have to impose on the customer.

A data scientist may mine the data for buying patterns, an approach known as “analytics,” or she may have a system to mine the data automatically, an approach known as “machine learning.”  You know you have good analytics when the system turns up an original and unanticipated buying pattern.  Maybe, for example, customers are more or less likely to buy appearance protection based on the color of their vehicle.

At the most basic level, predictive selling is about statistical inference.  Let’s say your data mining tells you that, of customers planning to keep the car more than five years, 75% have bought a service contract.  You may infer that the next such customer is 75% likely to follow suit, which makes the service contract a better pitch than some other product with a 60% track record.  One statistic per product hardly rises to the level of “analytics,” but it’s better than nothing.

Another thing to look at is the size of the database.  If our 75% rule for service contract is based on hundreds of deals, it’s probably pretty accurate.  If it’s based on thousands of deals, that’s better.  Our humble data scientist won’t see many used, leased, beige minivans unless she has “big data.”  Here is where a dealer group that can pool data across many stores, or an online selling site, has an advantage.

If you are implementing such a system, you not only have a challenge getting enough data, you also have to worry about contaminating the data you’ve got.  You see, pace Werner Heisenberg, using the data also changes the data.  Customers don’t arrive in F&I already familiar with the products, according to research from IHS.

Consider our service contract example.  Your statistics tell you to present it only for customers keeping their vehicle more than five years.  That now becomes a self-fulfilling prophecy.  Going forward, your database will fill up with service contract customers who meet that criterion because you never show it to anyone else.

You can never know when a customer is going to buy some random product.  This is why F&I trainers tell you to “present every product to every customer, every time.”  There is a technical fix, which is to segregate your sample data (also known as “training data” for machine learning) from your result data.  The system must flag deals where prediction was used to restrict the presentation, and never use these deals for statistics.

Doesn’t that mean you’ll run out of raw data?  It might, if you don’t have a rich supply.  One way to maintain fresh training data is periodically to abandon prediction, show all products, let the F&I manager do his job, and then put that deal into the pool of training data.

Customers complete a thinly disguised “survey” while they’re waiting on F&I, which their software uses to discern which products to offer and which ones the customer is most likely to buy based upon their responses.

Regulatory compliance is another reason F&I trainers tell you to present every product every time.  Try telling the CFPB that “my statistics told me not to present GAP on this deal.”  There’s not a technical fix for that.

One motivation for the interview approach, versus a four-column menu, is that it’s better suited to form factors like mobile and chat.  This is a strong inducement for the online selling sites.  In the F&I suite, however, the arguments are not as strong.  Trainers are uniformly against the idea that you can simply hand over the iPad and let it do the job for you.

No, I have not gone over to the Luddites.  This article offers advice to people developing (or evaluating) predictive selling systems, and most of the advantages accrue to the online people.  The “home court advantage” in the F&I suite is that you can do a four-column menu, and there is a professional there to present it.