Digital Root

Showing Numbers In Context

The Boston Globe demonstrates the value of showing numbers in context when building charts.  Often times, charts will casually depict numbers on different scales (e.g., Millions and Billions side by side) which obscures interpretation.

When creating visualizations, try to keep values on a relative scale whenever possible. If not, try to create a common measurement for context (as Shaq is below).  And when all else fails, try using a natural log version to illustrate the relative shape of the data.

 

But Her Chi-Square Is Really Cute

Repost from Andrew Gelman (Professor of Statistics @ Columbia University) :

Brian Mulford writes:

I [Mulford] ran across this blog post and found myself questioning the relevance of the test used.

I'd think Chi-Square would be inappropriate for trying to measure significance of choice in the manner presented here; irrespective of the cute hamster. Since this is a common test for marketers and website developers - I'd be interested in which techniques you might suggest?

For tests of this nature, I typically measure a variety of variables (image placement, size, type, page speed, "page feel" as expressed in a factor, etc) and use LOGIT, Cluster and possibly a simple Bayesian model to determine which variables were most significant (chosen). Pearson Chi-squared may be used to express relationships between variables and outcome but I've typically not used it to simply judge a 0/1 choice as statistically significant or not.

My reply:

I like the decision-theoretic way that the blogger (Jason Cohen, according to the webpage) starts:

If you wait too long between tests, you're wasting time. If you don't wait long enough for statistically conclusive results, you might think a variant is better and use that false assumption to create a new variant, and so forth, all on a wild goose chase! That's not just a waste of time, it also prevents you from doing the correct thing, which is to come up with completely new text to test against.

But I agree with Mulford that chi-square is not the way to go. I'd prefer a direct inference on the difference in proportions. Take that inference--the point estimate and its uncertainty, estimated using the usual (y+1)/(n+2) formulas--and then carry that uncertainty into your decision making. Balance costs and benefits, and all that.

Moving forward, you're probably making lots and lots of this sort of comparison, so put it into a hierarchical model and you'll get inferences that are more reasonable and more precise.

But . . . who knows? Maybe Cohen's advice is a net plus. Ignoring the chi-square stuff, the key message I take away from the above-linked blog is that, with small samples, randomness can be huge. And that's an important lesson--really, one of the key concepts in statistics. Don't overreact to small samples. If the silly old chi-square test is your way of coming to this conclusion, that's not so bad.

 

 

 

Simple Retargeting Beats Out Statistical-Models

Comscore published a report on the effective payoff for varying display campaigns earlier this week.

Unlike many ad-research reports, Comscore reviewed a variety of campaigns and placement strategies from broad network buys, to re-targeting and behavioral targeting and how they impacted overall conversion. 

Included in their analysis:

•  Impact from a search perspective – did display formats drive post-impression activity?

• Life-time impact of conversion – did a responder not only visit the website, but did they become a customer and continue beyond the first purchase?

•  Cost Performance - Relative cost/reach of each type of placement.

 

 Findings:

Surprisingly, the contextual, optimized and behavioral strategies had similar results to run-of-network and other less ‘intelligent’ methods. However re-targeting had an average performance 200%-300% better than every other placement, but obviously is  gated by reach (people are only re-targeted if they visited the site first – thus a smaller audience).

   

In my experience optimization can work well within a site, creating a contextual and meaningful interaction (think Netflix or Amazon) but less so for ad campaigns.  Re-targeting can be very powerful for high-intent consumers who browsed and left or abandoned a cart.  Behavioral can work, but I suspect the conditions are too variable campaign to campaign to be effectively measured in a study like this one.

 

Conclusions :

• While these finding’s aren’t applicable to every situation, they illustrate the diminishing returns on overly sophisticated display placements. In the end, intelligent media planning, mixed strategy and common-sense re-targeting garner the biggest payoffs. 

•  Search and post-impression continue to be undervalued but significant as a metric for behavior and campaign ROI.

 

Patterns in Thinking

I read an interesting blog post from Without Geometry Life is Pointless regarding the notion of problem solving and thinking.

The list, which references the works of many authors is a good example of techniques used in many disciplines.  I've personally used many of these in Marketing, Analytics and Computer Science, but suspect these will be useful in any applied context.

In particular, I find the order of these elements are important.  Start with bounding, then work backwards, and develop tests to validate your findings.

You'll notice that none of the entries is "read up or Google answers".  Thinking in abstract is often times best done entirely without external influence. Its astounding how often you'll find holes and opportunities in subjects that are well documented.

1.    Pattern Sniff

A.     On the lookout for patterns
“Ok.  We’ve been working on this staircase problem and it seems that you can’t write perfect squares as a sum of consecutive whole numbers.”

B.     On the lookout for shortcuts
“It would be nice if there were a faster way to do 57x34 than adding 57 to itself 34 times. Think we can find a way?”

2.    Experiment, Guess and Conjecture

A.     Can begin to work on a problem independently
“I’m not sure how to solve this problem, but I’m confident I can make some progress.”

B.     Estimates
“Without doing any calculations, I’m guessing that it will take him 30 seconds to walk up the down escalator.”

C.     Conjectures
“Based on my work, I think the following is true.”

D.    Healthy skepticism of experimental results
“Boy, it sure seems like this 4, 2, 1 thing always repeats but we don’t have a proof yet.”

E.     Determines lower and upper bounds
“I know it will take the people at least 10 minutes to cross the bridge because the 10 minute soldier has to cross the bridge.  I also found a solution that takes 19 minutes so I know the final answer is somewhere between 10 and 19 minutes.”

F.     Looks at small or large cases to find and test conjectures
“I made a table of the first 5 cases and I think I see a pattern.  I’m going to see if this pattern holds for the 100^th case.”

G.     Is thoughtful and purposeful about which case(s) to explore

H.    Keeps all but one variable fixed

“So I’m exploring the equation y=mx+b and I’m wondering how the graph changes as m and b change.  For now, I’m going to set m to 1 and just look at how the graph changes when I change b.”

I.      Varies parameters in regular and useful ways

(Even/odd example)

J.      Works backwards (guesses at a solution and see if it makes sense)

3.    Organize & Simplify

A.     Records results in a useful way
“I’m going to make a table.”

B.     Process, solutions and answers are detailed and easy to follow

C.     Looks at information about the problem or solution in different ways

D.    Determine whether the problem can be broken up into simpler pieces
“I think I can solve this problem by solving these other 2 simpler problems.”

E.     Considers the form of data (deciding when, for example, 1+2 is more helpful than 3)

“I’m going to leave my fraction as 6/36 because the 6 represents the number of ways you can roll a 7 with 2 standard dice and the 36 represents the total number of rolls.”

F.     Uses parity and other methods to simplify and classify cases
“Next time we play 21 Nim I’m going to keep track of whether the running sum is a multiple of 3, one more than a multiple of 3, or 2 more than a multiple of 3.”

4.    Describe

A.     Verbal/visual articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

B.     Written articulation of thoughts, results, conjectures, arguments, process, proofs, questions, opinions

C.     Can explain both how and why

“The algorithm for dividing fractions is simple.  Now I just need to work on making sense why this works.” 

D.    Creates precise problems

E.     Invents notation and language when helpful
“For the sugar weighing problem, I don’t want to have to write out every solution in words so I’m going to let the symbol 3w~3s stand for the act of putting the 3 pound weight on one side of the balance scale, measuring out 3 pounds of sugar on the other side of the scale, and then setting aside the sugar.”

F.     Ensures that this invented notation and language is precise
“I need to be careful that I am differentiating between sugar that I am measuring and sugar I am using as a weight.”

5.     Tinker & Invent
         A.   Creates variations

B.     Looks at simpler examples when necessary

C.     Looks at more complicated examples when necessary

D.    Creates extensions and generalizations

E.     Creates algorithms for doing things

F.     Looks at statements that are generally false to see when they are true

G.     Creates and alters rules of a game

H.    Creates axioms for a mathematical structure

I.      Invents new mathematical systems that are innovative, but not arbitrary

6.    Visualize

A.     Uses pictures to describe and solve problems

B.     Uses manipulatives to describe and solve problems

C.     Reasons about shapes

“I see how this structure is made.”

D.    Visualizes data

E.     Looks for symmetry

F.     Visualizes relationships (using tools such as Venn diagrams and graphs)

G.     Vizualizes processes (using tools such as graphic organizers)

H.    Visualizes changes

I.      Visualizes calculations (such as doing arithmetic mentally)

7.    Strategize, Reason and Prove

A.     Moves from data driven conjectures to theory based conjectures

B.     Tests conjectures using thoughtful cases

C.     Proves conjectures using reasoning

D.    Looks for mistakes or holes in proofs

8.    Connect

A.     Articulates how different skills and concepts are related

B.     Applies old skills and concepts to new material

C.     Describes problems and solutions using multiple representations

D.    Finds and exploits similarities between problems (invariants, isomorphisms)

9.    Listen & Collaborate

A.     Respectful to others when they are talking

B.     Asks for clarification when necessary

C.     Challenges others in a respectful way when there is disagreement

D.    Participates

E.     Ensures that everyone else has the chance to participate

F.     Willing to ask questions when needed

G.     Willing to help others when needed

H.    Shares work in an equitable way

I.      Gives others the opportunity to have “aha” moments

c    Contextualize, Reflect and Persevere

A.     Determines givens

B.     Eliminates unimportant information

C.     Makes and articulates reasonable assumptions

D.    Determines if answer is reasonable

E.     Determines if there are additional or easier explanations

F.     Continuously reflects on process

G.     Works on one problem for greater and greater lengths of time

H.    Spends more and more time stuck without giving up

Using Search To Predict Hits

There are over 100MM internet searches performed daily. Keywords, phrases and numbers are collected and stored within the major search providers vast databases.

What if that search data could be used to predict trends before they happen?

In 2008 and 2009 Google researchers showed some success in trending flu cases using search data alone. However,  most studies since have been inconclusive, or statistically irrelevant.

A few researchers working with Yahoo! recently leapfrogged prior studies by formulating a set of statistical models that were comparable to baseline models.  The intent was to see if by just using search data, could they get close to well-informed models.  The first step was to observe the impact of time on the models. In the case of Movies, Games and Music, build up towards release showed significant traffic spikes, signaling consumer intent (or interest).

The conclusion of the studies showed that search data alone - automatically generated models predicted revenue/success for  Games, Music, and Movie releases with an overall range of 74%-93% accuracy. 

The regression (log) scatterplot below depicts the model visually. The dots represent predictive values, the line actual baseline. The closer the dots are to the line, the more predictive the regression is said to be. In this case, with Theatrical releases, the model had a near 80% accuracy for success based solely on search volume data.

 

The same models can't be used universally for all topics, products or brands. And these models may be improved upon by adding more variables, or using traditional methods of measurement.

The big deal here is that using something as simple as query data - future outcomes are meaningfully predictable. The reduction in cost and time in rendering the models is staggering.

In addition to search, mining unstructured internet data, social media and sentiment often reveals trends that weren't visible through traditional studies. When combined with Direct-Survey & Consumer Insight analysis, the planning capacity grows dramatically and informs marketers in a whole new, and often surprising way.

Does Targeted Advertising Erode Profits?

Taken at face value the report from MIT Sloan Assistant Professor Alessandro Bonatti contradicts what every computational advertising network has been saying for years.

"...as technology keeps improving, more and more web sites can sell very narrow products to very specialized audiences... with lots of people targeting the same audience the profits to be made through specialized advertising become more and more spread out... instead of competing for one large pool... you will have price war in each targeted segment as the slice gets more and more narrow."  Bonatti concludes that, "... the better the technology, the lower the profits for advertisers... "

There's more to this story than a clumsy application of the economic theory of crowding out.

A key flaw in the report is that CPM is used as the benchmark for profitability. While CPM was the preeminent metric in the days before digital, where there was little or no ability to measure beyond the impression. Not to mention the inherent elasticity of CPM as a representation of cost and reach.

Digital advertising and marketing has enabled the use of Cost-Per-Action (CPA) and full life-cycle tracking of events from consumer impression through to conversion and re-engagement.

A second point is the focus on long-tail consumables. When applying the notion of crowding and diminishing value (or utility) it's important to qualify the volume and ratio of units considered.

In the long tail, low volume goods become marketable due to targeting. Without targeting, the cost and overhead associated with selling the items is unsustainable and generally removed from consumer choice.

In the digital world, the leveraging mass scale markets and digital targeting pairs highly distributed consumers and sparsely available goods together in a cost effective manner. Essentially bridging the production gap of industrialization expressed by economists for the last two centuries.

The observation of impressions and clicks representing a fraction of outcome behaviors is also worth noting.  More sophisticated research has revealed that impressions have a profound effect on sales, awareness and consideration - even if they don't result in a click.

While the report has merit for what it measures - CPM in a long tail environment, it is missing the point of its own supposition; Targeted Advertising reflects the changing of what measures matter and creation of a new class of utility.

The Science of Computational Advertising

Yahoo! Chief Scientists overview the science of Computational Advertising to Stanford's Management of Science & Engineering department.

The series of lectures were made public by Stanford and is surprisingly cogent and approachable for those unfamiliar with predictive statistics or interested in the mechanics of digital advertising.

  

What is Computational Advertising?

www.stanford.edu/class/msande239/lectures/lecture-01.pdf

  

Bidding and Price Mechanics for Ad Networks

(Use of Nash's famous Equilibrium Theory - Not the Christian Bale Movie) 

www.stanford.edu/class/msande239/lectures/lecture-02.pdf

Paid Search Placement Mechanics

How do they know which ad to put above another in search ads? www.stanford.edu/class/msande239/lectures/lecture-03.pdf

Sponsered Search - Part I 

www.stanford.edu/class/msande239/lectures/lecture-04.pdf

Sponsored Search - Part II 

www.stanford.edu/class/msande239/lectures/lecture-05.pdf

What Happens After The Click

www.stanford.edu/class/msande239/lectures/lecture-06.pdf

Page Ranking Method Comparison

www.stanford.edu/class/msande239/lectures/guest-lecture-ag.pdf

Targeting And Display Ads

www.stanford.edu/class/msande239/lectures/lecture-08.pdf

Data: Anonymous or Useful; Never Both

Anyone who works with consumer data understands that when sharing or using private data, individual identifiers must be "anonymized" for privacy concerns.

In the US this is typically accomplished by removing the Social Security number, name, address and account identifier.  Data elements such as ZIP code, birth date and gender are considered legally acceptable for sharing or publishing.

However, a growing body of research is proving that even generic identifiers present a serious risk.  For instance:

• In 2008, two UT researched reverse engineered Netflix's public data  using only a name and three movie ratings to identify every movie that individual rented.
• In 2006 AOL released search data for 650,000 users, but only after removing user names, IP addresses and anything considered identifying.  Within days, an individual was identified by nothing more than a generic users ID and search content such as “‘landscapers in Lilburn, GA,’.  The seemingly anonymous data was tracked down to Thelma Arnold, a 62-year-old widow from Lilburn, Georgia who  confirmed that searches such as “numb fingers”, “60 single men”, and “dog that urinates on everything." belonged to her.

With the growing body of publicly available data sets from sources such as the US Government  (here, here and here),  Amazon, or dataset sites, the risk of identification through cross reference increases dramatically.

True, the techniques used today to identity individuals is somewhat brute-force and manualized, the growing body of cross-referencing data will lower the bar over the next few years. And remember, the internet never forgets.

For those responsible for data sharing here's a few points of advice beyond standard anonymization techniques:

• Group age by ranges instead of providing DOB. E.g., 18-24 or even reported age.
• Instead of providing ZIP codes, use US Census FIPS codes to identify broader geographic areas.
• Avoid including gender where possible.

For consumers, be selective of where you record your information online. Its not possible today to exist online without providing at least some of the data, but for elective services such as facebook or youtube, approximate your DOB or Zip lest the organization share it down the line.

Its Hip To Be Quadratic

Around the turn of the century, a gifted linguist and student named Venkatraman Shastri abruptly left his studies to pursue spiritual enlightenment. Being one of the foremost scholars in Sanskrit at the time, he believed hidden truths lay dormant in the oldest of religious texts; the Vedas. 

The Vedas were originated between 1500 and 500 BC or possibly earlier. Roughly translating to "Wisdom", the Vedas are a culmination of India's historical and spiritual development.

After years of research and devotion to spiritual interests, Venkatraman claimed to have found an ancient algebraic language among sets of rhyming syllables within the verses. By translating the verses into a practical notion he discovered they touched everything from addition and subtraction to trigonometry.  There was even a sonnet of praise that encoded the calculation and value of pi/10 to 32 decimal places, within its rhythmic syllables.

Prior to his death, Venkatraman (having developed as an international spiritual leader and titled Swami) wrote a manuscript describing what he called the "16 Sutra's"; The body of an ancient Vedic math system.

While controversial, most agree that the system offers a powerful, yet simple structure for many common math problems that you encounter on a daily basis.

As an introduction to this system, an examination of Quadratic equations (quadratus is Latin for "square") serves as a good demonstration.

                                                                                                                                              

Pop Quiz:  Without the use of a calculator, take no more than 7 seconds on each problem:

43² =

 21² = 

57²  =

35² =

75² =

If you're anything like me, you felt the urge to open up Calc after the first few seconds.  The Vedic system provides a simple pattern to quickly solve these problems without having to memorize any math beyond basic 9x9 tables or require the use of your computer.

                                                                                                                                              

Demonstration:  We'll use 2-digit numbers (e.g., 43² or 35²) for simplicity's sake.  Any number can follow the pattern but involves continuing the steps beyond the practicality of the demonstration. 

Step 1: Multiply the last number by itself. Example, 3*3=9.

Step 2: Multiply the two numbers by each other, then by 2. If the previous step produce any number larger than 9, add the first integer to the sum.  Example: 4*3*2 = 24. 

Step 3: Multiply the first number by itself. Again, if the previous step produced any number greater than 9, add the first integer to the sum.  Example: 4*4 = 16.  24 is bigger than 9, so add the 2 to the sum of 16 - making it 18. 

Step 4: The answer is 1849

At first glance this may not seem easier - in fact it may be confusing. The following graphs demonstrate the concept visually to clarify the steps:

 

Each number in the equation (43²) is broken down into three simple steps. As values are calculated, anything larger than 9 carries to the next step, building the answer as you go.

Lets look at the example of 21². In this instance all of the steps produce numbers under 9, so there isn't anything to carry forward - just write down each number to build the answer. 

 

Want to try one more to be sure? Try 57²:

You'll notice above that the second and third steps required an extra 4 and 7 to be carried forward in order to reach the final answer. 

Now that you have the idea - here's a bonus for numbers ending in 5.  You can skip the first two steps and jump right to the third, almost.  Simply take the first number and multiply it by one more and follow it with 25.  Example: 35² - 3*4=12, then follow it with 25 for an answer of 1225.  Any number you want to square that ends in 5, always calculates to n,25.   

 

Try 75²  to get the hang of it:  Start with multiplying 7*8=56, then write 25 for your answer. 

This simple example of applied Vedic mathematics illustrates the inherent efficiency of the system. Using small logic steps to break down a larger problem, making it simple to tackle numbers and problems that we'd given-up trying to master. With modest practice using Vedic math, most of these problems become solvable only using your mind.

Could Pythagoras Hack Your Password?

“A Mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas.”- Magnus Gosta Mittag-Leffler.

What does a 1950’s-era book keeper, encryption and MySpace have in common? Aside from being topics of discussion on your worst-date-ever, they’re connected by a form of mathematics; Modulo Arithmetic (or Modular math if you prefer). In simple terms, a set of numbers are “Modular” if two quantities are considered congruent or equal. Digital Roots, the adding of a numbers digits until only a single digit remains, are a form of modular math (and in turn; Number Theory). Digital roots have been recorded for thousands of years, formalized by Pythagoras in 530BCE and even earlier in Indian Vedic math.

Modulo and digital roots are mentioned here as an interesting illustration of how underlying patterns exist behind the numbers we work with every day. Take the following for example:

Modulo 9 and Digital Roots Are Equal

1327 / 9 = 14.4

The Digital Root of 1327 is 4

(1+3+2+7=13, 1+3=4)

The remainders being equivalent (4) makes them “Modular” to one another. Any positive number divided by 9 will have a remainder that is equivalent to its digital root. Building on that simple observation, look at a slightly more complex pattern that emerges when applying digital roots to square numbers:

Digital Roots of Square Numbers

Square	Value	Digital Root
1²	1	1
2²	4	4
3²	9	9
4²	16	1+6 = 7
5²	25	2+5 = 7
6²	36	3+6 = 9
7²	49	4+9=13=1+3=4
8²	64	6+4=10, 1+0=1
9²	81	8+1=9
10²	100	1+0+0=1

The pattern of 1497-7941-9 (a nice little numerical palindrome you may notice) repeats for as long as you wish to sum up squares. This among the many patterns that are based digital roots; demonstrate an underlying symmetry in numbers. 

Digital roots can also reveal short-cuts in everyday math. For example, book keepers knew a trick to verify addition well before computers were commonplace. Try it for yourself! To verify the answer in an addition exercise, obtain the root of all the numbers to be added, and compare it to the digital root of the answer – if the roots match, your addition was correct.

For Example:

1327 = 4

+4256 =8 (4+8=12, 1+2=3)

----------------------------------

5583 = 3

More complicated uses of digital roots become predictive, enabling modern tools such as encryption (primes derived from roots), the density of concrete (roots of concrete measures relate to stress capacity formulae), or even databases that validate vast sums of information instantly (checksum algorithms are often derived from modular math). Some research has discovered that forms of encryption may be “cracked” by the use of roots and primes to unscramble your message, without having access to the password.

In our world of computing, it may not seem that important, but these numerical facts were viewed as evidence of divinity thousands of years ago. The reverences for these patterns are found modern and ancient in art forms. Pairing roots with images and colors, beautiful repeating patterns take shape, somehow collectively appealing in its familiarity.

This brief review of roots is meant to serve as small introduction to the breadth and beauty in numerical forms. But isn’t what’s important here. It’s the idea of seeing the underlying pattern in everyday objects and situations – in what we take for granted. 

In the end, math isn’t really about numbers, it’s about ideas. Pythagoras may not have been able to crack your password but I suspect if he were presented with the problem he’d have ideas on its attempt. What about you, can you find inspiration in the symmetry of numbers?