mathbabe

This week our guest lecturer for the Columbia Data Science class was Brian Dalessandro. Brian works at Media6Degrees as a VP of Data Science, and he’s super active in the research community. He’s also served as co-chair of the KDD competition.

Before Brian started, Rachel threw us a couple of delicious data science tidbits.

The Process of Data Science

First we have the Real World. Inside the Real World we have:

• Users using Google+
• People competing in the Olympics
• Spammers sending email

From this we draw raw data, e.g. logs, all the olympics records, or Enron employee emails. We want to process this to make it clean for analysis. We use pipelines of data munging, joining, scraping, wrangling or whatever you want to call it and we use tools such as:

• python
• shell scripts
• R
• SQL

We eventually get the data down to a nice format, say something with columns:

name event year gender event time

Note: this is where you typically start in a standard statistics class. But it’s not where we typically start in the real world.

Once you have this clean data set, you should be doing some kind of exploratory data analysis (EDA); if you don’t really know what I’m talking about then look at Rachel’s recent blog post on the subject. You may realize that it isn’t actually clean.

Next, you decide to apply some algorithm you learned somewhere:

• k-nearest neighbor
• regression
• Naive Bayes
• (something else),

depending on the type of problem you’re trying to solve:

• classification
• prediction
• description

You then:

• interpret
• visualize
• report
• communicate

At the end you have a “data product”, e.g. a spam classifier.

K-means

So far we’ve only seen supervised learning. K-means is the first unsupervised learning technique we’ll look into. Say you have data at the user level:

• G+ data
• survey data
• medical data
• SAT scores

Assume each row of your data set corresponds to a person, say each row corresponds to information about the user as follows:

age gender income Geo=state household size

Your goal is to segment them, otherwise known as stratify, or group, or cluster. Why? For example:

• you might want to give different users different experiences. Marketing often does this.
• you might have a model that works better for specific groups
• hierarchical modeling in statistics does something like this.

One possibility is to choose the groups yourself. Bucket users using homemade thresholds. Like by age, 20-24, 25-30, etc. or by income. In fact, say you did this, by age, gender, state, income, marital status. You may have 10 age buckets, 2 gender buckets, and so on, which would result in 10x2x50x10x3 = 30,000 possible bins, which is big.

You can picture a five dimensional space with buckets along each axis, and each user would then live in one of those 30,000 five-dimensional cells. You wouldn’t want 30,000 marketing campaigns so you’d have to bin the bins somewhat.

Wait, what if you want to use an algorithm instead where you could decide on the number of bins? K-means is a “clustering algorithm”, and k is the number of groups. You pick k, a hyper parameter.

2-d version

Say you have users with #clicks, #impressions (or age and income – anything with just two numerical parameters). Then k-means looks for clusters on the 2-d plane. Here’s a stolen and simplistic picture that illustrates what this might look like:

The general algorithm is just the same picture but generalized to d dimensions, where d is the number of features for each data point.

Here’s the actual algorithm:

• randomly pick K centroids
• assign data to closest centroid.
• move the centroids to the average location of the users assigned to it
• repeat until the assignments don’t change

It’s up to you to interpret if there’s a natural way to describe these groups.

This is unsupervised learning and it has issues:

• choosing an optimal k is also a problem although , where n is number of data points.
• convergence issues – the solution can fail to exist (the configurations can fall into a loop) or “wrong”
• but it’s also fast
• interpretability can be a problem – sometimes the answer isn’t useful
• in spite of this, there are broad applications in marketing, computer vision (partition an image), or as a starting point for other models.

One common tool we use a lot in our systems is logistic regression.

Thought Experiment

Brian now asked us the following:

How would data science differ if we had a “grand unified theory of everything”?

He gave us some thoughts:

• Would we even need data science?
• Theory offers us a symbolic explanation of how the world works.
• What’s the difference between physics and data science?
• Is it just accuracy? After all, Newton wasn’t completely precise, but was pretty close.

If you think of the sciences as a continuum, where physics is all the way on the right, and as you go left, you get more chaotic, then where is economics on this spectrum? Marketing? Finance? As we go left, we’re adding randomness (and as a clever student points out, salary as well).

Bottomline: if we could model this data science stuff like we know how to model physics, we’d know when people will click on what ad. The real world isn’t this understood, nor do we expect to be able to in the future.

Does “data science” deserve the word “science” in its name? Here’s why maybe the answer is yes.

We always have more than one model, and our models are always changing.

The art in data science is this: translating the problem into the language of data science

The science in data science is this: given raw data, constraints and a problem statement, you have an infinite set of models to choose from, with which you will use to maximize performance on some evaluation metric, that you will have to specify. Every design choice you make can be formulated as an hypothesis, upon which you will use rigorous testing and experimentation to either validate or refute.

Never underestimate the power of creativity: usually people have vision but no method. As the data scientist, you have to turn it into a model within the operational constraints. You need to optimize a metric that you get to define. Moreover, you do this with a scientific method, in the following sense.

Namely, you hold onto your existing best performer, and once you have a new idea to prototype, then you set up an experiment wherein the two best models compete. You therefore have a continuous scientific experiment, and in that sense you can justify it as a science.

Classifiers

Given

• data
• a problem, and
• constraints,

we need to determine:

• a classifier,
• an optimization method,
• a loss function,
• features, and
• an evaluation metric.

Today we will focus on the process of choosing a classifier.

Classification involves mapping your data points into a finite set of labels or the probability of a given label or labels. Examples of when you’d want to use classification:

• will someone click on this ad?
• what number is this?
• what is this news article about?
• is this spam?
• is this pill good for headaches?

From now on we’ll talk about binary classification only (0 or 1).

Examples of classification algorithms:

• decision tree
• random forests
• naive bayes
• k-nearest neighbors
• logistic regression
• support vector machines
• neural networks

Which one should we use?

One possibility is to try them all, and choose the best performer. This is fine if you have no constraints or if you ignore constraints. But usually constraints are a big deal – you might have tons of data or not much time or both.

If I need to update 500 models a day, I do need to care about runtime. these end up being bidding decisions. Some algorithms are slow – k-nearest neighbors for example. Linear models, by contrast, are very fast.

One under-appreciated constraint of a data scientist is this: your own understanding of the algorithm.

Ask yourself carefully, do you understand it for real? Really? Admit it if you don’t. You don’t have to be a master of every algorithm to be a good data scientist. The truth is, getting the “best-fit” of an algorithm often requires intimate knowledge of said algorithm. Sometimes you need to tweak an algorithm to make it fit your data. A common mistake for people not completely familiar with an algorithm is to overfit.

Another common constraint: interpretability. You often need to be able to interpret your model, for the sake of the business for example. Decision trees are very easy to interpret. Random forests, on the other hand, not so much, even though it’s almost the same thing, but can take exponentially longer to explain in full. If you don’t have 15 years to spend understanding a result, you may be willing to give up some accuracy in order to have it easy to understand.

Note that credit cards have to be able to explain their models by law so decision trees make more sense than random forests.

How about scalability? In general, there are three things you have to keep in mind when considering scalability:

• learning time: how much time does it take to train the model?
• scoring time: how much time does it take to give a new user a score once the model is in production?
• model storage: how much memory does the production model use up?

Here’s a useful paper to look at when comparing models: “An Empirical Comparison of Supervised Learning Algorithms”, from which we learn:

• Simpler models are more interpretable but aren’t as good performers.
• The question of which algorithm works best is problem dependent
• It’s also constraint dependent

At M6D, we need to match clients (advertising companies) to individual users. We have logged the sites they have visited on the internet. Different sites collect this information for us. We don’t look at the contents of the page – we take the url and hash it into some random string and then we have, say, the following data about a user we call “u”:

u = <xyz, 123, sdqwe, 13ms>

This means u visited 4 sites and their urls hashed to the above strings. Recall last week we learned spam classifier where the features are words. We aren’t looking at the meaning of the words. So the might as well be strings.

At the end of the day we build a giant matrix whose columns correspond to sites and whose rows correspond to users, and there’s a “1” if that user went to that site.

To make this a classifier, we also need to associate the behavior “clicked on a shoe ad”. So, a label.

Once we’ve labeled as above, this looks just like spam classification. We can now rely on well-established methods developed for spam classification – reduction to a previously solved problem.

Logistic Regression

We have three core problems as data scientists at M6D:

• feature engineering,
• user level conversion prediction,
• bidding.

We will focus on the second. We use logistic regression– it’s highly scalable and works great for binary outcomes.

What if you wanted to do something else? You could simply find a threshold so that, above you get 1, below you get 0. Or you could use a linear model like linear regression, but then you’d need to cut off below 0 or above 1.

What’s better: fit a function that is bounded in side [0,1]. For example, the logit function

wanna estimate

.

To make this a linear model in the outcomes , we take the log of the odds ratio:

The parameter keeps shape of the logit curve but shifts it back and forth. To interpret further, consider what we call the base rate, the unconditional probability of “1” (so, in the case of ads, the base rate would correspond to the click-through rate, i.e. the overall tendency for people to click on ads; this is typically on the order of 1%).

If you had no information except the base rate, the average prediction would be just that. In a logistical regression, defines the base rate. Specifically, the base rate is approximately equal to

The slope defines the slope of the logit function. Note in general it’s a vector which is as long as the number of features we are using for each data point.

Our immediate modeling goal is to use our training data to find the best choices for and We will use a maximum likelihood estimation or convex optimization to achieve this; we can’t just use derivatives and vector calculus like we did with linear regression because it’s a complicated function of our data.

The likelihood function is defined by:

where we are assuming the data points are independent and where

We then search for the parameters that maximize this having observed our data:

The probability of a single observation is

where is the modeled probability of a “1” for the binary outcome $Y_i.$ Taking the product of all of these we get our likelihood function which we want to maximize.

Similar to last week, we now take the log and get something convex, so it has to have a global maximum. Finally, we use numerical techniques to find it, which essentially follow the gradient like Newton’s method from calculus. Computer programs can do this pretty well. These algorithms depend on a step size, which we will need to adjust as we get closer to the global max or min – there’s an art to this piece of numerical optimization as well. Each step of the algorithm looks something like this:

where remember we are actually optimizing our parameters and to maximize the (log) likelihood function, so the you see above is really a vector of s and the function corresponds to our

“Flavors” of Logistic Regression for convex optimization.

The Newton’s method we described above is also called Iterative Reweighted Least Squares. It uses the curvature of log-likelihood to choose appropriate step direction. The actual calculation involves the Hessian matrix, and in particular requires its inversion, which is a kxk matrix. This is bad when there’s lots of features, as in 10,000 or something. Typically we don’t have that many features but it’s not impossible.

Another possible method to maximize our likelihood or log likelihood is called Stochastic Gradient Descent. It approximates gradient using a single observation at a time. The algorithm updates the current best-fit parameters each time it sees a new data point. The good news is that there’s no big matrix inversion, and it works well with huge data and with sparse features; it’s a big deal in Mahout and Vowpal Wabbit. The bad news is it’s not such a great optimizer and it’s very dependent on step size.

Evaluation

We generally use different evaluation metrics for different kind of models.

First, for ranking models, where we just want to know a relative rank versus and absolute score, we’d look to one of:

• area under the receiver operating curve (AUC)
• area under the cumulative lift curve

Second, for classification models, we’d look at the following metrics:

• lift: how much more people are buying or clicking because of a model
• accuracy: how often the correct outcome is being predicted
• f-score
• precision
• recall

Finally, for density estimation, where we need to know an actual probability rather than a relative score, we’d look to:

• mean absolute error
• mean squared error

In general it’s hard to compare lift curves but you can compare AUC (area under the receiver operator curve) – they are “base rate invariant.” In other words if you bring the click-through rate from 1% to 2%, that’s 100% lift but if you bring it from 4% to 7% that’s less lift but more effect. AUC does a better job in such a situation when you want to compare.

Density estimation tests tell you how well are you fitting for conditional probability. In advertising, this may arise if you have a situation where each ad impression costs $c and for each conversion you receive $q. You will want to target every conversion that has a positive expected value, i.e. whenever

But to do this you need to make sure the probability estimate on the left is accurate, which in this case means something like the mean squared error of the estimator is small. Note a model can give you good rankings but bad P estimates.

Similarly, features that rank highly on AUC don’t necessarily rank well with respect to mean absolute error. So feature selection, as well as your evaluation method, is completely context-driven.

In the third week of the Columbia Data Science course, our guest lecturer was Jake Hofman. Jake is at Microsoft Research after recently leaving Yahoo! Research. He got a Ph.D. in physics at Columbia and taught a fantastic course on modeling last semester at Columbia.

After introducing himself, Jake drew up his “data science profile;” turns out he is an expert on a category that he created called “data wrangling.” He confesses that he doesn’t know if he spends so much time on it because he’s good at it or because he’s bad at it.

Thought Experiment: Learning by Example

Jake had us look at a bunch of text. What is it? After some time we describe each row as the subject and first line of an email in Jake’s inbox. We notice the bottom half of the rows of text looks like spam.

Now Jake asks us, how did you figure this out? Can you write code to automate the spam filter that your brain is?

Some ideas the students came up with:

• Any email is spam if it contains Viagra references. Jake: this will work if they don’t modify the word.
• Maybe something about the length of the subject?
• Exclamation points may point to spam. Jake: can’t just do that since “Yahoo!” would count.
• Jake: keep in mind spammers are smart. As soon as you make a move, they game your model. It would be great if we could get them to solve important problems.
• Should we use a probabilistic model? Jake: yes, that’s where we’re going.
• Should we use k-nearest neighbors? Should we use regression? Recall we learned about these techniques last week. Jake: neither. We’ll use Naive Bayes, which is somehow between the two.

Why is linear regression not going to work?

Say you make a feature for each lower case word that you see in any email and then we used R’s “lm function:”

lm(spam ~ word1 + word2 + …)

Wait, that’s too many variables compared to observations! We have on the order of 10,000 emails with on the order of 100,000 words. This will definitely overfit. Technically, this corresponds to the fact that the matrix in the equation for linear regression is not invertible. Moreover, maybe can’t even store it because it’s so huge.

Maybe you could limit to top 10,000 words? Even so, that’s too many variables vs. observations to feel good about it.

Another thing to consider is that target is 0 or 1 (0 if not spam, 1 if spam), whereas you wouldn’t get a 0 or a 1 in actuality through using linear regression, you’d get a number. Of course you could choose a critical value so that above that we call it “1” and below we call it “0”. Next week we’ll do even better when we explore logistic regression, which is set up to model a binary response like this.

How about k-nearest neighbors?

To use k-nearest neighbors we would still need to choose features, probably corresponding to words, and you’d likely define the value of those features to be 0 or 1 depending on whether the word is present or not. This leads to a weird notion of “nearness”.

Again, with 10,000 emails and 100,000 words, we’ll encounter a problem: it’s not a singular matrix this time but rather that the space we’d be working in has too many dimensions. This means that, for example, it requires lots of computational work to even compute distances, but even that’s not the real problem.

The real problem is even more basic: even your nearest neighbors are really far away. this is called “the curse of dimensionality“. This problem makes for a poor algorithm.

Question: what if sharing a bunch of words doesn’t mean sentences are near each other in the sense of language? I can imagine two sentences with the same words but very different meanings.

Jake: it’s not as bad as it sounds like it might be – I’ll give you references at the end that partly explain why.

Aside: digit recognition

In this case k-nearest neighbors works well and moreover you can write it in a few lines of R.

Take your underlying representation apart pixel by pixel, say in a 16 x 16 grid of pixels, and measure how bright each pixel is. Unwrap the 16×16 grid and put it into a 256-dimensional space, which has a natural archimedean metric. Now apply the k-nearest neighbors algorithm.

Some notes:

• If you vary the number of neighbors, it changes the shape of the boundary and you can tune k to prevent overfitting.
• You can get 97% accuracy with a sufficiently large data set.
• Result can be viewed in a “confusion matrix“.

Naive Bayes

Question: You’re testing for a rare disease, with 1% of the population is infected. You have a highly sensitive and specific test:

• 99% of sick patients test positive
• 99% of healthy patients test negative

Given that a patient tests positive, what is the probability that the patient is actually sick?

Answer: Imagine you have 100×100 = 10,000 people. 100 are sick, 9,900 are healthy. 99 sick people test sick, and 99 healthy people do too. So if you test positive, you’re equally likely to be healthy or sick. So the answer is 50%.

Let’s do it again using fancy notation so we’ll feel smart:

Recall

and solve for

The denominator can be thought of as a “normalization constant;” we will often be able to avoid explicitly calculuating this. When we apply the above to our situation, we get:

This is called “Bayes’ Rule“. How do we use Bayes’ Rule to create a good spam filter? Think about it this way: if the word “Viagra” appears, this adds to the probability that the email is spam.

To see how this will work we will first focus on just one word at a time, which we generically call “word”. Then we have:

The right-hand side of the above is computable using enough pre-labeled data. If we refer to non-spam as “ham”, we only need and This is essentially a counting exercise.

Example: go online and download Enron emails. Awesome. We are building a spam filter on that – really this means we’re building a new spam filter on top of the spam filter that existed for the employees of Enron.

Jake has a quick and dirty shell script in bash which runs this. It downloads and unzips file, creates a folder. Each text file is an email. They put spam and ham in separate folders.

Jake uses “wc” to count the number of messages for one former Enron employee, for example. He sees 1500 spam, and 3672 ham. Using grep, he counts the number of instances of “meeting”:

grep -il meeting enron1/spam/*.txt | wc -l

This gives 153. This is one of the handful of computations we need to compute

Note we don’t need a fancy programming environment to get this done.

Next, we try:

• “money”: 80% chance of being spam.
• “viagra”: 100% chance.
• “enron”: 0% chance.

This illustrates overfitting; we are getting overconfident because of our biased data. It’s possible, in other words, to write an non-spam email with the word “viagra” as well as a spam email with the word “enron.”

Next, do it for all the words. Each document can be represented by a binary vector, whose jth entry is 1 or 0 depending whether jth word appears. Note this is a huge-ass vector, we will probably actually represent it with the indices of the words that actually do show up.

Here’s the model we use to estimate the probability that we’d see a given word vector given that we know it’s spam (or that it’s ham). We denote the document vector and the various entries , where the correspond to all the indices of in other words over all the words. For now we denote “is spam” by :

The theta here is the probability that an individual word is present in a spam email (we can assume separately and parallel-ly compute that for every word). Note we are modeling the words independently and we don’t count how many times they are present. That’s why this is called “Naive”.

Let’s take the log of both sides:

[It’s good to take the log because multiplying together tiny numbers can give us numerical problems.]

The term doesn’t depend on a given document, just the word, so let’s rename it Same with . The real weights that vary by document are the ‘s.

We can now use Bayes’ Rule to get an estimate of which is what we actually want. We can also get away with not computing all the terms if we only care whether it’s more likely to be spam or to be ham. Only the varying term needs to be computed.

Wait, this ends up looking like a linear regression! But instead of computing them by inverting a huge matrix, the weights come from the Naive Bayes’ algorithm.

This algorithm works pretty well and it’s “cheap to train” if you have pre-labeled data set to train on. Given a ton of emails, just count the words in spam and non-spam emails. If you get more training data you can easily increment your counts. In practice there’s a global model, which you personalize to individuals. Moreover, there are lots of hard-coded, cheap rules before an email gets put into a fancy and slow model.

Here are some references:

• “Idiot’s Bayes – not so stupid after all?” – the whole paper is about why it doesn’t suck, which is related to redundancies in language.
• “Naive Bayes at Forty: The Independence Assumption in Information“
• “Spam Filtering with Naive Bayes – Which Naive Bayes?“

Laplace Smoothing

Laplace Smoothing refers to the idea of replacing our straight-up estimate of the probability of seeing a given word in a spam email with something a bit fancier:

We might fix and for example, to prevents the possibility of getting 0 or 1 for a probability. Does this seem totally ad hoc? Well if we want to get fancy, we can see this as equivalent to having a prior and performing a maximal likelihood estimate.

If we denote by the maximal likelihood estimate, then we have:

In other words, we are asking the question, for what value of were the data D most probable? If we assume independent trials then we want to maximize

If you take the derivative, and set it to zero, we get

In other words, just what we had before. Now let’s add a prior. Denote by the maximum a posteriori likelihood:

This similarly asks the question, given the data I saw, which parameter is the most likely?

Use Bayes’ rule to get . This looks similar to above except for the , which is the “prior”. If I assume is of the form ; then we get the above, Laplacian smoothed version.

Sometimes and are called “pseudo counts”. They’re fancy but also simple. It’s up to the data scientist to set the values of these hyperparameters, and it gives us two knobs to tune. By contrast, k-nearest neighbors has one knob, namely k.

Note: In the last 5 years people have started using stochastic gradient methods to avoid the non-invertible (overfitting) matrix problem. Switching to logistic regression with stochastic gradient method helped a lot, and can account for correlations between words. Even so, Naive Bayes’ is pretty impressively good considering how simple it is.

Scraping the web: API’s

For the sake of this discussion, an API (application programming interface) is something websites provide to developers so they can download data from the website easily and in standard format. Usually the developer has to register and receive a “key”, which is something like a password. For example, the New York Times has an API here. Note that some websites limit what data you have access to through their API’s or how often you can ask for data without paying for it.

When you go this route, you often get back weird formats, sometimes in JSON, but there’s no standardization to this standardization, i.e. different websites give you different “standard” formats.

One way to get beyond this is to use Yahoo’s YQL language which allows you to go to the Yahoo! Developer Network and write SQL-like queries that interact with many of the common API’s on the common sites like this:

select * from flickr.photos.search where text=”Cat” and api_key=”lksdjflskjdfsldkfj” limit 10

The output is standard and I only have to parse this in python once.

What if you want data when there’s no API available?

Note: always check the terms and services of the website before scraping.

In this case you might want to use something like the Firebug extension for Firefox, you can “inspect the element” on any webpage, and Firebug allows you to grab the field inside the html. In fact it gives you access to the full html document so you can interact and edit. In this way you can see the html as a map of the page and Firebug is a kind of tourguide.

After locating the stuff you want inside the html, you can use curl, wget, grep, awk, perl, etc., to write a quick and dirty shell script to grab what you want, especially for a one-off grab. If you want to be more systematic you can also do this using python or R.

Other parsing tools you might want to look into:

• lynx and lynx –dump: good if you pine for the 1970’s. Oh wait, 1992. Whatever.
• Beautiful Soup: robust but kind of slow
• Mechanize (or here) super cool as well but doesn’t parse javascript.

Postscript: Image Classification

How do you determine if an image is a landscape or a headshot?

You either need to get someone to label these things, which is a lot of work, or you can grab lots of pictures from flickr and ask for photos that have already been tagged.

Represent each image with a binned RGB – (red green blue) intensity histogram. In other words, for each pixel, for each of red, green, and blue, which are the basic colors in pixels, you measure the intensity, which is a number between 0 and 255.

Then draw three histograms, one for each basic color, showing us how many pixels had which intensity. It’s better to do a binned histogram, so have counts of # pixels of intensity 0 – 51, etc. – in the end, for each picture, you have 15 numbers, corresponding to 3 colors and 5 bins per color. We are assuming every picture has the same number of pixels here.

Finally, use k-nearest neighbors to decide how much “blue” makes a landscape versus a headshot. We can tune the hyperparameters, which in this case are # of bins as well as k.