Sunday, June 18, 2017

Entropy


In my physics degree, we were asked to calculate the entropy of a chess board. Students smarter than me snorted at this silly exercise. Yet, entropy is just a statistical measure. You cannot measure it directly (there are no entropy-ometers) but it exists everywhere and can be used in odd places like machine learning (eg maximum entropy classifiers which "from all the models that fit our training data, selects the one which has the largest entropy").

Alternatively, you might want to find the configuration with the smallest entropy. An example is here where the quality of a clustering algorithm (k-means in this case) is assessed by looking at the entropy of the detected clusters. "As an external criteria, entropy uses external information — class labels in this case. Indeed, entropy measures the purity of the clusters with respect to the given class labels. Thus, if every cluster consists of objects with only a single class label, the entropy is 0. However, as the class labels of objects in a cluster become more varied, the entropy value increases."

For instance, say you are trying to find the parameters for an equation such that it best fits the data. "At the very least, you need to provide ... a score for each candidate parameter it tries. This score assignment is commonly called a cost function. The higher the cost, the worse the model parameter will be... The cost function is derived from the principle of maximum entropy." [1]

What is Entropy

I found this description of heads (H) and tails (T) from tossing a coin enlightening:

"If the frequency [f] of H is 0.5 and f(T) is 0.5, the entropy E, in bits per toss, is

-0.5 log2 0.5

for heads, and a similar value for tails. The values add up (in this case) to 1.0. The intuitive meaning of 1.0 (the Shannon entropy) is that a single coin toss conveys 1.0 bit of information.

Contrast this with the situation that prevails when using a "weighted" or unfair penny that lands heads-up 70% of the time. We know intuitively that tossing such a coin will produce less information because we can predict the outcome (heads), to a degree. Something that's predictable is uninformative. Shannon's equation gives

-0.7 log2 (0.7) = 0.3602

for heads and

-0.3 log2  (0.3) = 0.5211

for tails, for an entropy of 0.8813 bits per toss. In this case we can say that a toss is 11.87% [1.0 - 0.8813] redundant."

Here's another example:

X = 'a' with probability 0.5
    'b' with probability 0.25
    'c' with probability 0.125
    'd' with probability 0.125

The entropy of this configuration is:

H(X) = -0.5 log(0.5) - 0.25 log(0.25) - 0.125 log(0.125) - 0.125 log(0.125) = 1.75 bits

What does this actually mean? Well, if the average number of questions asked ("Is X 'a'? If not, is it 'b'? ...") then "the resulting expected number of binary questions required is 1.75" [2].

Derivation of entropy

Basically, we want entropy to be extensive. That is "parameters that scale with the system. In other words U(aS,aV,aN)=aU(S,V,N)".

So, if SX is the entropy of system X, then the combined entropy of two systems, A and B, would be:

SC = SA + SB

Second, we want it to be largest when all the states are equally probably. Let's call the function f then the average value is:

S = <f> = Σipif(pi)               Equation 1

Now, given two sub-systems, A and B, the system they make up C will have entropy:

SC = ΣiΣjpipjf(pi)f(pj)            Equation 2

that is, the we are summing probabilities over the states where A is in state i and B is in state j.

For equation 2 to conform to the form of equation 1, let's introduce the variable pij=pipj.Then:

SC = ΣiΣjpijf(pij)

For this to be true, f = C ln p since ln(ab) = ln(a) + ln(b).

This is the argument found here.

[1] Machine Learning with Tensor Flow.
[2] Elements of Information Theory.

Sunday, June 11, 2017

Scala Equality


The code:

Set(1) == List(1)

will always return false but will also always compile. This is pathological.

"Equality in Scala is a mess ... because the language designers decided Java interoperability trumped doing the reasonable thing in this case." (from here).

There are ways of solving this problem, the simplest being to use === that a number of libraries offer. Here are a three different ways of doing it:

  def scalazTripleEquals(): Unit = {
    import scalaz._
    import Scalaz._
//    println(List(1) === List("1")) // doesn't compile :)
  }

  def scalaUtilsTripleEquals(): Unit = {
    import org.scalautils.TypeCheckedTripleEquals._
//    println(List(1) === List("1")) // doesn't compile :)
  }

  def scalacticTripleEquals(): Unit = {
    import org.scalactic._
    import TypeCheckedTripleEquals._
//    println(List(1) === (List("1"))) // doesn't compile :)
  }

But what about incorporating it into the Scala language itself?

Scala creator, Martin Odersky's views can be found here here. He is proposing "it is opt-in. To get safe checking, developers have to annotate with @equalityClass ... So this means we still keep universal equality as it is in Scala now - we don’t have a choice here anyway, because of backwards compatibility."

Warning: ScalaTest

There is a horrible gotcha using Scalactic and ScalaTest (which is odd since they are stable mates). The problem is that you want compilation to fail for something likes this:

import org.scalatest.{FlatSpec, Matchers}

class MyTripleEqualsFlatSpec extends FlatSpec with Matchers {

  "triple equals" should "not compile" in {
    List(1)  should === (List("1"))
  }

}

Only it doesn't. It happily compiles! This is not what was expected at all given the code in scalacticTripleEquals() above. The solution can be found here. You must change the class signature to:

class MyTripleEqualsFlatSpec extends FlatSpec with Matchers with TypeCheckedTripleEquals {

for the compiler to detect this error.

Saturday, June 10, 2017

Either Mither


Either has changed. A superficial Google might suggest that Either represents just that - either A or B.

"You cannot, at least not directly, use an Either instance like a collection, the way you are familiar with from Option and Try. This is because Either is designed to be unbiased.

"Try is success-biased: it offers you map, flatMap and other methods that all work under the assumption that the Try is a Success, and if that’s not the case, they effectively don’t do anything, returning the Failure as-is." (from here)

And: "if you use Either for error reporting, then it is true that you want it to be biased to one side, but that is only one usecase of many, and hardcoding a single special usecase into a general interface smells of bad design" from here.

But then you see this in the Scala documentation: "Either is right-biased, which means that Right is assumed to be the default case to operate on. If it is Left, operations like map, flatMap, ... return the Left value unchanged".

This apparent contradiction arises as Scala 2.12 changed Either. It has become biased.

Let's demonstrate using ScalaTest. First, we define an Either and some functions to act on it:

    type LeftType   = List[Int]
    type RightType  = Int
    type EitherType = Either[LeftType, RightType]
    val left        = Left(List[Int]())
    val right       = Right(1)

    val rightFn: (RightType) => RightType = _ + 1
    val leftFn:  (LeftType)  => LeftType  = _ :+ 1

Then, the test looks like:

    "Either" should {
      "be agnostic if left or right has been explicitly stated on an Either that's a Left" in {
        val either: EitherType = left
        either.left.map(leftFn) shouldEqual Left(List(1))
        either.right.map(rightFn) shouldEqual left // unchanged
      }
      "be agnostic if left or right has been explicitly stated on an Either that's a Right" in {
        val either: EitherType = right
        either.right.map(rightFn) shouldEqual Right(2)
        either.left.map(leftFn) shouldEqual right // unchanged
      }
.
.

So far, so good. But this following code is new in 2.12 (it won't compile in earlier versions):

      "ignore the unbiased side" in {
        val either: EitherType = left
        either.map(rightFn) shouldEqual left
      }
      "map the biased side" in {
        val either: EitherType = right
        either.map(rightFn) shouldEqual Right(2)
      }

Either's new monadic functions cannot take anything other than a function of type (RightType) => ...

The mnemonic is that if you're using this for error reporting, then Right is the right answer.

Tuesday, May 30, 2017

Rocha Thatte Cycle Detection Algorithm


Flows of money and ownership in a network of businesses and individuals can indicate fraudulent behaviour. For instance, if there is a cycle in the network such that X owns Y who owns Z and Z audits X, you can quickly see that there is a conflict of interest. Such suspicions are of interest to us.

GraphX is very good at sniffing out these networks but you don't get cycle detection out-of-the-box. So, I rolled-my-own that happened to be similar to an algorithm somebody else has already discovered, the Rocha Thatte algorithm.

The algorithm

The Wikipedia page gives an excellent overview so I won't bore you with the details. Suffice to say that each vertex passes all the new paths going through it to its neighbouring vertex at each super-step.

The code is quite simple since GraphX does all the heavy lifting. Let's introduce a few type aliases:

import org.apache.spark.graphx.{VertexId, EdgeTriplet}

  type VertexPrg[T]   = (VertexId, T, T) => T
  type EdgePrg[T, ED] = (EdgeTriplet[T, ED]) => Iterator[(VertexId, T)]

No matter which algorithm we create, they have to implement these functions.

Now, we'll introduce some-domain specific aliases:

  type Path[T]  = Seq[T]
  type Paths[T] = Set[Path[T]]

and finally, the GraphX merge function (which is just (U,U)=>U) for us would look like:

  type MergeFn[T] = (Paths[T], Paths[T]) => Paths[T]

then the implementation of Rocha Thatte looks like this. The 'program' that runs on the vertex can be created here:

def vertexPrg[T](merge: MergeFn[T]): VertexPrg[Paths[T]] = { case (myId, myAttr, message) =>
  merge(myAttr, message)


and the 'program' running on edges looks like this:

type AddStepFn[T, ED] = (EdgeTriplet[Paths[T], ED]) = Paths[T]
def edgePrg[T, ED](add: AddStepFn[T, ED]): EdgePrg[Paths[T], ED] = { case edge =>
  import edge._
  val allPaths = add(edge)
  // TODO check attributes here for whatever business reasons you like
  if (allPaths == dstAttr) Iterator.empty else Iterator((dstId, allPaths))
}

(note: I've simplified the implementation for illustrative purposes. This code performs no checks.)

The problem

The shape of the graph is important. For instance, I ran this algorithm on a (disconnected) network with about 200 million edges, 400 million vertices and sub-graphs with a maximum diameter of 6. It ran in about 20 minutes on a cluster with 19 beefy boxes.

However, I ran it on a much smaller (connected) network of 26 thousand vertices, 175 thousand edges and a diameter of 10 with little success. I found that I could manage only 3 iterations before Spark executors started to die with (apparently) memory problems.

The problem was that this graph had regions that were highly interconnected (it actually represented all the business entities we had that were related to BlackRock Investment Management and Merrill Lynch, of which there are many). Let's say that a particular vertex has 100 immediate neighbours each with 100 of their own and each of them had 100. This quickly explodes into many possible paths through the original vertex (about 1 million) after only 3 super-steps.

It's not too surprising that this is an issue for us. After all, Spark's ScalaDocs do say "ideally the size of [the merged messages] should not increase."

For such a dense graph, super nodes are almost inevitable. For our purposes, we could ignore them but YMMV depending on your business requirements.

Monday, May 29, 2017

Good Hash


Since the feature hashing code that comes with Spark is based on a 32-bit hash giving us lots of collisions, we went for a 64-bit implementation. But what makes a good hash algorithm?

Rather than roll my own, I used this code. But how do I make sure it's good? This Stack Exchange answer gave me some clues by using the chi-squared test. Here, we basically compare with some fairly simple maths, our expected answer to our actual answer to indicate whether there is an unusually high number of collisions that our algorithm generates.

"The chi-squared test of goodness of fit is used to test the hypothesis that the distribution of a categorical variable within a population follows a specific pattern of proportions, while the alternative hypothesis is that the distribution of the variable follows some other pattern." [1]

The Chi-Square Test

"The values of column and rows totals are called marginals because they are on the margin of the table... The numbers within the table ... are called joint frequencies...

"If the two variables are not related, we would expect that the frequency of each cell would be the product of its marginals, divided by the sample size." [1]

This is because the probabilities of A and B are:

P(A ∩ B) = P(A) P(B) iff A ⊥ B

In other words, given a sample size of N, the probability of A is

P(A) = NA / N

and

P(B) = NB / N

so

P(A ∩ B) = NB NA / N2

so the expected frequency would be

N P(A ∩ B) = NB NA / N

exactly as [1] says.

The Chi-Square distribution then looks like this:

χ = Σi,j=1 (Oij - Eij)2/ Eij

where i and j are being summed over all the rows and columns.

The Results

Using this new hashing algorithm for our feature hashing resulted in no collisions while hashing all words in the corpus of text where previously we were getting about 70 million collisions with the 32-bit version.

Consequently, the number of records we then had to compare dropped by about 20 million (or about 2% of the total).

Unfortunately, although I am able to execute a chi-square test on the 32-bit algorithm, the 64-bit algorithm has such a large possible number of hash values, it's not practically possible to test it in such a manner.

[1] Statistics in a Nutshell, Sarah Boslaugh


Friday, May 26, 2017

The Probability Monad (part 1)


This is an interesting idea: probability distributions can be modeled as monads. The canonical description lives here but it's very Haskell-heavy. So, in an attempt to grok the probability monad, you might like to look at a Scala implementation here.

[Aside. I tried looking at the Haskell code but had to run
cabal install --dependencies-only --enable-tests
see here for more information.]

The monad in this Scala library is Distribution[T] where T can be, say, a Double such as in a Gaussian distribution:

  object normal extends Distribution[Double] {
    override def get = rand.nextGaussian()
  }

It could be something more interesting, for instance, here the Distribution monad in this particular case is parameterized with a List[Int].

  /**
   * You roll a 6-sided die and keep a running sum. What is the probability the
   * sum reaches exactly 30?
   */
  def dieSum(rolls: Int): Distribution[List[Int]] = {
    always(List(0)).markov(rolls)(runningSum => for {
      d <- die
    } yield (d + runningSum.head) :: runningSum)
  }
  def runDieSum = dieSum(30).pr(_ contains 30)

Simulation, simulation, simulation

The method pr will create a simulation where we sample an arbitrary number of monads (default of 10 000). We then filter them for those that contain a score of exactly 30 and calculate the subsequent probability.

Filtering the monads means that traversing the list of 10 000 and calling filter on each one to find ones with a score of 30. Each monad in the list is actually a recursive structure 30 deep (the number of  rolls of the dice; any more is pointless as the total will necessarily be greater than 30).

That's the high-level description. Let's drill down.

State monads again

This recursive structure is a state monad. The monads are created by the recursive calls to markov(). This method creates a new monad by calling flatMap on itself. The get method of this new, inner monad takes the value of its outer monad, passes it to the function that flatMap takes as an argument and in turn calls get on the result.

Having created this inner monad, markov() is called on it and we start the next level of recursion until we have done so 30 times. It is this chain of get calling get when the time comes that will build up the state.

Consequently, we have the outermost monad being a constant Distribution that holds List(0). This is what a call to the outermost get will return. However, get is not publicly accessible. We can only indirectly access it by calling the monad functions.

In short, we have what is a little like a doubly-linked list. The outermost monad contains the "seed", List(0), and a reference to the next monad. The inner monads contain a reference to the next monad (if there is one) and a reference to its outer monad's value via get.

Note that it is the innermost monad that is passed back to the call site calling dieSum, in effect turning the structure inside out.

Anyway, the next job is to filter the structure. This creates a new monad (referencing the erstwhile innermost monad) to do the job but remember monads are lazy so nothing happens yet. It's only when we call a sample method on this monad that something starts to happen. At this point, get is called and we work our way up the get-chain until we reach the outermost monad that contains List(0). Then we "pop" each monad, executing the runningSum => function on the results of the monad before. This is where we roll the die and add append the cumulative result to the List.

If the given of the filter monad is not met, then we keep trying the whole thing again until it is.

Finally, we count the results that meet our predicate dividing by total number of runs. Evidently, we've taken a frequentist approach to probabilities here.


Sunday, May 21, 2017

Lazy Scala


This might be elementary but when a simple error trips me more than once, I blog it.

You probably know that an Iterator can only be accessed once as it's stateful. "An Iterator can only be used once because it is a traversal pointer into a collection, and not a collection in itself" from here. (Iterable just means the implementing class can generate an Iterator).

The simple mistake I made was to check the size of the iterator in a temporary log statement before mapping over it. What was a little more interesting was that other collections behave the same way if you call them via their TraversableOnce interface.

To demonstrate, say we have a Set

    val aSet = Set(1, 2, 3)

and two functions that are identical other than the type of their argument:

  def mapSet[T](xs: Set[T]): TraversableOnce[String] = {
    val mapped = xs.map(_.toString)
    println(mapped.mkString("\n"))
    mapped
  }

  def mapTraversableOnce[T](xs: TraversableOnce[T]): TraversableOnce[String] = {
    val mapped = xs.map(_.toString)
    println(mapped.mkString("\n"))
    mapped
  }

then mapTraversableOnce will return an empty iterator while mapSet will return a Set of Strings. This will come as a surprise to anybody expecting Object Oriented behaviour.

Iterator also violates the functor laws. Take two otherwise identical methods:

  def isTraversableFunctor[T, U, V](xs: Traversable[T], f: T => U, g: U => V): Boolean = {
    val lhs = xs.map(f).map(g)
    val rhs = xs.map(g compose f)
    (lhs == rhs)
  }

  def isTraversableOnceFunctor[T, U, V](xs: TraversableOnce[T], f: T => U, g: U => V): Boolean = {
    val lhs = xs.map(f).map(g)
    val rhs = xs.map(g compose f)
    (lhs == rhs)
  }

and pass them aSet. The first will say it's a functor, the second says it's not.

This is somewhat trivial as the reason it fails is that the iterator has not been materialized. "TraversableOnce's map is not a functor map, but, then again, it never pretended to be one. It's design goals specifically precluded it from being a functor map" from Daniel C. Sobral.

Laziness

Iterator comes into its own when we want access to the underlying elements lazily. But there are other ways to do this like Stream "which implements all its transformer methods lazily."

Almost all collections are eager by default. Use the view method to make them lazy. Use force on these to make them eager again.

Finally, some methods are naturally lazy. For instance, exists terminates quickly if it finds what it's looking for.