Mean Imputation in Apache Spark

If you are interested in building predictive models on Big Data, then there is a good chance you are looking to use Apache Spark. Either with MLLib or one of the growing number of machine learning extensions built to work with Spark such as Elephas which lets you use Keras and Spark together.

For all its flexibility, you will find certain things about working with Spark a little tedious.

For one thing you will have to write a lot of boilerplate code to clean and prepare features before training models. MLLib only accepts numerical data, and will not handle NULL values. There are some great library functions for String indexing and One Hot Encoding. However, you will still need to explicitly apply all these functions yourself. Compared with using a product like H2O, which does all this via configuration options in the model, writing this code can be a chore.

As frustrating as it is, the flexibility of being able to build reusable Machine Learning Pipelines in which you can make your feature generation part of the Meta-Parameters, will make it all worthwhile in the end.

So, to this end, here is a little bit of re-usable code that will help with one common data preparation task: Mean Imputation of Missing Values.

If you are lucky enough to be on Spark 2.2.0 then there is a library function you can use to do this. This was lifted directly from this StackOverflow post:

import org.apache.spark.ml.feature._

def imputeMeans(df: org.apache.spark.sql.DataFrame, feet: Array[String]): (org.apache.spark.sql.DataFrame, Array[String]) = {
        val outcols = feet.map(c => s"${c}_imputed")
        val imputer = new Imputer().setInputCols(feet).setOutputCols(outcols).setStrategy("mean")
        val resdf = imputer.fit(df).transform(df)
        (resdf, outcols)
}

Which should be fairly self explanatory, you simply pass the function a dataframe and an array of column names. You will get back a new data frame with additional columns in which the '_imputed' suffix has been added.

If you are on an older version of Spark then you can do the following

def imputeToMean(df: org.apache.spark.sql.DataFrame, col: String): org.apache.spark.sql.DataFrame = {
        val meanVal = df.select(mean(df(col))).collectAsList().get(0).getDouble(0)
        df.na.fill(meanVal, Seq(col))
}

def imputeMeans(df: org.apache.spark.sql.DataFrame, cols: Array[String]): org.apache.spark.sql.DataFrame = {
        cols.foldLeft(df)( (dfin, col) => imputeToMean(dfin, col) )
}

Again the imputeMeans function takes a DataFrame and an array of column names. This time it just returns a new DataFrame in which the required columns have had their NULLs replaced with the column mean. This version can be time consuming to run, so I suggest that you cache it once it is done.

Hope that helps.

Logistic Regression with R

Logistic Regression

Regression is performed when you want to produce a function that will predict the value of something you don't know (the dependent variable) on the basis of a collection of things you do know (the independent variables).

The problem is that regression is typically done with a linear function and very few real world processes are linear. Hence, a great deal of statistics and machine learning research concerns methods for fitting non-linear functions, but controlling for the explosion in complexity that comes with it.

Logistic Regression is one of the methods that tries to solve this problem. In particular, Logistic Regression produces an output between 0 and 1 which can be interpreted as the probability of your target event happening.

Let's look at the form of Logistic Regression to get a better understanding:

You start with the goal of a function that approximates the probability of the target T for any input vector X :

p(T) = F(X)

In order to assure that F(X) takes the form of a valid probability (i.e. always between 0 and 1) we make us of the logistic function 1/(1+e^-K). If K is a big number the e^-K approaches 0 and hence the output of the logistic function approaches 1. If on the other hand K is a very small number the e^-K becomes very large and the output of the logistic function approaches 0.


So we are fitting the following function:

p(T) = 1 / [ 1 + e^-g(X) ]

We have added the function g(X) to afford us some flexibility in how we feed the input vector X into the logistic function. Here is where we place our usual linear regression function. We say

g(X) = B_0 + B_1 * X_1 + B_2 * X_2 + ...... + B_N * X_N

i.e. a linear function over all the dimensions of the input X.

Now, in order to perform our linear regression, we need to transform the function definition. You can do the transformation yourself if you like. What you will find is that with some re-arrangement you find that the function g(X) is equal to:

g(X) = - ln [ (1-p) / p ]

And by exploiting the properties of the logarithm you can further re-arrange to get the log odds ratio.

g(X) = ln [ p / (1-p) ]

An astute reader might notice a problem. For a target value of 1 (i.e p=1) the fraction is undefined. Luckily we can use the properties of the logarithm again and define our target as

 ln [ p / (1-p) ] =  ln(p) - ln(1-p)

...and this is the target value onto which you perform the linear regression.

In other words you fit the value of the parameters (the Bs) so that

B_0 + B_1*X_1 + B_2*X_2 + ...... + B_N*X_N = ln(p) - ln(1-p)


That is all well and good, how can we do that with R ? you might ask.

Well, I have gone ahead and converted some code from a bunch of different tutorials into a little R workbook that will take you through applied Logistic Regression in R. You can find the Logistic Regression Code Example in my GitHub account right here.

It all boils down to using the Generalised Linear Model function.

This R function will fit your Logistic Regression for you.

If you follow that code example to the end you will get a plot like the one below, which shows you the original data in green, the model fitted to that data in black, and some predictions for unseen parts of the input space in red.

Logistic Regression allows you a great deal of flexibility in your model. The parameterized linear model can be changed how you want, adding or removing independent variables. You can even add higher order combinations of the independent variables.

A common Machine Learning process is to experiment with different forms of this model and examine how the statistical significance of the fit changes.

Just be wary of the pernicious problem of over-fitting.

Machine Learning for Hackers

I recently read "Machine Learning for Hackers" by Drew Conway and John Myles White.

I'd picked it up because I heard it was a good way to get familiar with the data mining capabilities of R. I also expected the case study based approach to be a good way to see how they approach a broad array of machine learning problems. In these respects I was reasonably well rewarded. You will find a bunch of R code scraps that can be reused with a little effort. Unfortunately the explanation of what the code does (and how) is often absent. In this sense the book is true to its name: you will learn some recipes for tackling certain problems, but you may not understand how the code works, let alone the technique being applied.

The one issue I found unforgivable is that in the instances where the authors talk about machine learning theory, or use its terms, they are often wrong. One example is the application of naive Bayes to spam classification. The scoring function they use is the commonly used likelihood times the prior, leaving off the evidence divisor.

As a method of scoring in Bayesian methods this is appropriate because it is proportional to calculating the full posterior probability, and much more efficient to compute. However, the resulting score is not a probability, yet the authors continuously refer to it as one. This may seem minor, but to me it undermined my confidence in their ability to communicate necessary details about the techniques they are applying.

Another example: in the section on distance metrics the authors state that multiplying a matrix by its transpose computes “the correlation between every pair of columns in the original matrix.” This is also wrong. What they want to say is that it produces a matrix of scores that indicate the correlation between the rows. It is an approximation because the score depends on the length of the columns and whether they have been normalised. These values would not be comparable between matrices. What would be comparable between matrices is a correlation coefficient, but this is not what is being computed.

I am not suggesting that a hacker's guide to machine learning should include a thorough theoretical treatment of the subject. I think only that where terms and theory are introduced they should be used correctly. By this criteria this book is a failure. However, for my purposes (grabbing some code snippets for doing analysis with R) it was moderately successful. My largest disappointment was that given the mistakes I noticed regarding the topics about which I have reasonable knowledge, I have no confidence in their explanation of those areas where I am ignorant.

Maximum Likelihood Estimation

Maximum Likelihood Estimation is widely applicable method for estimating the parameters of a probabilistic model.

Developed by R.A.Fisher in the 1920s the principle behind it is that the ideal parameter settings of a model are the ones that make the observed data most likely.

It is applicable in any situation in which the model can be specified such that the probability of the desired variable y can be expressed as a parameterised function over the vector of observed variables (X).

P(y|X) = f(X,φ)

The parameters φ of the function f(X, φ) are what we want to estimate.

The model is designed to be a function in which the parameters are set and we get back a probability value for a given x. However, we need a process to determine these model parameters. The Likelihood function is defined to be equal to this function, but operating as a function over the parameter space of φ.

L(φ | y,X )= P(y|X,φ)

It is important to recognise that Likelihood is not the probability of the parameters, it is just equal to the probability of y given the parameters. As such it is not a probability distribution over φ.

If we have N observations in our data set, and we let D represent all N of these observations of X and y, then we can express the Likelihood function for this entire data set D as :

L(φ | D ) = ∏^N_i=1 P(y_i|X_i,φ )

Maximum Likelihood is then simply defined as Argmax φ over this function. Finding the value of φ that maximises this function can be done a number of ways.

To find an analytical solution to the Likelihood equation we find the partial derivative of the function with respect to each of the paramters. We then solve this series of equations for the parameter values such the the partial derivatives are equal to zero. This gives us a position that is either a max or min. We then find the second partial derivative with respect to each parameter and make sure it is negative at the points found in the first step. This will give us an analytical peak on the Likelihood surface.

The reality of maximising the Likelihood by searching the parameter space depends a great deal on the problem. Numerous tricks occur to simplify the problem. The natural logarithm of the Likelihood function is often taken because they are monotonically related, so the MLE can be obtained by maximising the log of the Likelihood. In addition, taking the log turns the product into a Sum and can improves the chance of finding an analytical solution, and improve the computational tractability of finding a numerical solution.

In the next post I will summarise the use of the Expectation Maximisation algorithm for situations in which the Likelihood function cannot be solved analytically.