For on line users, conversion generally refers to the user action that results in some tangible gain for a business e.g., an user opening an account or an user making his or her first purchase. Next to drawing large number of users to a web site, getting an user to convert is the most critical event in an user’s relationship with on line business. Being able to predict when an user will convert to become a customer should be an important tool that on line businesses should have at their disposal. A business could intiate targeted marketing campaign based on the prediction result.

In this post, I will be using user online behavior data to predict whether an user will convert using Markov Chain Classifier. The Hadoop based implementation is part of my OSS project *avenir*.

## Markov Chain Classification for Sequence Data

Depending on whether the data is instance or sequence , there two classes of learning algorithms. With sequence data, the order of the data points is considered an important aspect of the data and the learning algorithms will exploit the order in the data. Markov Chain Classification is a supervised learning algorithm for sequential data.

Sequence data with a temporal context is called time series data. For many learning problems, sequence data is more effective. When we use instance data, the order between the data points, temporal or something else, is lost.

For our example, each web site visit or session will be considered a data point in a sequence of data points. Session data ordered by time will be our sequence data. The data elements in the sequence i.e. session data will a vector of several attributes.

There are many relevant attributes for web session. I will be considering only the following. I am not implying that these are only feature attributes that matter.

*Time elapsed since the last visit**Time spent in the session*

Other feature attributes that could be relevant in the predictive model are as follows. It requires something called feature analysis to determine the relevant sub set of features for any predictive analytic problem.

*Referrer for the first visit**Hour of the day for visit**Day of the week for visit**Number of pages visited in a session*

## Building Markov Model

The first step towards building the complete solution is to build the Markov State Transition model. It’s a state transition probability matrix. If a problem is modeled with n states, the matrix will be n x n. A matrix element value in the matrix is the probability of transitioning from one state to another.

Going back to our problem, we are are considering only two session attributes. To keep the state transition matrix manageable, we will discretize the attributes into 3 levels; **H**igh, **M**edium and **L**ow. With two attributes, we will end up with 9 states in out problem. Each session will be characterized with two symbols, which stands for a state. For example, HM will imply that time elapsed since last session is high and time spent in the current session is medium.

Each line in our input will capture the data about all consecutive past sessions for a given user. It will consist of the following

*Cookie ID (or User ID)**Class variable indicating whether the user converted or not (***T**rue or**F**alse)*Sequence of session data where each element of the sequence is a 2 alphabet symbol*

This is essentially our training data. Here is some sample data

4F014156K07N,F,LL,ML,HH,HL,LL,HM,HL,LH,ML,HH,HL,LH G7C0M9H5SUZ1,F,HL,LM,HL,MH,HH,HH,ML,HL GWBX875AD31D,F,LL,HM,HL,HL,HM KRO2F24JUDE5,T,HL,HM,HM,HL,HM,MH,HM,HL,HL 3J0G4BB9BI1Q,F,LM,LH,LH,MH,LM,MH,LH

This data is ingested by the map reduce class *MarkovStateTransitionModel*. The output will consist of 2 matrices, each being a 9 x 9 state transition probability matrix. Here is the output. The first line contains the list of states.

LL,LM,LH,ML,MM,MH,HL,HM,HH classLabel:T 85,103,219,81,46,105,159,94,105 100,142,294,55,61,127,79,49,89 102,142,311,54,61,135,45,46,100 100,93,155,88,61,93,214,118,72 98,105,218,93,55,115,120,100,91 98,124,299,55,39,136,91,47,107 121,51,81,121,61,75,276,101,108 108,84,150,110,62,67,201,100,115 87,108,208,74,59,114,166,74,106 classLabel:F 106,51,50,129,66,53,300,130,110 128,53,69,124,60,57,276,125,102 105,59,89,109,52,62,296,111,113 108,46,46,130,59,54,289,147,116 108,49,51,154,43,57,294,125,115 115,64,55,120,58,60,276,131,115 111,52,41,135,64,55,288,137,111 112,52,43,133,54,55,288,136,122 117,48,50,143,62,56,270,137,112

The cell values in the matrix are not actual probability, but normalized frequency count. As we will see later, since we will take ratio of the matrix cell values, actual probability is not necessary.

## Predicting with Markov Model

Our test data format will be similar to the training data, except that the class variable is missing, which is going to be predicted. The test data and the Markov State Transition matrix produced by the first map reduce is processed by the map reduce class *MarkovModelClassifier*. For each sequence we calculate log odd ratio as below.

*logOdds = Σlog(tp(i,j) / tn(i,j))*

*where *

*tp(i,j) = transition probability for transition from state i to j for positive (T) class value *

*tn(i,j) = transition probability for transition from state i to j for negative (F) class value*

From the sequence, we take contiguous pairs of data points by sliding an window of size 2 over the sequence. For each state pair, we read two normalized frequency count values from the two matrices and calculate the log odds ratio as above. If the ratio is greater than 0, then the user is likely to convert.

In plain language, we determine whether the sequence of states for an user is more likely for positive class values or negative class values and make the prediction accordingly. The prediction score i.e. log odds ratio changes with time based on the sequence of states at any given time.

Instead of 0, a threshold value can be provided through a configuration parameter. For example, a small positive threshold value could be provided to keep us way from the class separation boundary and reduce the number of false positives. Here is some sample output, for a log odds threshold of 0.5. In this output snippet, one user has been predicted for conversion

6F1RIJZ882GA,F,-0.9419187552056059 R393GY9VWM9P,F,-1.0870003657324971 X573L7KOBRR0,F,-0.01828609983127566 KMUY5UN2M7TY,T,1.1076440756594925 2VBVQLCDDYJE,F,-4.542646122905596

All the predictive model does is that it identifies users that have the potential to convert. A business could target those users with marketing campaign, hopefully to turn that potential into reality.

The marketing could specifically target users that are close to the border line i.e., log odds ratio is slightly above the threshold. Those are the users that need to be nurtured into paying customers. Those with high log odds ratio, are more likely to convert any way and may not need any special attention.

## Meta Learning and Model Complexity

Meta learning is about understanding and learning about the learning process. In machine learning context, typically it involves learning about and selecting some higher order parameters that drive the learning algorithm performance. In our case, we are concerned about two meta learning parameters.

We would like find out the optimum sub set of feature attributes among all feature attributes listed earlier. We would also like to learn whether or not a higher order Markov Model will have more predictive accuracy and if so what order.

It’s good to remind ourselves that making a model more complex will not necessarily yield the best result, because of bias variance trade off. Generally there is an optimum level of complexity when the generalization error is minimum.

## Feature Set Selection

We could increase the complexity of our model by including more feature attributes.We have used only two session related variable, with each one having 3 possible values. We ended up with 9 x 9 state transition probability matrices.

If we included some of the other attributes mentioned earlier, the state transition probability matrices would have been much bigger.

These are some of the ways we could explore the feature space looking for an optimum sub set of features providing best prediction accuracy.

*Find relevance of each feature attribute with respect to the class atribute and other feature attributes**Brute force way of trying different feature sub sets with learning algorithm being used*

Details of the first approach can be find in my earlier post on feature sub set selection. The solution is based on calculating certain statistical parameters based entropy and mutual information.

## Higher Order Markov Model

We used first order Markov Model i.e., the state probability is only conditioned on the immediately earlier state. As a result, we have two dimensional state transition probability matrices. If the probability was conditional on n earlier states, we would have used (n +1) dimensional state transition probability matrices.

The question is how do we select the number conditioning data points n for best predictive accuracy. Here are are couple of approaches.

*For different values of n, construct higher order conditional probability matrix. Find the conditional entropy for each case. Choose n based on lowest conditional entropy.**Brute force way of trying different n with learning algorithm being used*

## Summing Up

We have explored Markov Model based predictive analytic technique that operates on sequence data. The tutorial for preparing data and executing the map reduce jobs are available.

This is a great breakdown, thank you

Hi Pranab,

You mention that the state transition probability matrix is a normalized frequency count. Is that just the total number of times a state appears across all customers? Wondering what you mean by “normalized frequency?”

As a follow up, doesn’t taking the frequency count of states somehow ignore the temporal information? That is, the model is no longer using information about customer visits early on vs. at a later time.

Normalized frequency is the count of the specific states upon total count.

It is a basic normalization process used in feature scaling.

State transition probabilities are across all customers, but there are 2 of them one for converting customers and one for non converting customers. The counts for each row is normalized i.e. divided by the row sum and multiplied by the constant

The state transition matrix captures the temporal information. A row is the current state and the column is the next state state. The corresponding cell value is the probability of being in the next state.

Since it’s a first order Markov chain, the complete sequence for all visits for an user is an aggregate of many first order transitions, involving 2 states, which are current and next.