[box type=”note” align=”” class=”” width=””]The following two-part tutorial is an excerpt from the book Mastering Machine Learning with Spark 2.x by Alex Tellez, Max Pumperla and Michal Malohlava. [/box]
When collecting real-world data between individual measures or events, there are usually very intricate and highly complex relationships to observe. The guiding example for this tutorial is the observation of click events that users generate on a website and its subdomains. Such data is both interesting and challenging to investigate. It is interesting, as there are usually many patterns that groups of users show in their browsing behavior and certain rules they might follow. Gaining insights about user groups, in general, is of interest, at least for the company running the website and might be the focus of their data science team. Methodology aside, putting a production system in place that can detect patterns in real time, for instance, to find malicious behavior, can be very challenging technically. It is immensely valuable to be able to understand and implement both the algorithmic and technical sides.
In this tutorial, we will look into doing pattern mining in Spark. The tutorial is split up into two main sections. In the first, we will first introduce the three available pattern mining algorithms that Spark currently comes with and then apply them to an interesting dataset. In particular, you will learn the following from this two-part tutorial:
- The basic principles of frequent pattern mining.
- Useful and relevant data formats for applications.
- Understanding and comparing three pattern mining algorithms available in Spark, namely FP-growth, association rules, and prefix span.
Frequent pattern mining
When presented with a new data set, a natural sequence of questions is:
- What kind of data do we look at; that is, what structure does it have?
- Which observations in the data can be found frequently; that is, which patterns or rules can we identify within the data?
- How do we assess what is frequent; that is, what are the good measures of relevance and how do we test for it?
On a very high level, frequent pattern mining addresses precisely these questions. While it’s very easy to dive head first into more advanced machine learning techniques, these pattern mining algorithms can be quite informative and help build an intuition about the data.
To introduce some of the key notions of frequent pattern mining, let’s first consider a somewhat prototypical example for such cases, namely shopping carts. The study of customers being interested in and buying certain products has been of prime interest to marketers around the globe for a very long time. While online shops certainly do help in further analyzing customer behavior, for instance, by tracking the browsing data within a shopping session, the question of what items have been bought and what patterns in buying behavior can be found applies to purely offline scenarios as well. We will see a more involved example of clickstream data accumulated on a website soon; for now, we will work under the assumption that only the events we can track are the actual payment transactions of an item.
Just this given data, for instance, for groceries shopping carts in supermarkets or online, leads to quite a few interesting questions, and we will focus mainly on the following three:
- Which items are frequently bought together? For instance, there is anecdotal evidence suggesting that beer and diapers are often brought together in one shopping session. Finding patterns of products that often go together may, for instance, allow a shop to physically place these products closer to each other for an increased shopping experience or promotional value even if they don’t belong together at first sight. In the case of an online shop, this sort of analysis might be the base for a simple recommender system.
- Based on the previous question, are there any interesting implications or rules to observe in shopping behavior?, continuing with the shopping cart example, can we establish associations such as if bread and butter have been bought, we also often find cheese in the shopping cart? Finding such association rules can be of great interest, but also need more clarification of what we consider to be often, that is, what does frequent mean.
- Note that, so far, our shopping carts were simply considered a bag of
items without additional structure. At least in the online shopping scenario, we can endow data with more information. One aspect we will focus on is that of the sequentiality of items; that is, we will take note of the order in which the products have been placed into the cart. With this in mind, similar to the first question, one might ask, which sequence of items can often be found in our transaction data? For instance, larger electronic devices bought might be followed up by additional utility items.
The reason we focus on these three questions, in particular, is that Spark MLlib comes with precisely three pattern mining algorithms that roughly correspond to the aforementioned questions by their ability to answer them. Specifically, we will carefully introduce FP- growth, association rules, and prefix span, in that order, to address these problems and show how to solve them using Spark. Before doing so, let’s take a step back and formally introduce the concepts we have been motivated for so far, alongside a running example. We will refer to the preceding three questions throughout the following subsection.
Pattern mining terminology
We will start with a set of items I = {a1, …, an}, which serves as the base for all the following concepts. A transaction T is just a set of items in I, and we say that T is a transaction of length l if it contains l item. A transaction database D is a database of transaction IDs and their corresponding transactions.
To give a concrete example of this, consider the following situation. Assume that the full item set to shop from is given by I = {bread, cheese, ananas, eggs, donuts, fish, pork, milk, garlic, ice cream, lemon, oil, honey, jam, kale, salt}. Since we will look at a lot of item subsets, to make things more readable later on, we will simply abbreviate these items by their first letter, that is, we’ll write I = {b, c, a, e, d, f, p, m, g, i, l, o, h, j, k, s}. Given these items, a small transaction database D could look as follows:
|
Transaction ID |
Transaction |
|
1 |
a, c, d, f, g, i, m, p |
|
2 |
a, b, c, f, l, m, o |
|
3 |
b, f, h, j, o |
|
4 |
b, c, k, s, p |
|
5 |
a, c, e, f, l, m, n, p |
Table 1: A small shopping cart database with five transactions
Frequent pattern mining problem
Given the definition of a transaction database, a pattern P is a transaction contained in the transactions in D and the support, supp(P), of the pattern is the number of transactions for which this is true, divided or normalized by the number of transactions in D:
supp(s) = suppD(s) = |{ s’ ∈ S | s < s’}| / |D|
We use the < symbol to denote s as a subpattern of s’ or, conversely, call s’ a superpattern of s. Note that in the literature, you will sometimes also find a slightly different version of support that does not normalize the value. For example, the pattern {a, c, f} can be found in transactions 1, 2, and 5. This means that {a, c, f} is a pattern of support 0.6 in our database D of five items.
Support is an important notion, as it gives us a first example of measuring the frequency of a pattern, which, in the end, is what we are after. In this context, for a given minimum support threshold t, we say P is a frequent pattern if and only if supp(P) is at least t. In our running example, the frequent patterns of length 1 and minimum support 0.6 are {a}, {b}, {c}, {p}, and {m} with support 0.6 and {f} with support 0.8. In what follows, we will often drop the brackets for items or patterns and write f instead of {f}, for instance.
Given a minimum support threshold, the problem of finding all the frequent patterns is called the frequent pattern mining problem and it is, in fact, the formalized version of the aforementioned first question. Continuing with our example, we have found all frequent patterns of length 1 for t = 0.6 already. How do we find longer patterns? On a theoretical level, given unlimited resources, this is not much of a problem, since all we need to do is count the occurrences of items. On a practical level, however, we need to be smart about how we do so to keep the computation efficient. Especially for databases large enough for Spark to come in handy, it can be very computationally intense to address the frequent pattern mining problem.
One intuitive way to go about this is as follows:
- Find all the frequent patterns of length 1, which requires one full database scan. This is how we started with in our preceding example.
- For patterns of length 2, generate all the combinations of frequent 1-patterns, the so-called candidates, and test if they exceed the minimum support by doing another scan of D.
- Importantly, we do not have to consider the combinations of infrequent patterns, since patterns containing infrequent patterns can not become frequent. This rationale is called the apriori principle.
- For longer patterns, continue this procedure iteratively until there are no more patterns left to combine.
This algorithm, using a generate-and-test approach to pattern mining and utilizing the apriori principle to bound combinations, is called the apriori algorithm. There are many variations of this baseline algorithm, all of which share similar drawbacks in terms of scalability. For instance, multiple full database scans are necessary to carry out the iterations, which might already be prohibitively expensive for huge datasets. On top of that, generating candidates themselves is already expensive, but computing their combinations might simply be infeasible. In the next section, we will see how a parallel version of an algorithm called FP-growth, available in Spark, can overcome most of the problems just discussed.
The association rule mining problem
To advance our general introduction of concepts, let’s next turn to association rules, as first introduced in Mining Association Rules between Sets of Items in Large Databases, available at http:/ /arbor. ee. ntu. edu. tw/~chyun/ dmpaper/agrama93. pdf. In contrast to solely counting the occurrences of items in our database, we now want to understand the rules or implications of patterns. What I mean is, given a pattern P1 and another pattern P2, we want to know whether P2 is frequently present whenever P1 can be found in D, and we denote this by writing P1 ⇒ P2. To make this more precise, we need a concept for rule frequency similar to that of support for patterns, namely confidence. For a rule P1 ⇒ P2, confidence is defined as follows:
conf(P1 ⇒ P2) = supp(P1 ∪ P2) / supp(P1)
This can be interpreted as the conditional support of P2 given to P1; that is, if it were to restrict D to all the transactions supporting P1, the support of P2 in this restricted database would be equal to conf(P1 ⇒ P2). We call P1 ⇒ P2 a rule in D if it exceeds a minimum confidence threshold t, just as in the case of frequent patterns. Finding all the rules for a confidence threshold represents the formal answer to the second question, association rule mining. Moreover, in this situation, we call P1 the antecedent and P2 the consequent of the rule. In general, there is no restriction imposed on the structure of either the antecedent or the consequent. However, in what follows, we will assume that the consequent’s length is 1, for simplicity.
In our running example, the pattern {f, m} occurs three times, while {f, m, p} is just present in two cases, which means that the rule {f, m} ⇒ {p} has confidence 2/3. If we set the minimum confidence threshold to t = 0.6, we can easily check that the following association rules with an antecedent and consequent of length 1 are valid for our case:
{a} ⇒ {c}, {a} ⇒ {f}, {a} ⇒ {m}, {a} ⇒ {p}
{c} ⇒ {a}, {c} ⇒ {f}, {c} ⇒ {m}, {c} ⇒ {p}
{f} ⇒ {a}, {f} ⇒ {c}, {f} ⇒ {m}
{m} ⇒ {a}, {m} ⇒ {c}, {m} ⇒ {f}, {m} ⇒ {p}
{p} ⇒ {a}, {p} ⇒ {c}, {p} ⇒ {f}, {p} ⇒ {m}
From the preceding definition of confidence, it should now be clear that it is relatively straightforward to compute the association rules once we have the support value of all the frequent patterns. In fact, as we will soon see, Spark’s implementation of association rules is based on calculating frequent patterns upfront.
[box type=”info” align=”” class=”” width=””]At this point, it should be noted that while we will restrict ourselves to the measures of support and confidence, there are many other interesting criteria available that we can’t discuss in this book; for instance, the concepts of conviction, leverage, or lift. For an in-depth comparison of the other measures, refer to http:/ / www. cse. msu. edu/ ~ptan/ papers/ IS. pdf.[/box]
The sequential pattern mining problem
Let’s move on to formalizing, the third and last pattern matching question we tackle in this chapter. Let’s look at sequences in more detail. A sequence is different from the transactions we looked at before in that the order now matters. For a given item set I, a sequence S in I of length l is defined as follows:
s = <s1, s2, …, sl>
Here, each individual si is a concatenation of items, that is, si = (ai1 … aim), where aij is an item in I. Note that we do care about the order of sequence items si but not about the internal ordering of the individual aij in si. A sequence database S consists of pairs of sequence IDs and sequences, analogous to what we had before. An example of such a database can be found in the following table, in which the letters represent the same items as in our previous shopping cart example:
|
Sequence ID |
Sequence |
|
1 |
<a(abc)(ac)d(cf)> |
|
2 |
<(ad)c(bc)(ae)> |
|
3 |
<(ef)(ab)(df)cb> |
|
4 |
<eg(af)cbc> |
Table 2: A small sequence database with four short sequences.
In the example sequences, note the round brackets to group individual items into a sequence item. Also note that we drop these redundant braces if the sequence item consists of a single item. Importantly, the notion of a subsequence requires a little more carefulness than for unordered structures. We call u = (u1, …, un) a subsequence of s = (s1, …, sl) and write u < s if there are indices 1 ≤ i1 < i2 < … < in ≤ m so that we have the following:
u1 < si1, …, un < sin
Here, the < signs in the last line mean that uj is a subpattern of sij. Roughly speaking, u is a subsequence of s if all the elements of u are subpatterns of s in their given order. Equivalently, we call s a supersequence of u. In the preceding example, we see that <a(ab)ac> and a(cb)(ac)dc> are examples of subsequences of <a(abc)(ac)d(cf)> and that <(fa)c> is an example of a subsequence of <eg(af)cbc>.
With the help of the notion of supersequences, we can now define the support of a sequence s in a given sequence database S as follows:
suppS(s) = supp(s) = |{ s’ ∈ S | s < s’}| / |S|
Note that, structurally, this is the same definition as for plain unordered patterns, but the < symbol means something else, that is, a subsequence. As before, we drop the database subscript in the notation of support if the information is clear from the context. Equipped with a notion of support, the definition of sequential patterns follows the previous definition completely analogously. Given a minimum support threshold t, a sequence s in S is said to be a sequential pattern if supp(s) is greater than or equal to t. The formalization of the third question is called the sequential pattern mining problem, that is, find the full set of sequences that are sequential patterns in S for a given threshold t.
Even in our little example with just four sequences, it can already be challenging to manually inspect all the sequential patterns. To give just one example of a sequential pattern of support 1.0, a subsequence of length 2 of all the four sequences is <ac>. Finding all the sequential patterns is an interesting problem, and we will learn about the so-called prefix span algorithm that Spark employs to address the problem in the following section.
Next time, in part 2 of the tutorial, we will see how to use Spark to solve the above three pattern mining problems using the algorithms introduced.
If you enjoyed this tutorial, an excerpt from the book Mastering Machine Learning with Spark 2.x by Alex Tellez, Max Pumperla and Michal Malohlava, check out the book for more.
