CS439: Introduction to Data Science

CS439: Introduction to Data Science

all 2024Problem Set 1

Due: 11:59pm Friday, October 11, 2024

Late Policy: The homework is due on 10/11 (Friday) at 11:59pm. We will release the solutionsof the homework on Canvas on 10/16 (Wednesday) 11:59pm. If your homework is submitted toCanvas before 10/11 11:59pm, there will no late penalty. If you submit to Canvas after 10/1111:59pm and before 10/16 11:59pm (i.e., before we release the solution), your score will bepenalized by 0.9k , where k is the number of days of late submission. For example, if yousubmitted on 10/14, and your original score is 80, then your final score will be 80*0.93 =58.32for 14-11=3 days of late submission. If you submit to Canvas after 10/16 11:59pm (i.e., after werelease the solution), then you will earn no score for the homework.General Instructions

Submission instructions: These questions require thought but do not require long answers.Please be as concise as possible. You should submit your answers as a writeup in PDF format,for those questions that require coding, write your code for a question in a single source codefile, and name the file as the question number (e.g., question_1.java or question_1.py), finally,put your PDF answer file and all the code files in a folder named as your Name and NetID (i.e.,Firstname-Lastname-NetID.pdf), compress the folder as a zip file (e.g., Firstname-LastnameNetID.zip), and submit the zip file via Canvas.For the answer writeup PDF file, we have provided both a word template and a latex templatefor you, after you finished the writing, save the file as a PDF file, and submit both the originalfile (word or latex) and the PDF file.

Questions

1. Map-Reduce (35 pts)Write a MapReduce program in Hadoop that implements a simple “People You Might Know”social network friendship recommendation algorithm. The key idea is that if two people have alot of mutual friends, then the system should recommend that they connect with each other.

Input: Use the provided input file hw1q1.zip.The input file contains the adjacency list and has multiple lines in the following format:

<User><TAB><Friends>Here, <User> is a unique integer ID corresponding to a unique user and <Friends> is a commaseparated list of unique IDs corresponding to the friends of the user with the unique ID <User>Note that the friendships are mutual (i.e., edges are undirected): if A is friend with B, then B isalso friend with A. The data provided is consistent with that rule as there is an explicit entry foreach side of each edge.

Algorithm: Let us use a simple algorithm such that, for each user U, the algorithm recommends

N = 10 users who are not already friends with U, but have the largest number of mutual friends

in common with U.

Output: The output should contain one line per user in the following format:

<User><TAB><Recommendations>

where <User> is a unique ID corresponding to a user and <Recommendations> is a comma

separated list of unique IDs corresponding to the algorithm’s recommendation of people that<User> might know, ordered by decreasing number of mutual friends. Even if a user hasewer than 10 second-degree friends, output all of them in decreasing order of the number ofmutual friends. If a user has no friends, you can provide an empty list of recommendations. Ifthere are multiple users with the same 代 写CS439: Introduction to Data Science  number of mutual friends, ties are broken by orderingthem in a numerically ascending order of their user IDs.Also, please provide a description of how you are going to use MapReduce jobs to solve thisproblem. We only need a very high-level description of your strategy to tackle this problem.

Note: It is possible to solve this question with a single MapReduce job. But if your solutionrequires multiple MapReduce jobs, then that is fine too.

What to submit:

(i) The source code as a single source code file named as the question number (e.g.,question_1.java).

(ii) Include in your writeup a short paragraph describing your algorithm to tackle this problem.

(iii) Include in your writeup the recommendations for the users with following user IDs:

924, 8941, 8942, 9019, 9020, 9021, 9022, 9990, 9992, 9993.

2. Association Rules (35 pts)

Association Rules are frequently used for Market Basket Analysis (MBA) by retailers to

understand the purchase behavior of their customers. This information can be then used formany different purposes such as cross-selling and up-selling of loyalty programs, store design, discount plans and many others.Evaluation of item sets: Once you have found the frequent itemsets of a dataset, you need to

choose a subset of them as your recommendations. Commonly used metrics for measuringsignificance and interest for selecting rules for recommendations are:

2a. Confidence (denoted as conf(A → B)): Confidence is defined as the probability of

occurrence of B in the basket if the basket already contains A:conf(A → B) = Pr(B|A),where Pr(B|A) is the conditional probability of finding item set B given that item set A is

present.2b. Lift (denoted as lift(A → B)): Lift measures how much more “A and B occur together” than“what would be expected if A and B were statistically independent”:????(? → ?) =

????(? → ??(?where ?(?) = !"##$%& * (() and N is the total number of transactions (baskets).

3. Conviction (denoted as conv(A→B)): it compares the “probability that A appears without B if

they were independent” with the “actual frequency of the appearance of A without B”:

????(? → ?) =

1 − ?(?)

1 − ????(? → ?)

(a) [5 pts]

A drawback of using confidence is that it ignores Pr(B). Why is this a drawback? Explain why lift

and conviction do not suffer from this drawback?

(b) [5 pts]

A measure is symmetrical if measure(A → B) = measure(B → A). Which of the measures

presented here are symmetrical? For each measure, please provide either a proof that themeasure is symmetrical, or a counterexample that shows the measure is not symmetrical.

c) [5 pts]A measure is desirable if its value is maximal for rules that hold 100% of the time (such rules arealled perfect implications). This makes it easy to identify the best rules. Which of the above

measures have this property? Explain why.

Product Recommendations: The action or practice of selling additional products or services to

existing customers is called cross-selling. Giving product recommendation is one of theexamples of cross-selling that are frequently used by online retailers. One simple method to

give product recommendations is to recommend products that are frequently browsedtogether by the customers.Suppose we want to recommend new products tothe customer based on the products thehave already browsed on the online website. Write a program using the A-priori algorithm tfind products which are frequently browsed together. Fix the support to s = 100 (i.e. product

pairs need to occur together at least 100 times to be considered frequent) and find itemsets of

size 2 and 3.

Use the provided browsing behavior dataset browsing.txt. Each line represents a browsing

session of a customer. On each line, each string of 8 characters represents the id of an item

browsed during that session. The items are separated by spaces.

Note: for the following questions (d) and (e), the writeup will require a specific rule ordering

but the program need not sort the output.

(d) [10pts]

Identify pairs of items (X, Y) such that the support of {X, Y} is at least 100. For all such pairs,

compute the confidence scores of the corresponding association rules: X ⇒ Y, Y ⇒ X. Sort therules in decreasing order of confidence scores and list the top 5 rules in the writeup. Break ties,

if any, by lexicographically increasing order on the left hand side of the rule.(e) [10pts]Identify item triples (X, Y, Z) such that the support of {X, Y, Z} is at least 100. For all such triples,compute the confidence scores of the corresponding association rules: (X, Y) ⇒ Z, (X, Z) ⇒ Y,

and (Y, Z) ⇒ X. Sort the rules in decreasing order of confidence scores and list the top 5 rules inthe writeup. Order the left-hand-side pair lexicographically and break ties, if any, by

lexicographical order of the first then the second item in the pair.

What to submit:Include your properly named code file (e.g., question_2.java or question_2.py), and include the

answers to the following questions in your writeup: (i) Explanation for 2(a).

(ii) Proofs and/or counterexamples for 2(b).(iii) Explanation for 2(c).

(iv) Top 5 rules with confidence scores for 2(d).(v) Top 5 rules with confidence scores for 2(e).

3. Locality-Sensitive Hashing (30 pts)

When simulating a random permutation of rows, as described in Sec 3.3.5 of MMDS textbook,we could save a lot of time if we restricted our attention to a randomly chosen k of the n rows,rather than hashing all the row numbers. The downside of doing so is that if none of the k rows

contains a 1 in a certain column, then the result of the min-hashing is “don’t know,” i.e., we get

no row number as a min-hash value. It would be a mistake to assume that two columns that

both min-hash to “don’t know” are likely to be similar. However, if the probability of getting

“don’t know” as a min-hash value is small, we can tolerate the situation, and simply ignore such

min-hash values when computing the fraction of min-hashes in which two columns agree.

(a) [10 pts]Suppose a column has m 1’s and therefore (n-m) 0’s. Prove that the probability we ge“don’t know” as the min-hash value for this column is at most ( ++ - )..

(b) [10 pts]Suppose we want the probability of “don’t know” to be at most ?,/0. Assuming n and m are

both very large (but n is much larger than m or k), give a simple approximation to the smallesvalue of k that will assure this probability is at most ?,/0. Hints: (1) You canuse ( +,+ - ). as theexact value of the probability of “don’t know.” (2) Remember that for large x, (1 − 1 / )1 ≈ 1/?.

(c) [10 pts]

Note: This question should be considered separate from the previous two parts, in that we areno longer restricting our attention to a randomly chosen subset of the rows.When min-hashing, one might expect that we could estimate the Jaccard similarity without

sing all possible permutations of rows. For example, we could only allow cyclic permutationsi.e., start at a randomly chosen row r, which becomes the first in the order, followed by rowr+1, r+2, and so on, down to the last row, and then continuing with the first row, second row,

and so on, down to row r−1. There are only n such permutations if there are n rows. However,these permutations are not sufficient to estimate the Jaccard similarity correctly.Give an example of two columns such that the probability (over cyclic permutations only) thattheir min-hash values agree is not the same as their Jaccard similarity. In your answer, pleaseprovide (a) an example of a matrix with two columns (let the two columns correspond to setsdenoted by S1 and S2) (b) the Jaccard similarity of S1 and S2, and (c) the probability that arandom cyclic permutation yields the same min-hash value for both S1 and S2.

What to submit:Include the following in your writeup:(i) Proof for 3(a)(ii) Derivation and final answer for 3(b)(iii) Example for 3(c)