COMP9318 ASSIGNMENT 1 Solved

Q1. (40 marks)
Consider the following base cuboid Sales with four tuples and the aggregate function SUM:

Location
Time
Item
Quantity
Sydney
2005
PS2
1400
Sydney
2006
PS2
1500
Sydney
2006
Wii
500
Melbourne
2005
XBox 360
1700
Location, Time, and Item are dimensions and Quantity is the measure. Suppose the system has built-in support for the value ALL.

(1)  List the tuples in the complete data cube of R in a tabular form with 4 attributes,

i.e., Location,Time,Item,SUM(Quantity)?

(2)  Write down an equivalent SQL statement that computes the same result (i.e., the cube). You can only use standard SQL constructs, i.e., no CUBE BY clause.

(3)  Consider the following ice-berg cube query:

SELECT Location, Time, Item, SUM(Quantity)

                   FROM        Sales

CUBE BY Location, Time, Item HAVING COUNT(*) 1

Draw the result of the query in a tabular form.

(4)  Assume that we adopt a MOLAP architecture to store the full data cube of R, with the following mapping functions:

= ‘Sydney’, = ‘Melbourne’,

                                                              0        if x = ALL.


if x = 2005, if x = 2006,
                                                                  0       if x = ALL.

            2                                                                          DUE ON 23:59 23 MAY, 2018 (WED)
                                                                          1          if x = ‘PS2’,

2

fItem(x) =

3

0
if x = ‘XBox 360’, if x = ‘Wii’, if x = ALL.
Draw the MOLAP cube (i.e., sparse multi-dimensional array) in a tabular form of (ArrayIndex,V alue). You also need to write down the function you chose to map a multi-dimensional point to a one-dimensioinal point.

Q2. (30 marks)
Consider binary classification where the class attribute y takes two values: 0 or 1. Let the feature vector for a test instance be a d-dimension column vector ~x. A linear classifier with the model parameter w (which is a d-dimension column vector) is the following function:

                                                                          (                      

                                                                            1         , if w x 0

y =

                                                                            0      , otherwise.

We make additional simplifying assumptions: x is a binary vector (i.e., each dimension of x take only two values: 0 or 1).

•   Prove that if the feature vectors are d-dimension, then a Na¨ıve Bayes classifier is a linear classifier in a d + 1-dimension space. You need to explicitly write out the vector w that the Na¨ıve Bayes classifier learns.

•   It is obvious that the Logistic Regression classifier learned on the same training dataset as the Na¨ıve Bayes is also a linear classifier in the same d + 1-dimension space. Let the parameter w learned by the two classifiers be wLR and wNB, respectively. Briefly explain why learning wNB is much easier than learning wLR.



Q3. (30 marks)
Consider a dataset consisting of n training data xi and the corresponding class label yi ∈ {0,1}.

(1) Consider the standard logistic regression model:

P[y = 1 | x] = σ(wx)

where σ is the sigmoid function.

The learning of the model parameter is to find w∗ that minimizes some function of w, commonly known as the loss function.

                                                    COMP9318 (18S1) ASSIGNMENT 1                                                                           3
Prove that the loss function for logistic regression is:



(2) Consider a variant of the logistic regression model:

P[y = 1 | x] = f(wx)

where f : < → [0,1] is a squashing function that maps a real value to a value between 0 and 1.

Write out its loss function.