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.
- List the tuples in the complete data cube of R in a tabular form with 4 attributes,
i.e., Location,Time,Item,SUM(Quantity)?
- 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
- 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.
- 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.
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(w>x)
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.
Prove that the loss function for logistic regression is:
(2) Consider a variant of the logistic regression model:
P[y = 1 | x] = f(w>x)
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.
