ASSIGNMENT 1 SOLUTION

Description

Q1. (40 marks)

Consider the following base cuboid Sales with four tuples and the aggregate function

SUM:

Location	T ime	Item	Quantity
Sydney	2005	PS2	1400
Sydney	2006	PS2	1500
Sydney	2006	Wii	500
Melbourne	2005	XBox 360	1700

Location, T ime, 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; T ime; 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:

8

>1 if x = ‘Sydney’;

<

f_Location(x) = 2 if x = ‘Melbourne’;

>

:0 if x = ALL:

8

>1 if x = 2005;

<

f_{T ime}(x) = 2 if x = 2006;

>

:0 if x = ALL:

2 DUE ON 23:59 14 APR, 2019 (SUN)

8

• if x = ‘PS2’;

>

>

>

<2 if x = ‘XBox 360’;

f_Item(x) =

>3 if x = ‘Wii’;

>

>

:0 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 classi cation 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 classi er with the model parameter w (which is a d-dimension column vector) is the following function:

(

_{y =} 1 , if w^>x > 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 ve Bayes classi er is a linear classi er in a d + 1-dimension space. You need to explicitly write out the vector w that the Na ve Bayes classi er learns.

It is obvious that the Logistic Regression classi er learned on the same training dataset as the Na ve Bayes is also a linear classi er in the same d + 1-dimension space. Let the parameter w learned by the two classi ers be w_LR and w_NB, re-spectively. Brie y explain why learning w_NB is much easier than learning w_LR.

i	log x	i	=	i	i	Hint .1 log
		P		x
						Q

Q3. (30 marks)

We have a sample of mixture of two chemical compound, S₁ and S₂. The (unknown) percentages of each chemical in the sample are denoted as q₁ and q₂ (whereas q₁ + q₂ = 1), respectively.

We have a device that can detect the percentages of m = 3 di erent components that are contained in both chemical compounds, albeit with di erent percentages. We denote the components as f O_j g^m_j=1. We list the percentages of each components in pure S_is in the following table:

pi;j	O1	O2	O3
S1	0.1	0.2	0.7
S2	0.4	0.5	0.1

After measuring the three components, we obtain their percentages as f u_j g^m_j=1.

COMP9318 (19T1) ASSIGNMENT 1

3

1. Write out the log likelihood function (as a function of q_i, p_i;j, and u_i).

2. If u₁ = 0:3; u₂ = 0:2; u₃ = 0:5, what are the MLE of q₁ and q₂? What are the expected percentage of each component under a model with the MLE parameters?

Submission

Please write down your answers in a le named ass1.pdf. You must write down your name and student ID on the rst page.

You can submit your le by

give cs9318 ass1 ass1.pdf

Late Penalty. -10% per day for the rst two days, and -20% for each of the following days.