Assignment 1 MNIST Database

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STAT 341: Assignment 1
54 Marks
Your assignment must be submitted by the due date listed at the top of this document, and it must be
submitted electronically in .pdf format via Crowdmark/LEARN. This means that your responses for different
questions should be in separate .pdf files. Your .pdf solution files must have been generated by R Markdown.
• For mathematical questions: your solutions must be produced by LaTeX (from within R Markdown).
Handwritten and scanned/photographed solutions will not be accepted and you will receive zero points.
• For computational questions: R code should always be included in your solution (via code chunks in R
Markdown). If code is required and you provide none, you will received zero points.
• For interpretation question: plain text (within R Markdown) is fine.
Organization and comprehensibility is part of a full solution. Consequently, points will be deducted for
solutions that are not organized and incomprehensible.
• You will submit your solutions in the form of one pdf file per question through LEARN For example,
for Q1 you should submit one pdf file containing the solution to the first question only. Failing to follow
the formatting instructions may result in your whole paper or individual questions receiving a grade of
Question One – 17 Marks – MNIST Database
Here we will use a subset of the Modified National Institute of Standards and Technology database (MNIST)
as an introduction to R and data exploration. A description of the data can be found at http://yann.lecun.
The MNIST database of handwritten digits, available from this page, has a training set of 60,000
examples, and a test set of 10,000 examples. It is a subset of a larger set available from NIST.
The digits have been size-normalized and centered in a fixed-size image.
It is a good database for people who want to try learning techniques and pattern recognition
methods on real-world data while spending minimal efforts on preprocessing and formatting.
a) [2 Marks] Write a function initializeDigitPlot that accepts no parameters and creates a blank
plot with x and y coordinate ranges of 0.5 to 28.5, no borders, and no axes. Test this function.
b) [2 Marks] Write a function drawBox that accepts three parameters (x, y, brightness) that draws a
solid-filled 1×1 box centred at (x, y) onto a preexisting plot with colour equal to the greyscale value.
The grayscale value is between zero and 1, where 0 is white, 1 is black and the values inbetween are
differents levels of grey. Test this function using the part a) and the parameters (1,1,0.5), (1,2,0)
and (1,3,1). Use the commands polygon and gray.
c) [2 Marks] Write a function drawDigit that accepts a numeric matrix of dimension 28 × 28 that calls
initializeDigitPlot and drawBox to draw the matrix. The values of the matrix are brightness values.
d) [2 Marks] Import the files one100.csv and two100.csv digits file. These files contain one hundred
1’s and one hundred 2’s with 28 × 28 = 784 columans. Each row has brightness values of a digit made
from the consecutive columns of a matrix. Use the drawDigit function to draw the first digit from
each of the files one100.csv and two100.csv side by side in a 1 × 2 grid.
e) [1 Mark] Plot all the digits from one100.csv in a 10 × 10 grid.
f) [1 Mark] Plot all the digits from two100.csv in a 10 × 10 grid.
g) [4 Marks] Summarize each of these populations by taking the average and median of the brightness
within each pixel. Then plot the four resulting 28 × 28 matrices in a 2 × 2 grid using your drawDigit
function. Compare and constrast using the average and median to summarize the images.
h) [3 Marks] Summarize the variability of these populations by taking the standard deviation of the
brightness within each pixel. Then plot the two resulting 28 × 28 matrices in a 1 × 2 grid using your
drawDigit function. Comment on the variability.
Question Two – 17 Marks – Properties of Pearson’s moment coefficient of skewness
Consider the population P = {y1, . . . , yN }. Pearson’s moment coefficient of skewness is
a(P) =
u∈P (yu − y)
[SDP (y)]3
and hence a measure of skewness. In this question you will investigate several of its properties.
• Note: Some marks in each part allocated to formating and organization.
(a) [3 points] Determine whether this skewness coefficient is location invariant, location equivariant, or
(b) [3 points] Determine whether this skewness coefficient is scale invariant, scale equivariant, or neither.
(c) [3 points] Determine whether this skewness coefficient is location-scale invariant, location-scale
equivariant, or neither.
(d) [3 points] Determine whether this skewness coefficient is replication invariant, replication equivariant,
or neither.
(e) [3 points] For the population below, plot the sensitivity curve of a (P) for y ∈ [−8, 8]. You may find
the sc() function from lecture useful.
pop <- rnorm(1001)
(f) [2 points] Given all that you have learned in parts (a) – (f), state one thing that is good about the
pearson’s moment coefficient of skewness attribute and one thing that is bad about the skewness
Question Three – 10 Marks – Finding and describing a population.
By searching the web, find a public dataset that constitutes a population. Provide the following:
• Give a description of the data in your own words. Then justify why the dataset is indeed a population
as opposed to a sample.
• A URL to access the data.
• Define what is an unit and describe two variate(s) that have been recorded.
• Give a single graphical display of the populaton.
• Describe some interesting attribute or feature of this population.
Some places you might consider looking:
• Kaggle
• UCI Machine Learning Repository
• r/datasets
• KDnuggets
Criteria Descriptor Marks
Data/URL Creativity: Was an interesting or unique dataset chosen, provided? /2
Description Clarity & Justification /2
Unit/Variate Description, Correctness and Graphic /4
Feature Description and Justification /2
Question Four – 10 Marks – Review
A student (Ryan Browne) was given the following question.
In your own words summarize the concept of sensitivity based on subsection 2.2.3 – Influence & Sensitivity.
• You are limited to 1 to 2 pages.
• Your solution should use a combination of formulas, full sentences and an example.
Criteria Descriptor Marks
Format Organization & LaTex /3
Writing Clarity & Grammar /2
Content Coverage, Depth, Relevant Terminology used and Example /5
• The student’s solution is provided on LEARN in the file “Sensitivity_Question.pdf”. Line numbers
have been added to each page to help refer to certain parts of the report.
a) [3 Marks] Give 3 examples of improper formatting, LaTex or grammar
b) [2 Marks] Give 2 examples of errors in content or gaps in knowledge.
c) [2 Marks] Did the student summarize the concept of sensitivity in their own words? Briefy explain
how and provide some evidence or examples.
d) [1 Mark] Give one suggestion on how to improve the report.
e) [2 Marks] Give this report a grade and briefy describe why this grade is appropriate.


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