assignment report pdf

Understanding Assignments

What this handout is about.

The first step in any successful college writing venture is reading the assignment. While this sounds like a simple task, it can be a tough one. This handout will help you unravel your assignment and begin to craft an effective response. Much of the following advice will involve translating typical assignment terms and practices into meaningful clues to the type of writing your instructor expects. See our short video for more tips.

Basic beginnings

Regardless of the assignment, department, or instructor, adopting these two habits will serve you well :

• Read the assignment carefully as soon as you receive it. Do not put this task off—reading the assignment at the beginning will save you time, stress, and problems later. An assignment can look pretty straightforward at first, particularly if the instructor has provided lots of information. That does not mean it will not take time and effort to complete; you may even have to learn a new skill to complete the assignment.
• Ask the instructor about anything you do not understand. Do not hesitate to approach your instructor. Instructors would prefer to set you straight before you hand the paper in. That’s also when you will find their feedback most useful.

Assignment formats

Many assignments follow a basic format. Assignments often begin with an overview of the topic, include a central verb or verbs that describe the task, and offer some additional suggestions, questions, or prompts to get you started.

An Overview of Some Kind

The instructor might set the stage with some general discussion of the subject of the assignment, introduce the topic, or remind you of something pertinent that you have discussed in class. For example:

“Throughout history, gerbils have played a key role in politics,” or “In the last few weeks of class, we have focused on the evening wear of the housefly …”

The Task of the Assignment

Pay attention; this part tells you what to do when you write the paper. Look for the key verb or verbs in the sentence. Words like analyze, summarize, or compare direct you to think about your topic in a certain way. Also pay attention to words such as how, what, when, where, and why; these words guide your attention toward specific information. (See the section in this handout titled “Key Terms” for more information.)

“Analyze the effect that gerbils had on the Russian Revolution”, or “Suggest an interpretation of housefly undergarments that differs from Darwin’s.”

Additional Material to Think about

Here you will find some questions to use as springboards as you begin to think about the topic. Instructors usually include these questions as suggestions rather than requirements. Do not feel compelled to answer every question unless the instructor asks you to do so. Pay attention to the order of the questions. Sometimes they suggest the thinking process your instructor imagines you will need to follow to begin thinking about the topic.

“You may wish to consider the differing views held by Communist gerbils vs. Monarchist gerbils, or Can there be such a thing as ‘the housefly garment industry’ or is it just a home-based craft?”

These are the instructor’s comments about writing expectations:

“Be concise”, “Write effectively”, or “Argue furiously.”

Technical Details

These instructions usually indicate format rules or guidelines.

“Your paper must be typed in Palatino font on gray paper and must not exceed 600 pages. It is due on the anniversary of Mao Tse-tung’s death.”

The assignment’s parts may not appear in exactly this order, and each part may be very long or really short. Nonetheless, being aware of this standard pattern can help you understand what your instructor wants you to do.

Interpreting the assignment

Ask yourself a few basic questions as you read and jot down the answers on the assignment sheet:

Why did your instructor ask you to do this particular task?

Who is your audience.

• What kind of evidence do you need to support your ideas?

What kind of writing style is acceptable?

• What are the absolute rules of the paper?

Try to look at the question from the point of view of the instructor. Recognize that your instructor has a reason for giving you this assignment and for giving it to you at a particular point in the semester. In every assignment, the instructor has a challenge for you. This challenge could be anything from demonstrating an ability to think clearly to demonstrating an ability to use the library. See the assignment not as a vague suggestion of what to do but as an opportunity to show that you can handle the course material as directed. Paper assignments give you more than a topic to discuss—they ask you to do something with the topic. Keep reminding yourself of that. Be careful to avoid the other extreme as well: do not read more into the assignment than what is there.

Of course, your instructor has given you an assignment so that they will be able to assess your understanding of the course material and give you an appropriate grade. But there is more to it than that. Your instructor has tried to design a learning experience of some kind. Your instructor wants you to think about something in a particular way for a particular reason. If you read the course description at the beginning of your syllabus, review the assigned readings, and consider the assignment itself, you may begin to see the plan, purpose, or approach to the subject matter that your instructor has created for you. If you still aren’t sure of the assignment’s goals, try asking the instructor. For help with this, see our handout on getting feedback .

Given your instructor’s efforts, it helps to answer the question: What is my purpose in completing this assignment? Is it to gather research from a variety of outside sources and present a coherent picture? Is it to take material I have been learning in class and apply it to a new situation? Is it to prove a point one way or another? Key words from the assignment can help you figure this out. Look for key terms in the form of active verbs that tell you what to do.

Key Terms: Finding Those Active Verbs

Here are some common key words and definitions to help you think about assignment terms:

Information words Ask you to demonstrate what you know about the subject, such as who, what, when, where, how, and why.

• define —give the subject’s meaning (according to someone or something). Sometimes you have to give more than one view on the subject’s meaning
• describe —provide details about the subject by answering question words (such as who, what, when, where, how, and why); you might also give details related to the five senses (what you see, hear, feel, taste, and smell)
• explain —give reasons why or examples of how something happened
• illustrate —give descriptive examples of the subject and show how each is connected with the subject
• summarize —briefly list the important ideas you learned about the subject
• trace —outline how something has changed or developed from an earlier time to its current form
• research —gather material from outside sources about the subject, often with the implication or requirement that you will analyze what you have found

Relation words Ask you to demonstrate how things are connected.

• compare —show how two or more things are similar (and, sometimes, different)
• contrast —show how two or more things are dissimilar
• apply—use details that you’ve been given to demonstrate how an idea, theory, or concept works in a particular situation
• cause —show how one event or series of events made something else happen
• relate —show or describe the connections between things

Interpretation words Ask you to defend ideas of your own about the subject. Do not see these words as requesting opinion alone (unless the assignment specifically says so), but as requiring opinion that is supported by concrete evidence. Remember examples, principles, definitions, or concepts from class or research and use them in your interpretation.

• assess —summarize your opinion of the subject and measure it against something
• prove, justify —give reasons or examples to demonstrate how or why something is the truth
• evaluate, respond —state your opinion of the subject as good, bad, or some combination of the two, with examples and reasons
• support —give reasons or evidence for something you believe (be sure to state clearly what it is that you believe)
• synthesize —put two or more things together that have not been put together in class or in your readings before; do not just summarize one and then the other and say that they are similar or different—you must provide a reason for putting them together that runs all the way through the paper
• analyze —determine how individual parts create or relate to the whole, figure out how something works, what it might mean, or why it is important
• argue —take a side and defend it with evidence against the other side

More Clues to Your Purpose As you read the assignment, think about what the teacher does in class:

• What kinds of textbooks or coursepack did your instructor choose for the course—ones that provide background information, explain theories or perspectives, or argue a point of view?
• In lecture, does your instructor ask your opinion, try to prove their point of view, or use keywords that show up again in the assignment?
• What kinds of assignments are typical in this discipline? Social science classes often expect more research. Humanities classes thrive on interpretation and analysis.
• How do the assignments, readings, and lectures work together in the course? Instructors spend time designing courses, sometimes even arguing with their peers about the most effective course materials. Figuring out the overall design to the course will help you understand what each assignment is meant to achieve.

Now, what about your reader? Most undergraduates think of their audience as the instructor. True, your instructor is a good person to keep in mind as you write. But for the purposes of a good paper, think of your audience as someone like your roommate: smart enough to understand a clear, logical argument, but not someone who already knows exactly what is going on in your particular paper. Remember, even if the instructor knows everything there is to know about your paper topic, they still have to read your paper and assess your understanding. In other words, teach the material to your reader.

Aiming a paper at your audience happens in two ways: you make decisions about the tone and the level of information you want to convey.

• Tone means the “voice” of your paper. Should you be chatty, formal, or objective? Usually you will find some happy medium—you do not want to alienate your reader by sounding condescending or superior, but you do not want to, um, like, totally wig on the man, you know? Eschew ostentatious erudition: some students think the way to sound academic is to use big words. Be careful—you can sound ridiculous, especially if you use the wrong big words.
• The level of information you use depends on who you think your audience is. If you imagine your audience as your instructor and they already know everything you have to say, you may find yourself leaving out key information that can cause your argument to be unconvincing and illogical. But you do not have to explain every single word or issue. If you are telling your roommate what happened on your favorite science fiction TV show last night, you do not say, “First a dark-haired white man of average height, wearing a suit and carrying a flashlight, walked into the room. Then a purple alien with fifteen arms and at least three eyes turned around. Then the man smiled slightly. In the background, you could hear a clock ticking. The room was fairly dark and had at least two windows that I saw.” You also do not say, “This guy found some aliens. The end.” Find some balance of useful details that support your main point.

You’ll find a much more detailed discussion of these concepts in our handout on audience .

The Grim Truth

With a few exceptions (including some lab and ethnography reports), you are probably being asked to make an argument. You must convince your audience. It is easy to forget this aim when you are researching and writing; as you become involved in your subject matter, you may become enmeshed in the details and focus on learning or simply telling the information you have found. You need to do more than just repeat what you have read. Your writing should have a point, and you should be able to say it in a sentence. Sometimes instructors call this sentence a “thesis” or a “claim.”

So, if your instructor tells you to write about some aspect of oral hygiene, you do not want to just list: “First, you brush your teeth with a soft brush and some peanut butter. Then, you floss with unwaxed, bologna-flavored string. Finally, gargle with bourbon.” Instead, you could say, “Of all the oral cleaning methods, sandblasting removes the most plaque. Therefore it should be recommended by the American Dental Association.” Or, “From an aesthetic perspective, moldy teeth can be quite charming. However, their joys are short-lived.”

Convincing the reader of your argument is the goal of academic writing. It doesn’t have to say “argument” anywhere in the assignment for you to need one. Look at the assignment and think about what kind of argument you could make about it instead of just seeing it as a checklist of information you have to present. For help with understanding the role of argument in academic writing, see our handout on argument .

What kind of evidence do you need?

There are many kinds of evidence, and what type of evidence will work for your assignment can depend on several factors–the discipline, the parameters of the assignment, and your instructor’s preference. Should you use statistics? Historical examples? Do you need to conduct your own experiment? Can you rely on personal experience? See our handout on evidence for suggestions on how to use evidence appropriately.

Make sure you are clear about this part of the assignment, because your use of evidence will be crucial in writing a successful paper. You are not just learning how to argue; you are learning how to argue with specific types of materials and ideas. Ask your instructor what counts as acceptable evidence. You can also ask a librarian for help. No matter what kind of evidence you use, be sure to cite it correctly—see the UNC Libraries citation tutorial .

You cannot always tell from the assignment just what sort of writing style your instructor expects. The instructor may be really laid back in class but still expect you to sound formal in writing. Or the instructor may be fairly formal in class and ask you to write a reflection paper where you need to use “I” and speak from your own experience.

Try to avoid false associations of a particular field with a style (“art historians like wacky creativity,” or “political scientists are boring and just give facts”) and look instead to the types of readings you have been given in class. No one expects you to write like Plato—just use the readings as a guide for what is standard or preferable to your instructor. When in doubt, ask your instructor about the level of formality they expect.

No matter what field you are writing for or what facts you are including, if you do not write so that your reader can understand your main idea, you have wasted your time. So make clarity your main goal. For specific help with style, see our handout on style .

Technical details about the assignment

The technical information you are given in an assignment always seems like the easy part. This section can actually give you lots of little hints about approaching the task. Find out if elements such as page length and citation format (see the UNC Libraries citation tutorial ) are negotiable. Some professors do not have strong preferences as long as you are consistent and fully answer the assignment. Some professors are very specific and will deduct big points for deviations.

Usually, the page length tells you something important: The instructor thinks the size of the paper is appropriate to the assignment’s parameters. In plain English, your instructor is telling you how many pages it should take for you to answer the question as fully as you are expected to. So if an assignment is two pages long, you cannot pad your paper with examples or reword your main idea several times. Hit your one point early, defend it with the clearest example, and finish quickly. If an assignment is ten pages long, you can be more complex in your main points and examples—and if you can only produce five pages for that assignment, you need to see someone for help—as soon as possible.

Tricks that don’t work

Your instructors are not fooled when you:

• spend more time on the cover page than the essay —graphics, cool binders, and cute titles are no replacement for a well-written paper.
• use huge fonts, wide margins, or extra spacing to pad the page length —these tricks are immediately obvious to the eye. Most instructors use the same word processor you do. They know what’s possible. Such tactics are especially damning when the instructor has a stack of 60 papers to grade and yours is the only one that low-flying airplane pilots could read.
• use a paper from another class that covered “sort of similar” material . Again, the instructor has a particular task for you to fulfill in the assignment that usually relates to course material and lectures. Your other paper may not cover this material, and turning in the same paper for more than one course may constitute an Honor Code violation . Ask the instructor—it can’t hurt.
• get all wacky and “creative” before you answer the question . Showing that you are able to think beyond the boundaries of a simple assignment can be good, but you must do what the assignment calls for first. Again, check with your instructor. A humorous tone can be refreshing for someone grading a stack of papers, but it will not get you a good grade if you have not fulfilled the task.

Critical reading of assignments leads to skills in other types of reading and writing. If you get good at figuring out what the real goals of assignments are, you are going to be better at understanding the goals of all of your classes and fields of study.

You may reproduce it for non-commercial use if you use the entire handout and attribute the source: The Writing Center, University of North Carolina at Chapel Hill

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Writing your assignment

The Writing your assignment resource is designed and monitored by Learning Advisers and Academic Librarians at UniSA.

The purpose of a report is to investigate an issue and 'report back' findings which allow people to make decisions or take action and depending on your course.  The report may require you to record, to inform, to instruct, to analyse, to persuade, or to make specific recommendations, so it is important to check your task instructions and identify the approach you are required to take.  Your completed report should consist of clear sections which are labelled with headings and sub-headings, and are logically sequenced, well developed and supported with reliable evidence . In this section you will learn more about writing a report, including process, structure and language use.  The report writing checklist at the end of this section can help you finalise your report.

• The main purpose of a report is usually to investigate an issue and report back with suggestions or recommendations to allow people to make decisions or take action.
• You will need to find information on the issue by reading through course materials and doing further research via the UniSA Library and relevant databases.
• Report writing requires you to plan and think, so give yourself enough time to draft and redraft, and search for more information before you complete the final version.
• The report is typically structured with an introduction, body paragraphs, a conclusion and a reference list.
• It usually has headings and subheadings to organise the information and help the reader understand  the issue being investigated, the analysis of the findings and the recommendations or implications that relate directly to those findings.
• A report can also include dot points or visuals such as graphs, tables or images to effectively present information.
• Always check the task instructions and feedback form as there might very specific requirements for the report structure.

Locate the task instructions in your course outline and/or on your course site, and use this activity to plan your approach.

• Reports overview  (pdf)
• Using headings in your writing  (pdf)
• Abstracts and introductions  (pdf)
• Writing introductions  (pdf)
• Writing paragraphs  (pdf)
• Literature reviews (pdf)
• Writing conclusions  (pdf) 
• Constructing graphs, tables and diagrams  (pdf)
• Psychology example report  (pdf)
• More example reports  (link)

Click through the slides below to learn about the key characteristics of academic writing. 

• Academic vocabulary and phrases  (pdf)
• Expressing yourself clearly and concisely  (pdf)
• Tentative language  (pdf)
• Writing objectively  (pdf)
• Academic phrasebank  - Courtesy: Uni of Manchester (link)

HW4: K-Means and Gaussian Mixture Models

Last modified: 2024-04-18 10:56

Due date: Thu Apr 18, 2024 at 11:59pm ET

Status: Released.

How to turn in: Submit PDF to https://www.gradescope.com/courses/712231/assignments/4329534/

Jump to: Problem 1   Problem 2   Problem 3   Problem 4

Questions?: Post to the hw4 topic on the Piazza discussion forums.

Instructions for Preparing your PDF Report

What to turn in: PDF of typeset answers via LaTeX. No handwritten solutions will be accepted, so that grading can be speedy and you get prompt feedback.

Please use provided LaTeX Template: https://github.com/tufts-ml-courses/cs136-24s-assignments/blob/main/unit4_HW/hw4_template.tex

Your PDF should include (in order):

• Cover page with your full name, estimate of hours spent, and Collaboration statement
• Problem 1a, 1b, 1c, 1d, 1e
• Problem 2a, 2b
• Problem 4a is OPTIONAL. Worth up to 6 points back on other parts. Total score for HW4 cannot go higher than 100%.

When you turn in the PDF to gradescope, mark each part via the in-browser Gradescope annotation tool

Problem 1: K-means walk-through

Recall that K-means minimizes the following cost function:

where each assignment variable \(r_n\) is a one-hot vector of size \(K\) .

The K-means algorithm is specified in pseudocode as:

• \(x_1, \ldots x_N\) : Training dataset
• \(\mu^0_{1:K} \) : Initial guess of cluster center locations

Consider running K-means on the following dataset of N=7 examples, in this (N, D)-shaped array

When the initial cluster locations are given by the following K=3 cluster locations, in this (K, D)-shaped array:

We'll denote these initial locations mathematically as \(\mu^0\) . Here and below, we'll use superscripts to indicate the specific iteration of the algorithm at which we ask for the value, and we'll assume that iteration 0 corresponds to the initial configuration.

We've visualized the 7 data examples (black squares) and the 3 initial cluster locations (crosses) in this figure:

Plot of the N=7 toy data examples (squares) and K=3 initial cluster locations (crosses).

Problem 1a: Find the optimal one-hot assignment vectors \(r^1\) for all \(N=7\) examples, when given the initial cluster locations \(\mu^0\) . This corresponds to executing step 1 of K-means algorithm. Report the value of the cost function \(J(x, r^1, \mu^0)\) .

Problem 1b: Find the optimal cluster locations \(\mu^1\) for all \(K=3\) clusters, using the optimal assignments \(r^1\) you found in 1a. This corresponds to executing step 2 of K-means algorithm. Report the value of the cost function \(J(x, r^1, \mu^1)\) .

Problem 1c: Find the optimal one-hot assignment vectors \(r^2\) for all \(N=7\) examples, using the cluster locations \(\mu^1\) from 1b. Report the value of the cost function \(J(x, r^2, \mu^1)\) .

Problem 1d: Find the optimal cluster locations \(\mu^2\) for all \(K=3\) clusters, using the optimal assignments \(r^2\) you found in 1c. Report the value of the cost function \(J(x, r^2, \mu^2)\) .

Problem 1e: What interesting phenomenon do you see happening in this example regarding cluster 2? How could you set cluster 2's location after part d above to better fulfill the goals of K-means (find K clusters that reduce cost the most)?

Problem 2: Relationship between GMM and K-means

Bishop's PRML textbook Sec. 9.3.2 describes a technical argument for how a GMM can be related to the K-means algorithm. In this problem, we'll try to make this argument concrete for the same toy dataset as in Problem 1.

To begin, given any GMM parameters, we can use Bishop PRML Eq. 9.23 to compute the posterior probability of assigning each example \(x_n\) to cluster \(k\) via the formula:

Now, imagine a GMM with the following concrete parameters:

• mixture weights \(\pi_{1:K}\) set to the uniform distribution over \(K=3\) clusters
• covariances \(\Sigma_{1:K}\) set to \(\epsilon I_D\) for all clusters, for some \(\epsilon > 0\)

We can leave the locations \(\mu_{1:K}\) at any valid values.

Problem 2a: Show (with math) that using the parameter settings defined above, the general formula for \(\gamma_{nk}\) will simplify to the following (inspired by PRML Eq. 9.42):

Problem 2b: What will happen to the vector \(\gamma_n\) as \(\epsilon \rightarrow 0\) ? How is this related to K-means?

Hint 2(i): Try it out concretely on the toy data from Problem 1 above.

Hint 2(ii): No need for a formal proof here. Just show you understand what happens in the limit, not why.

Problem 3: Covariances of mixtures

Background: Consider a continuous random variable \(x\) which is a vector in \(D\) -dimensional space: \(x \in \mathbb{R}^D\) .

We assume that \(x\) follows a mixture distribution with PDF \(p^{\text{mix}}\) , using \(K\) components indexed by integer \(k\) :

The \(k\) -th component has a mixture "weight" probability of \(\pi_k\) . Across all \(K\) components, we have a parameter \(\pi = [ \pi_1 ~ \pi_2 ~ \ldots ~ \pi_K]\) , whose entries are non-negative and sum to one.

The \(k\) -th component has a specific data-generating PDF \(f_k\) . We don't know the functional form of this PDF (it could be Gaussian, or something else), and the form could be different for every \(k\) . However, we do know that this PDF \(f_k\) takes two parameters, a vector \(\mu_k \in \mathbb{R}^D\) and a matrix \(\Sigma_k\) which is a \(D \times D\) symmetric, positive definite matrix. We further know that these parameters represent the mean and covariance of vector \(x\) under the pdf \(f_k\) :

Problem 3a: Prove that the covariance of vector \(x\) under the mixture distribution is given by:

where we define \(m = \mathbb{E}_{p^{\text{mix}(x)}}[x]\) .

Hint 3(i): We know a closed-form for m: \(m = \sum_{k=1}^K \pi_k \mu_k\) .

Hint 3(ii): For any random vector \(x\) , we know : \(\mathbb{E}[xx^T] = \text{Cov}(x) + \mathbb{E}[x] \mathbb{E}[x]^T\)

Problem 4: Jensen's Inequality and KL Divergence

Optional. Not required.

Background reading

Skim Bishop PRML's Sec. 1.6 ("Information Theory"), which introduces several key concepts useful for the EM algorithm, including:

• Jensen's inequality
• KL divergence

Background: Negative logarithms are convex

Consider the negative logarithm function: \(f(a) = - \log a\) , for inputs \(a > 0\) . Recall that \(f(a)\) is a convex function, because its second derivative is always positive:

Background: Jensen's inequality for negative logarithms

Now, suppose we have a random variable \(A\) that takes one of \(K\) possible values.

Define each candidate value \(a_k > 0\) , and let its associated probability be \(r_k \in [0, 1]\) . Writing the probabilities as a vector \(\mathbf{r} = [r_1, \ldots r_K]\) , we know these non-negative values must sum to one: \(\mathbf{r} \in \Delta^K\) .

We are interested in the expected value of \(f(A)\) , where \(f\) is the negative logarithm. We can derive the following bound using Jensen's inequality (see PRML textbook Eq. 1.115),

Expanding out these expectations and invoking \(f\) 's definition as the negative log (which only takes positive inputs), we have:

This bound holds for any positive vector \(\mathbf{a}\) and any probability vector \(\mathbf{r}\) .

We can visualize this Jensen bound in the following figure, using two selected points \(a_1 = 0.1\) and \(a_2 = 7.2\) .

Plot of our convex function of interest ("f", black) and its *linear interpolation* (magenta) between outputs that correspond to two inputs "a1" and "a2". Clearly, function f is a *lower bound* of its interpolation (magenta). In terms of probabilities, this means $\mathbb{E}[ f(A) ] \geq f( \mathbb{E}[A] )$

Notation setup

We'll use one-hot indicator vectors here. Let \(e_k\) denote the one-hot vector of size \(K\) where entry \(k\) is non-zero.

Define random variable \(z\) as a one-hot indicator vector of size \(K\) . So, the \(K\) possible values of \(z\) are \(\{ e_1, e_2, \ldots e_K\}\) .

Define two possible Categorical distributions over \(z\) , denoted \(q\) and \(p\) .

Each uses an all positive probability vector parameters \(\mathbf{r} \in \Delta^K_+\) and \(\mathbf{\pi} \in \Delta^K_+\) . Here \(\Delta_+^{K}\) denotes the set of \(K\) -length vectors whose sum is one and whose entries are all strictly positive .

The KL divergence from \(q\) to \(p\) is defined as:

Problem statement

Problem 4a: Consider any two Categorical distributions \(q(z)\) and \(p(z)\) that assign positive probabilities over the same size- \(K\) sample space. Show that their KL divergence is non-negative.

That is, show that \(KL( q(z) || p(z) ) \geq 0\) , or equivalently that

when \(\mathbf{r} \in \Delta^K_+\) and \(\mathbf{\pi} \in \Delta^K_+\) .

Hint: Expand the definition of KL as an expectation out so it is purely an evaluatable function of \(r\) and \(\pi\) , then use Jensen's inequality for negative logarithms.

Note: it is possible to prove the KL is non-negative even when some entries in \(r\) or \(\pi\) are exactly zero, but this requires taking some limits rather carefully, and we want you to avoid that burdensome detail. Thus, here we consider \(r\) and \(\pi\) as having all positive entries.