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PHYS3888 MATLAB

创建日期：2024-05-24 15:11:24

所属分类：物理Physical

标签： PHYS3888

MATLAB

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PHYS3888

How to do this exam
You may use any existing resource to solve the problems in this paper, including the lecture
notes, code you’ve already written, Matlab documentation (e.g., use doc if you are unsure of
syntax), and Google searches.
However, during the period in which this exam is active, YOU MAY NOT have any
communication with any other person about any aspect of it.
Make sure that you have downloaded the code and data that accompanies this exam.
Note that the difficulty of questions is unevenly distributed through the paper. It can be a good
strategy to skip questions you are initially stuck on.
Please leave yourself plenty of time (approximately 1 hour) to upload a pdf containing all of
your answers. You may use a mix of scanned images of pen-and-paper work and any typeset
results, or images.
You are encouraged to provide code where relevant (e.g., which can allow for partial marks to be
awarded when final results are incorrect). To aid readability, please present code in a fixed-width
font (e.g., like Monaco).
You have a total of 24 hours, so do not risk missing the deadline due to scanning/upload issues.
Statement of individual contribution
[Statement] (×0 multiplier on final score if incomplete)
To have a valid exam, you must accurately complete this question:
Copy out and sign the following statement: “The results submitted in this exam are my own
work only. I had no communication with any other person about any aspect of it.”
Setting
We are in Mordor: a land of orcs, ogres, and desolate evil. And yet, amidst the moral devastation,
there is one shining light: interdisciplinary physics.
1
Part A: Nazgul (25 marks)
Nazgul fly about Mordor, searching the land below for The One Ring.
We have obtained a dataset of their flight pattern, which you can load in as:
load('examData.mat','nazgulFlight');
The variable, nazgulFlight, contains x- and y- coordinates (columns 1 and 2, respectively)
corresponding to the set of locations that the Nazgul visited. Coordinates are measured in
kilometers, with Mount Doom at the origin. Locations were visited in the order presented down
the rows (e.g., the first location is the first row of nazgulFlight, the second location the second
row, etc.).
A1: Birds-eye view (5 marks)
[A1.1] (1 mark)
What are the (x, y) coordinates of the first location the Nazgul visited?
[A1.2] (1 mark)
How many other locations did the Nazgul subsequently visit?
[A1.3] (1 mark)
Run:
A = nazgulFlight(1:100,:);
State what the rows and columns of A correspond to.
[A1.4] (2 marks)
Plot the trajectory of Nazgul flight for the first 100 locations visited. Plot each location as its x
and y coordinates. Use a marker to denote locations and use lines to join consecutively visited
locations. Provide your plot.
A2: Power-laws in Nazgul flight (12 marks)
Compute a set of distances, dist, using the computeDistances function provided:
dist = computeDistances(nazgulFlight);
2
[A2.1] (2 marks)
By reading the lines of code in computeDistances, identify what information is stored in dist.
If N locations were visited, state why dist will be a vector of length N − 1.
[A2.2] (1 mark)
Notice the calculation sqrt(sum((y - x).ˆ2)) in computeDistances.m. Write down this
calculation in the form of a mathematical equation: d(x, y) = . . ..
[A2.3] (3 marks)
Using the function provided, binLogLog.m, plot the distribution of dist on logarithmic axes
with 10 bins.
[A2.4] (1 mark)
What feature of your plot indicates that the distances, dist, follow a power-law distribution,
p(x) ∝ x−γ?
[A2.5] (2 marks)
From the logarithmic binning above, estimate the power-law exponent, γ, to two significant
figures. Provide your code with your result.
[A2.6] (3 marks)
Speculate briefly on the scale-free distribution of distances in the context of Nazgul flight:
1. Explain qualitatively what the trajectory of Nazgul flight would look like if the distribution
of dist instead had a characteristic scale (was a unimodal distribution).
2. Speculate about a possible advantage of the scale-free distribution of dist for the Nazgul
to search for The One Ring.
A3: Nazgul are sooo random (8 marks)
In this section we aim to assess the structure of Nazgul flight patterns by comparing the distances,
dist, to those computed from random flights between the same set of locations.
Randomize the ordering of the locations visited as nazgulRand:
rng('default');
N = size(nazgulFlight,1);
nazgulRand = nazgulFlight(randperm(N),:);
[A3.1] (4 marks)
Use computeDistances to compute distances between the randomized locations, nazgulRand,
storing your result in the variable, distRand.
Plot the probability distribution of distRand as a histogram on linear axes, and again on
logarithmic axes.
Explain any relevant features of your plot, focusing on any important differences from the
power-law distribution of dist.
Use your results to assess whether the flight data in nazgulFlight are consistent with random
flights between the same set of locations.
[A3.2] (4 marks)
3
We can further investigate non-random structure in Nazgul flight by analyzing the angular
dependence of successive flight directions. Recall that for a random walk, the direction flown to
next location (from location xi → xi+1) is independent of any prior information, including the
direction from the previous location to the current location (xi−1 → xi).
Extract a vector of angles between successive locations in nazgulFlight, as θi (e.g., using the
atan2 function). Produce a scatter plot of θi+1 as a function of θi.
Provide your code and your plot.
Use your plot to assess whether angles between successive locations are independent.
Part B: Hebbian orc antics (21 marks)
The orcs guarding a key tower in Mordor enter 6× 6 binary patterns into the security control
pad shown below.
Recall that the binary Hopfield network sets network weights according to the Hebbian learning
rule,
wij =
N∑
n=1
x
(n)
i x
(n)
j , (1)
for neural states x ∈ {−1, 1} across a set of N memories.
Note that binary patterns across each key in the 6× 6 grid can be coded as a 36-long binary
vector, allowing passcodes to be represented as memories in a Hopfield network containing 36
neurons. Orcs, being quite forgetful, have trouble remembering their codes and rely on such a
Hopfield network to correct any errors in partial memories of their passcodes.
B1: The Hebbian Learning Rule (7 marks)
The Hebbian learning rule gives high positive weights, wij , to pairs of correlated neurons.
[B1.1] (1 mark)
In this context, describe what it means for two neurons to be correlated.
[B1.2] (1 mark)
Identify what it means about a pair of neurons, i and j, if wij < 0.
4
[B1.3] (2 marks)
If the learning rule, Eq. (1), were modified to take an average rather than a sum, as
wij =
1
N
N∑
n=1
x
(n)
i x
(n)
j ,
what effect would it have on the network weights and resulting behavior? Briefly explain your
answer.
[B1.4] (3 marks)
Recall the rule for updating neural states, x ∈ {−1, 1}, as a function of the neuron activation, a:
x(a) = 1 if a ≥ 0 ,
x(a) = −1 if a < 0.
Consider an alternative rule, in which neural states instead take values x ∈ {−1, 2}, and the
update rule is modified changed to (change shown bold):
x(a) = 2 if a ≥ 0 ,
x(a) = −1 if a < 0.
Briefly explain what effect (if any) this change would have on:
1. The computation of correlations, wij , and
2. The network’s dynamics.
B2: Samwise Sabotage (7 marks)
The hobbit, Samwise, watches carefully as the head orc, Murzush, retrieves his passcode. Samwise
notes down Murzush’s correct passcode as x(Murzush)i , with xi ∈ {−1, 1}. Samwise decides to
sabotage the recall device by entering a new memory that will make Murzush’s passcode less
stable in the network. His aim is to input the most deceptive possible memory using the strategy
of neutralizing any correlations that are informative of Murzush’s passcode, x(Murzush)i .
First consider the case where the network weights have been computed, from Eq. (1), using just
a single memory, x(Murzush)i . Samwise thus aims to construct a second memory, x
(Sam)
i , with the
aim of cancelling out the correlation structure of Murzush’s memory. The optimal such memory
would, after computing Eq. (1) across both memories, result in wij = 0 for all i, j.
[B2.1] (2 marks)
A physicist from UNSW suggests that the problem is trivial. To cancel out Murzush’s memory,
you should construct a new memory, x(Sam)i , that is the opposite of Murzush’s memory, as
x
(Sam)
i = −x(Murzush)i . Briefly explain why this would be a poor strategy. Include in your answer
whether any worse strategies exist (and, if so, describe them).
[B2.2] (3 marks)
Consider the first four elements of Murzush’s memory, shown below. The first two neurons are
‘on’ (x1 = 1, x2 = 1), and the next two neurons are ‘off’ (x3 = −1, x4 = −1). All pairwise
products, xixj , are annotated.
5
We can score the quality, Q, of possible states, x(Sam)i , as the number of unique weights, wij ,
i 6= j, for which wij = 0 (computed after both Murzush and Samwise’s memories).
Write down a set of the four states of x(Sam)i (i.e., for i = 1, . . . , 4), for which Q is maximal.
State whether it is possible to completely neutralize the correlations in this four-state system,
and describe whether your results provide some intuition about how the Hopfield network can
robustly store different memories.
[B2.3] (2 marks)
Consider an alternative setting in which, as well as Murzush’s passcode, the network weights were
computed from the passcodes of many other orcs. Samwise has knowledge of: (i) the resulting
weights, wij , computed from all the passcodes; and (ii) Murzush’s passcode.
Relative to having knowledge of just Murzush’s passcode, qualitatively describe how Samwise
could use knowledge of wij from the full set of orc memories to more effectively sabotage
Murzush’s passcode.
B3: The Weight is Over (7 marks)
Samwise sneaks up to the device and extracts the Hebbian weight matrix, computed using Eq.
(1) from the passcodes of many orcs.
Load the data as:
load('examData.mat','weightMatrix')
[B3.1] (2 marks)
Does weightMatrix encode a directed or undirected network? Briefly state why.

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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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