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CS 179: Introduction to Graphical Models
Creation date：2024-07-02 15:39:17
Belonging category：计算机computer science
Tag CS 179
Introduction to Graphical Models

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CS 179: Introduction to Graphical Models
Homework 
As a simple model of noise, suppose that whenever we transmit (or, store and retrieve) a bit, there is an
independent probability of error p that we will read the wrong bit value, i.e., if we stored a zero we read a one,
and vice versa. (This is called the “binary symmetric channel” model.)
Repetition codes. The simplest possible error correcting code is a repetition code, where we simply transmit
multiple copies of the same bit. If we mean to send a 0, we can send “000”, and for 1 send “111”. Then, if one bit
is transmitted incorrectly, we can still recover the correct value. The original “0” is called the data bit(s), while the
transmitted “000” is called the codeword. Note that any error correction forces us to send extra bits; the ratio of
the data length to the codeword length is called the rate, so this is a rate-1/3 repetition code. It is also easy to see
that we can correct one error (e.g., decode 010 → 0) by simple majority vote, but cannot correct two errors (e.g.,
110 → 1, since it is more probable this is 111 with a single bit error).
Hamming codes. A more sophisticated approach uses more carefully chosen parity bits to add redundancy. The
(7,4) Hamming code (for 4 data bits encoded into 7 code bits) is given by
0000 → 0000000 1000 → 1110000
0001 → 1101001 1001 → 0011001
where the blue digits indicate the redundant, parity check bit additions. See the table in https://en.wikipedia.
org/wiki/Hamming(7,4) code for the full values, and an elegant “venn diagram” illustration of how the data bits
and parity check bits are related. For the purpose of our problem, it is sufficient to know that (a) we transmit 7
bits for any block of 4 data bits, and (b) if we encounter an error in zero or one bits, we can recover from (correct
for) the error, while errors in two or more bits will make at least one error in the four data bits.
Problem 1: Probability
Let us analytically evaluate the probability of making an error using either no code, a repetition code, or a Hamming
code under various settings. We will send “blocks” of size B = 4 bits, and wish to recover these blocks error-free.
For the purposes of this problem, let the probability p of an error be p = 0.05.
(a) Compute the probability that, with no encoding at all, we will make at least one mistake in sending a block
of four bits.
Homework 5 UC Irvine 1/ 3
CS 179: Introduction to Graphical Models Spring 2020
(b) Now compute the probability that, with a rate 1/3 repetition code, we will make at least one error in the
block of four bits. Note that we are now sending 12 bits: 3 copies of each of the four data bits; any single
error in each of these four sets of 3 bits can be recovered from, but not two errors.
(c) Now compute the probability that, with a Hamming (7,4) code, we will make at least one error in the four
bits. (Again, we send 7 bits, and decode all 4 correctly if we have at most one error in the 7.)
You should find that the repetition code is slightly better, in terms of the probability of error; however, this
ignores the fact that it must transmit almost twice as many bits as the Hamming code. In order to make this a bit
more comparable, let us suppose that we are willing to use rate 1/2, but no lower – so, we are willing to use 8 bits
to send a block of size 4, but no more. Then, our repetition code can only afford to repeat two bits, and must send
the other two unencoded.
(d) Compute the probability of making at least one error if we use a repetition code for two bits, and send two
bits unencoded. (So: single errors in the first 3 bits and second 3 bits are recoverable, but single errors in
the last two bits are not.)
The difference between these two codes, intuitively, is that in the repetition code we introduce redundancy for
individual bits, while in the Hamming code we introduce redundancy that spans several bits. By distributing the
redundancy better across the data bits, we can improve our ability to correct for (rare) errors.
Problem 2: Simulations
Let us next set up a basic framework for encoding a data block, simulating bit errors, and then decoding and
checking for correctness. First, we will use it to confirm the Hamming result from the previous problem.
In the homework template, you will find a function encode(D, checks) that takes a data block D and a
set of parity checks checks and returns an encoded block T . Parity checks for a (7,4) Hamming code are also
provided; for example, T4 equals the parity of D0 ∧ D1 ∧ D2
, and so on. Finally, a simple greedy_decode function
can decode the codeword block. (It is written slightly more generally than necessary.)
(a) Using the provided functions, encode the data [0,1,1,0] , and verify that the resulting codeword decodes
correctly.
(b) Try decoding after introducing a single bit error, and again after introducing a second bit error. Comment on
your results.
(c) Finally, a small simulation loop (also provided) will estimate the probability of an error in blocks of size
B=4. Try this, and verify that it matches your analytic solution from Problem 1.
Problem 3: Encoding Larger Blocks
Elegant, hand designed codes like the Hamming code are used in many systems; compact disks use a class of
codes called Reed-Solomon codes. But for many systems, graphical models provide the best error correction
techniques. Convolutional codes are a type of code that works as a finite state machine, and is decoded using
dynamic programming on a Markov chain; but here we’ll look at one of the best types of codes for large blocks:
LDPC codes and loopy graphical models.
(a) Suppose that we have a large block (say, B = 100) that we wish to transmit without error. If we encode this
with 25 Hamming (7,4) codes, we will have an error if any of these 25 codes introduce an error. Making use
of your result from Problem 1(c), compute (analytically) the probability of this happening.
(b) Code provided in the template will generate a parity check array for this encoding. Run your simulation
procedure on this (larger) code and verify that your esimate matches your analytic solution.
LDPC codes use many parity checks randomly distributed through the data block in order to create a large
amount of interconnected redundancy. To ensure that every bit has some redundancy, we will generate one check
bit for each data bit, and connect it to two other randomly selected data bits. You can use the provided parity
check generator to create a random LDPC code
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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