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### chapter 1:measuring information

In the beginning was the Word...

情報理論：日本語，英語で隔年開講

- 今年度は日本語で授業を行うが，スライドは英語のものを使用
Information Theory: English and Japanese, alternate years

- the course will be taught in Japanesein this year
- video-recorded English classesLecture Archives 2011
- Slides are in English
- this slide can be found athttp://apal.naist.jp/~kaji/lecture/
- test questions are given in both of Japanese and English

Information Theory

Information Theory （情報理論）

- is founded by C. E. Shannon in 1948
- focuses on mathematical theory of communication
- gave essential impacts on today’s digital technology
- wired/wireless communication/broadcasting
- CD/DVD/HDD
- data compression
- cryptography, linguistics, bioinformatics, games, ...
In this class, we learn basic subjects of information theory.

(half undergraduate level + half graduate school level)

Claude E. Shannon

1916-2001

class plan

This class consists of four chapters(+ this introduction):

- chapter 0: the summary and the schedule of this course
(today)

- chapter 1: measuring information
- chapter 2: compact representation of information
- chapter 3: coding for noisy communication
- chapter 4: cryptography

what’s the problem?

To understand our problem, date back to 1940s...

- Teletype (電信) was widely used for communication.
- Morse code: dots ( ∙ ) and dashes ( − )

- dot = 1 unit long, dash = 3 units long
- 1 unit silence between marks
- 3 units silence between letters, etc.

10111000111000000010101010001110111011100011101110001

They already had “digital communication”.

machinery for information processing

No computers yet, but there were “machines”...

Teletype model 14-KTR, 1940

http://www.baudot.net/teletype/M14.htm

Enigma machine

http://enigma.wikispaces.com/

- They could do something complicated.
- The transmission/recording of messages were...
- inefficient...messages should be as short as possible
- unreliable...messages are often disturbed by noises
The efficiency and the reliability were two major problems.

the model of communication

A communication system can be modeled as;

C.E. Shannon, A Mathematical Theory of Communication,

The Bell System Technical Journal, 27, pp. 379–423, 623–656, 1948.

channel,

storage medium,

etc...

encoder,

modulator,

codec, etc...

what is the “efficiency”?

A communication is efficient if the size of B is small.

- subject to A = D, or A ≈ D
- with, or without noise (B ≠ C, or B = C)

A

B

C

D

problem one: efficiency

Example: You need to record the weather of Tokyo everyday.

- weather = {sunny, cloudy, rainy}
- You can use “0” and “1”, but you cannot use blank spaces.

- 2-bit record everyday
- 200 bits for 100 days

weather

sunny

cloudy

rainy

codeword

00

01

10

0100011000

Can we shorten the representation?

better code?

The code B gives shorter representation than the code A.

- Can we decode the code B correctly?
- Yes, as far as the sequence is processed from the beginning.

- Is there a code which is more compact than this code B?
- No, and Yes(→ next slide).

weather

sunny

cloudy

rainy

code A

00

01

10

code B

00

01

1

code A...0100011000

code B...010001100

think average

Sometimes, events are not equally likely...

weather

sunny

cloudy

rainy

probability

0.5

0.3

0.2

code A

00

01

10

code B

00

01

1

code C

1

01

00

- with the code A, 2.0 bit / event(always)
- with the code B,
20.5 + 20.3 + 10.2 = 1.8 bit / event in average

- with the code C,
10.5 + 20.3 + 20.2 = 1.5 bit/ event in average

the best code?

Can we represent information with 0.00000000001 bit per event?

...No, maybe.

- It is likely that there is a “limit” which we cannot get over.
- Shannon investigated the limit mathematically.
→ For this event set, we need 1.485 or more bit per event.

weather

sunny

cloudy

rainy

probability

0.5

0.3

0.2

This is the amount of information

which must be carried by the code.

class plan in April

- chapter 0: the summary and the schedule of this course
- chapter 1: measuring information
- We establish a mathematical means to measure
information in a quantitative manner.

- We establish a mathematical means to measure
- chapter 2: compact representation of information
- We learn several coding techniques which give compact representation of information.

- chapter 3: coding for noisy communication
- chapter 4: cryptography

what is the “reliability”?

A communication is reliable if A = Dor A ≈ D.

- the existence of noise is essential (B ≠ C)
- How small can we make the size of B?

A

B

C

D

problem two: reliability In the usual conversation, we sometimes use phonetic codes.

Communication is not always reliable.

- transmitted information ≠ received information

ABCADC

ABCABC

- Errors of this kind are unavoidable in real communication.

ABC

Alpha, Bravo, Charlie

あさひの「あ」

いろはの「い」

Alpha, Bravo, Charlie

ABC

phonetic code

- A phonetic code adds redundant information.
- The redundant part helps correcting possible errors.
→use this mechanism over 0-1 data, and we can correct errors!

Alpha

the real

information

redundant (冗長な) information

for correcting possible errors

redundancy

Q. Can we add “redundancy” to binary data?

A. Yes, use parity bits.

A parity bit is...

a binary digit which is to make the number of 1’s in data even.

- 00101 → 001010 (two 1’s → two 1’s)
- 11010 → 110101 (three 1’s → four1’s)
One parity bit may tell you that there are odd numbers of errors,

but not more than that.

to correct error(s)

basic idea: use several parity bits to correct errors

Example: Add five parity bits to four-bits data (a0,a1, a2, a3).

codeword =

(a0,a1, a2, a3, p0,p1, q0,q1, r)

a0

a1

p0

a2

a3

p1

This code corrects one-bit error,

but it is too straightforward.

q0

q1

r

class plan in May

- chapter 0: the summary and the schedule of this course
- chapter 1: measuring information
- chapter 2: compact representation of information
- chapter 3: coding for noisy communication
- We study practical coding techniques for finding and correcting errors.

- chapter 4: cryptography
- We review techniques for protecting information from intensive attacks.

schedule

- April

(Mon)

Tue

10

17

24

Thu

12

19

26

- report (quiz):
- will be assigned by
- the end of April

- May

×

01

08

15

22

29

03

10

17

24

31

×

×

- June

04

05

- test:
- questions given in English/Japanese

statistics in 2011: A ... 51 / B ... 20 / C ... 18 / did not pass ... 13

motivation

“To tell plenty of things, we need more words.”

...maybe true, but can you give the proof of this statement?

We will need to...

- measure informationquantitatively (定量的に測る)
- observe the relation between the amount of information
and its representation.

Chapter 1 focuses on the first step above.

the uncertainty (不確実さ)

Information tells what has happened at the information source.

- Before you receive information, there is much uncertainty.
- Afteryou receive information, the uncertainty becomes small.
the difference of uncertainty the amount of information

FIRST, we need to measure the uncertainty of information source.

this difference indicates

the amount of information

much

uncertainty

small

uncertainty

Before

After

the definition of uncertainty

The uncertainty is defined according to the statistics (統計量),

BUT,

we do not have enough time today....

In the rest of today’s talk,

we study two typical information sources.

- memoryless & stationary information source
- Markov information source

assumption

In this class, we assume that...

- an information source produces one symbol per unit time
(discrete-time information source)

- the set of possible symbols is finite and countable (有限可算)
(digital information source)

Note however that, in the real world,

there are continuous-time and/or analogueinformation sources.

- cf. sampling & quantization

Preliminary (準備)

- Assume a discrete-time digital information source S:
- M = {a1, ..., ak}... the set of symbols of S
(S is said to be a k-ary information source.)

- Xt...the symbol which S produces at time t
- The sequence X1, ..., Xn is called a message produced by S.
Example: S = fair dice

- M = {a1, ..., ak}... the set of symbols of S

if the message is

, then

memoryless & stationary information source

A memoryless & stationary information source satisfies...

- memorylesscondition:
“A symbol is chosen independently from past symbols.”

- stationary condition: for any t
“The probability distribution is time invariant.”

trial 1

trial 2

trial 3

:

123456...

ajcgea...

gajkfh...

wasdas...

:

- memoryless = 無記憶
- stationary = 定常

the same probability distribution

memoryless & stationary information source

Examples of memoryless & stationary information source:

- the “dice” example, coin toss, ...
information sources with memory:

- English text:
- wireless communication...burst noise
not-stationary information sources:

- weather...P(snow) is large in winter
- and more?

Markov information source

Markov information source

- a simple model of information source with memory
- The choice of the next symbol depends on
at most m previous symbols

(m-th order Markov source)

Andrey Markov

1856-1922

m = 0 memoryless source

m = 1 simple Markov source

Xt

S

R

1-bit register

Example of (simple) Markov sourceS ... memoryless & stationary source with P(0) = q, P(1) = 1 – q

- if Xt–1 = 0, then R = 0:
- S = 0 Xt = 0 ...PXt|Xt–1(0 | 0) = q
- S = 1 Xt = 1... PXt|Xt–1(1 | 0) = 1 – q

- if Xt–1 = 1, then R = 1:
- S = 0 Xt = 1... PXt|Xt–1(1 | 1) = q
- S = 1 Xt = 0... PXt|Xt–1(0 | 1) = 1 – q

Xt

S

R

1-bit register

1 / 1–q

0

1

0 / q

1 / q

0 / 1–q

Markov source as a finite state machinem-th order k-ary Markov source:

- The next symbols depends on previous m symbols.
- The model is having one of km internal states.
- The state changes when a new symbol is generated.
finite state machine

generated

symbol

probability

B

C

two important propertiesirreducible (既約) Markov source:

- We can move to any state from any state.

this example is NOT irreducible

- aperiodic (非周期的) Markov source:
- We have no periodical behavior (strict discussion needed...).

this example is NOT aperiodic

A

B

irreducible + aperiodic = regular

example of the regular Markov source

1/0.1

0/0.9

A

B

0/0.8

1/0.2

start from the state 0

start from the state 1

...

...

...

...

...

...

time

P (state=A)

P (state=B)

time

P (state=A)

P (state=B)

1

1.0

0.0

1

0.0

1.0

2

0.8

0.2

2

0.9

0.1

3

0.89

0.11

3

0.88

0.12

4

0.889

0.111

4

0.888

0.112

converge (収束する) to

thesame probabilities

stationary probabilities

0/0.9

A

B

0/0.8

1/0.2

computation of the stationary probabilities- t : P(state = A) at time t
- t : P(state = B) at time t
t+1 = 0.9t + 0.8t

t+1 = 0.1t + 0.2t

t+1+ t+1= 1

- If t and tconverge to and , respectively, then
- we can putt+1=t=and t+1=t=.
- = 0.9 + 0.8
- = 0.1 + 0.2
- += 1

=8/9, =1/9

0/0.9

A

B

0/0.8

1/0.2

Markov source as a stationary sourceAfter enough time has elapsed...

a regular Markov source can be regarded as a stationary source

=8/9, =1/9

0 will be produced with probabilityP(0) = 0.9 + 0.8 = 0.889

1 will be produced with probabilityP(1) = 0.1 + 0.2 = 0.111

summary of today’s class

- overview of this course
- motivation
- four chapters

- typical information sources
- memoryless & stationary source
- Markov source

1/0.6

0/0.4

0/0.5

1/0.2

B

C

0/0.8

1/0.5

exercise- Determine the stationary probabilities.
- Compute the probability that 010 is produced.

This is to check your understanding.

This is not a report assignment.

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