# 自然語言處理(上)

## Chap01 Introduction

### Applications of NLP

- Machine translation
- google translate

- Speech recognition
- Siri

- Smart input method
- ㄐㄅㄈㄏ → 加倍奉還

- Sentiment(情感) analysis
- Information retrieval
- Question Anwering
- Turing Test

- Optical character recognition (OCR)

### Critical Problems in NLP

- Ambiguity(不明確性)
**The most important thing in NLP**- Lexical(字辭)
`current`

: noun or adjective`bank`

(noun): money or river

- Syntactic(語法)
`[saw [the boy] [in the park]]`

`[saw [the boy in the park]]`

- Semantic(語義)
- “John kissed his wife, and so did Sam”. (Sam kissed John’s wife or his own?)
- agent(施事) vs. patient(受事)

- ill-form(bad form)
- typo
- grammatical errors

- Robustness
- various domain
- 網路語言：取材於方言俗語、各門外語、縮略語、諧音、甚至以符號合併以達至象形效果等等
- emoticon(表情符號)

### Main Topics in Large-Scale NLP Design

- Knowledge representation
- organize and describe linguistic knowledge

- Knowledge strategies
- use knowledge for efficient parsing, ambiguity resolution, ill-formed recovery

- Knowledge acquisition
- setup and maintain knowledge base systematically and cost-effectively

- Knowledge integration
- consider various knowledge sources effectively

### Models

用演算法來轉換文字結構，以產生最後結果

- State machines
- Rule-based approaches
- Logical formalisms
- Probabilistic models

### Approaches

NLP start from 1960, **statictics method** wins after 1995

Rule-based approach

- Advantages
- No need database
- Easy to incorporate with knowledge
- Better generalization to a unseen domain
- Explainable and traceable
- easy to understand

- Disadvantages
- Hard to maintain consistency (at different situation)
- Hard to handle uncertain knowledge (define uncertainty factor)
- irregular information

- Not easy to avoid redundancy
- Knowledge acquisition is time consuming

Corpus(語料庫)-based approach

- Advantages
- Knowledge acquisition can be automatically achieved by the computer
- Uncertain knowledge can be objectively quantified(知識可被量化)
**Consistency and completeness**are easy to obtain- Well established statistical theories and technique are available

- Disadvantages
- Generalization is poor for small-size database
- Unable to reasoning
- Hard to identify the effect of each parameter
- Build database is time consuming

- Corpus
- Brown Corpus (1M words),Birmingham Corpus (7.5M words), LOB Corpus (1M words), etc
- Corpora(語料庫(複數)) of special domains or style
- Newspaper, Bible, etc

- Information in Corpora
- pure-text corpus
- language usage of real world, word distribution, co-occurrence

- tagged corpus
- parts of speech, structures, features

- pure-text corpus

Hybrid approach

- Use rule-based approach when
- there are rules that have good coverage
- it can be governed by a small number of rules

- extensional knowledge is important to the system

- there are rules that have good coverage
- Use corpus-based approach when
- Knowledge needed to solve the problem is huge and intricate
- A good model or formulation exists

### Implementation

Natural Language Toolkit(NLTK): Open source Python modules, linguistic data and documentation for research and development in natural language processing

features

- Corpus readers
- Tokenizers
- whitespace, newline, blankline, word, treebank, sexpr, regexp, Punkt sentence segmenter

- Stemmers
- Porter, Lancaster, regexp

- Taggers
- regexp, n-gram, backoff, Brill, HMM, TnT

- Chunkers
- regexp, n-gram, named-entity

- Metrics
- accuracy, precision, recall, windowdiff, distance metrics, inter-annotator agreement coefficients, word association measures, rank correlation

- Estimation
- uniform, maximum likelihood, Lidstone, Laplace, expected likelihood, heldout, cross-validation, Good-Turing, Witten-Bell

- Miscellaneous
- unification, chatbots, many utilities

## Chap02 Overall Pictures

Knowledge Categories

- Phonology(聲音，資料來源)
- Morphology(詞性)
- Syntax(句構)
- Semantics(語義)
- Pragmatics(句子關聯，語用學)
- Discourse(篇章分析，話語)

### Morphology(Structure of words)

- part-of-speech(POS) tagging(詞性標註, lexical category)
- find the roots of words
- e.g., going → go, cats → cat

### Syntax(structure of sentences)

- Context-Free Grammars(CFG)
- Chomsky Normal Form(CNF)
- can only use following two rules
`non-terminal → terminal`

`non-terminal → non-terminal non-terminal`

- can only use following two rules
- dependency
- local dependency
- words near together would probably have the same syntax rule

- long-distance dependency
- wh-movement(疑問詞移位)
- What did Jennifer buy? → 什麼 (助動詞) 珍妮佛 買了

- 分裂句 Right-node raising
- [[she would have bought] and [he might sell]] shares

- Argument-cluster coordination
- I give [[you an apple] and [him a pear]]

**challenge for some statistical NLP approaches (like n-grams)**

- wh-movement(疑問詞移位)

- local dependency

### Semantics(meaning of individual sentences)

- semantic roles
- agent(主詞)
- patient(受詞)
- instrument(工具)
- goal(目標)
- Beneficiary(受益)
- He threw the book(patient) at me(goal)
- John sold the car for a friend(beneficiary)

- Subcategorizations(次分類)
- 及物、不及物動詞

- Semantics can be divided into two parts
- Lexical Semantics
- 上下位，同義(反義)，部分-整體

- Composition Semantics
- 合起來的意義與單一字意義不同

- Lexical Semantics
- Implementation
- WordNet®(large lexical database of English)
- Thesaurus(索引典)
- 同義詞詞林
- 廣義知網中文詞知識庫(E-HowNet)
- FrameNet

#### FrameNet

A dictionary of more than 10,000 word senses, 170,000 manually annotated sentences

Frame Semantics(Charles J. Fillmore)

- the meanings of most words can be more understood by semantic frame
- Including description of a type of event, relation, or entity and the participants in it
- Example:
`apply_heat`

frame- When one of these words appear, this frame will be applied
`Fry(炸)`

,`bake(烘)`

,`boil(煮)`

, a`nd broil(烤)

- Frame elements: Cook, Food, Heating_instrument and Container
- a person doing the cooking (Cook)
- the food that is to be cooked (Food)
- something to hold the food while cooking (Container)
- a source of heat (Heating_instrument)

- [
`Cook`

the boys] … GRILL [`Food`

fish] [`Heating_instrument`

on an open fire]

- When one of these words appear, this frame will be applied

### Pragmatics(how sentences relate to each other)

- explain what the speaker really expressed
- Understand the scope of
- quantifiers
- speech acts
- discourse analysis
- anaphoric relations(首語重複)

- Anaphora(首語重複) and Coreference(指代)
- 張三是老師,他教學很認真,同時,他也是一個好爸爸。
- Type/Instance: “老師”/“張三”, “一個好爸爸”/“張三”

- crucial to
**information extraction** - Dialogue Tagging

### Discourse Analysis

Example

- 1a: 佛羅倫斯哪個博物館在1993年的爆炸事件中受到破壞？
- 1b: 這個事件哪一天發生？
- 問句1b「這個事件」，指的是問句1a「1993年的爆炸事件」

### Summary

From The Three (and a Half) Futures of NLP

- NLP is
**Notation Transformation**(e.g. English → Chinese), with some information(POS, syntatic, senmatic…) added - Much NLP is engineering
- select and tuning learning performance

- Knowledge is crucial in language-related research areas, but providing a large scaleknowledge base is difficult and costly
- Knowledge Base
- WordNet
- FrameNet
- Wikipedia
- Dbpedia
- Freebase
- Siri
- Google Knowledge Graph

- Knowledge Base
- Hierarchy of transformations(由深至淺)
- pragmatics, writing style
- deeper semantics, discourse
- shallow semantics, co-reference
- syntax, POS(part-of-speech)

- shallow semantics, co-reference

- deeper semantics, discourse
- 分析時由淺至深

- pragmatics, writing style

#### Analysis

#### NLP progress by now

## Chap03 Collocations(搭配詞)

- 多個單字組合成一個有意義的語詞，其意義無法從各個單字中推得
- e.g. black market

- Subclasses of Collocations
- compound nouns
- telephone box and post office

- idioms
- kick the bucket(氣絕)

- Light verbs(輕動詞)
- 動詞失去其意義，需要和其他有實質意義的詞作搭配
- e.g. The man took a walk(walk, not take) vs The man took a radio(take)

- Verb particle constructions(語助詞) or Phrasal Verbs(詞組動詞, 短語動詞, V + 介系詞)
- take in = deceive, look sth. up

- proper names
- San Francisco

- Terminology(專有名詞)

- compound nouns
- Classification
- Fixed expressions
- in short (O)
- in shorter or in very short(X)

- Semi-fixed expressions(可用變化形)
- non-decomposable idioms
- kick the bucket (O)
- he kicks the bucket(O)
- the bucket was kicked (X)

- compound nominals
- car park, car parks

- Proper names

- non-decomposable idioms
- Syntactically-Flexible Expressions
- decomposable idioms
- let the cat out of the bag

- verb-particle constructions
- light verbs

- decomposable idioms
- Institutionalized Phrases (習慣用法)
- salt and pepper(○) pepper and salt(×)
- traffic light
- kindle excitement(點燃激情)

- Fixed expressions

### Collocation detection

- by Frequency
- by Mean and Variance of the distance between focal word (焦點詞) and collocating word(搭配詞)
- Hypothesis Testing
- Mutual Information

#### By Frequency

- 找出現機率大的bigrams
- not always significant
- 篩選可能為組合詞的詞性組合(Ex. adj+N)

- The collocations found
- What if a word have two possible collocations?(strong force, powerful force)

#### By Mean and Variance of the distance between focal word(焦點詞) and collocating word(搭配詞)

- many collocations consist of more flexible relationships
- frequency is not suitable

- Define a collocational window
- e.g., 3-4 words before/after

- assemble every word pair as a bigram
- e.g., A B C D → AB, AC, AD, BC, BD, CD

- computes the mean and variance of the offset between the two words
- 變異數愈低，代表兩個字之間的位置關聯愈固定

- z-score $z = {freq - \mu \over \sigma}$: the strength of a word pair

#### Hypothesis Testing(假設檢定)

- Even high frequency and low variance can be accidental
- null hypothesis(虛無假設, H0)
- 設 w1 and w2 is completely independent → w1 and w2 不是搭配詞
- P(w1w2) = P(w1)P(w2)

- 設 w1 and w2 is completely independent → w1 and w2 不是搭配詞
- 假設H0為真，計算這兩個字符合H0的機率P
- 若P太低則否決H0(→ 是搭配詞)

- Two issues
- Look for particular patterns in the data
- How much data we have seen

- 種類包括：t檢驗，Z檢驗，卡方檢驗，F檢驗

##### t-test

- Test whether
**distributions of two groups**are**statistically different**or not- H0 → (w1, w2) has no differnece with normal distribution
- considering
**variance**of the data - formula

- Calculate t by alpha level and degree of freedom
- alpha level
`α`

: confidence- in normal distribution，α = 95%落在mean±1.96std之間, α = 99%落在mean±2.576std之間
- If t-value is larger than 2.576, we say the two groups
**are different**with 99% confidence

- degree of freedom: number of sample-1
- total = number of two groups-2

- alpha level
**t↑ → more difference → more possible to reject null hypothesis → more possible to be collocation**- Example: “new” occurs 15,828 times, “companies” 4,675 times, “new companies” 8 times, total 14,307,668 tokens
**Null hypothesis: the occurrences of new and companies are independent(not collocation)**- H0 mean = P(new, companies) = P(new) x P(companies) = $\frac{15828 \times 4678}{14307668^2} = 3.615 \times 10^{-7}$
- H0 var = p(1-p) ~= p when p is small
- tvalue = $\frac{5.591 \times 10^7 - 3.615\times 10^7}{\sqrt{\frac{5.591 \times 10^7}{14307668}}} = 0.999932$
- 0.999932 < 2.576, we cannot reject the null hypothesis
- new company are not collocation

- the above words are possible collocations

Hypothesis testing of differences

- useful for lexicography
- which word(strong, powerful) is suitable to modify “computer”?

- T-test can be used for
**comparison of the means of two normal populations**- H0 is that the average difference is 0 (u = 0)
- v1 and v2 are the words we are comparing (e.g., powerful and strong), and w is the collocate of interest(e.g., computers)

##### Chi-Square test

- T-test assumes that probabilities are normally distributed
- not really

- Chi-Square: compare
**observed frequencies**with**expected frequencies**- If
**difference between observed and expected frequencies**is large, we can reject H0

- If
- Example
- expected frequency of “new companies”: $\frac{8+4667}{14307668} \times \frac{8+15820}{14307668} \times 14307668$ = 5.2
- chi-square value = χ^2
- When α=0.05, χ^2=3.841
- Because 1.55<3.841, we cannot reject the null hypothesis. new companies is not a good candidate for a collocation

- Comparison with T-test
- The 20 bigrams with the highest t scores in the test corpus are also the 20 bigrams with the highest χ^2 scores
- χ^2 is appropriate for large probabilities(t-test is not because of normality assumption)

- Application: Translation
- find similarity of word pairs

### Likelihood Ratios

- Advantage compared with Chi-Square test
- more appropriate for sparse data
- easier to interpret

Likelihood Ratios within single corpus

- examine two hypothesis
- H1: occurrence of w2 is independent of the previous occurrence of w1(null hypothesis)
- H2: occurrence of w2 is dependent of the previous occurrence of w1

- maximum likelihood estimate
- using binomial distribution
- $b(k;n, x) = \binom nk x^k \times (1-x)^{n-k}$
- only different at probability
- $L(H_1) = b(c_{12};c_1, p)b(c_2-c_{12}; N-c_1, p)$
- $L(H_2) = b(c_{12};c_1, p_1)b(c_2-c_{12}; N-c_1, p_2)$

- log of likelihood ratio λ
- use D = -2logλ to examine the significance of two words, which can asymptotically(漸近) chi-square distributed

Likelihood Ratios between two or more corpora

- useful for the discovery of
**subject-specific collocations**

### Mutual Information

- measure of
**how much one word tells us about the other**(information theory) - pointwise mutual information(PMI)
- MI是在獲得一個隨機變數的資訊之後，觀察另一個隨機變數所獲得的「資訊量」（單位通常為位元）

- mutual information = Expection(PMI)
- works bad in sparse environments
- As the perfectly dependent bigrams get rarer, their mutual information increases →
**bad measure of dependence**

- As the perfectly dependent bigrams get rarer, their mutual information increases →
**good measure of independence**- when perfect independence, I(x, y) = 0

- New formula: $C(w1w2)I(w1w2)$
- With MI, bigrams composed of low-frequency words will receive a higher score than bigrams composed of high-frequency words

Chain rule for entropy

- $H(X,Y) = H(Y|X) + H(X) = H(X|Y) + H(Y)$
- Conditional entropy $H(Y|X)$ expresses how much
**extra information**you still need to supply on average to communicate Y when X is known - $H(X)-H(X|Y) = H(Y)-H(Y|X)$
- This difference is called the
**mutual information between X and Y**(X, Y共同擁有的information) - MI is not similar to chi-square

- This difference is called the

Entropy

- Entropy: uncertainty of a variable
- Incorrect model’s cross entropy is larger than correct model’s
- 正確model和猜測model的差別：P(X)logP(X) ↔ P(X)logPM(X)

- Entropy Rate: Per-word entropy(= sentence entropy / N)
- Cross Entropy:
**average informaton**needed to**identify an event drawn from the set**between two probability distributions- 交叉熵的意義是用該模型對文本識別的難度，或者從壓縮的角度來看，每個詞平均要用幾個位來編碼

- Joint entropy H(X, Y): average information needed to
**specify both values**

### Case Study

Emotion Analysis

- Non-verbal Emotional Expressions
- text (raw) and emoticons(表情符號) (tag) form collection
- appearance of an emoticon is a good emotion indicator to sentences
- check the dependency of each word in sentences
- Evaluation
- Use top 200 lexiconentries as features
- Tag={Positive, Negative}
- LIBSVM

## Chap04 N-gram Model

N-grams are token sequences of length N

applications

- Automatic speech recognition
- Author Identification
- Spelling correction
- Grammatical Error Diagnosis
- Machine translation

### Counting

- Example:
*I do uh main-mainly business data processing*- Should we count “uh”(pause) as tokens?
- What about the repetition of “mainly”? Should such do-overs count twice or just once?(重複)
- The answers depend on the application
- “uh” is not needed for query
- “uh” is very useful in dialog management

- Corpora: Google Web Crawl
- 1,024,908,267,229 English tokens
- 13,588,391 wordform types
- even large dictionaries of English have only around 500k types. Why so many here?
- Numbers
- Misspellings
- Names
- Acronyms(縮寫)

### Language model

Language models assign a probability to a word sequence

Ex. `P(the mythical unicorn) = P(the) * P(mythical | the) * P(unicorn | the mythical)`

- Markov assumption: the probability of a word depends only on
**limited previous words**- Generalization: n previous words, like bigram, trigrams, 4-grams……
- As we increase the value of N, the accuracy of model increases

N-Gram probabilities come from a training corpus

overly narrow corpus: probabilities don’t generalize

overly general corpus: probabilities don’t reflect task or domain

maximum likelihood estimate

- maximizes the probability of the training set T given the model M
- Suppose the word “Chinese” occurs 400 times in a corpus
- MLE estimate is 400/1000000 = .004
- makes it most likely that “Chinese” will occur 400 times in a million word corpus

- P([s] I want englishfood [s]) = P(I|[s]) x P(want|I) x P(english|want) x P(food|english) x P([s]|food) = 0.000031$

Usage

capture some knowledge about language

- World Knowledge
- P(english food|want) = .0011
- P(chinese food|want) = .0065

- syntax
- P(to|want) = .66
- P(eat| to) = .28
- P(food| to) = 0

- discourse
`P(i|<s>)`

= .25

- World Knowledge
Shannon’s Method: use language model to generate random sentences

- Shakespeare as a Corpus
- 99.96% of the possible bigrams were never seen (have zero entries in the table)

- This is the biggest problem in language modeling

- Shakespeare as a Corpus

### Evaluating N-Gram Models

- Extrinsic(外在的) evaluation
- Compare performance of the application within two models
- time-consuming

- Intrinsic evaluation
- perplexity
- But get poor approximation unless the test data looks just like the training data

- not sufficient to publish

- perplexity

Standard Method

- Train → Test
- A dataset which is different from our training set, but both drawn from the same source
- use evaluation metric(Ex. perplexity)
- Example
- Create a fixed lexicon L, of size V
- At text normalization phase,
**any training word not in L changed to UNK**(unknown word token) **count UNK like a normal word**

- At text normalization phase,
- When testing, also use UNK counts for any word not in training
- The best language model is one that best predicts an unseen test set

- Create a fixed lexicon L, of size V

### perplexity(複雜度)

- Definition
- notion of surprise
- The more surprised the model is, the lower probability it assigned to the test set
**Minimizing perplexity is the same as maximizing probability**

- notion of surprise
- probability of a test set, as normalized by the number of words
- 物理意義是單詞的編碼大小
- 如果在某個測試語句上，語言模型的perplexity值為2^190，說明該句子的編碼需要190bits

- relate to entropy
- Perplexity(p, q) = $2^{H(p,q)}$
- p is the test sample distribution, and q is the distribution of language model
- do everything in log space to avoid underflow and calculate faster

### word entropy

- word entropy for English
- 11.82 bits per word [Shannon, 1951]
- 9.8 bits per word [Grignetti, 1964]

- word entropy in medical language
- 11.15 bits per word

## Chap05 Statistical Inference

- Statistical Inference：
**taking some data**(generated by unknown distribution) and then**making some inferences(推理，推測)**about this distribution - three issues
**Dividing the training data into equivalence classes****Finding a good statistical estimator for each equivalence class****Combining multiple estimators**

### Form Equivalence Class

- Classification Problem
**predict target feature**based on various**classificatory features**- reliability v.s. discrimination
- The more classes, the more discrimination, but estimation feature is not reliable

- Independent assumption
- assume data is nearly independent

- Statistical Language Modeling
- Language Model: P(W)
- LM does not depend on acoustics
- the acoutstics probability is constant(calculated by data)

- n-gram model
- assume equivalence classes are previous n-1 words
- Markov Assumption: Only the prior n-1 local context affects the next entry
- (n-1)th Markov Model or n-gram

**Building n-grams**- Remove punctuation(標點) and normalize text
- Map out-of-vocabulary words to unknown symbol(UNK)
- Estimate conditional probabilities by joint probabilities
- P(n | n-2, n-1) = P(n-2, n-1, n) / P(n-2, n-1)

### Finding statistical estimator

- Goal: derive
**probability estimate of target feature**based on observed data - Running Example
- From n-gram data P(w1,..,wn), predict P(wn|w1,..,wn-1)

- Solutions
- Maximum Likelihood Estimation
- Laplace’s, Lidstone’s and Jeffreys-Perks’ Laws
- Held Out Estimation
- Cross-Validation
- Good-Turing Estimation

- Model combination
- Combine models (unigram, bigram, trigram, …) to use the most precise model available
- interpolation(內插) and back-off(後退)
- use higher order models when model has enough data
- back off to lower order models when there isn’t enough data

Terminology

- Ex.
`[s] a b a b a`

- N = 5 (
`[s]a,ab,ba,ab,ba`

) - B = 3 (
`[s]a,ab,ba`

) - C(w1, w2…) = 某N-gram(Ex. ab)出現次數
- r = 某N-gram出現頻率
- Nr = 有幾個「出現r次的N-gram」
- Tr = 出現r次的N-gram，在test data出現的總次數

- N = 5 (

#### (1) Maximum Likelihood Estimation

- usually unsuitable for NLP
- sparseness of the data(a lot of word sequences with zero probabilities)

- Use Discounting or Smoothing technique to improve
- Smoothing
- Smoothing is like Robin Hood: Steal from the rich and give to the poor
- no word sequences has 0 probability

- Discounting
- assign some probability to unseen events

- Smoothing

#### (2) Laplace’s, Lidstone’s and Jeffreys-Perks’ Laws

- Laplace: add 1 to every count
- gives far too much probabilities to unseen events
- Usage: In domains where the number of zeros isn’t so huge
- pilot studies
- document classification

- Lidstone and Jeffreys-Perks: add a smaller value λ < 1
- B:number of bins
- Expected Likelihood Estimation (ELE)(Jeffreys-Perks Law)
- λ=1/2

#### (3) Held Out Estimation

- compute frequencies in training data and held out data
- Tr / Nr = Average frequency of training frequency r N-grams
- estimate frequency(value for validation)
- 計算出現在training corpus r次的bigrams，在held-out corpus出現的次數稱為Tr。 因為這種bigrams有Nr個，因此平均為Tr / Nr

- Tr / Nr = Average frequency of training frequency r N-grams
- Validation
- if the probabilities estimated on training data are close to those on held-out data, it’s a good language model
- 參考資料–validation in machine learning

- Prevent Overtraining(overfit)
- test on different data

Training portion and testing portion (5-10% of total data)

- Held out data (validation data)
- available training data: real training data(90%) + held out data(10%)

- Instead of presenting a single performance figure, testing result on each smaller sample
- Using t-test to reject the possibility of an accidental difference

#### (4) Cross-Validation

If data is not enough, use each part of the data both as training data and held out data

- Deleted Estimation
- $N_r^a$ = number of n-grams occurring r times in the
**a th part**of the training data - $T_r^{ab}$ = number of occurs in part b of 「bigrams occurs r times in part a」

- $N_r^a$ = number of n-grams occurring r times in the

- Split the training data into K sections
- For each section k: hold-out section k and compute counts from remaining K-1 sections; compute Tr(k)
- Estimate probabilities by averaging over all sections

estimate frequency of deleted estimation

#### (5) Good-Turing Estimation

- 用出現一次的來預測沒出現過的
- 若出現次數>k，不變，否則套用公式
- renormalize to sum = 1

- Simple Good-Turing
- replace any zeros in the sequence by linear regression:
`log(Nc) = a+blog(c)`

- replace any zeros in the sequence by linear regression:
- after good-turing

explaination from stanford NLP course

- when use leave-one-out validation, the possibilities of unseen validation data is $\frac{N_1}{N}$(when thing-saw-once is the validation data), the possibilities of validation data have been seen K times is $\frac{(k+1)N_{k+1}}{N}$
- Josh Goodman’s intuition: assume You are fishing, and caught 10 carp,3 perch,2 whitefish, 1 trout, 1 salmon, 1 eel = 18 fish
- P(unseen) = N1/N0 = N1/N = 3/18
- C(trout) = $2 \times N_2/N_1$ = $2 \times (1/3)$ = 2/3
- P(trout) = 2/3 / 18 = 1/27

- for large k, often get zero estimate, so do not change the count
- C(the) = 200000, C(a) = 190000, $C*(the) = (200001)N_{200001} / N_{200000} = 0 (because N_{200001} = 0)$

#### (6) Absolute Discounting

從所有非零N-gram中拿出λ，平均分配給所有未出現過的N-gram

### Combining Estimator

Combination Methods

- Simple Linear Interpolation(內插)(finite mixture models)
- Ex. trigram, bigram and unigram
- More generally, λ can be a function of (wn-2, wn-1, wn)

- use Expectation-Maximization (EM) algorithm to get weights

- Ex. trigram, bigram and unigram
- General Linear Interpolation
- general form for a linear interpolation model
- weights are a function of the history

- Katz’s Backing-Off
- choose proper order to train model (base on training data)
- If the n-gram appeared more than k times
- use MLE estimate and discount it

- If the n-gram appeared k times or less
- use an estimate from
**lower-order n-gram**

- use an estimate from

- If the n-gram appeared more than k times
- back-off probability
- $P_{Dis}(w_n|w_{n-2},w_{n-1})$ is specific discounted estimate. e.g., Good-Turing or Absolute Discounting
- unseen trigram is estimated by bigram and β
**β(wn-2, wn-1)**and**α**are chosen so that sum of probabilities = 1

- more genereal form

- choose proper order to train model (base on training data)

Most usual approach in large speech recognition: trigram language model, Good-Turing discounting, back-off combination

## Chap06 Hidden Markov Models(HMM)

- statistical tools that are useful for NLP
**part-of-speech-tagging**- We construct “Visible” Markov Models in training, but treat them as Hidden Markov Models when tagging new corpora

- model a
**state sequence**(perhaps through time)**of random variables**that have dependencies- 狀態(state)並不是直接可見的，但受狀態影響的某些變量(output symbol)則是可見的
- known value
**output symbols(words)**字詞- probabilistic function of state relation 和state相關的機率函式

- unknown value
**state(part-of-speech tags)**目前的state，即POS tag

- rely on 2 assumptions
- Let X=(X1, …, XT) be a sequence of random variables, X is markov chain if

- Limited Horizon
- a word’s tag only depends on
**previous**tag(state只受前一個state影響)

- a word’s tag only depends on
- Time Invariant
- the dependency does not change over time(轉移矩陣不變)

Description

- initial state π, state = Q, Observations = O, transition matrix = A, output(observation) matrix = B
- $a_{ij}$ = probability of state $q_i$ transition to state $q_j$
- $b_i(k)$ = probability of observe output symbol $O_k$ when state = $q_i$

### 3 problems of HMM

中文解說：隱馬可夫鏈

- 評估（Evaluation）：what is probability of the observation sequence given a model? (P(Observes|Model))
- Used in model improvement
- Used in classification
- Word spotting in speech recognition, language identification, speaker identification, author identification……
- Given an observation, compute P(O|model) for all models

- Use
**Forward algorithm**to solve it

- 解碼（Decoding）：Given an observation sequence and model, what is the
**most likely state sequence**? (P(States|Observes, Model)) 下一個state是什麼- Used in tagging (tags=hidden states)
- Use
**Viterbi algorithm**to solve it

- 學習（Learning）：Given an observation sequence, infer the best model parameters (argmax(Model) P(Model|Observes))
- 「fill in model parameters that make the observation sequence most likely」
- Used for building HMM Model from data
- Use
**EM(Baum-Welch, backward-forward algorithm)**to solve it

### Solutions of HMM problem

#### Forward

- simply sum of the probability of each possible state sequence
- Direct evaluation
- time complexity = $(2T+1) \times N^{T+1}$ -> too big

- Use dynamic programming
- record the probability of subpaths of the HMM
- The probability of longer subpaths can be calculated from shorter subpaths
- similar to Viterbi: viterbi use MAX() instead of SUM()

Forward Algorithm

- $α_t(i)$ = probability of state = qi at time = t
- α的求法：將time = t-1 的 α 值，乘上在time = t時會在qi state的機率，並加總
- 順向推出所有可能的state sequence會產生此observation的機率和, 即為此model會產生此observation的機率
- Σ P($O_1, O_2, O_3$ | possible state sequence) = P($O_1, O_2, O_3$ | Model)

Example:Urn(甕)

- genie has two urns filled with red and blue balls
- genie selects an urn and then draws a ball from it
- The urns are hidden
- The balls are observed

- After a lot of draws
- know the distribution of colors of balls in each urn(B matrix)
- know the genie’s preferences in draw from one urn or the next(A matrix)

- assume output (observation) is Blue Blue Red (BBR)
- Forward: P(BBR|model) = 0.0792 (SUM of all possible states’ probability)

Viterbi

compute

**the most possible path**$v_t(i)$ =

**most possible path probability**from time = 0 to time = t, and state = qi at time = tViterbi in Urn example

def viterbi(obs, states, start_p, trans_p, emit_p):

V = [{}]

path = {}

# Initialize base cases (t == 0)

for y in states:

V[0][y] = start_p[y] * emit_p[y][obs[0]]

path[y] = [y]

# Run Viterbi for t > 0

for t in range(1,len(obs)):

V.append({})

newpath = {}

for y in states:

(prob, state) = max([(V[t-1][y0] * trans_p[y0][y] * emit_p[y][obs[t]], y0) for y0 in states])

# ↑ find the most possible state transitting to given state y at time=t

V[t][y] = prob

newpath[y] = path[state] + [y]

# newpath(at time t) can overwrite path(at time t-1)

path = newpath

(prob, state) = max([(V[len(obs) - 1][y], y) for y in states])

return (prob, path[state])

#### Backward

- Useful for parameter estimation

Description

- Backward variables β, which are the total probability of seeing the rest of the observation sequence($O_t to O_T$) given state qi at time t
- 初始化β：令t=T時刻所有狀態的β為1

- 由後往前計算
- 如果要計算某observation的概率，只需將t=1的後向變量相加

#### Backward-Forward

- We can locally maximize model parameter λ, by an iterative hill-climbing known as Baum-Welch algorithm(=Forward-Backward) (by EM Algorithm structure)

Forward-Backward Algorithm

- find which** state transitions(A matrix)** and
**symbol observaions(B matrix)**were**probably used the most** - By
**increasing the probability of those**, we can get a better model which gives a higher probability to the observation sequence - transition probabilities and path probabilities are both require each other to calculate
- use A matrix to calculate path probabilities
- need path probabilities to update A matrix
- use EM algorithm

EM algorithm (Expectation-Maximization)

- 迭代算法，它的最大優點是簡單和穩定，但容易陷入局部最優
- (隨機)選擇參數λ0，找出在λ0下最可能的狀態，計算每個訓練樣本的可能結果的概率，再
**重新估計新的參數λ**。經過多次的迭代，直至某個收斂條件滿足為止 - Urn Example
- update transition matrix A ($a_{12}, a_{11}$ … )
- P(1→2) = 0.0414
- P(1→1) = 0.0537
- normalize: P(1→2)+P(1→1) = 1, P(1→2) = 0.435 …

- 若state數目多的時候，計算量過大…
- 用backward, forward
- 前面用forward, 後面用backward

- update transition matrix A ($a_{12}, a_{11}$ … )

Combine forward and backward

- Let ξt be the probability of being in state i at time t and state j at time t+1,
**given observation and model λ** - use not-quite-ξ to get ξ because
- P(O|λ) → problem1 of HMM 的答案 → 用forward解

- 見上方backward, forward同時使用之圖
- ξ可用來計算transition matrix

Summary of Forward-Backward

- Initialize λ=(A,B)
- Compute α, β, ξ using observations
- Estimate new λ’=(A,B)
- Replace λ with λ’
- If not converged go to 2

## Chap07 Part-of-Speech Tagging

alias: **parts-of-speech**, **lexical categories**, **word classes**, **morphological classes**, **lexical tags**

- Noun, verb, adjective, preposition, adverb, article, interjection, pronoun, conjunction
- preposition(P)
- of, by, to

- pronoun(PRO)
- I, me, mine

- determiner(DET)
- the, a, that, those

Usage

Speech synthesis

Tag before parsing

Information extraction

Finding names, relations, etc.

Machine Translation

Closed class

- the class that is hard to add new words
- Usually function words (short common words which play a role in grammar)
- prepositions: on, under, over,…
- particles: up, down, on, off, …
- determiners: a, an, the, …
- pronouns: she, who, I, …
- conjunctions: and, but, or, …
- auxiliary verbs: can, may should, …
- numerals: one, two, three, third, …

Open class

- new ones can be created all the time
- For English: Nouns, Verbs, Adjectives, Adverbs

Choosing Tagset: Ex. “Penn TreeBank tagset”, 45 tag

Methods for POS Tagging

- Rule-based tagging
- ENGTWOL: ENGlish TWO Level analysis

- Stochastic: Probabilistic sequence models
- HMM (Hidden Markov Model)
- MEMMs (Maximum Entropy Markov Models)

- Transformation-Based Tagger (Brill)

### Rule-Based Tagging

- Assign all possible tags to each word
- Remove tags according to set of rules
- Typically more than 1000 hand-written rules

### Hidden Markov Model tagging

- special case of Bayesian inference
- Foundational work in computational linguistics

- related to the “noisy channel” model that’s the basis for ASR, OCR and MT
- Decoding view
- Consider all possible sequences of tags
- choose the tag sequence which is most possible given the observation sequence of n words w1…wn

- Generative view
- This sequence of words must have resulted from some hidden process
- A sequence of tags (states), each of which emitted a word

- $t^n_1$(t hat), which is the most possible tag
- use viterbi to get tag

Evaluation

- Overall error rate with respect to a gold-standard test set
- Error rates on particular tags/words
- Tag confusions, Unknown words…
- Typically accuracy reaches 96~97%

Unknown Words

- Simplest model
- Unknown words can be of any part of speech, or only in any open class

- Morphological and other cues
- ~ed: past tense forms or past participles

### Maximum entropy Markov model (MEMM)

Maximum Entropy Model

- MaxEnt: multinomial(多項式) logistic regression
- Used for sequence classification/sequence labeling
- Maximum entropy Markov model (MEMM)
- a common MaxEnt classifier

Exponential(log-linear) classifiers

- Combine features linearly
- Use the sum as an exponent
- Example

Maximum Entropy Markov Model

- MaxEnt model
- classifies
**a**observation into**one**of discrete classes

- classifies
- MEMM
- augmentation(增加) of the basic MaxEnt classifier
**assign a class to each element in a sequence**

POS tagging from MaxExt to MEMM

- include some source of knowledge into the tagging process
- The simplest approach
- run the local classifier and
**feature is classifier from the previous word** - Flaw
- It makes a hard decision on each word before moving on the next word
- cannot use information from the later words

- run the local classifier and

**discriminative model**(判別模型)

- Compute the posterior P(Tag|Word) directly to decide tag
- 求解條件機率分佈 P(y|x) 預測 y → 求P(tag|word)來取得tag
- 不考慮聯合機率分佈 P(x, y)
- 對於諸如分類和回歸問題，由於不考慮聯合機率分佈，採用判別模型可以取得更好的效果

HMM and MEMM(順推和逆推的差別)

- Unlike HMM, MEMM can condition on any
**useful feature of observation**- HMM: state is the fiven value
- MEMM: observation is the given value

- viterbi function for MEMM

### Transformation-Based Learning of Tags

- Tag each word with its most frequent tag
- Construct a list of transformations that
**improve the initial tag** - trigger environment: at the limited number of words before/after

- Trigger by tags
- Trigger by word
- Trigger by morphology(詞法學)

<! – ====================分水嶺：尚未分類========================== –>

### Zipf’s Law (long tail phenomenon)

a word’s frequency is approximately inversely proportional to its rank in the word distribution list

單詞出現的頻率與它在頻率表裡的排名成反比:

頻率最高的單詞出現的頻率大約是出現頻率第二位的單詞的2倍

#### Jelinek-Mercer Smoothing

interpolate(插值) between bigram and unigram

because if p(eat the) = 0 and p(eat thou) = 0

it still must consider that p(eat the) > p(eat thou)

because p(the) > p(thou)

so p(eat the) = N * p(the | eat) + (1-N) * p(the | thou)

## Language Model: Applications

### Query Likelihood Model

given a query 𝑞, rank the probability 𝑝(𝑑|q)

So the following arguments are equivalent:

1.𝑝𝑑𝑞: find the document 𝑑 that is most likely to be relevant to 𝑞

2.𝑝𝑞𝑑: find the document 𝑑 that is most likely to generate the query 𝑞

Typically, unigram LMs are used in IR(information retrieval)

Retrieval does not depend that much on sentence structure

### Dependence Language Model

Relax the independence assumption of unigram LMs

Do not assume that the dependency only exist between **adjacent** words

Introduce a hidden variable: “linkage” 𝐿

Ex.

skipped….

### Proximity Language Model

Proximity: how close the query terms appear in a document

the closer they are, the more likely they are describing the same topic or concept

### Positional Language Model

Position: define a LM for each position of a document, instead of the entire document

Words closer to a position will contribute more to the language model of this position

### Speech Recognition

- The “origin” of language models
- used to restrict the search space of possible word sequences
- requires higher order models: knowing previous acoustic is important!
- Speed is important!
- N-gram LM with modified Kneser-Ney smoothing is extensively used

### Machine Translation (MT)

- Decoding: given the probability model(s), find the best translation
- Similar role as in speech recognition:
**eliminate unlikely word sequences** - Higher order Kneser-Ney smoothed n-gram LM is widely used
- NNLM-style models tend to outperform standard back-off LMs
- Also significantly speeded up in (Delvin et al, 2014)

## 參考資料

- HHChen 課堂講義
- SDLin 講義
- Stanford NLP course
- 52nlp