From 35d79a2164808bec5391c3b217e44f676f7538a6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pelletier?= Date: Sat, 19 Oct 2019 21:28:39 -0400 Subject: [PATCH] =?UTF-8?q?ajout=20POS,=20rendu=20=C3=A0=20Viterbi?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Makefile | 3 +- examen_partiel.md | 270 +++++++++++++++++++++++++++++++++++++++++++++- hmm.dot | 71 ++++++++++++ hmm.png | Bin 0 -> 151897 bytes 4 files changed, 340 insertions(+), 4 deletions(-) create mode 100644 hmm.dot create mode 100644 hmm.png diff --git a/Makefile b/Makefile index a63ccc4..72f08df 100644 --- a/Makefile +++ b/Makefile @@ -1,4 +1,5 @@ dot_compile: - dot -Tpng fsa_sheep.dot -o fsa_sheep.png + dot -Tpng -o fsa_sheep.png fsa_sheep.dot + dot -Tpng -o hmm.png hmm.dot build: examen_partiel.md dot_compile pandoc examen_partiel.md -o examen_partiel.pdf diff --git a/examen_partiel.md b/examen_partiel.md index 5e3500c..000897c 100644 --- a/examen_partiel.md +++ b/examen_partiel.md @@ -745,17 +745,281 @@ Ce type de modèle va discriminer parmi les classes possibles $P(c|doc)$ ## Régression logistique +Forme générale de la régression: + +$$ +P(y=c|x) = \sum_{i=1}^N{w_{i}f_{i}+b} +$$ + +Classificateur binaire: + +$$ +\log \left( \frac{P(y=0|x)}{P(y=1|x)} \right) = \sum_{i=1}^N{w_{i}f_{i}} +$$ + +En utilisant la fonction sigmoïde: + +$$ +\begin{aligned} +P(y=1) &= \sigma(w \cdot x + b) \\ +&= \frac{1}{1+e^{-(w \cdot x + b)}} \\ +P(y=0) &= 1- \sigma(w \cdot x + b) \\ +&= \frac{e^{-(w \cdot x + b)}}{1+e^{-(w \cdot x + b)}} +\end{aligned} +$$ + +On obtient une fonction de décision: + +$$ +\begin{aligned} +\hat{y} = +\begin{cases} +1 &, P(y=1|x)>0.5 \\ +0 &, sinon +\end{cases} +\end{aligned} +$$ + +Pour la régression multiple, on utilise la fonction softmax: + +$$ +p(y=c|x) = \frac{e^{w_c \cdot x + b_c}}{\sum_{j=1}^k e^{w_j \cdot x + b_j}} +$$ + +### Construction des attributs + +- Variables indicatrices sur une caractéristique du document +- Compteurs de mots ou de polarités +- Logarithme de la longueur du document + +Chaque attribut a un poids positif ou négatif + +### Évaluation + +Pour choisir la classe la plus probable, on peut simplifier le problème de régression: + +$$ +\begin{aligned} +\hat{c} &= arg\,max_{c \in C} P(c|x) \\ +&=arg\,max_{c \in C} \frac{\exp({\sum_{i=1}^N w_{i}f_{i}(c,x)})}{\sum_{c^{\prime} \in C}{\exp({\sum_{i=1}^N w_{i}f_{i}(c,x)})}} \\ +&=arg\,max_{c \in C} \exp({\sum_{i=1}^N w_{i}f_{i}(c,x)}) \\ +&=arg\,max_{c \in C} \sum_{i=1}^N w_{i}f_{i}(c,x) \\ +\end{aligned} +$$ + +On ignore le dénominateur et la fonction exponentielle tout en conservant l'ordre. + +### Apprentissage + +Trouver les paramètres $w_i$ et $b$ qui minimisent les erreurs du classificateur. On veut minimiser une fonction de perte ou maximiser une fonction de vraisemblance. + +Pour la régression logistique: on maximise la vraisemblance: + +$$ +\begin{aligned} +\log(P(y|x)) &= \log(\hat{y}^y(1-\hat{y}^{1-y})) +\end{aligned} +$$ + +Ce qui revient à minimiser la fonction de perte suivante: + +$$ +\begin{aligned} +L_{CE}(\hat{y},y) &= -\log(P(y|x)) \\ +&= -[y \log(\hat{y})+(1-y)\log(1-\hat{y})] \\ +L_{CE}(w,b) &= -[y \log(\sigma(w \cdot x + b))+(1-y)\log(1-\sigma(w \cdot x + b))] +\end{aligned} +$$ + +La fonction de coût est: + +$$ +Cost(w,b) = \frac{1}{m} \sum_{i=1}^m L_{CE}(w,b) +$$ + +On minimise la fonction de perte: + +$$ +\begin{aligned} +\frac{\partial L_{CE}(w,b)}{\partial w_j} &= [\sigma(w \cdot x+b)-y]x_j +\end{aligned} +$$ + +Algorithme utilisé: Descente de gradient stochastique ou avec mini-batch. + +## Conclusion + +||Régression logistique|Naive Bayes| +|---|---|---| +|Hypothèse d'indépendance|Non|Oui| +|Ajustement pour corrélation|Oui|Non| +|Interprétabilité|Oui|Non| +|Taille de l'échantillon|Grand|Petit| +|Entraînement|Algorithme complexe|Simple et rapide| # Analyse grammaticale -## Classe de mots +**Analyse grammaticale**: Attribuer une étiquette lexicale à chaque mot d’un texte + +Usages: + +- Synthèse vocale (prononciation) +- Analyse syntaxique (nom ou verbe) +- Extraction d’information (noms, relations) +- Traduction automatique (prendre le bon sens) + +## Classes de mots + +- Classes fermées + - Nombre fixe de membres: + - Prépositions et conjonctions: de, en, vers, et, mais, car,... + - Pronoms: je, tu, elle, le mien, ... + - Verbes auxiliaires: *may*, *can*, *will*, *been*, ... + - Déterminants et nombres: un, la, le, ses, mon, ... + - Particules (anglais): up, down, off, on +- Classes ouvertes + - On peut ajouter de nouveaux mots + - Noms (propres, communs, comptables, massifs) + - Verbes (action, processus) + - Adjectifs + - Adverbes (manière, temps, quantité, lieu, ...) ## POS Tags -## Analyseur +- Plusieurs ensemble d'étiquettes disponibles + - Haut niveau (N, V, ADJ, ADV, DET, PRE, PRO) + - Plus détaillé (Penn Treebank: 45 étiquettes) + +|Tag |Description |Example | +|-----|-----------------------------------------|---------------------------| +|CC |conjunction, coordinating |and, or, but | +|CD |cardinal number |five, three, 13% | +|DT |determiner |the, a, these | +|EX |existential there |there were six boys | +|FW |foreign word |mais | +|IN |conjunction, subordinating or preposition|of, on, before, unless | +|JJ |adjective |nice, easy | +|JJR |adjective, comparative |nicer, easier | +|JJS |adjective, superlative |nicest, easiest | +|LS |list item marker | | +|MD |verb, modal auxillary |may, should | +|NN |noun, singular or mass |tiger, chair, laughter | +|NNS |noun, plural |tigers, chairs, insects | +|NNP |noun, proper singular |Germany, God, Alice | +|NNPS |noun, proper plural |we met two Christmases ago | +|PDT |predeterminer |both his children | +|POS |possessive ending |'s | +|PRP |pronoun, personal |me, you, it | +|PRP$ |pronoun, possessive |my, your, our | +|RB |adverb |extremely, loudly, hard | +|RBR |adverb, comparative |better | +|RBS |adverb, superlative |best | +|RP |adverb, particle |about, off, up | +|SYM |symbol |% | +|TO |infinitival to |what to do? | +|UH |interjection |oh, oops, gosh | +|VB |verb, base form |think | +|VBZ |verb, 3rd person singular present |she thinks | +|VBP |verb, non-3rd person singular present |I think | +|VBD |verb, past tense |they thought | +|VBN |verb, past participle |a sunken ship | +|VBG |verb, gerund or present participle |thinking is fun | +|WDT |wh-determiner |which, whatever, whichever | +|WP |wh-pronoun, personal |what, who, whom | +|WP$ |wh-pronoun, possessive |whose, whosever | +|WRB |wh-adverb |where, when | +|. |punctuation mark, sentence closer |.;?* | +|, |punctuation mark, comma |, | +|: |punctuation mark, colon |: | +|( |contextual separator, left paren |( | +|) |contextual separator, right paren |) | + +Problème: Les mots ont souvent plus d'une étiquette, il faut alors déterminer la bonne. + +## Modèle HMM + +**HMM**: Hidden Markov Models + +- Approche probabiliste +- Cas spécial d'intérence bayésienne +- Modèle génératif: + - On détermine la structure de la phrase + - On choisit le mot associé à chaque étiquette + - La première étape est une chaîne de Markov et la seconde, l'extension cachée + +### Définitions + +- **Chaîne de Markov**: Automate avec des probabilités associées aux transitions. Chaque état ne dépend que du précédent. +- **Séquence d'entrée**: États suivis par l'automate. +- **Espace d'états**: Ensemble fermé d'états +- **Probabilité de transition**: Probabilité de passer de l'état $i$ à l'état $j$: $P(q_i=j|q_{i-1}=i)$ +- **Probabilité initiale**: $\pi_i$: probabilité de démarrer dans l'état $q_i$ + +### Application au POS tagging + +- Les états correspondent aux étiquettes lexicales. Ils sont cachés, ce qu'on observe, ce sont les mots. D'où le besoin d'une extension. +- Probabilité de transition $A=\lbrace a_{ij} \rbrace$: modélise la séquence d'étiquettes +- Probabilité d'émission $B=\lbrace b_i(k) \rbrace$: associe les mots aux étiquettes + +![](hmm.png) + +Quelle est la meilleure séquence d'étiquettes qui correspond à la séquence d'observations ? + +### Approche probabiliste + +- Considérer toutes les séquences possibles $t_1, t_2, \ldots, t_n$ +- Sélectionner celle qui est la plus probable: maximiser $P(t_1, t_2, \ldots, t_n|w_1,\ldots,w_n)$ +- Problème d'optimisation: + +$$ +\hat{t}_1^n = arg\,max_{t_1^n} \frac{P(w_1^n|t_1^n)P(t_1^n)}{P(w_1^n)} +$$ + +Comme le dénominateur est une constante dans cette optimisation, on peut simplifier le problème: + +$$ +\hat{t}_1^n = arg\,max_{t_1^n} \overbrace{P(w_1^n|t_1^n)}^{\text{vraisemblance}}\overbrace{P(t_1^n)}^{\text{a priori}} +$$ + +- Vraisemblance: les mots correspondent aux étiquettes +- a priori: probabilité de retrouver une séquence d'étiquettes + +Estimation: + +- Présence d'une étiquette ne dépend que de la précédente: bigrammes +$$ +P(t_1^n) \approx \prod_{i=1}^n P(t_i|t_{i-1}) +$$ +- Exemple: +$$ +P(\text{DT JJ NN} \ldots) = P(\text{DT}|\text{})P(\text{JJ}|\text{DT})P(\text{NN}|\text{JJ})\ldots +$$ +- Présence d'un mot ne dépend que de son étiquette +$$ +P(w_1^n|t_1^n) \approx \prod_{i=1}^n P(w_i|t_i) +$$ +- Exemple: +$$ +P(\text{le petit chien}|\text{DT JJ NN}) = P(\text{le}|\text{DT})P(\text{petit}|\text{JJ})P(\text{chien}|\text{NN}) +$$ +- Résultat: +$$ +\hat{t}_1^n = arg\,max_{t_1^n}P(w_1^n|t_1^n)P(t_1^n) \approx \prod_{i=1}^n \overbrace{P(w_i|t_i)}^{\text{émission}}\overbrace{P(t_i|t_{i-1})}^{\text{transition}} +$$ + +### Estimation des probabilités + +À l'aide d'un corpus étiqueté + +- Transition: $$P(t_i|t_{i-1}) = \frac{C(t_{t-1}|t_i)}{C(t_{i-1})}$$ +- Émission: $$P(w_i|t_i)=\frac{C(t_i,w_i)}{C(t_i)}$$ + +## Décodage: Algorithme de Viterbi + +Comme il n'est pas pratique d'énumérer tous les chemins possibles et d'estimer leurs probabilités, on utilise la programmation dynamique. + -## Algorithme de Viterbi ## MEMM diff --git a/hmm.dot b/hmm.dot new file mode 100644 index 0000000..fa7ca23 --- /dev/null +++ b/hmm.dot @@ -0,0 +1,71 @@ +digraph G { +ratio=1.2 +mincross = 2.0 + subgraph cluster_A { + q_0[label=0>]; + q_1[label=1>]; + q_2[label=2>]; + q_3[label=3>]; + q_4[label=4>]; + q_0->q_1[label="a01"]; + q_0->q_2[label="a02"]; + q_0->q_3[label="a03"]; + q_1->q_1[label="a11"]; + q_1->q_2[label="a12"]; + q_1->q_3[label="a13"]; + q_1->q_4[label="a14"]; + q_2->q_1[label="a21"]; + q_2->q_2[label="a22"]; + q_2->q_3[label="a23"]; + q_2->q_4[label="a24"]; + q_3->q_1[label="a31"]; + q_3->q_2[label="a32"]; + q_3->q_3[label="a33"]; + q_3->q_4[label="a34"]; + } + subgraph cluster_B1 { + B1[ + shape=plaintext + label=< + + + + + + + +
B1
P("aardvark"|MD)
P("will"|MD)
P("the"|MD)
P("black"|MD)
P("zebra"|MD)
+ >]; + } + subgraph cluster_B2 { + B2[ + shape=plaintext + label=< + + + + + + + +
B2
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P("black"|VB)
P("zebra"|VB)
+ >]; + } + subgraph cluster_B3 { + B3[ + shape=plaintext + label=< + + + + + + + +
B3
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