Classification Essay

177 words - 1 page

Ο απλός (naïve) Bayes ταξινομητής

Πλεονεκτήματα : Λειτουργεί με μια μικρή ποσότητα δεδομένων, χειρίζεται πολλαπλές κλάσεις
Μειονεκτήματα : Είναι ευαίσθητος ως προς τον τρόπο που έχουν προετοιμαστεί τα δεδομένα εισόδου
Συμβατός με : Ονομαστικές τιμές

Ο απλός Bayes είναι υποσύνολο της Bayesian θεωρίας αποφάσεων, οπότε θα αναφερθούμε πρώτα στη θεωρία αποφάσεων Bayes.
Έστω, P η κατανομή πιθανότητας, D ένα σύνολο στιγμιότυπων που γνωρίζουμε σε ποια κλάση ανήκουν και h μια υπόθεση (για παράδειγμα η υπόθεση τα δεδομένα D να ανήκουν σε μια συγκεκριμένη κλάση c). Ένα γνωρίζουμε την εκ των προτέρων πιθανότητα η υπόθεση h να είναι σωστή P(h), την πιθανότητα να παρατηρηθούν τα δεδομένα D, P(D) , και την εκ των υστέρων πιθανότητα να παρατηρηθούν τα δεδομένα D με την προϋπόθεση να είναι σωστή η υπόθεση h, P(D/h), τότε εφαρμόζοντας το θεώρημα Bayes μπορούμε να υπολογίσουμε την εκ των υστέρων πιθανότητα P(h/D), δηλαδή την πιθανότητα να είναι σωστή η υπόθεση h υπό την προϋπόθεση ότι παρατηρούνται τα δεδομένα D.

Ο τύπος υπολογισμού είναι :
P(h⁄D)=(P(D⁄h)∙P(h))/(P(D))
Η δυσκολία να εφαρμοστεί το θεώρημα Bayes στην μάθηση έγκειται στο γεγονός ότι πρέπει να γνωρίζουμε πολλές τιμές πιθανοτήτων. Για αυτό τον λόγο χρησιμοποιούμε μια απλουστευμένη εκδοχή της μάθησης κατά Bayes, τον απλό Bayes. Στον απλό Bayes γίνεται η παραδοχή πως όλα τα χαρακτηριστικά είναι ανεξάρτητα μεταξύ τους. Θα περιγράψουμε πως λειτουργεί ο απλός ταξινομητής Bayes. Κάθε στιγμιότυπο Χ του προβλήματος αποτελείται από ένα σύνολο γνωρισμάτων x1, x2¬,…,xm. Σε ένα πρόβλημα m κλάσεων c1, c2, …, cm, όταν δίνεται ένα άγνωστο στιγμιότυπο Χ για το οποίο δεν γνωρίζουμε σε ποια κλάση ανήκει, ο ταξινομητής προβλέπει ότι το Χ ανήκει στην κλάση με την μεγαλύτερη εκ των υστέρων πιθανότητα. Ο ταξινομητής αναθέτει το στιγμιότυπο Χ στην κλάση ci, αν και μόνο αν ισχύει :
P(ci / X) > P(cj / X) για 1≤j≤m ,j≠i

Έτσι μεγιστοποιείται η πιθανότητα P(ci / X), η οποία με βάση το θεώρημα Bayes δίνεται από τη σχέση:
P(c_i⁄X)=(P(X⁄c_i )∙P(c_i))/(P(X))
Στον παραπάνω τύπο το Ρ(Χ) είναι σταθερό για όλα τα στιγμιότυπα, οπότε το μόνο που χρειάζεται να μεγιστοποιηθεί είναι ο αριθμητής. Όμως ο υπολογισμός του όρου P(X/ci) είναι δαπανηρός, οπότε για να μειώσουμε το υπολογιστικό κόστος υποθέτουμε ότι υπάρχει μια ανεξαρτησία ως προς την κατανομή των κλάσεων. Δηλαδή, για την κλάση κάποιου στιγμιότυπου θεωρούμε πως οι τιμές των γνωρισμάτων του στιγμιότυπου είναι ανεξάρτητες μεταξύ τους. Προκύπτει η παρακάτω σχέση:
P(X⁄c_i )=∏_(k=1)^n▒〖P(x_k 〗∕c_i)
Οι πιθανότητες P(X/ci) μπορούν να υπολογιστούν κατευθείαν από τα στιγμιότυπα εκπαίδευσης. Προκειμένου να υπολογίσουμε ένα νέο στιγμιότυπο Χ, υπολογίζουμε την πιθανότητα P(X⁄c_i )∙P(c_i) για κάθε κλάση ci. Το στιγμιότυπο ανατίθεται στην κλάση ci , αν και μόνο αν ισχύει :
P(X/c¬i)∙P(ci) > P(X/cj)∙P(cj) για 1≤j≤m ,j≠i
Συνεπώς, για το στιγμιότυπο Χ επιλέγεται η κλάση με την μεγαλύτερη πιθανότητα.

Γενική προσέγγιση του απλού Bayes
1)Συλλογή : Με οποιαδήποτε μέθοδο
2) Προετοιμασία: Οι...

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