Geometrie und Bedeutung: Kap 5

Transkript

1 : Kap November 2011

2 Übersicht Der Begriff des Vektors Ähnlichkeits Distanzfunktionen für Vektoren Skalarprodukt Eukidische Distanz im R n

3 What are vectors I Domininic: Maryl: Dollar Po Euro Yen Dm = (6, 20, 15, 100) Dollar Po Euro Yen Mm = (15, 16, 20, 0)

4 What are vectors II Addition: Dm + Mm = (56, 36, 35, 100) Skalare Multiplikation: 1.2(Dm + Mm) = (67.2, 43.2, 42, 120)

5 What are vectors III Definition A mathematical operator is said to be commutative if a b = b a for all possible a and b.

6 What are vectors IV Kombination von Addition skalarer Multiplikation 1.2Dm + 1.2Mm = 1.2(Dm + Mm)

7 Koordinaten Dimesionen I (a 1, a 2, a 3, a 4 ) Dollar a 1, Po a 2, Euro a 3, Yen a 4... This key telling us which coordinate is which is called a basis for the vector space. For example, the basis for our currency vectors is the set {Dollar, Po, Euro, Yen}

8 Koordinaten Dimesionen II Definition The dimension of a vector space is the number of coordinates needed to specify a given point uniquely.

9 Problems Many possible meanings of the different terms are absent from this fragment bank can also refer to a river bank, and most of the time cream is more likely to mean dairy product than a 1960 s rock band. The term bass does have more than one meaning in our 3 documents it means a kind of musical instrument in Document 1 and a kind of fish in Document 2. If a user is only interested in one of these meanings (and in this case, it s unlikely that they d be interested in both at the same time), how are we to enable users to serch for only documents containing this meaning of bass. The term guitar appears only once in Document 1 but it s very possible that a user searching for the term guitar would also find articles containing the term Les Paul relevant, since Les Paul is a make of guitar.

10 Example Document 1 Document 2 Document 3 bank bass commercial cream guitar fishermen money bank = (0, 0, 4) bass = (2, 4, 0) money = (0, 1, 2)

11 Cosine Similarity I sim(bank, money) = (0 0) + (0 1) + (4 2) = 4 sim(bass, money) = (2 0) + (4 1) + (0 2) = 4 sim(bank, bass) = (0 2) + (0 4) + (4 0) = 0

12 Vector Space R n Definition The vector space R n is made up of all lists of the form (a 1, a 2,..., a n ) where each of these entries is a real number. By definition each point in R n has n coordinates, so the dimension of R n is the number n

13 Scalar Product Definition Let a = (a 1,..., a n ) and b = (b 1,..., b n ) be vectors in R n. Their scalar product a b is given by the formula a b = a 1 b 1 + a 2 b a n b n a b = a i b i

14 Euclidian Distance I a = (a 1, a 2, a 3 ), b = (b 1, b 2, b 3 ) p = b 1 a 1, q = b 2 a 2, r = b 3 a 3 d(a, b) = (b 1 a 1 ) 2 + (b 2 a 2 ) 2 + (b 3 a 3 ) 2 d(a, b) = ( (b i a i ) 2

15 Euclidian Distance II bank = (0, 0, 4) guitar = (1, 0, 0) money = (0, 1, 2) d(bank, money) = (0 0) 2 + (0 1) 2 + (4 2) 2 = d(bank, guitar) = (1 + 16) =

16 Euclidian Distance III But d(guitar, bass) 4.12 d(guitar, fishermen) 3.16

17 Norms and Unit I Definition d(0, a) = (ai ) 2 The norm or the length of the vector a is written a and is defined to be a = (ai ) 2 = a a

18 Norms and Unit II Document 1 Document 2 Document 3 bank bass commercial cream guitar fishermen money cos(guitar, cream) = = 1 cos(guitar, bass) = = cos(guitar, fishermen) = = 0

19 Norms and Unit III cos(a, b) = cos(guitar, bass) = a b a b guitar bass guitar bass =??? cos(guitar, fishermen) =??? guitar = (1, 0, 0) bass = (2, 4, 0) fishermen = (0, 3, 0)

20 Norms and Unit IV bank bass cmmercial cream guitar fishermen money bank bass commercial cream guitar fishermen money

21 Document I Doc 1 = (1, 0, 0) Doc 2 = (0, 1, 0) Doc 3 = (0, 0, 1) cos(bass, Doc 1 ) = cos(bass, Doc 2 ) = cos(bass, Doc 3 ) = 0

22 Document II bass+guitar = (0.477, 0.894, 0)+(1, 0, 0) = (1.477, 0.894, 0) Normalisiert: (0.855, 0.518, 0) cos(bass + guitar, Doc 1 ) = cos(bass + guitar, Doc 2 ) =?? cos(bass + guitar, Doc 3 ) =??