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# 7. Sequence Mining - PowerPoint PPT Presentation

7. Sequence Mining. Sequences and Strings Recognition with Strings MM & HMM Sequence Association Rules. Sequences and Strings. A sequence x is an ordered list of discrete items, such as a sequence of letters or a gene sequence Sequences and strings are often used as synonyms

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### 7. Sequence Mining

Sequences and Strings

Recognition with Strings

MM & HMM

Sequence Association Rules

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• A sequence x is an ordered list of discrete items, such as a sequence of letters or a gene sequence

• Sequences and strings are often used as synonyms

• String elements (characters, letters, or symbols) are nominal

• A type of particularly long string text

• |x| denotes the length of sequence x

• |AGCTTC| is 6

• Any contiguous string that is part of x is called a substring, segment, or factor of x

• GCT is a factor of AGCTTC

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• String matching

• Given x and text, determine whether x is a factor of text

• Edit distance (for inexact string matching)

• Given two strings x and y, compute the minimum number of basic operations (character insertions, deletions and exchanges) needed to transform x into y

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Given |text| >> |x|, with characters taken from an alphabet A

• A can be {0, 1}, {0, 1, 2,…, 9}, {A,G,C,T}, or {A, B,…}

• A shift s is an offset needed to align the first character of x with character number s+1 in text

• Find if there exists a valid shift where there is a perfect match between characters in x and the corresponding ones in text

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Given A, x, text, n = |text|, m = |x|

s = 0

whiles ≤ n-m

ifx[1 …m] = text [s+1 … s+m]

then print “pattern occurs at shift” s

s = s + 1

• Time complexity (worst case): O((n-m+1)m)

• One character shift at a time is not necessary

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• See StringMatching.ppt and do not use the following alg

• Given A, x, text, n = |text|, m = |x|

F(x) = last-occurrence function

G(x) = good-suffix function; s = 0

whiles ≤ n-m

j = m

while j>0 andx[j] = text [s+j]

j = j-1

if j = 0

then print “pattern occurs at shift” s

s = s + G(0)

else s = s + max[G(j), j-F(text[s+j0])]

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• ED between x and y describes how many fundamental operations are required to transform x to y.

• Fundamental operations (x=‘excused’, y=‘exhausted’)

• Substitutions e.g. ‘c’ is replaced by ‘h’

• Insertions e.g. ‘a’ is inserted into x after ‘h’

• Deletions e.g. a character in x is deleted

• ED is one way of measuring similarity between two strings

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Nearest-neighbor algorithm can be applied for pattern recognition.

• Training: data of strings with their class labels stored

• Classification (testing): a test string is compared to each stored string and an ED is computed; the nearest stored string’s label is assigned to the test string.

• The key is how to calculate ED.

• An example of calculating ED

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Markov Model: transitional states

• Hidden Markov Model: additional visible states

• Evaluation

• Decoding

• Learning

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• The Markov property:

• given the current state, the transition probability is independent of any previous states.

• A simple Markov Model

• State ω(t) at time t

• Sequence of length T:

• ωT = {ω(1), ω(2), …, ω(T)}

• Transition probability

• P(ωj(t+1)| ωi(t)) = aij

• It’s not required that aij =aji

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Visible states

• VT = {v(1), v(2), …, v(T)}

• Emitting a visible state vk(t)

• P(v k(t)| ωj(t)) = bjk

• Only visible states vk (t) are accessibleand states ωi (t) are unobservable.

• A Markov model is ergodic if every state has a nonzero prob of occuring give some starting state.

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Evaluation

• Given an HMM, complete with transition probabilities aij and bjk. Determine the probability that a particular sequence of visible states VT was generated by that model

• Decoding

• Given an HMM and a set of observations VT. Determine the most likely sequence of hidden states ωT that led to VT.

• Learning

• Given the number of states and visible states and a set of training observations of visible symbols, determine the probabilities aij and bjk.

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Sequence alignment (homology) and sequence assembly (genome sequencing)

• Trend analysis

• Trend movement vs. cyclic variations, seasonal variations and random fluctuations

• Sequential pattern mining

• Various kinds of sequences (weblogs)

• Various methods: From GSP to PrefixSpan

• Periodicity analysis

• Full periodicity, partial periodicity, cyclic association rules

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• Full periodic pattern

• ABCABCABC

• Partial periodic pattern

• Pattern hierarchy

• ABC ABC ABC DE DE DE DE ABC ABC ABC DE DE DE DE ABC ABC ABC DE DE DE DE

Sequences of transactions

[ABC:3|DE:4]

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• SPADE (Sequential Pattern Discovery using Equivalence classes)

• Constrained sequence mining (SPIRIT)

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

• R.O. Duda, P.E. Hart, and D.G. Stork, 2001. Pattern Classification. 2nd Edition. Wiley Interscience.

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

a11

a22

a12

1

2

a21

a13

a32

a23

a31

3

a33

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

v1

v1

a11

a22

a12

b21

b11

v2

b22

v2

1

2

b12

b23

a21

v3

b24

b13

v3

b14

a32

a13

a23

v4

v4

a31

v3

v1

3

b33

b31

b34

b32

v2

a33

v4

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

vk

1(2)

b2k

1

1

1

1

1

…………

a12

2(2)

a22

2

2

2

2

2

…………

a32

3(2)

3

3

3

3

3

…………

.

.

.

ac2

.

.

.

.

.

.

.

.

.

.

.

.

c(2)

c

c

c

c

c

…………

t =

1

2

3

T-1

T

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

v3

v1

v3

v2

v0

0

0

0

0

0.0011

0

0.2 x 2

1

0.09

0.0052

0.0024

0

0.3 x 0.3

1

0.1 x 0.1

0

0.01

0.0077

0.0002

0

2

0.4 x 0.5

0.2

0

0.0057

0.0007

0

3

t =

0

1

2

3

4

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

3

4

5

6

7

1

2

0

/v/

/i/

/t/

/e/

/r/

/b/

/i/

/-/

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

max(T)

0

0

0

0

0

0

…………

max(1)

1

1

1

1

1

1

…………

max(3)

max(T-1)

2

2

2

2

2

2

…………

max(2)

3

3

3

3

3

3

…………

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

c

c

c

c

c

c

…………

t =

1

2

3

4

T-1

T

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)

v3

v1

v3

v2

v0

0

0

0

0

0.0011

0

1

0.09

0.0052

0.0024

0

1

0

0.01

0.0077

0.0002

0

2

0.2

0

0.0057

0.0007

0

3

t =

0

1

2

3

4

Data Mining – Sequences

H. Liu (ASU) & G Dong (WSU)