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### 7. Sequence Mining

Sequences and Strings

Recognition with Strings

MM & HMM

Sequence Association Rules

Data Mining by H. Liu, ASU

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 interchangeably
- String elements (characters, letters, or symbols) are nominal
- A particularly long string is called 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 by H. Liu, ASU

Recognition with Strings

- String matching
- Given x and text, determine whether x is a factor of text

- Edit distance
- Given two strings x and y, compute the min number of basic operations (character insertions, deletions and exchanges) needed to transform x into y

Data Mining by H. Liu, ASU

String Matching

- Given |text| >> |x|, each discrete character is 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 each character in x and the corresponding one in text

Data Mining by H. Liu, ASU

Naïve String Matching

- Given alphabet 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 by H. Liu, ASU

Boyer-Moore String Matching

- 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 by H. Liu, ASU

Edit Distance

- ED between x and y describes how many fundamental operations are required to transform x to y.
- Fundamental operations (x=‘excused’, y=‘exhausted’)
- Substitutions, ‘c’ is replaced by ‘h’
- Insertions, ‘a’ is inserted into x after ‘h’
- Deletions, a character in x is deleted

- ED is one way of measuring similarity between two strings

Data Mining by H. Liu, ASU

Classification using ED

- 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 by H. Liu, ASU

Hidden Markov Model

- Markov Model: transitional states
- Hidden Markov Model: additional visible states
- Evaluation
- Decoding
- Learning

Data Mining by H. Liu, ASU

Markov Model

- 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 by H. Liu, ASU

Hidden Markov Model

- 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 by H. Liu, ASU

Three Key Issues with HMM

- 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 by H. Liu, ASU

Sequence Association Rule Mining

- SPADE (Sequential Pattern Discovery using Equivalence classes)
- Constrained sequence mining (SPIRIT)

Data Mining by H. Liu, ASU

Bibliography

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

Data Mining by H. Liu, ASU

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