probabilistic suffix trees
DESCRIPTION
Probabilistic Suffix Trees. CMPUT 606. Maria Cutumisu. October 13, 2004. Goal. Provide efficient prediction for protein families Probabilistic Suffix Trees (PSTs) are variable length Markov models (VMMs). Conceptual Map. Background. PSTs were introduced by Ron, Singer, Tishby - PowerPoint PPT PresentationTRANSCRIPT
Probabilistic Suffix Trees
Maria Cutumisu
CMPUT 606
October 13, 2004
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Goal Provide efficient prediction for
protein families Probabilistic Suffix Trees (PSTs) are
variable length Markov models (VMMs)
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Conceptual Map
Probabilistic Suffix Trees
ePST
Suffix TreesVariable Length Markov Model
bPST
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Background PSTs were introduced by Ron, Singer,
Tishby Bejerano, Yona made further
improvements (bPST) Poulin – efficient PSTs (ePSTs) PSTs a.k.a. prediction suffix trees
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Higher Order Markov Models A k-order Markov chain: history of
length k for conditional probabilities Exponential storage requirements Order of the chain increases, amount
of training data increases to improve estimation accuracy
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Variable Length Markov Models (VMMs) Space and parameter-estimation
efficient variable length of the history sequence
for prediction only needed parameters are stored
Created from less training data
>T1 Test sequenceAHGSGYMNAB
Training sequences
Is T1 in the training set?
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VMMs P(sequence) = product of the
probabilities of each amino acid given those that precede it
Conditional probability based on the context of each amino acid
A context function k(·) can select the history length based on the context x1 . . . xi−1 xi
VMMs were first introduced as PSTs
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PSTs VMMs for efficient prediction Pruned during training to contain
only required parameters bPST: represents histories ePST: represents sequences
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bPST Used to represent the histories for
prediction instead of the training sequences
The possible histories are the reversed strings of all the substrings of the training sequences
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Prediction with bPSTs The conditional probabilities P(xi|xi-1…)
are obtained for each position by tracing a path from the root that matches the preceding residues
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Construction bPST We add histories for the training data Nodes: parameters that estimate the
conditional probabilities γhistory(a) = P(a|history)
PbPST (xi|xi−1, . . . , x1) = γx1...xi−1(xi) if in bPST
else γx2...xi−1(xi) if in bPST etc.
else γ(xi)
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bPST created and pruned using 010010010011110101100010111
P(01001) = P(0)P(1|0)P(0|01)P(0|010)P(1|0100) = γ(0) γ0(1) γ01(0) γ0
*(0) γ00*(1)
= (13/27)(8/13)(5/8)(5/13)(4/5) = 10400/182520 = 0.057
Bre
tt P
oulin
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Complexity bPST bPST building process requires O(Ln2)
time L is the length limit of the tree n is the total length of the training set.
bPST building requires all training sequences at once (in order to get all the reverse substrings) and cannot be done online (the bPST cannot be built as the training data is encountered)
Prediction: O(mL), m = sequence length
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Improved bPST Idea: tree with training sequences n length of all training sequences m length of tested sequence Result (theoretical):
linear time building O(n) linear time prediction O(m).
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Efficient PST (ePST) Used for predicting protein function ePST represents sequences Linear construction and prediction
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Example ePST
Bre
tt P
oulin
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Prediction with ePSTs The probabilities for a substring are
obtained for each position by tracing the path representing the sequence from the root
If the entire sequence is not found in the tree, suffix links are followed
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Construction ePST ePSTs gain efficiency by representing
the training sequences in the PST Nodes store counts of the
subsequence occurrences in the training data (with respect to the complete tree)
Conditional probabilities deducted from the counts are stored as well
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Example ePST - AYYYA
Bre
tt P
oulin
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Complexity ePST Linear time and space with regards to
the combined length of the training sequences O(n)
Linear prediction time O(m)
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Advantages and Disadvantages Avoid exponential space
requirements and parameter estimation problems of higher order Markov chains
Pruned during training to contain only required parameters
bPSTs for local predictions: more accurate prediction than global
Loss in classification performance: Pfarm, SCOP
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Conclusions PSTs require less training and
prediction time than HMMs Despite some loss in classification
performance, PSTs compete with HMMs due to PSTs reduced resource demands
PSTs take advantage of VMMs higher order correlations
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References Brett Poulin, Sequence-based Protein
Function Prediction, Master Thesis, University of Alberta, 2004
G Bejerano, G Yona, Modeling protein families using probabilistic suffix trees, RECOMB’99
G Bejerano, Algorithms for variable length markov chain modeling, Bioinformatics Applications Note, 20(5):788–789, 2004
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PSTs and HMMs “HMMs do not capture any higher-order
correlations. An HMM assumes that the identity of a particular position is independent of the identity of all other positions.” [1]
PSTs are variable length Markov models for efficient prediction. The prediction uses the longest available context matching the history of the current amino acid.
For protein prediction in general, “the main advantage of PSTs over HMMs is that the training and prediction time requirements of PSTs are much less than for the equivalent HMMs.” [1]
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Suffix Trees (ST)
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oulin
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bPST Histories added to the tree must
occur more frequently than a threshold Pmin
The substrings are added in order of length from smallest to largest
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bPST vs ST The string s is only added to the tree if the
resulting conditional probability at the node to be created will be greater than the minimum prediction probability γmin + α and the probability for the prefix of the string is different (with some ratio r) from the probability assigned to the next shortest substring suf(s) (which is already in the tree). After all the substrings are added to the tree, the probabilities are smoothed according to the parameter γmin.
The smoothing (as calculated by the equation below) prevents any probability from being less than γmin
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New!