BioMarkovChains.BMC Type

BMC

Alias for the type BioMarkovChain.

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BioMarkovChains.BioMarkovChain Type

struct BioMarkovChain{A<:Alphabet} <: AbstractBioMarkovChain

A BioMarkovChain represents a Markov chain used in biological sequence analysis. It contains a transition probability matrix (tpm) and an initial distribution of probabilities (inits) and also the order of the Markov chain.

Fields

• tpm::Matrix{Float64}: The transition probability matrix.

• inits::Vector{Float64}: The initial distribution of probabilities.

• n::Int: The order of the Markov chain.

Constructors

• BioMarkovChain{A}(tpm::Matrix{Float64}, inits::Vector{Float64}, n::N=1) where {A}: Constructs a BioMarkovChain object with the provided transition probability matrix, initial distribution, and order.

• BioMarkovChain{A}(seq::SeqOrView{A}, n::Int64=1) where {A}: Constructs a BioMarkovChain object based on the DNA sequence and transition order.

Example

seq = LongDNA{4}("ACTACATCTA")

model = BioMarkovChain(seq, 2)

BioMarkovChain of DNA alphabet and order 2:
  - Transition Probability Matrix -> Matrix{Float64}(4 × 4):
   0.4444  0.1111  0.0     0.4444
   0.4444  0.4444  0.0     0.1111
   0.0     0.0     0.0     0.0
   0.1111  0.4444  0.0     0.4444
  - Initial Probabilities -> Vector{Float64}(4 × 1):
   0.3333  0.3333  0.0     0.3333

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BioMarkovChains.initials Method

initials(sequence::SeqOrView{A}) where A

Calculate the estimated initial probabilities for a Markov chain based on a given sequence.

This function takes a sequence of states and calculates the estimated initial probabilities of each state in the sequence for a Markov chain. The initial probabilities are estimated by counting the occurrences of each state at the beginning of the sequence and normalizing the counts to sum up to 1.

$$\begin{array}{rl}\pi i& =P(X_i=i),i\in T\\ \sum _{i=1}^N{\pi }_i& =1\end{array}$$

Now using the dinucleotides counts estimating the initials would follow:

$$\widehat{{\pi }_i}=c_i\sum _k{c}_k$$

Arguments

• sequence::SeqOrView{A}: The sequence of states representing the Markov chain.

Returns

An Vector{Flot64} of estimated initial probabilities for each state in the sequence.

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BioMarkovChains.log_odds_ratio_matrix Method

log_odds_ratio_matrix(model1::BioMarkovChain, model2::BioMarkovChain)

Calculates the log-odds ratio between the transition probability matrices of two BioMarkovChain models.

$$\beta =\log \frac{P(x|{\mathcal{m}}_1)}{P(x|{\mathcal{m}}_2)}$$

Where ${\mathcal{m}}_1$ and ${\mathcal{m}}_2$ are the two models transition probability matrices.

Arguments

• model1::BioMarkovChain: The first BioMarkovChain model.

• model2::BioMarkovChain: The second BioMarkovChain model.

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BioMarkovChains.log_odds_ratio_score Method

log_odds_ratio_score(sequence::SeqOrView{A}; modela::BioMarkovChain, modelb::BioMarkovChain, b::Number = 2)

Compute the log odds ratio score between a given sequence and a BioMarkovChain model.

$$S(x)=\sum _{i=1}^L{\beta }_{x_{i}x}=\sum _{i=1}\log \frac{a_{i-1}^{{\mathcal{m}}_1}{x}_i}{a_{i-1}^{{\mathcal{m}}_2}{x}_i}$$

Arguments

• sequence::SeqOrView{A}: A sequence of elements of type A.

• modela::BioMarkovChain: A BioMarkovChain model.

• modelb::BioMarkovChain: A BioMarkovChain model.

• b::Number = 2: The base of the logarithm used to compute the log odds ratio.

Returns

The log odds ratio score between the sequence and the models.

Example

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BioMarkovChains.markovprobability Method

markovprobability(sequence::LongNucOrView{4}, model::BioMarkovChain)

Compute the probability of a given sequence using a transition probability matrix and the initial probabilities distributions of a BioMarkovModel.

$$P(X_1=i_1,\dots ,X_T=i_T)={\pi }_{i_1}^{T-1}\prod _{t=1}^{T-1}{a}_{i_t,i_{t+1}}$$

Arguments

• sequence::LongNucOrView{4}: The input sequence of nucleotides.

Keywords

• model::BioMarkovChain=ECOLICDS: A given BioMarkovChain model.

• logscale::Bool=false: If true, the function will return the log2 of the probability.

• b::Number=2: The base of the logarithm used to compute the log odds ratio.

Returns

• probability::Float64: The probability of the input sequence given the model.

Example

seq = LongDNA{4}("CGCGCGCGCGCGCGCGCGCGCGCGCG")
   
markovprobability(seq, model=CPGPOS, logscale=true)
    -45.073409957110556

markovprobability(seq, model=CPGNEG, logscale=true)
    -74.18912168395339

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BioMarkovChains.perronfrobenius Method

perronfrobenius(sequence::SeqOrView{A}, n::Int64=1) where A

Compute the Perron-Frobenius matrix, a column-stochastic version of the transition probability matrix (TPM), for a given nucleotide sequence.

The Perron-Frobenius matrix captures the asymptotic probabilities of transitioning between nucleotides in the sequence over a specified number of steps n. It provides insight into the long-term behavior of a Markov chain or a dynamical system associated with the sequence.

Arguments

• sequence::SeqOrView{A}: A nucleotide sequence represented as a NucleicSeqOrView{A} object.

• n::Int64=1: The number of steps to consider for the transition probability matrix. Default is 1.

Returns

A copy of the Perron-Frobenius matrix. Each column of this matrix corresponds to the probabilities of transitioning from the current nucleotide state to all possible nucleotide states after n steps.

Example

sequence = LongSequence{DNAAlphabet{4}}("ACGTCGTCCACTACGACATCAGC")  # Replace with an actual nucleotide sequence
n = 2
pf = perronfrobenius(sequence, n)

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BioMarkovChains.transition_count_matrix Method

transition_count_matrix(sequence::LongSequence{DNAAlphabet{4}})

Compute the transition count matrix (TCM) of a given DNA sequence.

Arguments

• sequence::LongSequence{DNAAlphabet{4}}: a LongSequence{DNAAlphabet{4}} object representing the DNA sequence.

Returns

A Matrix object representing the transition count matrix of the sequence.

Example

seq = LongDNA{4}("AGCTAGCTAGCT")

tcm = transition_count_matrix(seq)

4×4 Matrix{Int64}:
 0  0  3  0
 0  0  0  3
 0  3  0  0
 2  0  0  0

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BioMarkovChains.transition_probability_matrix Method

transition_probability_matrix(sequence::LongSequence{DNAAlphabet{4}}, n::Int64=1)

Compute the transition probability matrix (TPM) of a given DNA sequence. Formally it construct $\hat{\mathcal{M}}$ where:

$${\mathcal{m}}_{ij}=P(X_t=j\mid X_{t-1}=i)=\frac{P(X_{t-1}=i,X_t=j)}{P(X_{t-1}=i)}$$

The transition matrices of DNA and Amino-Acids are arranged sorted and in row-wise matrices:

First the DNA matrix:

$${\mathcal{M}}_{DNA}=\left[\begin{array}{cccc}{}_{AA}& {}_{AC}& {}_{AG}& {}_{AT}\\ {}_{CA}& {}_{CC}& {}_{CG}& {}_{CT}\\ {}_{GA}& {}_{GC}& {}_{GG}& {}_{GT}\\ {}_{TA}& {}_{TC}& {}_{TG}& {}_{TT}\end{array}\right]$$

And then, the Aminoacids:

$${\mathcal{M}}_{AA}=\left[\begin{array}{ccccc}{}_{AA}& {}_{AC}& {}_{AD}& \dots & {}_{AW}\\ {}_{CA}& {}_{CC}& {}_{CD}& \dots & {}_{CW}\\ {}_{DA}& {}_{DC}& {}_{DD}& \dots & {}_{DW}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {}_{WA}& {}_{WC}& {}_{WD}& \dots & {}_{WW}\end{array}\right]$$

Arguments

• sequence::LongNucOrView{4}: a LongNucOrView{4} object representing the DNA sequence.

• n::Int64=1: The order of the Markov model. That is the ${\hat{M}}^n$

Keywords

• extended_alphabet::Bool=false: If true will pass the extended alphabet of DNA to search

Returns

A Matrix object representing the transition probability matrix of the sequence.

Example

seq = dna"AGCTAGCTAGCT"

tpm = transition_probability_matrix(seq)

4×4 Matrix{Float64}:
 0.0  0.0  1.0  0.0
 0.0  0.0  0.0  1.0
 0.0  1.0  0.0  0.0
 1.0  0.0  0.0  0.0

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