# Logarithm of the probabilities of state sequences

### Description

Compute the logarithm of the probability of each state sequence obtained from a state transition model. The probability of a sequence is equal to the product of each state probability of the sequence. There are several methods to compute a state probability.

### Usage

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### Arguments

`seqdata` |
The sequence to compute the probabilities. |

`prob` |
either the name ( |

`time.varying` |
Logical. If |

`begin` |
Model used to compute the probability of the first state. Either |

`weighted` |
Logical. If |

### Details

The sequence likelihood *P(s)* is defined as the product of the probability with which each of its observed successive state is supposed to occur at its position.
Let *s=s_1s_2 ... s_l* be a sequence of length *l*. Then

*
P(s)=P(s_1, 1) * P(s_2, 2) * ... * P(s_l, l)
*

with *P(s_t,t)* the probability to observe state *s_t* at position *t*.

The question is how to determinate the state probabilities *P(s_t,t)*. Several methods are available and can be set using the `prob`

argument.

One commonly used method for computing them is to postulate a Markov model, which can be of various order. We can consider probabilities derived from the first order Markov model, that is, each *P(s_t,t)*, *t>1* is set as the transition rate *p(s_t|s_(t-1))*. This is available in `seqlogp`

by setting `prob="trate"`

.

The transition rates may be considered constant over time/positions (`time.varying=FALSE`

), that is estimated across sequences from the observations at positions *t* and *t-1* for all *t* together. Time varying transition rates may also be considered (`time.varying=TRUE`

), in which case they are computed separately for each position, that is estimated across sequences from the observations at positions *t* and *t-1* for each *t*, yielding an array of transition matrices. The user may also specify his own transition rates array or matrix.

Another method is to use the frequency of a state at each position to set *P(s_t,t)* (`prob="freq"`

). In the latter case, the probability of a sequence is independent of the probability of the transitions. Here again, the frequencies can be computed all together (`time.varying=FALSE`

) or separately for each position *t* (`time.varying=TRUE`

).
For *t=1*, we set *P(s_1,1)* to the observed frequency of the state *s_1* at position 1. Alternatively, the `begin`

argument allows to specify the probability of the first state.

The likelihood *P(s)* being generally very small, `seqlogp`

return *-log(P(s))*. The latter quantity is minimal when *P(s)* is equal to *1*.

### Value

A vector containing the logarithm of each sequence probability.

### Author(s)

Matthias Studer and Alexis Gabadinho (with Gilbert Ritschard for the help page)

### Examples

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