Package {hmmTensor}

Backward Algorithm for HMM

Description

Computes backward probabilities \beta_t(i) = P(Y_{t+1}, \ldots, Y_T | X_t = i) using the same scaling factors from the Forward algorithm.

Usage

Backward(Y, T_mat, O, scale)

Arguments

Value

Matrix (K x T_len) of scaled backward probabilities

Baum-Welch Algorithm (EM) for HMM

Description

Estimates HMM parameters (T, O, pi) from observation sequences using the Expectation-Maximization algorithm.

Usage

BaumWelch(
  Y,
  K,
  N,
  initT = NULL,
  initO = NULL,
  initPi = NULL,
  num.iter = 100L,
  thr = 1e-06,
  verbose = FALSE
)

Arguments

Value

A list with components:

T_mat

Estimated transition matrix (K x K)

O

Estimated emission matrix (K x N)

pi0

Estimated initial distribution (length K)

loglik

Vector of log-likelihoods per iteration

converged

Logical

iter

Number of iterations

Forward Algorithm for HMM

Description

Computes forward probabilities \alpha_t(i) = P(Y_1, \ldots, Y_t, X_t = i) with log-scaling to avoid underflow.

Usage

Forward(Y, T_mat, O, pi0)

Arguments

Value

A list with components:

alpha

Matrix (K x T_len) of scaled forward probabilities

loglik

Log-likelihood of the observation sequence

scale

Vector of scaling factors (length T_len)

HMM Parameter Estimation via Matrix/Tensor Decomposition

Description

Estimates HMM parameters by decomposing the observation co-occurrence matrix/tensor. Supports multiple decomposition solvers.

Usage

HMM(
  Y,
  K,
  N = NULL,
  solver = c("symNMF", "SVD", "CP", "TT"),
  Beta = 2,
  order = 2L,
  lag = 1L,
  smooth = 1e-10,
  num.iter = 100L,
  thr = 1e-10,
  verbose = FALSE
)

Arguments

Details

The pipeline:

1. Convert observations to co-occurrence matrix \Omega = P_{2,1} via Seq2Prob

2. Decompose \Omega using the chosen solver

3. Recover HMM parameters (T, O, pi) from the decomposition

For NMF-based methods, \Omega \approx M \Theta M^\top where M relates to the emission matrix O and \Theta to diag(pi) %*% T.

Value

A list with components:

T_mat

Estimated transition matrix (K x K)

O

Estimated emission matrix (K x N)

pi0

Estimated initial distribution (length K)

Omega

Co-occurrence matrix/tensor used

decomp

Raw decomposition result

solver

Solver used

RecError

Reconstruction error (if available)

Examples

set.seed(42)
toy <- toyModel(type = "simple")
result <- HMM(toy$Y, K = toy$K, N = toy$N, solver = "symNMF")
result$T_mat

Convert Observation Sequences to Co-occurrence Matrix/Tensor

Description

Constructs empirical co-occurrence statistics from observation sequences. These form the basis for matrix/tensor decomposition approaches to HMM.

Usage

Seq2Prob(Y, N = NULL, order = 2L, lag = 1L, smooth = 0)

Arguments

Value

For order = 2: an N x N matrix (normalized). For order = 3: an rTensor Tensor object of dimension N x N x N.

Examples

Y <- c(1, 2, 3, 1, 2, 3, 1, 2, 3)
P2 <- Seq2Prob(Y, N = 3, order = 2)
P3 <- Seq2Prob(Y, N = 3, order = 3)

Viterbi Algorithm for HMM

Description

Finds the most likely state sequence given the observations using the Viterbi algorithm (log-space).

Usage

Viterbi(Y, T_mat, O, pi0)

Arguments

Value

A list with components:

path

Integer vector of most likely states (length T_len)

loglik

Log-likelihood of the best path

Generate Toy HMM Data

Description

Creates synthetic HMM data with visually clear structure for demonstrations and testing.

Usage

toyModel(type = c("simple", "weather", "leftright"), T_len = 500L, seed = NULL)

Arguments

Value

A list with components:

Y

Integer vector of observations

X

Integer vector of true hidden states

T_mat

True transition matrix

O

True emission matrix

pi0

True initial distribution

K

Number of hidden states

N

Number of observation symbols

type

Model type

Examples

toy <- toyModel("simple", T_len = 200, seed = 42)
table(toy$X)
table(toy$Y)