ActiveLearning4SPM: Active Learning for Process Monitoring

Description

Implements the methodology introduced in Capezza, Lepore, and Paynabar (2025) doi:10.1080/00401706.2025.2561744 for process monitoring with limited labeling resources. The package provides functions to (i) simulate data streams with true latent states and multivariate Gaussian observations as done in the paper, (ii) fit partially hidden Markov models (pHMMs) using a constrained Baum-Welch algorithm with partial labels, and (iii) perform stream-based active learning that balances exploration and exploitation to decide whether to request labels in real time. The methodology is particularly suited for statistical process monitoring in industrial applications where labeling is costly.

Author(s)

Maintainer: Christian Capezza christian.capezza@unina.it

Authors:

• Antonio Lepore

• Kamran Paynabar

References

Capezza, C., Lepore, A., and Paynabar, K. (2025). Stream-Based Active Learning for Process Monitoring. Technometrics. <doi:10.1080/00401706.2025.2561744>

Stream-Based Active Learning with a Partially Hidden Markov Model (pHMM)

Description

Implements the stream-based active learning strategy of Capezza, Lepore, and Paynabar (2025) for process monitoring with partially observed states. At each time step, the method fits a pHMM to the available data, and balances between exploitation (reducing predictive uncertainty) and exploration (detecting potential out-of-control shifts) to decide whether to request the true label of the current observation. Labeling requests are constrained by a user-defined budget.

Usage

active_learning_pHMM(
  y,
  true_x,
  T0,
  B = 0.1,
  weight_exploration = 0.5,
  lambda_MEWMA = 0.3,
  verbose = FALSE
)

Arguments

Details

The exploitation criterion is based on the entropy of the state sequence, while the exploration criterion uses a multivariate exponentially weighted moving average (MEWMA) statistic. The two criteria are combined with a user-defined weighting, and labeling stops when the budget is exhausted or at the end of the data stream.

Value

A list with components:

• decision: character vector indicating the action taken at each time ("label_exploitation", "label_exploration", or predicted state).

• xlabeled: updated state sequence including acquired labels.

• xhat: final predicted state sequence.

• scores: classification performance metrics (accuracy, precision, recall, F1, AUC) computed against the true states.

Note

This function is intended for simulation studies where the entire observation sequence y and the corresponding true states x are available in advance. The function uses these to evaluate the active learning strategy under a given budgets. In real-time applications, data and labels would arrive sequentially, and labels would only be obtained if requested by the strategy.

References

Capezza, C., Lepore, A., & Paynabar, K. (2025). Stream-Based Active Learning for Process Monitoring. Technometrics. <doi:10.1080/00401706.2025.2561744>.

Examples

library(ActiveLearning4SPM)
set.seed(123)
dat <- simulate_stream(T0 = 50, TT = 100, T_min_IC = 20, T_max_IC = 30)
out <- active_learning_pHMM(y = dat$y,
                            true_x = dat$x,
                            T0 = 50,
                            B = 0.1)
table(out$decision)
out$scores$f1

Fit a Partially Hidden Markov Model (pHMM)

Description

Fits a partially hidden Markov model (pHMM) to multivariate time series observations y with partially observed process states x, using a constrained Baum-Welch algorithm. The function allows the user to provide custom initial parameters, and supports constraints on known means and/or covariances, as well as equal or diagonal covariance structures.

Usage

fit_pHMM(
  y,
  xlabeled,
  nstates,
  ppi_start = NULL,
  A_start = NULL,
  mean_start,
  covariance_start = NULL,
  known_mean = NULL,
  known_covariance = NULL,
  equal_covariance = FALSE,
  covariance_structure = "full",
  max_iter = 200,
  tol = 0.001,
  verbose = FALSE
)

Arguments

Value

A list with components:

• y, xlabeled: the input data.

• log_lik, log_lik_vec: final and trace of log-likelihood.

• iter: number of EM iterations performed.

• logB, log_alpha, log_beta, log_gamma, log_xi: posterior quantities from the Baum-Welch algorithm.

• logAhat, mean_hat, covariance_hat, log_pi_hat: estimated model parameters.

• AIC, BIC: information criteria for model selection.

References

Capezza, C., Lepore, A., & Paynabar, K. (2025). Stream-Based Active Learning for Process Monitoring. Technometrics. <doi:10.1080/00401706.2025.2561744>.

Examples

library(ActiveLearning4SPM)
set.seed(123)
dat <- simulate_stream(T0 = 100, TT = 500)
y <- dat$y
xlabeled <- dat$x
d <- ncol(dat$y)
xlabeled[sample(1:600, 300)] <- NA
out <- fit_pHMM(y = y,
                xlabeled = xlabeled,
                nstates = 3,
                mean_start = list(rep(0, d), rep(1, d), rep(-1, d)),
                equal_covariance = TRUE)
out$AIC

Automatic Initialization and Fitting of a Partially Hidden Markov Model (pHMM)

Description

Fits a partially hidden Markov model (pHMM) to multivariate time series observations y with partially observed states x, using the constrained Baum-Welch algorithm. Unlike fit_pHMM, this function does not require user-specified initial parameters. Instead, it implements a customized initialization strategy designed for process monitoring with highly imbalanced classes, as described in the supplementary material of Capezza, Lepore, and Paynabar (2025).

Usage

fit_pHMM_auto(
  y = y,
  xlabeled = xlabeled,
  tol = 0.001,
  max_nstates = 5,
  ntry = 10
)

Arguments

Details

The initialization procedure addresses the multimodality of the likelihood and the sensitivity of the Baum-Welch algorithm to starting values:

1. A one-state model (in-control process) is first fitted using robust estimators of location and scatter.

2. To introduce an additional state, candidate mean vectors are selected from observations that are least well represented by the current model. This is achieved by computing moving averages of the data over window lengths k = 1, \ldots, 9, and then calculating the Mahalanobis distances of these smoothed points to existing state means.

3. The ntry observations with the largest minimum distances are retained as candidate initializations for the new state's mean.

4. For each candidate, a pHMM is initialized with:

   • Existing means fixed to their previous estimates.

   • The new state's mean set to the candidate vector.

   • A shared covariance matrix fixed to the robust estimate from the in-control state.

   • Initial state distribution \pi concentrated on the IC state.

   • Transition matrix with diagonal entries 1 - 0.01 (N-1) and off-diagonal entries 0.01.

5. Each initialized model is fitted with the Baum-Welch algorithm, and the one achieving the highest log-likelihood is retained.

6. This process is repeated until up to max_nstates states are considered.

This strategy leverages prior process knowledge (dominant in-control regime) and focuses the search on under-represented regions of the data space, which improves convergence and reduces sensitivity to random initialization.

Value

A list with the same structure as returned by fit_pHMM:

• y, xlabeled: the input data.

• log_lik, log_lik_vec: final and trace of log-likelihood.

• iter: number of EM iterations performed.

• logB, log_alpha, log_beta, log_gamma, log_xi: posterior quantities from the Baum-Welch algorithm.

• logAhat, mean_hat, covariance_hat, log_pi_hat: estimated model parameters.

• AIC, BIC: information criteria for model selection.

References

Capezza, C., Lepore, A., & Paynabar, K. (2025). Stream-Based Active Learning for Process Monitoring. Technometrics. <doi:10.1080/00401706.2025.2561744>.

Supplementary Material, Section B: Initialization of the Partially Hidden Markov Model. Available at <https://doi.org/10.1080/00401706.2025.2561744>.

Examples

library(ActiveLearning4SPM)
set.seed(123)
dat <- simulate_stream(T0 = 100, TT = 500)
y <- dat$y
xlabeled <- dat$x
d <- ncol(dat$y)
xlabeled[sample(1:600, 300)] <- NA
obj <- fit_pHMM_auto(y = y,
                     xlabeled = xlabeled,
                     tol = 1e-3,
                     max_nstates = 5,
                     ntry = 10)
obj$AIC

Simulate Process Monitoring Data for Stream-Based Active Learning

Description

Generate a sequence of latent states and corresponding multivariate Gaussian observations for process monitoring. The process has three possible states:

1. state 1: in-control (IC),

2. state 2: out-of-control (OC),

3. state 3: out-of-control (OC).

Usage

simulate_stream(
  d = 10,
  TT = 500,
  T0 = 100,
  T_min_IC = 60,
  T_max_IC = 85,
  T_OC = 5,
  mean = NULL,
  covariance = NULL
)

Arguments

Details

The first T0 observations are fixed in state 1 (IC). Then, in the following TT observations, only state 2 appears in the first half, and only state 3 appears in the second half. Within each half, runs of state 1 (IC) of random length between T_min_IC and T_max_IC alternate with fixed-length runs of the corresponding OC state of length T_OC.

Value

A list with elements:

Examples

library(ActiveLearning4SPM)
sim <- simulate_stream()
table(sim$x)