ActiveLearning4SPM

The ActiveLearning4SPM package implements the methodology of Capezza, Lepore, and Paynabar (2025) for
stream-based active learning for process monitoring:

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

The package provides tools to:

• Simulate multivariate data streams with true hidden states.
  
• Fit partially hidden Markov models (pHMMs) with user-specified or automatically initialized parameters.
  
• Perform stream-based active learning, balancing exploration and exploitation when deciding whether to acquire new labels under a budget constraint.

The methodology is motivated by process monitoring in industrial applications where obtaining labels is costly.

Installation

You can install the development version from GitHub:

# install.packages("devtools")
devtools::install_github("unina-sfere/ActiveLearning4SPM")

Once on CRAN, you’ll be able to install with:

install.packages("streamALpHMM")

Example

Simulate a data stream

library(ActiveLearning4SPM)

set.seed(123)
dat <- simulate_stream(T0 = 100, TT = 500, d = 10)
str(dat)

## List of 2
##  $ x: num [1:600] 1 1 1 1 1 1 1 1 1 1 ...
##  $ y: num [1:600, 1:10] 0.756 -0.872 -0.279 1.43 0.618 ...

Fit a pHMM with user-defined initialization

y <- dat$y
d <- ncol(y)
xlabeled <- dat$x
xlabeled[sample(1:600, 400)] <- NA  # partially labeled

fit <- fit_pHMM(
  y = y,
  xlabeled = xlabeled,
  nstates = 3,
  mean_start = list(rep(0, d), rep(1, d), rep(-1, d)),
  equal_covariance = TRUE
)
fit$AIC

## [1] 12711.52

Fit a pHMM with automatic initialization

fit_auto <- fit_pHMM_auto(y = y, xlabeled = xlabeled, max_nstates = 3)
fit_auto$AIC

## [1] 12711.52

Perform stream-based active learning

y <- dat$y[1:200, ]
true_x <- dat$x[1:200]
out <- active_learning_pHMM(y = y,
                            true_x = true_x,
                            T0 = 100, 
                            B = 0.1,
                            verbose = TRUE)

## t=101 | Available labels: 10 | Explor. p-value = 0.212 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=102 | Available labels: 10 | Explor. p-value = 0.314 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=103 | Available labels: 10 | Explor. p-value = 0.696 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=104 | Available labels: 10 | Explor. p-value = 0.744 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=105 | Available labels: 10 | Explor. p-value = 0.254 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=106 | Available labels: 10 | Explor. p-value = 0.174 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=107 | Available labels: 10 | Explor. p-value = 0.064 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=108 | Available labels: 10 | Explor. p-value = 0.571 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=109 | Available labels: 10 | Explor. p-value = 0.046 | Exploit. p-value = 1.000 | True state = 1 | Decision = label_exploration

## t=110 | Available labels: 9 | Explor. p-value = 0.062 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=111 | Available labels: 9 | Explor. p-value = 0.221 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=112 | Available labels: 9 | Explor. p-value = 0.167 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=113 | Available labels: 9 | Explor. p-value = 0.872 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=114 | Available labels: 9 | Explor. p-value = 0.799 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=115 | Available labels: 9 | Explor. p-value = 0.489 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=116 | Available labels: 9 | Explor. p-value = 0.234 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=117 | Available labels: 9 | Explor. p-value = 0.567 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=118 | Available labels: 9 | Explor. p-value = 0.392 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=119 | Available labels: 9 | Explor. p-value = 0.648 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=120 | Available labels: 9 | Explor. p-value = 0.959 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=121 | Available labels: 9 | Explor. p-value = 0.902 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=122 | Available labels: 9 | Explor. p-value = 0.519 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=123 | Available labels: 9 | Explor. p-value = 0.524 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=124 | Available labels: 9 | Explor. p-value = 0.506 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=125 | Available labels: 9 | Explor. p-value = 0.695 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=126 | Available labels: 9 | Explor. p-value = 0.891 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=127 | Available labels: 9 | Explor. p-value = 0.928 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=128 | Available labels: 9 | Explor. p-value = 0.841 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=129 | Available labels: 9 | Explor. p-value = 0.736 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=130 | Available labels: 9 | Explor. p-value = 0.899 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=131 | Available labels: 9 | Explor. p-value = 0.828 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=132 | Available labels: 9 | Explor. p-value = 0.178 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=133 | Available labels: 9 | Explor. p-value = 0.086 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=134 | Available labels: 9 | Explor. p-value = 0.275 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=135 | Available labels: 9 | Explor. p-value = 0.483 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=136 | Available labels: 9 | Explor. p-value = 0.851 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=137 | Available labels: 9 | Explor. p-value = 0.879 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=138 | Available labels: 9 | Explor. p-value = 0.090 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=139 | Available labels: 9 | Explor. p-value = 0.008 | Exploit. p-value = 1.000 | True state = 1 | Decision = label_exploration

## t=140 | Available labels: 8 | Explor. p-value = 0.271 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=141 | Available labels: 8 | Explor. p-value = 0.578 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=142 | Available labels: 8 | Explor. p-value = 0.295 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=143 | Available labels: 8 | Explor. p-value = 0.400 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=144 | Available labels: 8 | Explor. p-value = 0.477 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=145 | Available labels: 8 | Explor. p-value = 0.491 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=146 | Available labels: 8 | Explor. p-value = 0.416 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=147 | Available labels: 8 | Explor. p-value = 0.783 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=148 | Available labels: 8 | Explor. p-value = 0.503 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=149 | Available labels: 8 | Explor. p-value = 0.160 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=150 | Available labels: 8 | Explor. p-value = 0.309 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=151 | Available labels: 8 | Explor. p-value = 0.163 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=152 | Available labels: 8 | Explor. p-value = 0.180 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=153 | Available labels: 8 | Explor. p-value = 0.243 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=154 | Available labels: 8 | Explor. p-value = 0.700 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=155 | Available labels: 8 | Explor. p-value = 0.960 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=156 | Available labels: 8 | Explor. p-value = 0.503 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=157 | Available labels: 8 | Explor. p-value = 0.518 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=158 | Available labels: 8 | Explor. p-value = 0.217 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=159 | Available labels: 8 | Explor. p-value = 0.934 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=160 | Available labels: 8 | Explor. p-value = 0.883 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=161 | Available labels: 8 | Explor. p-value = 0.791 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=162 | Available labels: 8 | Explor. p-value = 0.570 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=163 | Available labels: 8 | Explor. p-value = 0.237 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=164 | Available labels: 8 | Explor. p-value = 0.035 | Exploit. p-value = 1.000 | True state = 1 | Decision = label_exploration

## t=165 | Available labels: 7 | Explor. p-value = 0.240 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=166 | Available labels: 7 | Explor. p-value = 0.378 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=167 | Available labels: 7 | Explor. p-value = 0.374 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=168 | Available labels: 7 | Explor. p-value = 0.234 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=169 | Available labels: 7 | Explor. p-value = 0.251 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=170 | Available labels: 7 | Explor. p-value = 0.344 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=171 | Available labels: 7 | Explor. p-value = 0.789 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=172 | Available labels: 7 | Explor. p-value = 0.608 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=173 | Available labels: 7 | Explor. p-value = 0.482 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=174 | Available labels: 7 | Explor. p-value = 0.497 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=175 | Available labels: 7 | Explor. p-value = 0.706 | Exploit. p-value = 1.000 | True state = 1 | Decision = 1

## t=176 | Available labels: 7 | Explor. p-value = 0.093 | Exploit. p-value = 1.000 | True state = 2 | Decision = label_exploration

## t=177 | Available labels: 6 | Explor. p-value = 0.008 | Exploit. p-value = 0.500 | True state = 2 | Decision = label_exploration

## t=178 | Available labels: 5 | Explor. p-value = 0.353 | Exploit. p-value = 0.300 | True state = 2 | Decision = 2

## t=179 | Available labels: 5 | Explor. p-value = 0.722 | Exploit. p-value = 0.200 | True state = 2 | Decision = 2

## t=180 | Available labels: 5 | Explor. p-value = 0.949 | Exploit. p-value = 0.650 | True state = 2 | Decision = 2

## t=181 | Available labels: 5 | Explor. p-value = 0.947 | Exploit. p-value = 0.250 | True state = 1 | Decision = 2

## t=182 | Available labels: 5 | Explor. p-value = 0.029 | Exploit. p-value = 0.650 | True state = 1 | Decision = label_exploration

## t=183 | Available labels: 4 | Explor. p-value = 0.018 | Exploit. p-value = 0.100 | True state = 1 | Decision = label_exploitation

## t=184 | Available labels: 3 | Explor. p-value = 0.105 | Exploit. p-value = 0.455 | True state = 1 | Decision = 1

## t=185 | Available labels: 3 | Explor. p-value = 0.060 | Exploit. p-value = 0.381 | True state = 1 | Decision = label_exploration

## t=186 | Available labels: 2 | Explor. p-value = 0.401 | Exploit. p-value = 0.033 | True state = 1 | Decision = label_exploitation

## t=187 | Available labels: 1 | Explor. p-value = 0.861 | Exploit. p-value = 0.804 | True state = 1 | Decision = 1

## t=188 | Available labels: 1 | Explor. p-value = 0.618 | Exploit. p-value = 0.058 | True state = 1 | Decision = 1

## t=189 | Available labels: 1 | Explor. p-value = 0.513 | Exploit. p-value = 0.146 | True state = 1 | Decision = 1

## t=190 | Available labels: 1 | Explor. p-value = 0.154 | Exploit. p-value = 0.182 | True state = 1 | Decision = 1

## t=191 | Available labels: 1 | Explor. p-value = 0.602 | Exploit. p-value = 0.525 | True state = 1 | Decision = 1

## t=192 | Available labels: 1 | Explor. p-value = 0.441 | Exploit. p-value = 0.056 | True state = 1 | Decision = 1

## t=193 | Available labels: 1 | Explor. p-value = 0.582 | Exploit. p-value = 0.469 | True state = 1 | Decision = 1

## t=194 | Available labels: 1 | Explor. p-value = 0.502 | Exploit. p-value = 0.857 | True state = 1 | Decision = 1

## t=195 | Available labels: 1 | Explor. p-value = 0.803 | Exploit. p-value = 0.792 | True state = 1 | Decision = 1

## t=196 | Available labels: 1 | Explor. p-value = 0.371 | Exploit. p-value = 0.200 | True state = 1 | Decision = 1

## t=197 | Available labels: 1 | Explor. p-value = 0.703 | Exploit. p-value = 0.700 | True state = 1 | Decision = 1

## t=198 | Available labels: 1 | Explor. p-value = 0.676 | Exploit. p-value = 0.500 | True state = 1 | Decision = 1

## t=199 | Available labels: 1 | Explor. p-value = 0.611 | Exploit. p-value = 0.400 | True state = 1 | Decision = 1

## t=200 | Available labels: 1 | Explor. p-value = 0.182 | Exploit. p-value = 0.150 | True state = 1 | Decision = label_exploitation

table(out$xhat, true_x)

##    true_x
##       1   2
##   1 194   0
##   2   1   5

out$scores

## $accuracy
## [1] 0.99
## 
## $precision
## [1] 0.8333333
## 
## $recall
## [1] 1
## 
## $f1
## [1] 0.9090909
## 
## $auc
## [1] 0.9947368