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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

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.