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Markov Stability Analysis
Nestimate provides two functions for Markov-chain stability analysis
of transition networks:
passage_time() – computes the full
matrix of mean first passage times (MFPT). Entry M[i, j] is the expected
number of steps to travel from state i to state j for
the first time. The diagonal equals the mean recurrence time 1/pi.markov_stability() – computes
per-state stability metrics: persistence, stationary probability, return
time, sojourn time, and mean accessibility.
Both functions accept a netobject,
cograph_network, tna object, row-stochastic
matrix, or a raw wide sequence data frame.
Data
trajectories contains 138 learners recorded at 15
time-steps each with three engagement states: Active, Average, and
Disengaged.
dim(trajectories)
#> [1] 138 15
table(as.vector(trajectories), useNA = "always")
#>
#> Active Average Disengaged <NA>
#> 703 813 354 200
Selecting mostly-active learners
We keep learners who were Active more than half the
time (> 7 of 15 steps).
df <- as.data.frame(trajectories)
active_count <- rowSums(df == "Active", na.rm = TRUE)
mostly_active <- active_count > ncol(df) / 2
cat(sum(mostly_active), "of", nrow(df), "learners qualify\n")
#> 42 of 138 learners qualify
sub <- df[mostly_active, ]
Sequence plots
state_pal <- c(Active = "#1a7a1a", Average = "#E69F00", Disengaged = "#CC79A7")
sequence_plot(
df,
type = "heatmap",
sort = "lcs",
k = 3,
k_color = "white",
k_line_width = 2,
state_colors = state_pal,
na_color = "grey88",
main = "Full sample - all 138 learners",
time_label = "Time-step",
y_label = "Learner",
legend = "bottom"
)
sequence_plot(
sub,
type = "heatmap",
sort = "lcs",
k = 2,
k_color = "white",
k_line_width = 2,
state_colors = state_pal,
na_color = "grey88",
main = "Mostly-active learners - Active > 7 of 15 steps (n = 42)",
time_label = "Time-step",
y_label = "Learner",
legend = "bottom"
)
Transition networks
net_all <- build_network(df, method = "relative")
net_sub <- build_network(sub, method = "relative")
round(net_all$weights, 3)
#> Active Average Disengaged
#> Active 0.698 0.267 0.035
#> Average 0.204 0.610 0.186
#> Disengaged 0.120 0.397 0.483
round(net_sub$weights, 3)
#> Active Average Disengaged
#> Active 0.819 0.164 0.017
#> Average 0.683 0.288 0.029
#> Disengaged 0.643 0.143 0.214
In the mostly-active group nearly every state transitions
predominantly back to Active, and the probability of entering Disengaged
is almost zero.
Mean First Passage Times
pt_all <- passage_time(net_all)
pt_sub <- passage_time(net_sub)
print(pt_all, digits = 2)
#> Mean First Passage Times (3 states)
#>
#> Active Average Disengaged
#> Active 2.69 3.63 10.36
#> Average 5.51 2.26 7.97
#> Disengaged 6.16 2.78 5.41
#>
#> Stationary distribution:
#> Active Average Disengaged
#> 0.3719 0.4431 0.1850
print(pt_sub, digits = 2)
#> Mean First Passage Times (3 states)
#>
#> Active Average Disengaged
#> Active 1.27 6.12 51.99
#> Average 1.47 5.36 51.29
#> Disengaged 1.54 6.28 41.75
#>
#> Stationary distribution:
#> Active Average Disengaged
#> 0.7895 0.1866 0.0240
Active to Disengaged rises from 10.4 to ~40 steps –
mostly-active learners are nearly four times harder to disengage. The
diagonal equals the mean recurrence time (how many steps between
consecutive visits to the same state).
plot(pt_all, title = "Full sample (n = 138)")
plot(pt_sub, title = "Mostly-active learners (n = 42)")
In the mostly-active heatmap the Active column is uniformly dark
(quickly reachable from any state) and the Disengaged column is
uniformly pale (nearly unreachable).
Stationary distribution
#> State Full sample Mostly active
#> 1 Active 37.2% 78.9%
#> 2 Average 44.3% 18.7%
#> 3 Disengaged 18.5% 2.4%
In the mostly-active group ~79% of long-run time is spent Active (vs
37% in the full sample).
Markov Stability
ms_all <- markov_stability(net_all)
ms_sub <- markov_stability(net_sub)
print(ms_all)
#> Markov Stability Analysis
#>
#> state persistence stationary_prob return_time sojourn_time
#> Active 0.6976 0.3719 2.69 3.31
#> Average 0.6099 0.4431 2.26 2.56
#> Disengaged 0.4831 0.1850 5.41 1.93
#> avg_time_to_others avg_time_from_others
#> 6.99 5.84
#> 6.74 3.20
#> 4.47 9.16
print(ms_sub)
#> Markov Stability Analysis
#>
#> state persistence stationary_prob return_time sojourn_time
#> Active 0.8191 0.7895 1.27 5.53
#> Average 0.2885 0.1866 5.36 1.41
#> Disengaged 0.2143 0.0240 41.75 1.27
#> avg_time_to_others avg_time_from_others
#> 29.06 1.50
#> 26.38 6.20
#> 3.91 51.64
The stability data frame contains:
persistence |
P[i,i] – probability of staying at the next step |
stationary_prob |
Long-run proportion of time in this state |
return_time |
Expected steps between consecutive visits (1/pi) |
sojourn_time |
Expected consecutive steps before leaving (1/(1-P[i,i])) |
avg_time_to_others |
Mean MFPT leaving this state to all others |
avg_time_from_others |
Mean MFPT from all other states arriving here |
plot(ms_all, title = "Full sample")
plot(ms_sub, title = "Mostly-active learners")
Active leads on persistence (0.81) and sojourn time (5.3 steps) –
once a learner is active they stay active. Average has the lowest
avg_time_from_others (1.6 steps) – the most accessible hub
state. Disengaged has avg_time_from_others of ~39 steps –
effectively unreachable.
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.