Scope. Add a bounded confidence lane beside a classical sensor KPI from turbofan run-to-failure sequences; values remain unchanged (phi((m,a)) = m).
Signals. Multivariate time series per engine unit with operating settings (op1..op3) and sensors (s1..s21). Choose any KPI (e.g., s7) and optionally a second cue (e.g., s14) for fusion.
Knobs (this run, generic). eps_a = 1e-6, eps_w = 1e-12. Primary cue via entropy on diff(m) with win = 51, bins = 16, k = 2.0. Optional second cue via residual-entropy on s14 with win = 31, bins = 16, k = 2.0. Fusion weights {w1=2, w2=1}.
Lane math (primary; entropy with explicit conventions).z := diff(m)p_i := count(z in bin_i) / N with the convention 0*log(0) := 0 (bins with p_i=0 contribute 0).
Binning policy (normative): equal-width bins over the window’s min..max of z.
Log base: natural log.H := -sum_i p_i*log(p_i) ; Hn := H/log(bins) ; a := tanh( k * (1 - Hn) ) ; then a := clamp(a, -1+eps_a, +1-eps_a).
Residual mapper (secondary cue).residual(x; win_r) := x - movavg(x, win_r) (simple causal moving average).
Apply the same entropy mapping to diff(residual(x)) with its own win, bins, k and the same 0*log(0)/binning/log conventions; then clamp.
Fusion (order-invariant).t := ( w1*atanh(a1) + w2*atanh(a2) ) / max(w1+w2, eps_w) ; a_out := tanh(t).
Conformance (observed).
• Collapse parity: True (value lane equals the original sensor to machine precision).
• Lane bounded/finite: True.
• Bands represent evidence strength only; the KPI value is never altered.
Observed bands (executive summary, dataset-neutral).
Single-cue (entropy on s7, short window) concentrates time in lower-confidence bands (A-/A--) during turbulent segments; adding a second cue (s14 residual-entropy, weights {2,1}) shifts share toward A0/A+ where cues corroborate. Exact distributions are provided in the repository link.
Reproduce (concept steps).
- Parse FD001 into rows
(unit, time, op1..op3, s1..s21). - Pick a unit and sensor for the value lane
m; computeawith the entropy mapper (conventions above). - (Optional) Map a second cue via
residual(s14; win=31), then entropy ondiff(residual), and fuse with the{U,W}rule. - Verify: collapse parity, finite/bounded
a, and band distribution.
Dataset acknowledgment (high-level).
Turbofan Engine Degradation Simulation Data (C-MAPSS, NASA Ames Research Center). This blog does not embed third-party data or CSVs. See the repository link on this site for scripts, stamped runs, CSVs, and the dataset’s license details.
Integrity & manifest (copy-ready skeletons).
Entropy on s7 (single cue).
[SSMH_MANIFEST]
profile = turbofan.s7_entropy
build_id = ssmh-turbofan-entropy
date_utc = <YYYY-MM-DD>
dataset = C-MAPSS
subset = FD001
unit = <unit_id>
kpi = s7
eps_a = 1e-6
eps_w = 1e-12
# Bands (alias note: App==A++, Ap==A+, Amm==A--)
bands = {App=0.75, Ap=0.50, A0=0.25, Am=0.10, Amm=-inf}
# Mapper (explicit conventions)
map_primary = entropy(diff(s7))
win = 51
bins = 16
k = 2.0
entropy_log_base = e
binning_policy = equal_width_minmax
zero_prob_convention = 0*log(0)=0
fusion = none
notes = Values unchanged; lane beside s7; U/W available for fusion
manifest_checksum = <sha256>
conformance_checksum = <sha256>
[/SSMH_MANIFEST]
Fused s7 entropy + s14 residual-entropy (weights {2,1}).
[SSMH_MANIFEST]
profile = turbofan.s7_entropy__s14_residual_fused
build_id = ssmh-turbofan-fused
date_utc = <YYYY-MM-DD>
dataset = C-MAPSS
subset = FD001
unit = <unit_id>
kpi = s7_plus_s14
eps_a = 1e-6
eps_w = 1e-12
# Bands (alias note: App==A++, Ap==A+, Amm==A--)
bands = {App=0.75, Ap=0.50, A0=0.25, Am=0.10, Amm=-inf}
# Cue 1: entropy on diff(s7)
map_primary = entropy(diff(s7))
win1 = 51
bins1 = 16
k1 = 2.0
# Cue 2: residual-entropy on s14 (explicit residual + entropy settings)
residual_def = movavg
map_secondary = entropy(diff(residual(s14)))
win_residual = 31
bins2 = 16
k2 = 2.0
entropy_log_base = e
binning_policy = equal_width_minmax
zero_prob_convention = 0*log(0)=0
# Fusion (order-invariant U/W on rapidities)
fusion = U/W
weights = {w1=2.0, w2=1.0}
notes = Order-invariant fusion; values unchanged; disagreement visible in bands
manifest_checksum = <sha256>
conformance_checksum = <sha256>
[/SSMH_MANIFEST]
Why this works (one sentence).phi((m,a)) = m keeps numbers identical, while {U,W} fusion (U += w*atanh(a), W += w, a_out := tanh(U/max(W,eps_w))) and atanh/tanh composition yield bounded, order-invariant evidence that highlights fragile intervals without hiding the KPI.
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Disclaimer
The contents in the Shunyaya Symbolic Mathematical Hardware (SSMH) materials are research/observation material. They are not engineering advice, not a safety standard or certification, and not operational guidance. Do not use for safety-critical, medical, legal, or financial decisions. Use at your own discretion; no warranties are provided; results depend on correct implementation and inputs.