SSMT – Mini-Metrics Library: Anomaly, Stability, and Phase Dwell (6.1–6.3)

Portable KPIs you can lift into dashboards, audits, and reports — without touching raw °C/°F.

Purpose.
Shunyaya Symbolic Mathematical Temperature (SSMT) does not stop at e_T and a_phase. It also defines clean, unitless KPIs that can be compared across devices, facilities, regions, or even environments with different physics. These KPIs answer questions like: “Is this location unusually hot for this hour?” “How volatile has this system been today?” “How long are we sitting right at the dangerous phase boundary?”

These KPIs live entirely in symbol space. They are built on e_T, a_phase / a_phase_fused, and hysteresis memory — not on raw °C/°F. That makes them consistent, governance-friendly, and safe for ML.

6.1 Notation and defaults

To keep everything readable and repeatable, SSMT defines notation that any analytics team can carry forward:

e_T(t)              : unitless contrast at time t
a_phase_star(t)     : chosen phase dial at time t
                      (use a_phase for single pivot
                       or a_phase_fused for multi-pivot)

lst(t)              : local solar time (or local clock hour)

L                   : window length
                      (default 24 h for hourly data)

Q                    : quantile operator
median is Q_0.5

P_t(•)              : fraction of time in [t−L, t]
                      for which a predicate holds

MAD(x)              : median absolute deviation
                      = median(|x − median(x)|)

Defaults.

  • Diurnal window = 24 h
  • Volatility window = 24 h
  • Rolling medians = 30 days (same hour-of-day)
  • You pick L to match cadence (hourly, 10-min, etc.), but once declared, you keep it published and stable.

Why this matters: you can now publish dashboards and compliance summaries without exposing raw temperature feeds, and another party can still reproduce every KPI exactly.

6.2 Core anomaly and normalization

This is how SSMT replaces “today felt hot” with something that can be audited.

Anomaly vs seasonal or diurnal norm.
Instead of asking “Is 32 °C bad here?”, SSMT asks:
“How far is the symbolic contrast e_T(t) from what is normal for this hour, at this location, in recent history?”

A_T(t) :=
  e_T(t)
  − median_{d in past 30 days, same hour}( e_T(d, hour(lst(t))) )

That gives a localized, time-of-day-aware anomaly score in symbol space.

Diurnal-normalized anomaly.
If you maintain an explicit diurnal profile, you can express a normal-corrected contrast:

e_T_star(t) :=
  e_T(t)
  − mu_diurnal( hour(lst(t)) )

This is powerful for systems with predictable heat cycles (for example machinery that always runs warmer mid-day). You don’t raise alarms just because the schedule is doing what it always does.

Robust z-score.
You can also build a dimensionless stability index that is robust to spikes:

Z_T(t) :=
  ( e_T(t)
    − median_{30d}( e_T | same hour )
  )
  / MAD_{30d}( e_T | same hour )

Optionally, you can scale MAD by k_MAD = 1.4826 if you want it to approximate a standard deviation under Gaussian assumptions. The key point is: this is a reproducible anomaly score that:

  • Ignores raw °C/°F,
  • Knows the time of day,
  • Respects local historical context,
  • And can be reviewed across multiple locations using the same logic.

Now “above normal” is a formal, replayable statement — not an argument.

6.3 Volatility and stability

After anomaly, the next question is: “How noisy is this system right now?” Volatility tells you about instability. Phase-band dwell tells you how long you’re sitting in a danger band.

Rolling volatility over a window L.
This is physical stress in motion form — not just “high” or “low,” but “choppy” vs “calm.”

V_T(t) :=
  sqrt( Var_{tau in [t−L, t]}( e_T(tau) ) )
  • High V_T(t) means the thermal situation is jumping around.
  • Low V_T(t) means conditions are smooth and predictable.
  • You pick L (default 24 h for hourly cadence), publish it, and hold it steady so trends are comparable across assets.

This is extremely useful for maintenance and uptime forecasting. High volatility often means seal stress, fatigue cycles, or unstable control behavior — even if absolute temperature is “acceptable.”

Phase-band dwell near pivot.
Sometimes the real concern isn’t “how hot overall,” it’s “how long are we flirting with the danger edge?”

SSMT turns that into a percentage of time spent hugging the danger zone:

TimeInPhaseBand :=
  fraction_{tau in [t−L, t]}(
    |a_phase_star(tau)| <= phi_star
  )

Where:

  • a_phase_star(t) is your chosen phase dial (a_phase for a single pivot like freezing, or a_phase_fused if multiple pivots matter),
  • phi_star is an agreed “near the edge” tolerance band,
  • Typical default: phi_star = Phi_freeze.

Interpretation:

  • High TimeInPhaseBand means “we are hovering right at the structural / survival limit and not leaving it.”
  • Low TimeInPhaseBand means “we either stayed clearly safe or clearly unsafe, but we didn’t hover and chatter.”

This directly supports operational review: “Did we actually sit at the freeze boundary for 3 hours?” is now a measurable quantity, not a dispute.

Why these KPIs matter.

  • They are unitless and symbolic.
  • They can be applied to different assets, different climates, even different physical domains, without rewriting the math.
  • They are built around repeatable windows and quantile logic, not fragile single-sensor spikes.
  • They are simple enough to include in compliance dashboards and reliability reports.

Once A_T(t), Z_T(t), V_T(t), and TimeInPhaseBand exist, temperature stops being “a raw number someone eyeballed” and becomes auditable evidence about stress, abnormality, and exposure.

Navigation
Previous: SSMT – CI, Compliance, and Upgrade Safety (5.4–5.5)
Next: SSMT – Symbolic Excursion Budgets, Flicker KPIs, and Governance Signals (6.4–6.7)

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SSMT – Table of Contents