Sigmoid Function Implementation in Environmental KPIs

Scientific Justification

Environmental KPIs use sigmoid functions because they accurately model the non-linear relationship between environmental conditions and human thermal perception. This approach aligns with established thermal comfort standards ASHRAE 55 and ISO 7730, which are based on the Predicted Mean Vote (PMV) model that inherently produces S-shaped response curves when mapping human thermal sensation from -3 (cold) to +3 (hot) against environmental parameters.

Mathematical Foundation

Formula: 100/(1+e^(-k*(mean-threshold)))

The sigmoid function reflects how humans experience thermal comfort through three key characteristics:

• Non-linear response: Small environmental changes have disproportionate impact near uncomfortable thresholds
• Gradual transitions: Smooth progression from comfortable to uncomfortable states
• Bounded output: Scores remain within 0-100% range with clear performance plateaus

Function Variants

Normal Sigmoid: Used when higher values indicate better performance

• Applied to comfort metrics where increased comfort hours improve scores
• Curve rises steeply as conditions approach optimal range

Inverted Sigmoid: Used when lower values indicate better performance

• Applied to stress metrics where reduced stress hours improve scores
• Curve descends steeply as stress conditions increase

PMV Model Alignment

The PMV model by P. Ole Fanger predicts average thermal sensation across a seven-point scale. Its characteristic curve shows:

• Flat central zone representing neutral comfort (PMV ≈ 0)
• Steep transitions indicating rapid perception changes away from neutral
• Plateaus at extremes (-3/+3) representing sensation limits

Sigmoid KPIs mathematically mirror this established human response pattern, providing scientific justification for the metric design.

Practical Implementation

• Comfort KPIs: Normal sigmoid rewards achievement of neutral thermal conditions (PMV near 0) with higher scores, penalizing deviation toward discomfort extremes
• Stress KPIs: Inverted sigmoid penalizes departure from comfortable conditions, with scores declining as thermal stress increases toward PMV extremes

This mathematical alignment ensures KPIs reflect validated human thermal perception research while providing quantifiable performance metrics for urban design evaluation.

Assessment Framework

These KPI thresholds reflect the enhanced sensitivity and consistency of the refined calculation system using uniform 0.2 steepness for sigmoid functions. The targets maintain realistic expectations while providing meaningful differentiation between design performance levels, grounded in established thermal comfort research and human perception models.

The sigmoid KPI approach has solid scientific grounding: it applies validated psychophysical relationships (from Fanger) as implemented in real building studies (de Dear & Brager) to create performance metrics.

The sigmoid function approach used in these KPIs is scientifically grounded in the established PMV model’s characteristic S-shaped response curves. It reflects the non-linear relationship between environmental conditions and human thermal perception, as documented in Fanger’s foundational work and validated through decades of thermal comfort research.

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