QSLS Correlation and Confidence Values: Embedding Uncertainty in Non-Linear Relationships

Abstract

The Quantifying Systems Level of Support (QSLS) framework utilizes correlation and confidence values to quantify relationships between systems engineering mechanisms and quality attributes. This paper examines how these seemingly linear metrics inherently embed uncertainty related to non-linear relationships in their computations, enabling a nuanced representation of complex system interactions despite using simple numerical scales.

Introduction

In systems engineering, the relationships between components, mechanisms, and quality attributes are rarely purely linear. Complex systems exhibit emergent behaviors, threshold effects, and multi-dimensional interactions that defy simple linear characterization. The QSLS framework addresses this challenge by employing a dual-metric approach: correlation values (0.0 to 1.0) and confidence levels (0.0 to 1.0), which together capture both the strength of relationships and the uncertainty inherent in their assessment.

Non-Linearity in System Relationships

Non-linear relationships in systems engineering manifest in various forms:

1. Threshold Effects: Where small changes in input produce disproportionate output changes

2. Emergent Properties: System behaviors not predictable from individual components

3. Feedback Loops: Circular relationships that amplify or dampen effects

4. Contextual Dependencies: Relationships that vary based on environmental conditions

5. Temporal Dynamics: Time-dependent relationships that evolve non-linearly

Embedding Uncertainty in Correlation Values

Correlation Value Synthesis

The QSLS correlation value (0.0 to 1.0) represents a synthesis of multiple factors:

1. Conceptual Relevance: Theoretical alignment between elements

2. Functional Interdependence: Practical impact on system performance

3. Contextual Alignment: Consistency with system architecture

4. Empirical Evidence: Historical and experimental data

5. Trade-offs: Competing priorities and constraints

These factors inherently include non-linear aspects. For example, functional interdependence may exhibit threshold behaviors where a component has minimal impact until certain conditions are met, then suddenly becomes critical.

Deep Analysis Requirements

The QSLS guidelines mandate deep analysis for each correlation, requiring evaluators to:

• Consider edge cases and special conditions

• Evaluate both positive and negative interaction effects

• Examine emergent properties from specific combinations

• Account for implementation complexity variations

• Assess non-deterministic behaviors

This deep analysis process embeds knowledge of non-linearities into the final correlation value, effectively compressing complex relationships into a simple metric.

Confidence Levels as Uncertainty Indicators

Multi-Dimensional Uncertainty

QSLS confidence levels incorporate uncertainty from multiple sources:

1. Evidence Strength (0.0-0.2 points)

2. Domain Knowledge Depth (0.0-0.2 points)

3. Consensus Level (0.0-0.2 points)

4. Definition Clarity (0.0-0.2 points)

5. Historical Validation (0.0-0.2 points)

Each dimension captures different aspects of uncertainty, including:

• Incomplete understanding of non-linear dynamics

• Variability in expert opinions about complex behaviors

• Limited historical data on edge cases

• Ambiguity in defining system boundaries and interactions

Non-Linear Uncertainty Propagation

The confidence calculation process itself introduces non-linearity:

1. Cumulative Effects: Points from each criterion sum to create the overall confidence

2. Threshold Behaviors: Confidence below 0.6 triggers additional review

3. Context Sensitivity: Same correlation may have different confidence levels in different contexts

4. Expert Judgment: Human evaluators apply non-linear reasoning when assigning points

Mathematical Framework

While the output values are linear (0.0 to 1.0), the underlying process can be represented as:

Correlation = f(conceptual_relevance, functional_interdependence, contextual_alignment, empirical_evidence, trade_offs)

Where f is a non-linear function that synthesizes multiple inputs through deep analysis.

Similarly, for confidence:

Confidence = g(evidence_strength, domain_knowledge, consensus_level, definition_clarity, historical_validation)

Where g represents the weighted combination and threshold effects in the confidence calculation.

Practical Implications

Interpretation Guidelines

The dual-metric approach enables several interpretations:

1. High Correlation, High Confidence: Strong linear relationship with minimal uncertainty

2. High Correlation, Low Confidence: Potentially non-linear relationship with significant uncertainty

3. Low Correlation, High Confidence: Weak relationship (possibly due to non-linear cancellation)

4. Low Correlation, Low Confidence: Unknown relationship requiring further investigation

Decision Support

System designers can use these metrics to:

1. Identify areas requiring deeper investigation

2. Prioritize uncertainty reduction efforts

3. Recognize potential non-linear effects in system design

4. Make risk-informed decisions based on both correlation and confidence

Case Studies

Example 1: SWAP Optimization

When correlating mechanisms with Size, Weight, and Power (SWAP) optimization, non-linear effects are common:

• A mechanism may provide linear benefits up to a threshold, then show diminishing returns

• Interactions between space, weight, and power constraints often exhibit non-linear trade-offs

• Confidence levels reflect uncertainty about these threshold points and interaction effects

Example 2: Security Attributes

Security mechanisms often exhibit non-linear behaviors:

• Anti-hacking measures may be effective only above certain thresholds

• Security effectiveness can degrade non-linearly under specific attack vectors

• Confidence values capture uncertainty about these non-linear vulnerabilities

Conclusion

The QSLS correlation and confidence values, while presented as simple linear metrics, effectively embed complex non-linear relationships through their evaluation methodology. This approach provides system engineers with a practical tool for quantifying relationships while acknowledging the inherent uncertainty in complex systems. The dual-metric system enables nuanced decision-making in the face of non-linear behaviors, supporting more robust system design and architecture.

References

1. Systems Engineering Handbook, INCOSE (2015)

2. ISO/IEC/IEEE 15288:2015 Systems and software engineering

Prepared by: QSLS Engineering Inc. Date: April 2025 Patent Pending: Case Number 18/925,529