Performance Causal Inversion Principle

• Performance Causal Inversion Principle is a framework that inverts causal mappings to compute all valid input adjustments for achieving targeted performance outputs.
• It integrates nonparametric symbolic regression with interval arithmetic-based set inversion to learn and certify causal relationships from observed data.
• The approach facilitates system tuning, diagnostic interventions, and robust optimization even when traditional mechanistic models are unavailable.

The Performance Causal Inversion Principle refers to a class of methodologies that explicitly invert the direction of cause-effect mapping in order to determine which system inputs (“adjustments”) are required to achieve a specified output (“performance”). This principle operationalizes causal manipulation: rather than predict outputs from inputs, it systematically computes the preimage of a desired output set within the input domain—thus identifying all adjustment configurations that will realize the targeted outcome, even when the internal model of the system is unknown. Central to many instantiations of this principle are two steps: (1) “learning” the causal mechanism by symbolic regression (where no parametric assumptions are made about the input–output mapping), and (2) applying set inversion techniques (typically leveraging interval arithmetic) to compute the solution set, which can be recursed and certified to yield only valid adjustments. The PCI principle has broad applicability, including system tuning, diagnostic interventions, and design optimization, especially where explicit causal models are difficult or impossible to obtain from first principles.

1. Mathematical Formulation and Core Workflow

Let $f: \mathbb{R}^n \to \mathbb{R}^p$ denote the unknown mapping from adjustment vector $x$ to performance vector $y$. Given only finitely many observed data pairs $(x_i, y_i)$, the objective is to identify a region of adjustment space $\mathcal{S} \subset \mathbb{R}^n$ such that all $x \in \mathcal{S}$ generate performances $f(x) \in \mathcal{P} \subset \mathbb{R}^p$, where $\mathcal{P}$ specifies the set of acceptable/targeted outputs.

The two-step methodology is:

1. Symbolic Regression via Genetic Programming (SRvGP):
   • Construct a nonparametric symbolic approximation $\tilde{f}$ to the true function $f$ using evolutionary programming, searching through expression trees without committing to any fixed model class.
   • The symbolic expression $\tilde{f}(x)$ converges (in practice, often uniformly) to $f(x)$ on the sampled input region $\mathcal{R}$; e.g., $\tilde{f}(x) \approx f(x)$ for $x \in \mathcal{R} \subset \mathbb{R}^n$.
2. Set Inversion Using Interval Arithmetic Algorithms:
   • Compute the preimage set

   $\mathcal{S} = f^{-1}(\mathcal{P}) \cap \mathcal{R}$

• Employ algorithms such as SIvIA and its probabilistic variant, the $\psi$-algorithm, which use interval arithmetic to subdivide the adjustment space into boxes $\mathcal{X}$.

• For each interval $\mathcal{X}$, calculate the inclusion function $[f](\mathcal{X})$ and the acceptance probability

  $$p(\mathcal{X}) = \frac{\mathrm{mes}([f](\mathcal{X}) \cap \mathcal{P})}{\mathrm{mes}([f](\mathcal{X}))}$$

  where $\mathrm{mes}(\cdot)$ is the Lebesgue measure (volume). Boxes with $p(\mathcal{X}) = 1$ are accepted, those with $p(\mathcal{X}) = 0$ are rejected, and ambiguous cases are split for recursion.

This inversion process identifies all valid system configurations that guarantee the targeted performance, even in the absence of a physical or mechanistic model.

2. Symbolic Regression for Causal Structure Identification

Symbolic regression via genetic programming operates by evolving populations of mathematical expression trees under selection pressure to optimize the fit between predicted and actual outputs. Unlike parametric regression, SRvGP does not impose a predefined functional form; instead, it can recover complex expressions, such as $f(x) = \sin(5x) \cdot \exp(-x^2)$, directly from data—even with high levels of observational noise.

SRvGP’s evolutionary search traverses noisy landscapes and can distill interpretable formulas that serve as generative causal models. The symbolic representation $\tilde{f}$ “opens the black box,” providing insight into variable relationships and enabling explicit sensitivity analysis and system understanding.

Key features include:

• Nonparametric, model-free search: No bias toward specific function classes.

• Noise filtration: Capable of extracting the correct structure under moderate noise conditions.

• Interpretability: Yields compact symbolic formulas revealing causal dependencies.

3. Set Inversion Algorithms and Interval Arithmetic

Set inversion is performed on the learned symbolic model using interval arithmetic to guarantee coverage of all possible values within each box. In particular, the SIvIA and $\psi$-algorithms:

• Partition the adjustment space $\mathcal{R}$ into multidimensional intervals (boxes).

• For each box $\mathcal{X}$, evaluate the range $[f](\mathcal{X})$ using interval extensions of $\tilde{f}$, rigorously accounting for all possible outputs due to variable dependencies and rounding errors.

• Acceptance probability $p(\mathcal{X})$ quantifies the proportion of outputs within the targeted performance region $\mathcal{P}$.

  • $p(\mathcal{X}) = 1$: All $x \in \mathcal{X}$ are guaranteed to yield performances in $\mathcal{P}$ (box is accepted).
  • $p(\mathcal{X}) = 0$: No $x \in \mathcal{X}$ yields an acceptable performance (box is rejected).
  • $0 < p(\mathcal{X}) < 1$: Ambiguous; box is split and recursed for finer resolution.

The process recurses until boxes are classified or reach a minimal diameter threshold.

4. Causal Manipulation and System Adjustment

By implementing set inversion on the symbolic model, the practitioner “inverts” the causal relationship, systematically determining all adjustment configurations compatible with a specified set of outcomes. This constitutes the core of the Performance Causal Inversion Principle: optimization is performed not by searching for the best adjustment, but by mapping desired outputs back to feasible inputs.

Practical implications:

• Control synthesis: Enables direct computation of feasible parameter sets for system tuning.
• Diagnostic intervention: Identifies which configuration changes are required to correct or improve performance.
• Robust optimization: Systematic enumeration—and certification—of all acceptable solutions within the input space.

5. Computational Complexity, Limitations, and Implementation Considerations

While theoretically rigorous, set inversion algorithms face several computational challenges:

• Curse of dimensionality: The number of interval boxes grows exponentially with dimensionality of the adjustment space, necessitating parallelization and adaptive bisection strategies.
• Interval arithmetic overestimation: Dependency effects (where variables appear multiple times in $\tilde{f}$) can lead to overly conservative interval ranges.
• Noise and model error: Symbolic regression may struggle with highly non-smooth or heavily noisy functions; preprocessing (e.g., Fourier or wavelet filtering) may be required to ensure recoverability.
• Overfitting risk: Expressions inferred outside the sampled domain may not generalize; interval arithmetic does not capture extrapolation uncertainty.

Implementation best practices include:

• Prior filtering and denoising of experimental data.
• Utilization of free–algebra–based interval arithmetic extensions to mitigate dependency inflation.
• Distributed computation for handling high–dimensional adjustment spaces.
• Application of adaptive box splitting to focus computational effort on ambiguous regions.

6. Applications and Extensions

The Performance Causal Inversion Principle applies broadly to systems where the causal law is unknown or partially observed, including:

• Multi-disciplinary experimental systems requiring targeted adjustment.
• Network performance and control, where inverse mapping of interventions to outcomes is needed.
• Engineering design and optimization problems subjected to complex, data-driven causal mechanisms.
• Diagnostic repair in automated or robotic systems, aligning adjustment actions with specified performance goals.

Its algorithmic structure is compatible with sequential extension (e.g., recursively in causal graphs) and can be augmented with probabilistic reasoning for uncertainty quantification.

7. Summary Table: Core Elements of the PCI Principle

Component	Mathematical Role	Implementation Considerations
Symbolic Regression (SRvGP)	Learns $\tilde{f}(x) \approx f(x)$	Requires sufficient data, noise handling
Set Inversion (SIvIA/$\psi$)	Computes $\mathcal{S} = f^{-1}(\mathcal{P})$	Interval arithmetic, box partition, complexity
Acceptance Probability	$$p(\mathcal{X}) = \frac{\mathrm{mes}([f](\mathcal{X}) \cap \mathcal{P})}{\mathrm{mes}([f](\mathcal{X}))}$$	Recursion on ambiguous/partial boxes

This organizing principle enables rigorous, interpretable, and certifiable adjustment of causal systems toward and within performance specifications, fully operating under data-driven, model-free regimes.