---
title: Correlation Impact Ratio (CIR)
url: https://www.emergentmind.com/topics/correlation-impact-ratio-cir
type: topic
---

# Correlation Impact Ratio (CIR)

The Correlation Impact Ratio (CIR) is a correlation-aware feature attribution score designed for explainable AI (XAI) in complex models and large or heterogeneous datasets. CIR quantifies the sign-aligned co-movement between features and model outputs after robust centering, delivering single-pass, scalable, and computationally efficient global explainability. The ExCIR framework introduces rigorous invariance and stability properties while extending naturally to groupwise attributions via BlockCIR, which addresses double-counting in highly correlated feature clusters. CIR's design allows lightweight transfer protocols, recreating full-model rankings with a fraction of data, making it suited to edge, streaming, and real-world deployment scenarios [2511.16482].

## 1. Formal Definition and Mathematical Construction

Let $X \in \mathbb{R}^{n \times d}$ be the observed feature matrix ($n$: samples, $d$: features), with feature columns $\bm{x}_j \in \mathbb{R}^n$. Let $\bm{y}=f(X) \in \mathbb{R}^n$ be the vector of model outputs (e.g., logits, predictions). Both inputs and outputs are robustly centered using the mid-mean operator:
\[
m(\bm{a}) = \tfrac{1}{2}(Q_{0.25}(\bm{a}) + Q_{0.75}(\bm{a})),
\]
where $Q_\alpha(\bm{a})$ denotes the empirical $\alpha$-quantile of $\bm{a}$. The robustly centered values are
\[
\tilde{x}_{ij} = x_{ij} - m(\bm{x}_j),\qquad \tilde{y}_i = y_i - m(\bm{y}).
\]
For a feature set $G \subseteq \{1, ..., d\}$, define per-sample quantities:
\[
p_{iG} = \sum_{j\in G} \tilde{x}_{ij} \tilde{y}_i,\qquad u_{iG} = \sum_{j\in G} |\tilde{x}_{ij} \tilde{y}_i|.
\]
Aggregate across samples to get the total signed co-movement ($N_G$) and co-movement mass ($D_G$):
\[
N_G = \sum_{i=1}^n p_{iG},\qquad D_G = \sum_{i=1}^n u_{iG}.
\]
The Correlation Impact Ratio for set $G$ is then
\[
\mathrm{CIR}(G) =
\begin{cases}
\tfrac{1}{2} \bigl(1 + \frac{N_G}{D_G} \bigr), & D_G > 0,\\
\tfrac{1}{2}, & D_G = 0.
\end{cases}
\]
For individual features, $\mathrm{CIR}_j = \mathrm{CIR}(\{j\})$ [2511.16482].

## 2. Theoretical Properties and Interpretation

CIR measures the fraction of co-movement mass that is sign-consistent; it can equivalently be written as
\[
\mathrm{CIR}(G) = \frac{\sum_i s_{iG} u_{iG}}{\sum_i u_{iG}},
\]
where $s_{iG} = \tfrac{1}{2}(1 + \mathrm{sign}(p_{iG}))$. Key invariance properties include:

- **Translation and positive-scale invariance**: Additive or positive multiplicative constants to $X$ or $\bm{y}$ do not affect CIR due to centering and self-cancellation.
- **Sign symmetry**: Flipping $X$ or $\bm{y}$ complements the score, as $N_G \to -N_G$ but $D_G$ is invariant.
- **Monotonicity**: Increasing the magnitude of aligned co-movement or reducing anti-aligned terms increases CIR.

CIR down-weights features with co-movement that fluctuates in sign, while rewarding consistently aligned (positive/negative) associations between features and outputs, even under feature correlation. The construction ensures post-hoc, single-pass computation, requiring only the model's outputs and feature data after training [2511.16482].

## 3. Algorithmic Procedure and Complexity

CIR and its group extension are computed using a lightweight single-pass protocol, particularly advantageous when $n$ is large or in streaming/edge scenarios:

- **Random sub-sampling**: Select a random subset of $f n$ rows ($f \in (0,1]$), keeping the model, hyperparameters, seed, and validation split fixed. With $f \approx 0.2$–$0.4$, the method empirically recovers global ranking structure at $3$–$9\times$ speed-up.
- **Computation**:
    - Quantile-based centering: $\Theta(n)$ (streaming) or $\Theta(n\log n)$ (sort-based).
    - Aggregation: $\Theta(n d)$ time, $\Theta(d)$ space for single features, and $\Theta(|\mathcal{G}|)$ for groupwise collections.

**Pseudocode Outline** (*LaTeX-style*):
- Compute robust centers for features and outputs on the sampled rows.
- Accumulate $N_j$, $D_j$ (and optionally $N_G$, $D_G$ for groups).
- Compute $\mathrm{CIR}_j$ as $\tfrac{1}{2}(1+N_j/D_j)$ or $1/2$ if $D_j=0$.

This protocol permits reproducibility and transferability with partial data, compatible with quantile sketches for streaming [2511.16482].

## 4. Groupwise Extension: BlockCIR and Double-Counting Mitigation

BlockCIR generalizes ExCIR to collective feature attribution for predefined or data-driven sets $G$:
\[
\mathrm{BlockCIR}_G = \mathrm{CIR}(G) = \tfrac{1}{2} \left(1 + \frac{\sum_i \sum_{j \in G} \tilde{x}_{ij} \tilde{y}_i}{\sum_i \sum_{j \in G} |\tilde{x}_{ij} \tilde{y}_i|}\right).
\]
BlockCIR aggregates aligned co-movement over sets, mitigating the double-counting effect present in collinear or redundant groups (e.g., synonyms, duplicated sensors, highly correlated gene clusters). Group construction may be:

- **Domain-driven**: using prior taxonomies such as medical code groupings or sensor channels.
- **Data-driven**: via hierarchical clustering or correlation thresholding to extract feature clusters.
- **Model-driven**: based on learned embeddings, heads, or structure-specific representations.

By scoring groups as single units, BlockCIR provides more stable, interpretable global rankings in strongly correlated feature regimes [2511.16482].

## 5. Empirical Evaluation and Comparative Results

Across 29 benchmark tasks (text, tabular, image, signal, remote sensing, and synthetic datasets using logistic regression and XGBoost backbones), ExCIR delivers:

- **High agreement** with established global attribution baselines regarding top-$k$ feature overlap (default $k=8$), with $>90\%$ retention at $f=0.2$–$0.4$ sub-sampling.
- **Runtime reductions**: $3$–$9\times$ wall-clock improvement (e.g., 18.3s $\rightarrow$ 4.1s for har6 at $f=0.2$).
- **Score robustness**: High Spearman ($\rho>0.98$) between mid-mean and median centering; mean centering degrades under outliers.
- **BlockCIR effectiveness**: Preserves top-marked features in collinear blocks, increases top-$k$ overlap, and avoids diluting importance across co-moving features.

Evaluation metrics include the Jaccard overlap, Spearman’s $\rho$, Kendall’s $\tau$, Orthogonal Procrustes residual, and symmetric KL divergence. Compared to SHAP/LIME/PFI, CIR/BlockCIR forgo model perturbations, scales linearly, and respects feature correlation structure—properties lacking in gradient-based or kernel-based (HSIC, MI) alternatives [2511.16482].

| Baseline            | Perturbation-Free | Correlation-Aware | Linear Scalability |
|---------------------|------------------|-------------------|--------------------|
| ExCIR/BlockCIR      | Yes              | Yes               | Yes                |
| SHAP/LIME/PFI       | No               | No                | No                 |
| Gradients           | Yes              | No                | Yes                |
| Kernel-based (HSIC) | Yes              | No                | No                 |

## 6. Applications, Limitations, and Open Problems

CIR and BlockCIR address global feature ranking in large tabular/text corpora, streaming scenarios with quantile sketches, and group-level attribution (multi-sensor, multi-channel) across vision and NLP (via class-conditioned extensions $\mathrm{CIR}^{(c)}$).

**Limitations:**
- CIR captures correlation, not causality; latent confounding can affect interpretations.
- It is less sensitive to nonlinear higher-order feature–output dependencies; features only interacting nonlinearly (e.g., sinusoidal relationships) yield near-zero CIR.
- In extremely heavy-tailed noise, robust centering may be insufficient, requiring further trimming or robustification.

**Open Directions:**
- **Conditional ExCIR (cCIR):** Isolating the unique effect of features after accounting for others.
- **Mutual-ExCIR (mCIR):** Extending to capture nonlinear or higher-order interactions by, for example, kernelizing co-movements.
- **Adaptive grouping:** Learning optimal feature-set partitions via graph or attention-based approaches.
- **Sample complexity theory:** Establishing the number of samples required to reach a target top-$k$ agreement under sub-sampling [2511.16482].

CIR and BlockCIR provide correlation-aware, efficient, and robust global explainability suitable for large-scale deployment and resource-constrained settings, complementing but not replacing methods sensitive to causal and nonlinear feature effects.

Source: https://www.emergentmind.com/topics/correlation-impact-ratio-cir