# Kernel-based sensitivity analysis on excursion sets

Christophette Blanchet-Scalliet, Noé Fellmann, Céline Helbert, Adrien Spagnol, Delphine Sinoquet

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## To cite this version:

Christophette Blanchet-Scalliet, Noé Fellmann, Céline Helbert, Adrien Spagnol, Delphine Sinoquet. Kernel-based sensitivity analysis on excursion sets. MASCOT-NUM2023, Apr 2023, Le Croisic, France. hal-04098720

HAL Id: hal-04098720

https://hal.science/hal-04098720

Submitted on 17 May 2023

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Distributed under a Creative Commons Attribution 4.0 International License
## Introduction

**Goal:** Quantifying the influence of uncertain inputs on feasible sets associated to a constrained robust optimization problem.

$$ \boldsymbol{x}^* = \arg \min_{\boldsymbol{x} \in \mathcal{X}} \mathbb{E}_U[f(\boldsymbol{x}, \boldsymbol{U})] \quad \text{s.t.} \quad \mathbb{P}_U[g(\boldsymbol{x}, \boldsymbol{U}) \le 0] \ge \alpha. $$

**Answer:** Performing Sensitivity analysis on excursion sets $\Gamma_U$ using kernel-based methods. We propose a kernel $k_{set}$ with which we compute HSIC-ANOVA indices.

$$ (U_1, ..., U_p) \mapsto \Gamma = \{x \in \mathcal{X}, g(x, U) \le 0\} $$

**Example:**

$$ \Gamma = \{x \in [-5, 5]^2, x_1^2 + 5x_2 - U_1 + U_2^2 - 1 \le 0\} \\ \rightarrow U_2 \text{ seems more influential than } U_1 \text{ on the feasible sets.} $$

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## Kernel-based Sensitivity Analysis

<figure><img src="image_4.png" /><figcaption>Fig. 1: Kernel mean embedding [6]</figcaption></figure>

With $K = k_X \otimes k_Y$, the Hilbert Schmidt Independence Criterion (HSIC) is given by:

$$ \begin{aligned} \text{HSIC}_{k_X, k_Y}(X, Y) &= \|\mu_K(X, Y) - \mu_{k_X}(X) \otimes \mu_{k_Y}(Y)\|_{\mathcal{H}_K}^2 \\ &= \mathbb{E}[k_X(X, X')k_Y(Y, Y')] \\ &\quad + \mathbb{E}[k_X(X, X')]\mathbb{E}[k_Y(Y, Y')] \\ &\quad - 2\mathbb{E}[\mathbb{E}[k_X(X, X')|X]\mathbb{E}[k_Y(Y, Y')|Y]]. \end{aligned} $$

When K is characteristic (injectivity of the mean embedding),

$$ \text{HSIC}_{k_X, k_Y}(X, Y) = 0 \quad \text{iif } X \perp Y. \\ \rightarrow \text{screening} $$

Assuming that the inputs are independent and that the input kernels are ANOVA,

$$ \text{HSIC}(\mathbf{X}, Y) = \sum_{A \subseteq \{1, ..., d\}} \sum_{B \subseteq A} (-1)^{|A|-|B|} \text{HSIC}(\mathbf{X}_B, Y). $$

HSIC-ANOVA indices [3] are then defined as:

$$ S_i^{\text{HSIC}} := \frac{\text{HSIC}(X_i, Y)}{\text{HSIC}(\mathbf{X}, Y)}, $$

$$ S_{T_i}^{\text{HSIC}} := 1 - \frac{\text{HSIC}(\mathbf{X}_{-i}, Y)}{\text{HSIC}(\mathbf{X}, Y)} $$

and are suited for ranking (and screening [5]).

## HSIC for set-valued outputs

With $A \Delta B = A \cup B - B \cap A$ and $\mu$ the Lebesgue measure, we define a kernel on closed sets by:

$$ \forall \Gamma_1, \Gamma_2 \in \mathcal{F}(\mathcal{X}), k_{\text{set}}(\Gamma_1, \Gamma_2) = \exp \left( - \frac{\mu(\Gamma_1 \Delta \Gamma_2)}{2\sigma^2} \right). $$

**Proposition 1**

$k_{set}$ is a kernel [1] and is characteristic.

Given an input $U_l$ and an associated ANOVA kernel $k_l$, the HSIC on sets that we call $H_{set}$ is defined by:

$$ \begin{aligned} \widehat{H}_{\text{set}}(U_l, \Gamma) &:= \text{HSIC}_{k_l, k_{\text{set}}}(U_l, \Gamma) \\ &= \mathbb{E}[(k_l(U_l, U_l') - 1)k_{\text{set}}(\Gamma, \Gamma')]. \end{aligned} $$

It can be estimated with its U-statistic:

$$ \widehat{H}_{\text{set}}(U_l, \Gamma) = \frac{2}{n(n-1)} \sum_{i<j}^{n} (k_l(U_l^{(i)}, U_l^{(j)}) - 1) k_{\text{set}}(\Gamma^{(i)}, \Gamma^{(j)}). $$

$k_{set}$ is rarely exactly known and requires to be estimated. Given $X^{(1)}, ..., X^{(m)}$ an iid sample of $X \sim \mathcal{U}(\mathcal{X})$, we propose the estimator:

$$ \widehat{k}_{\text{set}}(\Gamma^{(i)}, \Gamma^{(j)}) = e^{-\frac{\mu(\mathcal{X})}{2\sigma^2} \frac{1}{m} \sum_{k=1}^{m} \mathbb{1}_{\Gamma^{(i)} \Delta \Gamma^{(j)}}(X^{(k)})}, $$

leading finally to the nested $H_{set}$ estimator:

$$ \widehat{H}_{\text{set}}(U_l, \Gamma) = \frac{2}{n(n-1)} \sum_{i<j}^{n} (k_l(U_l^{(i)}, U_l^{(j)}) - 1) e^{-\frac{\mu(\mathcal{X})}{2\sigma^2} \frac{1}{m} \sum_{k=1}^{m} \mathbb{1}_{\Gamma^{(i)} \Delta \Gamma^{(j)}}(X^{(k)})}. $$

**Proposition 2**

The quadratic risk of the nested estimator $\widehat{H}_{set}$ verifies:

$$ \mathbb{E} \left( \widehat{H}_{\text{set}}(U_l, \Gamma) - H_{\text{set}}(U_l, \Gamma) \right)^2 \le 2 \left( \frac{2\sigma_1^2}{n(n-1)} + \frac{4(n-2)\sigma_2^2}{n(n-1)} + \frac{K^2\sigma_3^2}{m} \right). $$

HSIC-ANOVA indices on sets are then denoted $S_i^{\widehat{H}_{set}}$ and $S_{T_i}^{\widehat{H}_{set}}$.

## Toy case 1 [4]

$$ \forall x, u \in [-5, 5]^2 \times [-5, 5]^3 \quad g_1(x_1, x_2, u_1, u_2, u_3) = -x_1^2 + 5x_2 - u_1 + u_2^2 - 1 $$

<figure><img src="image_5.png" /><figcaption>Fig. 2: p-value of $\widehat{H}_{set}$ for $n = m = 100$</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3: $S_i^{\widehat{H}_{set}}$ for $n = m = 100$</figcaption></figure>

## Toy case 2 [2]

$$ g_2(x_1, x_2, u_1, u_2, u_p, u_{r_1}, u_{r_2}) = u_{r_2} - \max_{t \in [0, T]} \mathcal{Y}''(x_1 + u_1, x_2 + u_2, u_p; t), $$

with $\mathcal{Y}(x_1 + u_1, x_2 + u_2, u_p; t)$ the solution of the harmonic oscillator defined by:

$$ (x_1 + u_1)\mathcal{Y}''(t) + u_p\mathcal{Y}'(t) + (x_2 + u_2)\mathcal{Y}(t) = \eta(t). $$

<figure><img src="image_7.png" /><figcaption>Fig. 4: p-value of $\widehat{H}_{set}$ for $n = m = 100$</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5: $S_i^{\widehat{H}_{set}}$ for $n = m = 100$</figcaption></figure>

## Outlook

* Testing the indices on industrial applications (as viability kernels, or air pollutant concentration maps) and compared to other indices defined for set-valued output models.

* Incorporating this method inside a robust optimization methodology: reducing the uncertain input space dimension to get cheaper meta-models.

## References

[1] P. Balança, E. Herbin. A set-indexed Ornstein-Uhlenbeck process. 2012.

[2] A. Cousin et al. A two-step procedure for time-dependent reliability-based design optimization involving piece-wise stationary Gaussian processes. 2022.

[3] S. da Veiga. Kernel-based ANOVA decomposition and Shapley effects - Application to global sensitivity analysis. 2021.

[4] R. El Amri et al. A sampling criterion for constrained Bayesian optimization with uncertainties. 2021.

[5] G. Sarazin et al. Test d'indépendance basé sur les indices HSIC-ANOVA d'ordre total. 2022.

[6] D. Sejdinovic. Learning with Approximate Kernel Embeddings. RegML Workshop. 2017.