Quantile Based Global Sensitivity Measures

Notes on the paper Quantile Based Global Sensitivity Measures by S. Kucherenko, S. Song, L. Wang

Sensitivity Measures

Sobol’ sensitivity indices

Deterministic approach

ANOVA (Analysis of Variance) Decomposition

$f: \mathbb{R}^p \rightarrow \mathbb{R}$ may be decomposed as

\[\begin{eqnarray*} f(x)&=&f_0+\sum_{i=1}^{p}f_i(x_i)+\sum_{1 \le i < j \le p}f_{i,j}(x_i,x_j)+ \dots +f_{1,2,\dots ,p}(x_1,x_2,\dots,x_p)\\ &=& f_0 + \sum_{k=1}^{p} \sum_{|u|=k} f_u(x_u) \end{eqnarray*}\]

where
$f_0=\mathbb{E}[f(x)]$
$f_i(x_i)=\mathbb{E}[f(x)|x_i]-f_0$
$f_{i,j}(x_i,x_j)=\mathbb{E}[f(x)|x_i,x_j]-f_i(x_i)-f_j(x_j)-f_0$

this hierarchical decomposition exists $\forall f$ such that $f (x) \in L^2$
if $x_1,x_2,\dots,x_p$ are statistically independent then
$\mathbb{E}[f_u(x_u)f_v(x_v)]=0 \quad u \ne v$ $\Longrightarrow Var(f(x))=\sum_{k=1}^{p} \sum_{|u|=k} Var(f_u(x_u))$

Sobol’ indices

\[S_u=\frac{Var(f_u(x_u))}{Var(f(x))}\]

$S_u$ measures the relative contribution of $x_u$ to $Var(f_u(x_u))$
there are $2^p-1$ Sobol’s indices

Total Sobol’ indices

\[T_u=\sum_{v \cap u \ne \emptyset} S_v\]

First Order and Total Sobol’ Indices

\[\{S_k, T_k \}_{k=1}^p\]

$S_k, T_k \in [0,1]$ measures the importance of $x_k$
$S_k$ only measures the contribution of $x_k$
$T_k$ measures the contribution of all interactions involving $x_k$
$S_k \le T_k\ \forall k \in {1,2,…,p}$
$T_u \le \sum_{K \in u} T_k\ \forall u \in {1,2,…,p}$

Sensitivity Indices for Subsets of Variable

model function: $Y=f(x_1, \dots x_d)$
input vector: $x=(x_1, \dots x_d)$
PDF: $\rho(x_1, \dots x_d)$
subset of variables: $y=(x_{i_1}, \dots x_{i_s})=(x_i), 1 \le s< d, 1 \le i \le d$
complementary subset $z=(x_{i_1}, \dots x_{i_{d-s}})=x_{\sim i}$, so that $x=(y,z)$
total variance: $V= \sum_{s=1}^d \sum_{i_i < \dots < i_s} V_{i_i \dots i_s}$
partial variance: $V_{i_{1} \ldots i_{s}}=\int_{0}^{1} f_{i_{1} \ldots i_{s}}^{2}\left(x_{i_{1}}, \ldots, x_{i_{s}}\right) d x_{i_{1}}, \ldots, x_{i_{s}}$

The variance corresponding to $y$

\[V_y= \sum_{m=1}^s \sum_{(i_1 < \dots < i_m) \in K} V_{i_1, \dots ,i_m}\]

Total variance

\[V_y^{tot}=V-V_z\]

Global sensitivity indices

\[S_y=\frac{V_y}{V}\] \[S_y^{tot}=\frac{V_y^{tot}}{V}\]

Probabilistic approach

Total variance of $f(x_1, \dots x_d)$

\[V=V_{X_i}(E_{X_{\sim i}}(Y|X_i))+E_{X_i}(V_{X_{\sim i}}(Y|X_i))\] \[1=\frac{V_{X_i}(E_{X_{\sim i}}(Y|X_i))}{V}+\frac{E_{X_i}(V_{X_{\sim i}}(Y|X_i))}{V}\]

First order sensitivity index

\[S_i=\frac{V_{X_i}(E_{X_{\sim i}}(Y|X_i))}{V}\] \[S_i^{tot}=\frac{E_{X_{\sim i}}(V_{X_i}(Y|X_{\sim i}))}{V}\]

Quantile Based Sensitivity Measure

The αth quantile of the output CDF $q_Y(\alpha)$

\[\begin{eqnarray*} \alpha&=&\int_{-\infty}^{q_{Y}(\alpha)} \rho_{Y}(y) d y=P\left\{Y \leq q_{Y}(\alpha)\right\} \\ &=&\int_{Y \leq q_{Y}(\alpha)} \rho_{X}(\mathbf{x}) d \mathbf{x}=\int_{R^{d}} I_{Y \leq q_{Y}(\alpha)}[Y(\mathbf{x})] \rho_{X}(\mathbf{x}) d \mathbf{x} \end{eqnarray*}\]

More formally,

\[q_Y(\alpha)=F_Y^{-1}(\alpha)=\inf \left\{y|F(Y \le y) \ge \alpha \right\}\]

$\rho_{Y}(y)$ and $F_Y(y)$ are PDF and CDF of the output Y, respectively
$I(x)$ is an indicator function

New quantile-based sensitivity measures

\[\begin{eqnarray*} \bar{q}_i^{(1)}(\alpha)&=&E_{x_i}(|q_Y(\alpha)-q_{Y \mid X_{i}}(\alpha)|) \\ &=&\int |q_Y(\alpha)-q_{Y \mid X_{i}}(\alpha)|dF_{x_i} \end{eqnarray*}\] \[\begin{eqnarray*} \bar{q}_i^{(2)}(\alpha)&=&E_{x_i}[(q_Y(\alpha)-q_{Y \mid X_{i}}(\alpha))^2] \\ &=& \int (q_Y(\alpha)-q_{Y \mid X_{i}}(\alpha))^2 dF_{x_i} \end{eqnarray*}\]

Normalized versions of quantile-based sensitivity measures

\[Q_i^{(1)}(\alpha)=\frac{\bar{q}_i^{(1)}(\alpha)}{\sum_{j=1}^d \bar{q}_j^{(1)}(\alpha)}\] \[Q_i^{(2)}(\alpha)=\frac{\bar{q}_i^{(2)}(\alpha)}{\sum_{j=1}^d \bar{q}_j^{(2)}(\alpha)}\]

Other Moment Independent Measures

Chun et al.

Chun M, Han S, Tak N. An uncertainty importance measure using a distance metric for the change in a cumulative distribution function. Reliab Eng Syst Saf 2000;70:313–21

\[\begin{eqnarray*} CHT_{i}&=&\left[\int\left(q_{Y \mid X_{i}}(\alpha)-q_{Y}(\alpha)\right)^{2} d \alpha\right]^{1 / 2} / E[Y] \\ &=&\left[E_{\alpha}\left(q_{Y \mid X_{i}}(\alpha)-q_{Y}(\alpha)\right)^{2} d \alpha\right]^{1 / 2} / E[Y] \end{eqnarray*}\]

$q_{Y \mid X_i}$ is $\alpha$th quantile of CDF of the output conditional on the input variable $x_i$ being fixed at $x_i=X_i$

Borgonovo

Borgonovo E. A new uncertainty importance measure. Reliab Eng Syst Saf 2007;92:771–84.

\[\delta_{i}=\frac{1}{2} E_{x_{i}}\left[s\left(x_{i}\right)\right]=\frac{1}{2} \int \left[ \int \rho_{Y}(y)-\rho_{Y \mid X_{i}}(y) \mid d y \right] d F_{x_i}\]

where $\rho_{Y \mid X_{i}}(y)$ is the conditional on $x_i=X_i$ PDF of the output Y.

Linear Model with Normally Distributed Variables

\[Y=a_1x_1+a_2x_2+ \cdots +a_dx_d\]

$x_i \sim N(\mu_i,\sigma_i^2)$
$Y \sim N(\sum_{i=1}^d a_i \mu_i, \sum_{i=1}^d a_i^2 \sigma_i^2 )$
$Y \mid X_i \sim N(a_i x_i+\sum_{j=1,j \ne i}^d a_j \mu_j, \sum_{j=1,j \ne i}^d a_j^2 \sigma_j^2 )$

Sobol’s sensitivity indice

\[S_i=S_i^{tot}=\frac{a_i^2 \sigma_i^2}{\sum_{j=1}^d a_j^2 \sigma_j^2}\]

For this model

\[q_Y(\alpha)=\sum_{i=1}^d a_i \mu_i + \Phi^{-1}(\alpha) \sqrt{\sum_{i=1}^d a_i^2 \sigma_i^2}\] \[q_{Y|X_i=x_i}(\alpha)=a_i x_i + \sum_{j=1,j \ne i}^d a_j \mu_j + \Phi^{-1}(\alpha) \sqrt{\sum_{j=1,j \ne i}^d a_j^2 \sigma_j^2}\]

where $\Phi^{-1}(\alpha)$ is the inverse error function.

Hence

\[q_Y(\alpha)-q_{Y|X_i=x_i}(\alpha)=a_i(\mu_i - x_i)+\Phi^{-1}(\alpha) \left(\sqrt{\sum_{i=1}^d a_i^2 \sigma_i^2}-\sqrt{\sum_{j=1,j \ne i}^d a_j^2 \sigma_j^2} \right)\] \[\begin{eqnarray*} q_i^{(2)}(\alpha)&=&E_{x_i}[(q_Y(\alpha)-q_{Y \mid X_{i}}(\alpha))^2] \\ &=&E_{x_i} \left[ \left(a_i(\mu_i - x_i)+\Phi^{-1}(\alpha) \left(\sqrt{\sum_{i=1}^d a_i^2 \sigma_i^2}-\sqrt{\sum_{j=1,j \ne i}^d a_j^2 \sigma_j^2} \right) \right)^2 \right] \\ &=&a_i^2 \sigma_i^2+[\Phi^{-1}(\alpha)]^2 \left(\sqrt{\sum_{i=1}^d a_i^2 \sigma_i^2}-\sqrt{\sum_{j=1,j \ne i}^d a_j^2 \sigma_j^2} \right)^2 \end{eqnarray*}\]

Theorem 1. For the linear additive model with normally distributed variables
$\qquad \qquad \qquad \qquad \qquad \qquad Q_i^{(2)}(\alpha=0.5)=S_i$

Proof:
$\Phi^{-1}(\alpha=0.5)=0$
$q_i^{(2)}=a_i^2 \sigma_i^2$
$Q_i^{(2)}(\alpha)=\frac{\bar{q}_i^{(2)}(\alpha)}{\sum_{j=1}^d \bar{q}_j^{(2)}(\alpha)}=\frac{a_i^2 \sigma_i^2}{\sum_{j=1}^d a_j^2 \sigma_j^2}=S_i$

Monte Carlo estimators

The brute force estimator

Sampling N point $X^l=(x_1^l, \dots , x_d^l)$
Using MC or QMC sampling methods and estimating CDF

\[F_Y^{(N)}(y)=\frac{1}{N} \sum_{l=1}^N I(Y(x_1^l, \dots , x_d^l)<y)\] \[F_{Y \mid X_i}^{(N)}(y)=\frac{1}{N} \sum_{l=1}^N I(Y(x_1^l, \dots , x_i = X_i, \dots , x_d^l)<y)\]

Once CDFs are computed, MC/QMC estimates of quantiles can be computed as

\[q_Y^{(N)}(\alpha)=[F_Y^{(N)}]^{-1}(\alpha)=\inf \left\{ y \mid F_Y^{(N)}(Y \le y) \ge \alpha \right\}\] \[q_{Y \mid X_i}^{(N)}(\alpha)=[F_{Y \mid X_i}^{(N)}]^{-1}(\alpha)=\inf \left\{ y \mid F_{Y \mid X_i}^{(N)}(Y \le y \mid x_i=X_i) \ge \alpha \right\}\]

The Monte Carlo (MC) or Quasi-Monte Carlo (QMC) estimators

\[q_i^{(1)}(\alpha)=\frac{1}{N} \sum_{j=1}^N \mid q_Y^{(N)}(\alpha)-q_{Y \mid X_i^{j}}^{(N)}(\alpha) \mid\] \[q_i^{(2)}(\alpha)=\frac{1}{N} \sum_{j=1}^N \left( q_Y^{(N)}(\alpha)-q_{Y \mid X_i^{j}}^{(N)}(\alpha) \right)^2\]

The total number of sampled points and function evaluations for a fixed $i$ is equal to $N_T^i=NN+N$. To compute all $d$ estimates $N_T=dNN+N=N(dN+1)$.

Double loop reordering (DLR) approach

$N$ points $x^{(j)},j=1,2, \dots, N$ are generated from the joint PDF
Values of the output$\left\{ Y^{(j)}\right\}$ are found at this sampled set $\left\{ x^{(j)}\right\}$
CDF $F_Y(y)$ of the out put Y is found as before using

\[F_Y^{(N)}(y)=\frac{1}{N} \sum_{l=1}^N I(Y(x_1^l, \dots , x_d^l)<y)\]

For each random variable $x_i$, the sample set $\left\{ x^{(j)}\right\},j=1,2, \dots, N$ with corresponding values of $\left\{ Y^{(j)}\right\}$ is sorted in ascending order with respect to the values of $x_i$ and subdivided in $M$ equally populated partitions(bins) with $N_m=N/M$ points in each bin $(M < N)$. Within each bin the CDF $F_{Y \mid X_i}(y)$ of the output conditional on the input variable $x_i$ being fixed at $x_i=X_i^l, l=1, \dots ,N_m$ is estimated as

\[F_{Y \mid X_i}^{(N_m)}(y)=\frac{1}{N_m} \sum_{l=1}^{N_m} I(Y(x_1^l, \dots , x_i^l = X_i^l, \dots , x_d^l)<y)\]

Once CDFs are computed, MC/QMC estimates of quantiles can be computed as before

Finally, the MC/QMC estimators of integrals have the form:

\[\tilde{q_i}^{(1)}(\alpha)=\frac{1}{M} \sum_{j=1}^M \mid q_Y^{(N_m)}(\alpha)-q_{Y \mid X_i^{j}}^{(N_m)}(\alpha) \mid\] \[\tilde{q_i}^{(2)}(\alpha)=\frac{1}{M} \sum_{j=1}^M \left( q_Y^{(N_m)}(\alpha)-q_{Y \mid X_i^{j}}^{(N_m)}(\alpha) \right)^2\]

The total number of sampled points and function evaluations required for computing $\tilde{q_i}^{(1)}(\alpha)$ and $\tilde{q_i}^{(2)}(\alpha)$ for all $i=1,2, \dots ,d$ is $N_T=N$

Application

Value at Risk

Consider tomorrow’s price $P$ for a portfolio which depends on a set of $d$ random variables
Tomorrow’s values $\left\{x_i\right\}, i=1, \dots, d$ for a set of “risk factors”
The portfolio contains $m$ contracts whose values tomorrow are $\left\{C_k\right\}, k=1, \dots, m$
Portfolio price function

\[P(x_1,x_2, \dots ,x_d)=\sum_{k=1}^m C_k(x_1,x_2, \dots ,x_d)\]

We are interested in the value of the α-quantile of the profit and loss (P&L) distribution function of the portfolio which we denote $f_Y(y)$, where $Y=\Delta P$ is a change in the portfolio value over a specific defined time period $T$.
VaR is an α-quantile of model output as defined by

\[q_Y(\alpha)=F_Y^{-1}(\alpha)=\inf \left\{y|F(Y \le y) \ge \alpha \right\}\]

Quantile measures based on VaR for eisk management problems:

\[Q_i^{Var}=\frac{E_{x_i}(d[VaR_Y(\alpha)-VaR_{Y \mid X_i}(\alpha)])}{\sum_{j=1}^d E_{x_i}(d[VaR_Y(\alpha)-VaR_{Y \mid X_j}(\alpha)])}\]

$d[]$: $C_1$ distance and $L_2$ distance

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