394 lines
17 KiB
Markdown
394 lines
17 KiB
Markdown
%===========================================================
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\section{Uncertainty Quantification and Statistical Inference}
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\label{sec:uq}
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%===========================================================
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This section extends the deterministic continuous-time framework in
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Sections~\ref{sec:model_formulation}--\ref{sec:numerics} by modeling future
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usage inputs as continuous-time stochastic processes and propagating the
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resulting uncertainty through the mechanistic battery model. The objective is
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to obtain a \emph{distribution} of time-to-end (TTE) rather than a single-point
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estimate, and to quantify the global sensitivity of TTE to key parameters via
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variance-based indices. Importantly, the underlying electro-thermal-aging
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dynamics and the constant-power-load (CPL) closure are unchanged; randomness
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enters only through exogenous inputs and (optionally) uncertain parameters.
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%-----------------------------------------------------------
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\subsection{Motivation and Model Choices for Random Inputs (OU / Regime Switching)}
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\label{subsec:uq_motivation}
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%-----------------------------------------------------------
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Smartphone usage is intrinsically uncertain beyond a short forecasting horizon:
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screen brightness $L(t)$, CPU load $C(t)$, network activity $N(t)$, and signal
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quality $\Psi(t)$ exhibit mean-reverting fluctuations, cross-correlations, and
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occasional abrupt changes (e.g., screen-off $\to$ gaming; good $\to$ poor
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coverage). A purely deterministic extrapolation of $\mathbf{u}(t)$ therefore
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tends to understate variability and cannot support probabilistic statements
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(e.g., ``runtime exceeds $t$ with $90\%$ probability'').
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We model the future input vector
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\begin{equation}
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\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top
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\end{equation}
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as a continuous-time stochastic process, while preserving the mechanistic
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mapping
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$\mathbf{u}(t)\mapsto P_{\mathrm{tot}}(t)\mapsto I(t)\mapsto \dot{\mathbf{x}}(t)$.
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Two choices are considered:
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\begin{enumerate}
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\item \textbf{Bounded multivariate OU (Option U1).}
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A multivariate Ornstein--Uhlenbeck process provides mean reversion and
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cross-correlation in a continuous-time setting. Smooth bounding transforms
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ensure physical admissibility ($L,C,N\in[0,1]$ and $\Psi$ in a prescribed
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range).
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\item \textbf{Regime-switching OU (Option U2).}
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A continuous-time Markov chain $r(t)$ captures discrete ``modes'' (idle,
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browsing, video, gaming; good/poor coverage). Within each regime, an OU process
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drives the latent inputs. This yields bursty but still continuous trajectories.
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\end{enumerate}
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Both options are mechanism-compatible and avoid black-box regression: the
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battery physics remain deterministic conditional on the sampled input path.
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%-----------------------------------------------------------
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\subsection{Mathematical Definitions and Bounding Maps}
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\label{subsec:uq_definitions}
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%-----------------------------------------------------------
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\paragraph{Option U1: Bounded multivariate OU.}
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Let $\mathbf{y}(t)\in\mathbb{R}^4$ denote latent (unbounded) Gaussian processes
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associated with $[L,C,N,\Psi]$. We define
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\begin{equation}
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d\mathbf{y}(t)=\mathbf{K}\big(\boldsymbol{\mu}-\mathbf{y}(t)\big)\,dt
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+\mathbf{\Sigma}\,d\mathbf{W}(t),
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\label{eq:mvou}
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\end{equation}
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where $\mathbf{K}\succ 0$ controls correlation times, $\boldsymbol{\mu}$ is the
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long-run mean, $\mathbf{\Sigma}$ sets diffusion intensity, and $\mathbf{W}(t)$
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is a standard $4$-dimensional Brownian motion. Cross-channel correlations are
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encoded in $\mathbf{\Sigma}\mathbf{\Sigma}^\top$.
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Ambient temperature is modeled separately as a scalar OU process:
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\begin{equation}
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dT_a(t)=k_a\big(\mu_a-T_a(t)\big)\,dt+\sigma_a\,dW_a(t).
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\label{eq:ou_ta}
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\end{equation}
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To enforce physical bounds, we map latent variables to admissible inputs using
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a smooth logistic transform $\sigma(s)=(1+e^{-s})^{-1}$:
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\begin{align}
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L(t)&=\sigma\!\big(y_L(t)\big),\qquad
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C(t)=\sigma\!\big(y_C(t)\big),\qquad
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N(t)=\sigma\!\big(y_N(t)\big), \label{eq:bound_lcn}\\
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\Psi(t)&=\Psi_{\min}+(\Psi_{\max}-\Psi_{\min})\,\sigma\!\big(y_\Psi(t)\big).
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\label{eq:bound_psi}
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\end{align}
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This choice yields continuous trajectories and avoids nonphysical discontinuous
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jumps that could artificially trigger the CPL infeasibility condition.
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\paragraph{Option U2: Regime-switching OU.}
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Let $r(t)\in\{1,\dots,R\}$ be a continuous-time Markov chain with generator
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matrix $\mathbf{Q}=[q_{ij}]$, where $q_{ij}\ge 0$ for $j\neq i$ and
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$q_{ii}=-\sum_{j\neq i}q_{ij}$. Conditional on $r(t)$, we define
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\begin{equation}
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d\mathbf{y}(t)=\mathbf{K}_{r(t)}\big(\boldsymbol{\mu}_{r(t)}-\mathbf{y}(t)\big)\,dt
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+\mathbf{\Sigma}_{r(t)}\,d\mathbf{W}(t),
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\label{eq:rsou}
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\end{equation}
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and map $\mathbf{y}(t)$ to $\{L,C,N,\Psi\}$ using
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Eqs.~\eqref{eq:bound_lcn}--\eqref{eq:bound_psi}. Ambient temperature can also be
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regime dependent:
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\begin{equation}
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dT_a(t)=k_{a,r(t)}\big(\mu_{a,r(t)}-T_a(t)\big)\,dt+\sigma_{a,r(t)}\,dW_a(t).
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\label{eq:rsou_ta}
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\end{equation}
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This formulation captures abrupt mode changes while keeping inputs continuous
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between switching times.
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%-----------------------------------------------------------
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\subsection{Discrete-Time Input Generation (Update Equations)}
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\label{subsec:uq_generation}
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%-----------------------------------------------------------
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For Monte Carlo simulation, we require discrete-time updates over time step
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$\Delta t$. For a scalar OU process
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\begin{equation}
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dy=k(\mu-y)\,dt+\sigma\,dW,
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\label{eq:ou_scalar}
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\end{equation}
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the exact (in distribution) update is
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\begin{equation}
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y_{n+1}=\mu+(y_n-\mu)e^{-k\Delta t}
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+\sigma\sqrt{\frac{1-e^{-2k\Delta t}}{2k}}\,\xi_n,
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\qquad \xi_n\sim\mathcal{N}(0,1).
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\label{eq:ou_exact}
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\end{equation}
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For the multivariate OU \eqref{eq:mvou}, one may use the matrix-exponential form
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\begin{equation}
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\mathbf{y}_{n+1}=\boldsymbol{\mu}+\mathbf{A}\big(\mathbf{y}_n-\boldsymbol{\mu}\big)
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+\mathbf{B}\,\boldsymbol{\xi}_n,
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\qquad \boldsymbol{\xi}_n\sim\mathcal{N}(\mathbf{0},\mathbf{I}),
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\label{eq:mvou_exact}
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\end{equation}
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where $\mathbf{A}=e^{-\mathbf{K}\Delta t}$ and $\mathbf{B}$ satisfies
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$\mathbf{B}\mathbf{B}^\top=\int_0^{\Delta t}e^{-\mathbf{K}s}\mathbf{\Sigma}\mathbf{\Sigma}^\top e^{-\mathbf{K}^\top s}\,ds$.
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In practice, choosing $\mathbf{K}$ diagonal yields a simple componentwise
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update using \eqref{eq:ou_exact}, while correlations can be retained through
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$\mathbf{\Sigma}$.
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For regime switching, over sufficiently small $\Delta t$ we approximate
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\begin{equation}
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\mathbb{P}\big(r_{n+1}=j\,\big|\,r_n=i\big)\approx
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\begin{cases}
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q_{ij}\Delta t, & j\neq i,\\
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1+q_{ii}\Delta t, & j=i,
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\end{cases}
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\label{eq:ctmc_step}
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\end{equation}
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then update $\mathbf{y}$ using \eqref{eq:mvou_exact} with parameters associated
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with the realized regime $r_n$.
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%-----------------------------------------------------------
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\subsection{Monte Carlo Propagation and TTE Distribution}
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\label{subsec:uq_mc}
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%-----------------------------------------------------------
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Let $\omega$ denote the randomness driving $\mathbf{u}(t,\omega)$ (and, if
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included, uncertain parameters). For each sampled input path $\omega_m$, the
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battery dynamics are integrated using the deterministic solver from
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Section~\ref{sec:numerics}: RK4 with nested CPL current evaluation at each
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substep, including low-SOC OCV protection $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$,
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nonnegative polarization heating $v_p^2/R_1$, and the lightweight current cap
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$I=\min(I_{\mathrm{CPL}},I_{\max}(T_b))$.
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The runtime endpoint is defined by
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\begin{equation}
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\mathrm{TTE}(\omega)=\inf\left\{t>0:\; V_{\mathrm{term}}(t,\omega)\le V_{\mathrm{cut}}
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\ \text{or}\ z(t,\omega)\le 0\right\}.
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\label{eq:tte_uq}
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\end{equation}
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(Optionally, the CPL infeasibility risk time
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$t_{\Delta}=\inf\{t>0:\Delta(t,\omega)\le 0\}$ may be recorded as a separate
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diagnostic.)
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Given $M$ independent sample paths $\{\omega_m\}_{m=1}^M$, we obtain
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$\mathrm{TTE}_m=\mathrm{TTE}(\omega_m)$ and form the empirical CDF
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\begin{equation}
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\widehat{F}_{\mathrm{TTE}}(t)=\frac{1}{M}\sum_{m=1}^M \mathbf{1}\{\mathrm{TTE}_m\le t\}.
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\label{eq:emp_cdf}
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\end{equation}
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The empirical mean and variance are
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\begin{equation}
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\widehat{\mu}_{\mathrm{TTE}}=\frac{1}{M}\sum_{m=1}^M \mathrm{TTE}_m,\qquad
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\widehat{\sigma}^2_{\mathrm{TTE}}=\frac{1}{M-1}\sum_{m=1}^M(\mathrm{TTE}_m-\widehat{\mu}_{\mathrm{TTE}})^2.
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\label{eq:tte_moments}
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\end{equation}
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\paragraph{Monte Carlo error.}
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For standard Monte Carlo estimators of smooth functionals of TTE, the
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statistical error decays as $O(M^{-1/2})$. We therefore increase $M$ until key
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summaries (mean and selected quantiles) stabilize under doubling $M$.
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%-----------------------------------------------------------
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\subsection{Confidence Intervals, Quantiles, and Survival Curves}
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\label{subsec:uq_inference}
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%-----------------------------------------------------------
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\paragraph{Confidence interval for the mean.}
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By the central limit theorem, an approximate $95\%$ confidence interval for the
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mean TTE is
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\begin{equation}
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\widehat{\mu}_{\mathrm{TTE}}\ \pm\ 1.96\,\frac{\widehat{\sigma}_{\mathrm{TTE}}}{\sqrt{M}}.
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\label{eq:ci_mean}
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\end{equation}
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When $M$ is moderate and the distribution is skewed, a nonparametric bootstrap
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over $\{\mathrm{TTE}_m\}$ can be used to obtain robust confidence bounds.
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\paragraph{Quantiles.}
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Let $\mathrm{TTE}_{(1)}\le \cdots \le \mathrm{TTE}_{(M)}$ denote the ordered
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samples. The empirical $p$-quantile is
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\begin{equation}
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\widehat{q}_p=\mathrm{TTE}_{(\lceil pM\rceil)},\qquad p\in(0,1).
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\label{eq:quantile}
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\end{equation}
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In particular, the median is $\widehat{q}_{0.5}$, and the lower-tail quantile
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$\widehat{q}_{0.1}$ supports conservative ``guaranteed runtime'' statements.
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\paragraph{Survival function.}
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A reliability-style summary is the survival curve
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\begin{equation}
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\widehat{S}(t)=\mathbb{P}(\mathrm{TTE}>t)\approx 1-\widehat{F}_{\mathrm{TTE}}(t).
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\label{eq:survival}
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\end{equation}
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This directly answers: ``what is the probability the phone remains operational
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beyond time $t$?''
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%-----------------------------------------------------------
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\subsection{Variance-Based Global Sensitivity (Sobol Indices)}
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\label{subsec:uq_sobol}
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%-----------------------------------------------------------
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We quantify global parameter importance via variance-based sensitivity indices
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for the scalar quantity of interest (QoI)
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\begin{equation}
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Y=\mathrm{TTE}.
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\end{equation}
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Let $\boldsymbol{\xi}=(\xi_1,\dots,\xi_d)$ denote uncertain factors (e.g.,
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$k_L,\gamma,k_N,\kappa,\mu_a$ and other parameters as needed), assumed
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independent with prescribed prior distributions. Because usage randomness
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$\omega$ also contributes variance, we recommend defining the QoI as the
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\emph{conditional expectation} over usage paths:
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\begin{equation}
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Y(\boldsymbol{\xi})=\mathbb{E}_{\omega}\big[\mathrm{TTE}(\boldsymbol{\xi},\omega)\big],
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\label{eq:qoi_condexp}
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\end{equation}
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which yields stable and actionable sensitivities to design/physics parameters.
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In computations, \eqref{eq:qoi_condexp} is approximated by an inner Monte Carlo
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average over $M_{\omega}$ usage realizations.
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The first-order Sobol index of factor $\xi_i$ is defined as
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\begin{equation}
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S_i=\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid \xi_i]\big)}{\mathrm{Var}(Y)},
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\label{eq:sobol_first}
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\end{equation}
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and the total-effect index is
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\begin{equation}
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S_{T_i}=1-\frac{\mathrm{Var}\big(\mathbb{E}[Y\mid \boldsymbol{\xi}_{\sim i}]\big)}{\mathrm{Var}(Y)},
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\label{eq:sobol_total}
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\end{equation}
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where $\boldsymbol{\xi}_{\sim i}$ denotes all factors except $\xi_i$. Large
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$S_i$ indicates a strong main effect, while a large gap $S_{T_i}-S_i$ indicates
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substantial interaction and/or nonlinearity (expected here due to CPL feedback
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and electro-thermal coupling).
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%-----------------------------------------------------------
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\subsection{Saltelli Sampling and Estimation}
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\label{subsec:uq_saltelli}
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%-----------------------------------------------------------
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We employ the Saltelli sampling scheme for efficient estimation of Sobol
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indices. Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{N\times d}$ be two independent
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sample matrices of $\boldsymbol{\xi}$. For each $i\in\{1,\dots,d\}$, construct
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$\mathbf{A}^{(i)}_B$ by replacing the $i$-th column of $\mathbf{A}$ with the
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$i$-th column of $\mathbf{B}$. Denote the corresponding model evaluations by
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\begin{equation}
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Y_A^{(n)}=Y(\mathbf{A}_n),\quad
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Y_B^{(n)}=Y(\mathbf{B}_n),\quad
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Y_{A^{(i)}_B}^{(n)}=Y(\mathbf{A}^{(i)}_{B,n}),
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\qquad n=1,\dots,N.
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\end{equation}
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We estimate $\mathrm{Var}(Y)$ from the pooled samples and compute Sobol
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estimators in the following commonly used form:
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\begin{equation}
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\widehat{S}_i=
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\frac{\frac{1}{N}\sum_{n=1}^N Y_B^{(n)}\left(Y_{A^{(i)}_B}^{(n)}-Y_A^{(n)}\right)}
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{\widehat{\mathrm{Var}}(Y)},
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\label{eq:saltelli_first}
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\end{equation}
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\begin{equation}
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\widehat{S}_{T_i}=
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\frac{\frac{1}{2N}\sum_{n=1}^N \left(Y_A^{(n)}-Y_{A^{(i)}_B}^{(n)}\right)^2}
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{\widehat{\mathrm{Var}}(Y)}.
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\label{eq:saltelli_total}
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\end{equation}
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\paragraph{Nested averaging over usage paths.}
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Each $Y(\cdot)$ above is computed as
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\begin{equation}
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Y(\boldsymbol{\xi})\approx \frac{1}{M_{\omega}}\sum_{m=1}^{M_{\omega}}
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\mathrm{TTE}(\boldsymbol{\xi},\omega_m),
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\label{eq:nested_mc}
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\end{equation}
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where $\{\omega_m\}$ are i.i.d.\ usage realizations generated by
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Option~U1/U2. This inner average reduces the Monte Carlo noise in $Y$ so that
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the outer Saltelli estimators converge reliably in $N$.
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%-----------------------------------------------------------
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\subsection{Optional: Variance Reduction (LHS / Quasi-Monte Carlo)}
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\label{subsec:uq_varred}
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%-----------------------------------------------------------
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While plain Monte Carlo converges at rate $O(M^{-1/2})$, variance reduction can
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improve efficiency when computational budgets are tight.
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\paragraph{Latin hypercube sampling (LHS).}
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For estimating the TTE distribution under uncertain inputs/parameters, LHS can
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replace i.i.d.\ sampling of low-dimensional uncertain parameters
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$\boldsymbol{\xi}$ to reduce estimator variance without changing the model. LHS
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is especially effective when the dominant uncertainty is parameter-driven.
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\paragraph{Quasi-Monte Carlo (QMC).}
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For Sobol estimation (outer sampling), low-discrepancy sequences (e.g., Sobol
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sequences) can improve convergence of integral estimates in moderate
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dimensions. In this work, QMC can be applied to generate $\mathbf{A},\mathbf{B}$
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before constructing $\mathbf{A}^{(i)}_B$. Because our QoI involves a nested
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average \eqref{eq:nested_mc}, QMC primarily benefits the outer parameter
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integration, while the inner usage randomness still scales as
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$O(M_\omega^{-1/2})$.
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\paragraph{Control variates (conceptual).}
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If a simplified surrogate (e.g., the same model with fixed $T_b=T_a$ or without
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aging) is available, it may serve as a control variate to reduce variance of
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$\mathrm{TTE}$. We do not rely on this technique in the baseline pipeline.
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%-----------------------------------------------------------
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\subsection{Optional: Unified Two-Level Uncertainty (Inputs and Parameters)}
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\label{subsec:uq_twolevel}
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%-----------------------------------------------------------
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When both usage inputs and physical/power parameters are uncertain, the full
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QoI can be viewed hierarchically as
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\begin{equation}
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\mathrm{TTE}=\mathrm{TTE}(\boldsymbol{\xi},\omega),
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\end{equation}
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with $\boldsymbol{\xi}$ representing uncertain parameters (e.g., $k_L,\gamma,
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k_N,\kappa,\mu_a,hA$) and $\omega$ representing stochastic input realizations
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from Option~U1/U2. Two complementary summaries are useful:
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\paragraph{Unconditional runtime distribution.}
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The overall distribution integrates over both sources of uncertainty:
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\begin{equation}
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F_{\mathrm{TTE}}(t)=\mathbb{P}(\mathrm{TTE}\le t)=
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\int \mathbb{P}\!\left(\mathrm{TTE}(\boldsymbol{\xi},\omega)\le t\ \big|\ \boldsymbol{\xi}\right)
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\,p(\boldsymbol{\xi})\,d\boldsymbol{\xi}.
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\label{eq:unconditional}
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\end{equation}
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This is estimated by outer sampling of $\boldsymbol{\xi}$ and inner sampling of
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$\omega$.
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\paragraph{Sensitivity of conditional mean runtime.}
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For design guidance, sensitivities are computed for
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$Y(\boldsymbol{\xi})=\mathbb{E}_{\omega}[\mathrm{TTE}(\boldsymbol{\xi},\omega)]$
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as in \eqref{eq:qoi_condexp}, yielding Sobol indices that reflect how parameter
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variation shifts \emph{expected} runtime under random usage.
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\paragraph{Practical computation.}
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A computationally efficient compromise is to (i) propagate usage uncertainty
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with a large $M$ at nominal parameters to obtain $F_{\mathrm{TTE}}$, and (ii)
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compute Sobol indices with moderate inner averaging $M_\omega$ and outer sample
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size $N$ to rank parameter importance.
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%-----------------------------------------------------------
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\subsection*{Algorithmic Summary}
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%-----------------------------------------------------------
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For completeness, the full UQ pipeline used in subsequent sections can be
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summarized as follows:
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\begin{itemize}
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\item Generate stochastic input paths $\mathbf{u}(t,\omega)$ using
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Eqs.~\eqref{eq:mvou}--\eqref{eq:bound_psi} (Option~U1) or
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Eqs.~\eqref{eq:rsou}--\eqref{eq:rsou_ta} (Option~U2), with discrete updates
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given by \eqref{eq:ou_exact}--\eqref{eq:ctmc_step}.
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\item For each path, solve the mechanistic battery model using RK4 with nested
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CPL current evaluation (Section~\ref{sec:numerics}) and record
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$\mathrm{TTE}$ from \eqref{eq:tte_uq}.
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\item Construct the empirical distribution \eqref{eq:emp_cdf}, compute moments
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\eqref{eq:tte_moments}, confidence intervals \eqref{eq:ci_mean}, quantiles
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\eqref{eq:quantile}, and survival curve \eqref{eq:survival}.
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\item For global sensitivity, evaluate $Y(\boldsymbol{\xi})$ via nested averaging
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\eqref{eq:nested_mc} and estimate Sobol indices with Saltelli sampling
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\eqref{eq:saltelli_first}--\eqref{eq:saltelli_total}.
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\end{itemize}
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