layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # 📡 Non-Parametric Spatial Correlation Estimation @luk036 👨‍💻 · 2026 📅 --- ## 📝 Abstract This lecture explores non-parametric spatial correlation estimation and its significance in analyzing the variability in semiconductor devices. The intra-die variation within these devices may have spatially correlated patterns that necessitate precise statistical analysis during the design phase. Anisotropic models are utilized to account for variations in gate length, which demonstrate greater correlation in the horizontal direction compared to the vertical direction. Non-parametric methods are more appropriate for correlation functions, as previous studies utilizing parametric forms were restricted by presuppositions about the correlation function. The lecture proceeds to elaborate on random fields and the characteristics of correlation functions, followed by discussions on problem formulation and the usage of maximum likelihood estimation and least squares estimation for solutions. Lastly, convexity of the problems is examined. --- ### Overview 📋 .pull-left[ - 🎯 Motivation: - Why is spatial correlation important? 🤔 - Why anisotropic models? 🧐 - Why do non-parametric approaches make sense? 💡 - 📝 Problem Formulation - 📊 Non-parametric estimation - Least squares estimation 📏 - Maximum Likelihood estimation 🔍 - 🔢 Numerical experiment - 🏁 Conclusion ] .pull-right[  ] --- ### ❓ What does "Test Chip" mean? - 🧩 A test chip is a type of integrated circuit (IC) that is designed specifically for testing and validation purposes. It is used to evaluate the performance and reliability of new manufacturing processes, design methodologies, and DFM techniques. - 🛠️ Test chips typically contain a variety of test structures and functional blocks that are used to measure various aspects of the IC's performance, such as timing, power consumption, and signal integrity. They may also include structures that are used to evaluate the impact of process-induced variation on the IC's performance, such as ring oscillators, parametric test structures, and statistical analysis circuits. - 🔬 The use of test chips is important because it allows designers to evaluate the performance and reliability of new manufacturing processes and design methodologies before they are used in production. This can help identify potential issues and optimize the process or design to improve the performance and yield of the IC. Test chips can also be used to validate the effectiveness of DFM techniques in improving the manufacturability and yield of the IC. --- ### 🔍 Why Spatial Correlation? - 📉 As the minimum feature size of semiconductor devices continues to shrink, - Process variations are inevitable. It is desirable to develop more accurate statistical analysis during the design stage. - 🔄 Intra-die variation exceeds inter-die variation - Becomes dominant over total process variation - Often exhibits spatially correlated patterns. - 🛠️ Applications: - Statistical timing analysis -> Clock Skew Scheduling ⏱️ - Power/leakage minimization ⚡ --- ### 📐 Why Anisotropic Model? - 🔄 Isotropic assumption assumes that the correlation depends only on the distance between two random variables. It was made to simplify the computation. - ↔️ Certain variations, such variations in gate length, exhibit significantly stronger correlation in the horizontal direction than in the vertical direction. --- ### 🔄 Why Non-Parametric Approaches? - 📜 In earlier studies, the parametric form of the correlation function was simple, such as an exponential, Gaussian or Matérn function: - ✔️ Pros: guaranteed to be **positive definite**. - ❌ Cons: - non-convex; may be stuck in a local minimum 🕳️ - The actual correlation function may not necessarily be of this form. - isotropic model 🔄 --- ### 🔬 Related research 🔬🧪 - 📉 Piecewise linearization method (imprecise, not positive definite) - 📊 Parametric method (non-convex, too smooth, isotropic) - Exponential function 📈 - Gaussian function 🔔 - Matérn function 🌀 - 🛠️ Non-parametric method - Polynomial fitting - B-spline 🎚️ --- ### 🔄 Parametric vs Non-Parametric Estimation .mermaid[ <pre> graph TD A(("Measured Data")) --> B{Estimation Type} B -->|Parametric| C[Exponential / Gaussian / Matérn] B -->|Non-Parametric| D[B-Spline / Polynomial] C --> E[LSE or MLE] D --> E E --> F[Correlation Function] style A fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style B fill:#fff3e0,stroke:#e65100,stroke-width:3px style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px style E fill:#fff3e0,stroke:#e65100,stroke-width:3px style F fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px </pre> ] --- ### 🎲 Random Field - 🌀 Random field is an indexed family of random variables denote as $\{{\color{darkblack}\tilde{z}}(s): s \in D\}$, where ${\color{salmon}D} \subseteq \mathrm{R}^d$ - 📊 Covariance ${\color{purple}C}(s_i, s_j)$ = $\text{cov}({\color{darkblack}\tilde{z}}(s_i),{\color{darkblack}\tilde{z}}(s_j))$ = $\mathrm{E}[({\color{darkblack}\tilde{z}}(s_i) - \mathrm{E}[{\color{darkblack}\tilde{z}}(s_i)]) ({\color{darkblack}\tilde{z}}(s_j) - \mathrm{E}[{\color{darkblack}\tilde{z}}(s_j)])]$ - 🔗 Correlation ${\color{purple}R}(s_i, s_j) = {\color{purple}C}(s_i, s_j)/\sqrt{{\color{purple}C}(s_i, s_i) {\color{purple}C}(s_j, s_j)}$ - 🏠 The field is stationary, or homogeneous, if the distribution is unchanged when the point set is translated. - 🔄 The field is isotropic if the distribution is invariant under any rotation. - 📐 In HIF, let ${\color{brown}d} = \| s_i - s_j \|_2$: - ${\color{purple}C}(s_i, s_j) = {\color{purple}C}({\color{brown}d}) = {\color{green}\sigma^2} {\color{purple}\rho}({\color{brown}d})$ - ${\color{purple}R}(s_i, s_j) = {\color{purple}C}({\color{brown}d})/{\color{purple}C}(0)$ --- ### 📏 Properties of Correlation Function - ↔️ Even function, i.e. ${\color{purple}\rho}({\color{brown}\vec{h}}) = {\color{purple}\rho}(-{\color{brown}\vec{h}}) \implies$ its Fourier transform is real. - ✔️ Positive definiteness (PD) $\implies$ its Fourier transform is positive (Bochner's theorem). - 📉 Monotonicity: correlations are decreasing against ${\color{brown}h}$ 🤔 - ➕ Nonnegativeness: no negative correlation 🤔 - ⚡ Discontinuity at the origin: nugget effect. The nugget effect refers to the discontinuity at the origin in the correlation function of spatially correlated patterns. It indicates the presence of a small, non-zero correlation value between points that are very close to each other. In other words, it represents the variance component that cannot be explained by spatial correlation and is attributed to purely random variation. --- ### 📝 Problem Formulation - 🔄 Intra-die variation ${\color{darkblack}\tilde{z}} = {\color{darkblack}z}_{det} + {\color{darkblack}\tilde{z}}_{cor} + {\color{darkblack}\tilde{z}}_{rnd}$ - ${\color{darkblack}z}_{det}$: deterministic component - ${\color{darkblack}\tilde{z}}_{cor}$: correlated random component - ${\color{darkblack}\tilde{z}}_{rnd}$: purely random component - 📊 Given ${\color{coral}M}$ samples $({\color{darkblack}z}_1, {\color{darkblack}z}_2, \ldots, {\color{darkblack}z}_M) \in \mathbb{R}^n$. - 📏 Measured covariance matrix ${\color{red}Y}$: - ${\color{red}Y} = (1/{\color{coral}M}) \sum_{i=1}^{{\color{coral}M}} {\color{darkblack}z}_i {\color{darkblack}z}_i^\mathsf{T}$ (unlikely PD) - 💻 In MATLAB, simply call `cov(Zs',1)` to obtain ${\color{red}Y}$. - 🐍 In Python, simple call `np.cov(Zs, bias=True)` to obtain ${\color{red}Y}$. --- ### 🎯 Nearest PD Matrix Problem - 🔵 Given ${\color{red} Y}$. Find a nearest matrix ${\color{darkmagenta} \Sigma}$ that is positive definite. $$\begin{array}{ll} \text{minimize} & \| {\color{darkmagenta} \Sigma} - {\color{red} Y} \|_F \\ \text{subject to} & {\color{darkmagenta} \Sigma} \succ 0 \end{array}$$ where $\| {\color{darkmagenta} \Sigma} - {\color{red} Y} \|_F$ denotes the Frobenius norm, ${\color{darkmagenta}A} \succ 0$ denotes ${\color{darkmagenta}A}$ is positive definite. - 👉 Note: 1. the problem is convex 😃 2. the problem can be solved easily using CVX 😃 --- ### 📈 Maximum Likelihood Estimation - 🔍 Maximum likelihood estimation (MLE): $$\begin{array}{ll} \text{maximize} & \log \det {\color{darkmagenta} \Sigma}^{-1} - \mathrm{Tr}({\color{darkmagenta} \Sigma}^{-1} {\color{red} Y}) \\ \text{subject to} & {\color{darkmagenta} \Sigma} \succ 0 \end{array}$$ where $\mathrm{Tr}({\color{darkmagenta}A})$ denotes the trace of ${\color{darkmagenta}A}$. - 👉 Note: 1st term is concave 😭, 2nd term is convex --- ### 📈 Maximum Likelihood Estimation (cont'd) - Having ${\color{darkmagenta}S} = {\color{darkmagenta}\Sigma}^{-1}$, the problem becomes convex 😃: $$\begin{array}{ll} \text{minimize} & -\log \det {\color{darkmagenta} S} + \mathrm{Tr}({\color{darkmagenta} S} {\color{red} Y}) \\ \text{subject to} & {\color{darkmagenta} S} \succ 0 \end{array}$$ - 👉 Note: the problem can be solved easily using MATLAB with the CVX package, or using Python with the cvxpy package. --- ### 💻 Matlab Code of CVX ```matlab function Sig = log_mle_solver(Y); ndim = size(Y,1); cvx_quiet(false); cvx_begin sdp variable S(ndim, ndim) symmetric maximize(log_det(S) - trace(S*Y)) subject to S >= 0; cvx_end Sig = inv(S); ``` --- ### 🐍 Python Code ```python from cvxpy import * from scipy import linalg def mle_corr_mtx(Y): ndim = len(Y) S = Semidef(ndim) prob = Problem(Maximize(log_det(S) - trace(S*Y))) prob.solve() if prob.status != OPTIMAL: raise Exception('CVXPY Error') return linalg.inv(S.value) ``` --- ### 📊 Correlation Function (I) - Let ${\color{purple}\rho}({\color{brown}h}) = \sum_i^m {\color{darkblack}p}_i {\color{purple}\Psi_i}({\color{brown}h})$, where - ${\color{darkblack}p}_i$'s are the unknown coefficients to be fitted - ${\color{purple}\Psi_i}$'s are a family of basis functions. - Let $\{{\color{red}F_k}\}_{i,j} ={\color{purple}\Psi_k}( \| s_i - s_j \|_2)$. - The covariance matrix ${\color{red}\Omega}({\color{darkblack}p})$ can be recast as: $${\color{red}\Omega}({\color{darkblack}p}) = {\color{darkblack}p}_1 {\color{red}F_1} + \cdots + {\color{darkblack}p}_m {\color{red}F_m}$$ - Note 1: affine transformation preserved convexity - Note 2: inverse of matrix unfortunately **cannot** be expressed in convex form. --- ### 📊 Correlation Function (II) - Choice of ${\color{purple}\Psi_i}({\color{brown}h})$: - Polynomial ${\color{purple}P_i}({\color{brown}h})$: - Easy to understand 👍 - No guarantee of monotonicity; unstable for higher-order polynomials. - B-spline function ${\color{purple}B_i}({\color{brown}h})$ - Shapes are easier to control 👍 - No guarantee of positive definite 👎 --- ### 📊 Correlation Function (III) - To ensure that the resulting function is PD, additional constraints can be imposed according to Bochner's theorem, e.g.: - real(FFT($\{{\color{purple}\Psi_i}({\color{brown}h_k})\}$)) $\geq 0$ Bochner's theorem states that a continuous function is a valid covariance function if and only if its Fourier transform is a non-negative measure. In other words, a function can be a valid covariance function if and only if its Fourier transform is positive definite. This theorem is important in spatial statistics because it provides a way to check whether a given covariance function is valid or not. --- ### 📊 Non-Parametric Estimation - 📏 Least squares estimation $$\begin{array}{ll} \min_{{\color{darkmagenta}\kappa}, {\color{darkblack}p}} & \| {\color{red}\Omega}({\color{darkblack}p}) + {\color{darkmagenta}\kappa} I - {\color{red}Y} \|_\mathrm{F} \\ \text{s.t.} & {\color{red}\Omega}({\color{darkblack}p}) \succ 0, {\color{darkmagenta}\kappa} \geq 0 \end{array}$$ 👉 Note: convex problem 😃 - 🔍 Maximum likelihood estimation (MLE): $$\begin{array}{ll} \min_{{\color{darkmagenta}\kappa}, {\color{darkblack}p}} & \log \det ({\color{red}\Omega}({\color{darkblack}p}) + {\color{darkmagenta}\kappa} I) + \mathrm{Tr}(({\color{red}\Omega}({\color{darkblack}p}) + {\color{darkmagenta}\kappa} I)^{-1} {\color{red}Y}) \\ \text{s.t.} & {\color{red}\Omega}({\color{darkblack}p}) \succ 0, {\color{darkmagenta}\kappa} \geq 0 \end{array}$$ 👉 Note: - The 1st term is concave 😭, the 2nd term is convex - However, the problem is **geodesically convex**. - If enough samples are available, then ${\color{red}Y} \succ 0$. Furthermore, the MLE is a convex problem in $0 \preceq {\color{red}\Omega}({\color{darkblack}p}) + {\color{darkmagenta}\kappa} I \preceq 2{\color{red}Y}$ --- ### 📊 Isotopic Case I .pull-left[  : Data Sample ] .pull-right[  : Least Square Result ] --- ### 📊 Isotopic Case II .pull-left[  : Data Sample ] .pull-right[  : Least Square Result ] --- ### 📈 Convex Concave Procedure - Let ${\color{darkmagenta}\Sigma} = {\color{red}\Omega} + {\color{darkmagenta}\kappa} I$. Log-likelihood function is: - $\log \det {\color{darkmagenta}\Sigma}^{-1} - \mathrm{Tr}({\color{darkmagenta}\Sigma}^{-1}{\color{red}Y})$ - Convexify the first term using the fact: - $\log \det {\color{darkmagenta}\Sigma}^{-1} \geq \log \det {\color{darkmagenta}\Sigma}_0^{-1} + \mathrm{Tr}({\color{darkmagenta}\Sigma}_0^{-1} ({\color{darkmagenta}\Sigma} - {\color{darkmagenta}\Sigma}_0))$ - minimize: $-\log \det {\color{darkmagenta}\Sigma}_0^{-1} + \mathrm{Tr}({\color{darkmagenta}\Sigma}_0^{-1} ({\color{darkmagenta}\Sigma} - {\color{darkmagenta}\Sigma}_0)) + \mathrm{Tr}({\color{darkmagenta}\Sigma}^{-1}{\color{red}Y})$ - At each iteration ${\color{coral}k}$, the following convex problem is solved: $$\begin{array}{ll} \min & \mathrm{Tr}({\color{darkmagenta}\Sigma}_k^{-1} ({\color{darkmagenta}\Sigma} - {\color{darkmagenta}\Sigma}_k)) + \mathrm{Tr}({\color{darkmagenta}S} {\color{red}Y}) \\ \text{s.t.} & \left( \begin{array}{cc} {\color{darkmagenta}\Sigma} & I_n \\ I_n & {\color{darkmagenta}S} \end{array} \right) \succ 0, {\color{darkmagenta}\kappa} \geq 0 \end{array} $$ 👉 Note: Convergence to an optimal solution is not guaranteed, but is practically good. --- ### 💻 MATLAB Code ```matlab % Geometric anisotropic parameters alpha = 2; % scaling factor theta = pi/3; % angle Sc = [1 0; 0 alpha]; R = [sin(theta) cos(theta); -cos(theta) sin(theta)]; T = Sc*R; Sig = ones(n,n); for i=1:n-1, for j=i+1:n, dt = s(j,:)' - s(i,:)'; d = T*dt; % become isotropic after the location transformation Sig(i,j) = exp(-0.5*(d'*d)/(sdkern*sdkern)/2); Sig(j,i) = Sig(i,j); end end ``` --- ### 📊 Anisotopic Data  --- ### 📊 Isotropic Result  --- ### 📊 Anisotropic Result  --- count: false class: nord-dark, middle, center .pull-left[ ## Q&A 🎤 ] .pull-right[  ]