regression analysis
**Regression Analysis**
Semiconductor fabrication involves hundreds of sequential process steps, each governed by dozens of parameters. Regression analysis serves critical functions:
- Process Modeling: Understanding relationships between inputs and quality outputs
- Virtual Metrology: Predicting measurements from real-time sensor data
- Run-to-Run Control: Adaptive process adjustment
- Yield Optimization: Maximizing device performance and throughput
- Fault Detection: Identifying and diagnosing process excursions
Core Mathematical Framework
Ordinary Least Squares (OLS)
The foundational linear regression model:
$$
\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\varepsilon}
$$
Variable Definitions:
- $\mathbf{y}$ — $n \times 1$ response vector (e.g., film thickness, etch rate, yield)
- $\mathbf{X}$ — $n \times (k+1)$ design matrix of process parameters
- $\boldsymbol{\beta}$ — $(k+1) \times 1$ coefficient vector
- $\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \sigma^2\mathbf{I})$ — error term
OLS Estimator:
$$
\hat{\boldsymbol{\beta}} = (\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top\mathbf{y}
$$
Variance-Covariance Matrix of Estimator:
$$
\text{Var}(\hat{\boldsymbol{\beta}}) = \sigma^2(\mathbf{X}^\top\mathbf{X})^{-1}
$$
Unbiased Variance Estimate:
$$
\hat{\sigma}^2 = \frac{\mathbf{e}^\top\mathbf{e}}{n - k - 1} = \frac{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}{n - k - 1}
$$
Response Surface Methodology (RSM)
Critical for semiconductor process optimization, RSM uses second-order polynomial models.
Second-Order Model
$$
y = \beta_0 + \sum_{i=1}^{k}\beta_i x_i + \sum_{i=1}^{k}\beta_{ii}x_i^2 + \sum_{i n$)
- Addresses multicollinearity
- Captures latent variable structures
- Simultaneously models X and Y relationships
NIPALS Algorithm
1. Initialize: $\mathbf{u} = \mathbf{y}$
2. X-weight:
$$\mathbf{w} = \frac{\mathbf{X}^\top\mathbf{u}}{\|\mathbf{X}^\top\mathbf{u}\|}$$
3. X-score:
$$\mathbf{t} = \mathbf{X}\mathbf{w}$$
4. Y-loading:
$$q = \frac{\mathbf{y}^\top\mathbf{t}}{\mathbf{t}^\top\mathbf{t}}$$
5. Y-score update:
$$\mathbf{u} = \frac{\mathbf{y}q}{q^2}$$
6. Iterate until convergence
7. Deflate X and Y, extract next component
Model Structure
$$
\mathbf{X} = \mathbf{T}\mathbf{P}^\top + \mathbf{E}
$$
$$
\mathbf{Y} = \mathbf{T}\mathbf{Q}^\top + \mathbf{F}
$$
Where:
- $\mathbf{T}$ — score matrix (latent variables)
- $\mathbf{P}$ — X-loadings
- $\mathbf{Q}$ — Y-loadings
- $\mathbf{E}, \mathbf{F}$ — residuals
Spatial Regression for Wafer Maps
Wafer-level variation exhibits spatial patterns requiring specialized models.
Zernike Polynomial Decomposition
General Form:
$$
Z(r,\theta) = \sum_{n,m} a_{nm} Z_n^m(r,\theta)
$$
Standard Zernike Polynomials (first few terms):
| Index | Name | Formula |
|-------|------|---------|
| $Z_0^0$ | Piston | $1$ |
| $Z_1^{-1}$ | Tilt Y | $r\sin\theta$ |
| $Z_1^{1}$ | Tilt X | $r\cos\theta$ |
| $Z_2^{-2}$ | Astigmatism 45° | $r^2\sin 2\theta$ |
| $Z_2^{0}$ | Defocus | $2r^2 - 1$ |
| $Z_2^{2}$ | Astigmatism 0° | $r^2\cos 2\theta$ |
| $Z_3^{-1}$ | Coma Y | $(3r^3 - 2r)\sin\theta$ |
| $Z_3^{1}$ | Coma X | $(3r^3 - 2r)\cos\theta$ |
| $Z_4^{0}$ | Spherical | $6r^4 - 6r^2 + 1$ |
Orthogonality Property:
$$
\int_0^1 \int_0^{2\pi} Z_n^m(r,\theta) Z_{n'}^{m'}(r,\theta) \, r \, dr \, d\theta = \frac{\pi}{n+1}\delta_{nn'}\delta_{mm'}
$$
Gaussian Process Regression (Kriging)
Prior Distribution:
$$
f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}'))
$$
Common Kernel Functions:
*Squared Exponential (RBF)*:
$$
k(\mathbf{x}, \mathbf{x}') = \sigma^2 \exp\left(-\frac{\|\mathbf{x} - \mathbf{x}'\|^2}{2\ell^2}\right)
$$
*Matérn Kernel*:
$$
k(r) = \sigma^2 \frac{2^{1-
u}}{\Gamma(
u)}\left(\frac{\sqrt{2
u}r}{\ell}\right)^
u K_
u\left(\frac{\sqrt{2
u}r}{\ell}\right)
$$
Where $K_
u$ is the modified Bessel function of the second kind.
Posterior Predictive Mean:
$$
\bar{f}_* = \mathbf{k}_*^\top(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{y}
$$
Posterior Predictive Variance:
$$
\text{Var}(f_*) = k(\mathbf{x}_*, \mathbf{x}_*) - \mathbf{k}_*^\top(\mathbf{K} + \sigma_n^2\mathbf{I})^{-1}\mathbf{k}_*
$$
Mixed Effects Models
Semiconductor data has hierarchical structure (wafers within lots, lots within tools).
General Model
$$
y_{ijk} = \mathbf{x}_{ijk}^\top\boldsymbol{\beta} + b_i^{(\text{tool})} + b_{ij}^{(\text{lot})} + \varepsilon_{ijk}
$$
Random Effects Distribution:
- $b_i^{(\text{tool})} \sim N(0, \sigma_{\text{tool}}^2)$
- $b_{ij}^{(\text{lot})} \sim N(0, \sigma_{\text{lot}}^2)$
- $\varepsilon_{ijk} \sim N(0, \sigma^2)$
Matrix Notation
$$
\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\mathbf{b} + \boldsymbol{\varepsilon}
$$
Where:
- $\mathbf{b} \sim N(\mathbf{0}, \mathbf{G})$
- $\boldsymbol{\varepsilon} \sim N(\mathbf{0}, \mathbf{R})$
- $\text{Var}(\mathbf{y}) = \mathbf{V} = \mathbf{Z}\mathbf{G}\mathbf{Z}^\top + \mathbf{R}$
REML Estimation
Restricted Log-Likelihood:
$$
\ell_{\text{REML}}(\boldsymbol{\theta}) = -\frac{1}{2}\left[\log|\mathbf{V}| + \log|\mathbf{X}^\top\mathbf{V}^{-1}\mathbf{X}| + \mathbf{r}^\top\mathbf{V}^{-1}\mathbf{r}\right]
$$
Where $\mathbf{r} = \mathbf{y} - \mathbf{X}\hat{\boldsymbol{\beta}}$.
Physics-Informed Regression Models
Arrhenius-Based Models (Thermal Processes)
Rate Equation:
$$
k = A \exp\left(-\frac{E_a}{RT}\right)
$$
Linearized Form (for regression):
$$
\ln(k) = \ln(A) - \frac{E_a}{R} \cdot \frac{1}{T}
$$
Parameters:
- $k$ — rate constant
- $A$ — pre-exponential factor
- $E_a$ — activation energy (J/mol)
- $R$ — gas constant (8.314 J/mol·K)
- $T$ — absolute temperature (K)
Preston's Equation (CMP)
Basic Form:
$$
\text{MRR} = K_p \cdot P \cdot V
$$
Extended Model:
$$
\text{MRR} = K_p \cdot P^a \cdot V^b \cdot f(\text{slurry}, \text{pad})
$$
Where:
- MRR — material removal rate
- $K_p$ — Preston coefficient
- $P$ — applied pressure
- $V$ — relative velocity
Lithography Focus-Exposure Model
$$
\text{CD} = \beta_0 + \beta_1 E + \beta_2 F + \beta_3 E^2 + \beta_4 F^2 + \beta_5 EF + \varepsilon
$$
Variables:
- CD — critical dimension
- $E$ — exposure dose
- $F$ — focus offset
Bossung Curve: Plot of CD vs. focus at various exposure levels.
Virtual Metrology Mathematics
Predicting quality measurements from equipment sensor data in real-time.
Model Structure
$$
\hat{y} = f(\mathbf{x}_{\text{FDC}}; \boldsymbol{\theta})
$$
Where $\mathbf{x}_{\text{FDC}}$ is Fault Detection and Classification sensor data.
EWMA Run-to-Run Control
Exponentially Weighted Moving Average:
$$
\hat{T}_{n+1} = \lambda y_n + (1-\lambda)\hat{T}_n
$$
Properties:
- $\lambda \in (0,1]$ — smoothing parameter
- Smaller $\lambda$ → more smoothing
- Larger $\lambda$ → faster response to changes
Kalman Filter Approach
State Equation:
$$
\mathbf{x}_{k} = \mathbf{A}\mathbf{x}_{k-1} + \mathbf{w}_k, \quad \mathbf{w}_k \sim N(\mathbf{0}, \mathbf{Q})
$$
Measurement Equation:
$$
y_k = \mathbf{H}\mathbf{x}_k + v_k, \quad v_k \sim N(0, R)
$$
Update Equations:
*Predict*:
$$
\hat{\mathbf{x}}_{k|k-1} = \mathbf{A}\hat{\mathbf{x}}_{k-1|k-1}
$$
$$
\mathbf{P}_{k|k-1} = \mathbf{A}\mathbf{P}_{k-1|k-1}\mathbf{A}^\top + \mathbf{Q}
$$
*Update*:
$$
\mathbf{K}_k = \mathbf{P}_{k|k-1}\mathbf{H}^\top(\mathbf{H}\mathbf{P}_{k|k-1}\mathbf{H}^\top + R)^{-1}
$$
$$
\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k(y_k - \mathbf{H}\hat{\mathbf{x}}_{k|k-1})
$$
Classification and Count Models
Logistic Regression (Binary Outcomes)
For pass/fail or defect/no-defect classification:
Model:
$$
P(Y=1|\mathbf{x}) = \frac{1}{1 + \exp(-\mathbf{x}^\top\boldsymbol{\beta})} = \sigma(\mathbf{x}^\top\boldsymbol{\beta})
$$
Logit Link:
$$
\text{logit}(p) = \ln\left(\frac{p}{1-p}\right) = \mathbf{x}^\top\boldsymbol{\beta}
$$
Log-Likelihood:
$$
\ell(\boldsymbol{\beta}) = \sum_{i=1}^{n}\left[y_i \log(\pi_i) + (1-y_i)\log(1-\pi_i)\right]
$$
Newton-Raphson Update:
$$
\boldsymbol{\beta}^{(t+1)} = \boldsymbol{\beta}^{(t)} + (\mathbf{X}^\top\mathbf{W}\mathbf{X})^{-1}\mathbf{X}^\top(\mathbf{y} - \boldsymbol{\pi})
$$
Where $\mathbf{W} = \text{diag}(\pi_i(1-\pi_i))$.
Poisson Regression (Defect Counts)
Model:
$$
\log(\mu) = \mathbf{x}^\top\boldsymbol{\beta}, \quad Y \sim \text{Poisson}(\mu)
$$
Probability Mass Function:
$$
P(Y = y) = \frac{\mu^y e^{-\mu}}{y!}
$$
Model Validation and Diagnostics
Goodness of Fit Metrics
Coefficient of Determination:
$$
R^2 = 1 - \frac{\text{SSE}}{\text{SST}} = 1 - \frac{\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}{\sum_{i=1}^{n}(y_i - \bar{y})^2}
$$
Adjusted R-Squared:
$$
R^2_{\text{adj}} = 1 - (1-R^2)\frac{n-1}{n-k-1}
$$
Root Mean Square Error:
$$
\text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_i)^2}
$$
Mean Absolute Error:
$$
\text{MAE} = \frac{1}{n}\sum_{i=1}^{n}|y_i - \hat{y}_i|
$$
Cross-Validation
K-Fold CV Error:
$$
\text{CV}_{(K)} = \frac{1}{K}\sum_{k=1}^{K}\text{MSE}_k
$$
Leave-One-Out CV:
$$
\text{LOOCV} = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{y}_{(-i)})^2
$$
Information Criteria
Akaike Information Criterion:
$$
\text{AIC} = 2k - 2\ln(\hat{L})
$$
Bayesian Information Criterion:
$$
\text{BIC} = k\ln(n) - 2\ln(\hat{L})
$$
Diagnostic Statistics
Variance Inflation Factor:
$$
\text{VIF}_j = \frac{1}{1-R_j^2}
$$
Where $R_j^2$ is the $R^2$ from regressing $x_j$ on all other predictors.
Rule of thumb: VIF > 10 indicates problematic multicollinearity.
Cook's Distance:
$$
D_i = \frac{(\hat{\mathbf{y}} - \hat{\mathbf{y}}_{(-i)})^\top(\hat{\mathbf{y}} - \hat{\mathbf{y}}_{(-i)})}{k \cdot \text{MSE}}
$$
Leverage:
$$
h_{ii} = [\mathbf{H}]_{ii}
$$
Where $\mathbf{H} = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top$ is the hat matrix.
Studentized Residuals:
$$
r_i = \frac{e_i}{\hat{\sigma}\sqrt{1 - h_{ii}}}
$$
Bayesian Regression
Provides full uncertainty quantification for risk-sensitive manufacturing decisions.
Bayesian Linear Regression
Prior:
$$
\boldsymbol{\beta} | \sigma^2 \sim N(\boldsymbol{\beta}_0, \sigma^2\mathbf{V}_0)
$$
$$
\sigma^2 \sim \text{Inverse-Gamma}(a_0, b_0)
$$
Posterior:
$$
\boldsymbol{\beta} | \mathbf{y}, \sigma^2 \sim N(\boldsymbol{\beta}_n, \sigma^2\mathbf{V}_n)
$$
Posterior Parameters:
$$
\mathbf{V}_n = (\mathbf{V}_0^{-1} + \mathbf{X}^\top\mathbf{X})^{-1}
$$
$$
\boldsymbol{\beta}_n = \mathbf{V}_n(\mathbf{V}_0^{-1}\boldsymbol{\beta}_0 + \mathbf{X}^\top\mathbf{y})
$$
Predictive Distribution
$$
p(y_*|\mathbf{x}_*, \mathbf{y}) = \int p(y_*|\mathbf{x}_*, \boldsymbol{\beta}, \sigma^2) \, p(\boldsymbol{\beta}, \sigma^2|\mathbf{y}) \, d\boldsymbol{\beta} \, d\sigma^2
$$
For conjugate priors, this is a Student-t distribution.
Credible Intervals
95% Credible Interval for $\beta_j$:
$$
\beta_j \in \left[\hat{\beta}_j - t_{0.025,
u}\cdot \text{SE}(\hat{\beta}_j), \quad \hat{\beta}_j + t_{0.025,
u}\cdot \text{SE}(\hat{\beta}_j)\right]
$$
Design of Experiments (DOE)
Full Factorial Design
For $k$ factors at 2 levels:
$$
N = 2^k \text{ runs}
$$
Fractional Factorial Design
$$
N = 2^{k-p} \text{ runs}
$$
Resolution:
- Resolution III: Main effects aliased with 2-factor interactions
- Resolution IV: Main effects clear; 2FIs aliased with each other
- Resolution V: Main effects and 2FIs clear
Central Composite Design (CCD)
Components:
- $2^k$ factorial points
- $2k$ axial (star) points at distance $\alpha$
- $n_0$ center points
Rotatability Condition:
$$
\alpha = (2^k)^{1/4}
$$
D-Optimal Design
Maximizes the determinant of the information matrix:
$$
\max_{\mathbf{X}} |\mathbf{X}^\top\mathbf{X}|
$$
Equivalently, minimizes the generalized variance of $\hat{\boldsymbol{\beta}}$.
I-Optimal Design
Minimizes average prediction variance:
$$
\min_{\mathbf{X}} \int_{\mathcal{R}} \text{Var}(\hat{y}(\mathbf{x})) \, d\mathbf{x}
$$
Reliability Analysis
Cox Proportional Hazards Model
Hazard Function:
$$
h(t|\mathbf{x}) = h_0(t) \cdot \exp(\mathbf{x}^\top\boldsymbol{\beta})
$$
Where:
- $h(t|\mathbf{x})$ — hazard at time $t$ given covariates $\mathbf{x}$
- $h_0(t)$ — baseline hazard
- $\boldsymbol{\beta}$ — regression coefficients
Partial Likelihood
$$
L(\boldsymbol{\beta}) = \prod_{i: \delta_i = 1} \frac{\exp(\mathbf{x}_i^\top\boldsymbol{\beta})}{\sum_{j \in \mathcal{R}(t_i)} \exp(\mathbf{x}_j^\top\boldsymbol{\beta})}
$$
Where $\mathcal{R}(t_i)$ is the risk set at time $t_i$.
Challenge-Method Mapping
| Manufacturing Challenge | Mathematical Approach |
|------------------------|----------------------|
| High dimensionality | PLS, LASSO, Elastic Net |
| Multicollinearity | Ridge regression, PCR, VIF analysis |
| Spatial wafer patterns | Zernike polynomials, GP regression |
| Hierarchical data | Mixed effects models, REML |
| Nonlinear processes | RSM, polynomial models, transformations |
| Physics constraints | Arrhenius, Preston equation integration |
| Uncertainty quantification | Bayesian methods, bootstrap, prediction intervals |
| Binary outcomes | Logistic regression |
| Count data | Poisson regression |
| Real-time control | Kalman filter, EWMA |
| Time-to-failure | Cox proportional hazards |
Equations Quick Reference
Estimation
$$
\hat{\boldsymbol{\beta}}_{\text{OLS}} = (\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top\mathbf{y}
$$
$$
\hat{\boldsymbol{\beta}}_{\text{Ridge}} = (\mathbf{X}^\top\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^\top\mathbf{y}
$$
Prediction Interval
$$
\hat{y}_0 \pm t_{\alpha/2, n-k-1} \cdot \sqrt{\text{MSE}\left(1 + \mathbf{x}_0^\top(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{x}_0\right)}
$$
Confidence Interval for $\beta_j$
$$
\hat{\beta}_j \pm t_{\alpha/2, n-k-1} \cdot \text{SE}(\hat{\beta}_j)
$$
Process Capability
$$
C_p = \frac{\text{USL} - \text{LSL}}{6\sigma}
$$
$$
C_{pk} = \min\left(\frac{\text{USL} - \mu}{3\sigma}, \frac{\mu - \text{LSL}}{3\sigma}\right)
$$
Reference
| Symbol | Description |
|--------|-------------|
| $\mathbf{y}$ | Response vector |
| $\mathbf{X}$ | Design matrix |
| $\boldsymbol{\beta}$ | Coefficient vector |
| $\hat{\boldsymbol{\beta}}$ | Estimated coefficients |
| $\boldsymbol{\varepsilon}$ | Error vector |
| $\sigma^2$ | Error variance |
| $\lambda$ | Regularization parameter |
| $\mathbf{I}$ | Identity matrix |
| $\|\cdot\|_1$ | L1 norm (sum of absolute values) |
| $\|\cdot\|_2$ | L2 norm (Euclidean) |
| $\mathbf{A}^\top$ | Matrix transpose |
| $\mathbf{A}^{-1}$ | Matrix inverse |
| $|\mathbf{A}|$ | Matrix determinant |
| $N(\mu, \sigma^2)$ | Normal distribution |
| $\mathcal{GP}$ | Gaussian Process |