Free SOA Exam SRM (Statistics for Risk Modeling) Formula Sheet (2026)

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All Exam SRM Formulas

Basics of Statistical Learning 10 items
Bias-variance tradeoff
E[(y−f^(x))2]=Bias2(f^)+Var(f^)+σε2E[(y-\hat{f}(x))^2]=\text{Bias}^2(\hat{f})+\text{Var}(\hat{f})+\sigma^2_\varepsilon
Irreducible error σε2\sigma^2_\varepsilon cannot be reduced
k-fold cross-validation error
CV(k)=1k∑j=1kMSEjCV_{(k)}=\dfrac{1}{k}\sum_{j=1}^k \text{MSE}_j
Each fold serves as validation once
Expected test MSE decomposition
E[(y0−f^(x0))2]=Var(ε)+[Bias(f^(x0))]2+Var(f^(x0))E[(y_0 - \hat{f}(x_0))^2] = \text{Var}(\varepsilon) + [\text{Bias}(\hat{f}(x_0))]^2 + \text{Var}(\hat{f}(x_0))
Var(ε) = irreducible noise, Bias² = squared model bias, Var(f̂) = model variance
Mallows Cp statistic
Cp=1n(RSS+2dσ^2)C_p = \tfrac{1}{n}(\text{RSS} + 2d\hat{\sigma}^2)
n = sample size, RSS = residual sum of squares, d = number of predictors, σ^2\hat{\sigma}^2 = residual variance estimate
Mean squared error (regression)
MSE=1n∑i=1n(yi−f^(xi))2\text{MSE} = \frac{1}{n}\sum_{i=1}^{n}(y_i - \hat{f}(x_i))^2
n = sample size, yiy_i = actual, f^(xi)\hat{f}(x_i) = predicted response
Misclassification rate
Err=1n∑i=1nI(yi≠y^i)\text{Err} = \frac{1}{n}\sum_{i=1}^{n} I(y_i \ne \hat{y}_i)
n = sample size, I = indicator function, yiy_i = actual class, y^i\hat{y}_i = predicted class
LOOCV closed-form estimate for least-squares linear regression
CV(n)=1n∑i=1n(yi−y^i1−hi)2\text{CV}_{(n)} = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{y_i - \hat{y}_i}{1 - h_i} \right)^{2}
n = sample size, yiy_i = observed, ŷ_i = fitted, hih_i = leverage of obs i
Logistic regression logit model
log⁡[p/(1−p)]=Xβ\log[p/(1-p)] = X\beta
p = P(Y=1|X), X = predictor matrix, β = coefficient vector, log-odds linear in X
Linear regression model
Y=Xβ+εY = X\beta + \varepsilon
Y = numeric response, X = predictor matrix, β = coefficient vector, ε = error term with mean 0
Expected prediction error (squared loss)
E[(Y−f^(X))2]E[(Y - \hat{f}(X))^2]
Y = true response, f^(X)\hat{f}(X) = predicted response, E = expectation over (X, Y)
Linear Models 37 items
R-squared
R2=1−SSresSStot=1−∑(yi−y^i)2∑(yi−yˉ)2R^2 = 1 - \dfrac{SS_{\text{res}}}{SS_{\text{tot}}} = 1 - \dfrac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}
Adjusted R-squared
Rˉ2=1−(1−R2)(n−1)n−p−1\bar{R}^2 = 1 - \dfrac{(1-R^2)(n-1)}{n-p-1}
pp=number of predictors (excludes intercept)
F-statistic (regression)
F=(SStot−SSres)/pSSres/(n−p−1)F=\dfrac{(SS_{\text{tot}}-SS_{\text{res}})/p}{SS_{\text{res}}/(n-p-1)}
Tests H0:H_0: all slope coefficients are zero
Variance inflation factor (VIF)
VIFj=11−Rj2\text{VIF}_j = \dfrac{1}{1-R_j^2}
Rj2R_j^2=R2R^2 from regressing XjX_j on all other predictors
VIF>5–10 indicates multicollinearity
AIC (ordinary linear model)
AIC=1n(RSS+2dσ^2)\text{AIC} = \dfrac{1}{n}\left(\text{RSS} + 2d\hat{\sigma}^2\right)
dd=model size; σ^2\hat{\sigma}^2 from the full model, held fixed; lower is better
SRM scope: ordinary linear models only. The form (-2ln L + 2d) is out of scope.
BIC (ordinary linear model)
BIC=1n(RSS+ln⁡(n) d σ^2)\text{BIC} = \dfrac{1}{n}\left(\text{RSS} + \ln(n)\, d\, \hat{\sigma}^2\right)
Same dd and σ^2\hat{\sigma}^2 as AIC. Penalizes complexity harder once n>7n > 7
SRM scope: ordinary linear models only. The form (-2ln L + dln n) is out of scope.
Ridge regression penalty
Minimize: ∑(yi−y^i)2+λ∑j=1pβj2\sum(y_i-\hat{y}_i)^2+\lambda\sum_{j=1}^p\beta_j^2
L2L_2 penalty; shrinks but does not zero out coefficients
LASSO penalty
Minimize: ∑(yi−y^i)2+λ∑j=1p∣βj∣\sum(y_i-\hat{y}_i)^2+\lambda\sum_{j=1}^p|\beta_j|
L1L_1 penalty; produces sparse solutions (exact zeros)
Elastic net penalty
Minimize: ∑(yi−y^i)2+λ1∑∣βj∣+λ2∑βj2\sum(y_i-\hat{y}_i)^2+\lambda_1\sum|\beta_j|+\lambda_2\sum\beta_j^2
Combines LASSO (L1L_1) and Ridge (L2L_2)
Studentized residual
ri=eiσ^1−hiir_i = \frac{e_i}{\hat{\sigma}\sqrt{1 - h_{ii}}}
e = raw residual, σ̂ = residual SE, h = leverage; |r| > 2 flags outlier
Durbin-Watson statistic
DW=∑i=2n(ei−ei−1)2∑i=1nei2≈2(1−ρ^)DW = \frac{\sum_{i=2}^{n}(e_i - e_{i-1})^2}{\sum_{i=1}^{n} e_i^2} \approx 2(1 - \hat{\rho})
e = residual, n = sample size, ρ̂ = lag-1 residual autocorrelation
Cook's distance
Di=ri2p+1⋅hii1−hiiD_i = \frac{r_i^2}{p+1} \cdot \frac{h_{ii}}{1 - h_{ii}}
r = studentized residual, h = leverage, p = number of predictors; D > 1 is influential
High-leverage cutoff
hii>2(p+1)nh_{ii} > \frac{2(p+1)}{n}
h = leverage (hat matrix diagonal), p = number of predictors, n = sample size
Tweedie variance function
V(μ)=ϕ μpV(\mu) = \phi\, \mu^p
φ = dispersion, μ = mean, p = power parameter (p=0 normal, p=1 Poisson, 1<p<2 compound Poisson-gamma, p=2 gamma)
GLM link function and linear predictor
g(μ)=η=Xβg(\mu) = \eta = X\beta
g = link function, μ = mean response, η = linear predictor, X = design matrix, β = coefficient vector
GLM mean and variance from b(theta)
E[Y]=b′(θ)=μ, Var(Y)=b′′(θ) a(ϕ)E[Y] = b'(\theta) = \mu,\ \mathrm{Var}(Y) = b''(\theta)\, a(\phi)
θ = natural parameter, φ = dispersion, μ = mean, b'' a(φ) = V(μ)·a(φ)
Exponential family density form
f(y;θ,ϕ)=exp⁡{[yθ−b(θ)]/a(ϕ)+c(y,ϕ)}f(y;\theta,\phi) = \exp\{[y\theta - b(\theta)]/a(\phi) + c(y,\phi)\}
y = data, θ = natural parameter, φ = dispersion, a/b/c = family-specific functions
Exponential family density (GLM random component)
f(y;θ,ϕ)=exp⁡{[yθ−b(θ)]/a(ϕ)+c(y,ϕ)}f(y; \theta, \phi) = \exp\{[y\theta - b(\theta)]/a(\phi) + c(y, \phi)\}
θ = canonical parameter, φ = dispersion, b(θ) sets mean and variance
OLS coefficient estimator (closed form)
β^=(XTX)−1XTY\hat{\beta} = (X^{T}X)^{-1} X^{T} Y
X = design matrix of predictors, Y = response vector, β^\hat{\beta} = least-squares coefficient vector
GLM variance structure
Var(Y)=ϕ V(μ)\text{Var}(Y) = \phi\, V(\mu)
φ = dispersion parameter, V(μ) = variance function (μ for Poisson, μ(1−μ) for binomial, μ² for gamma)
GLM deviance (goodness-of-fit)
D=2[ℓ(saturated)−ℓ(μ^)]D = 2[\ell(\text{saturated}) - \ell(\hat{\mu})]
ℓ = log-likelihood, saturated = perfect-fit model, μ^\hat{\mu} = fitted means; analog of RSS
Box-Cox transformation
Y(λ)=(Yλ−1)/λY^{(\lambda)} = (Y^\lambda - 1)/\lambda
Y = positive response, λ = power parameter; λ=0 gives log transform, λ=1 gives no transform
Marginal effect of a predictor with an interaction term
∂η/∂X1=β1+β12X2\partial \eta / \partial X_1 = \beta_1 + \beta_{12} X_2
β1β_1 = main effect of X1X_1, β12β_{12} = interaction coefficient, X2X_2 = moderator value
t-statistic for a regression coefficient
t=β^j/SE(β^j)t = \hat{\beta}_j / SE(\hat{\beta}_j)
β̂_j = estimated coefficient, SE = standard error, compared to t-distribution with n-p-1 degrees of freedom
Likelihood ratio test statistic
Λ=2(log⁡LF−log⁡LR)∼χp−q2\Lambda = 2(\log L_F - \log L_R) \sim \chi^2_{p-q}
LFL_F = full model log-likelihood, LRL_R = reduced model log-likelihood, p-q = parameters dropped
K-nearest neighbors classification probability
Pr⁡(Y=j∣X=x0)=1K∑xi∈NK(x0)I(yi=j)\Pr(Y = j \mid X = x_0) = \frac{1}{K}\sum_{x_i \in N_K(x_0)} I(y_i = j)
j = class, K = neighbors, I(·) = indicator function, NKN_K(x₀) = K nearest training points
Ridge regression effective degrees of freedom
df(λ)=tr[X(XTX+λI)−1XT]df(\lambda) = tr[X(X^TX + \lambda I)^{-1}X^T]
X = design matrix, λ = ridge tuning parameter, I = identity matrix, tr = matrix trace
K-nearest neighbors regression prediction
f^(x0)=1K∑xi∈NK(x0)yi\hat{f}(x_0) = \frac{1}{K}\sum_{x_i \in N_K(x_0)} y_i
x₀ = query point, K = neighborhood size, NKN_K(x₀) = K nearest training points, yiy_i = neighbor responses
Ridge regression closed-form estimator
β^ridge=(XTX+λI)−1XTy\hat{\beta}^{ridge} = (X^TX + \lambda I)^{-1} X^T y
X = design matrix, y = response vector, λ = ridge tuning parameter, I = identity matrix
Small log-coefficient percent-change approximation
eβ^j−1≈β^je^{\hat{\beta}_j} - 1 \approx \hat{\beta}_j for ∣β^j∣<0.2|\hat{\beta}_j| < 0.2
βjβ_j = coefficient on log(Y); approximate percent change in Y per unit increase in XjX_j is 100βjβ_j%
Effective slope with interaction term
slopeX1=β^1+β^3X2\text{slope}_{X_1} = \hat{\beta}_1 + \hat{\beta}_3 X_2
β1β_1 = main effect on X1X_1, β3β_3 = coefficient on X1X_1 X2X_2 interaction, X2X_2 = value of interacting variable
Log-log elasticity interpretation
%ΔY≈β^j⋅%ΔXj\%\Delta Y \approx \hat{\beta}_j \cdot \%\Delta X_j
βjβ_j = coefficient when both XjX_j and Y are log-transformed; βjβ_j is the elasticity of Y with respect to XjX_j
Log-response multiplicative effect of a predictor
E[Y∣Xj+1]=E[Y∣Xj]⋅eβ^jE[Y \mid X_j + 1] = E[Y \mid X_j] \cdot e^{\hat{\beta}_j}
βjβ_j = coefficient in log(Y) model, XjX_j = predictor, e^{β_j} = multiplicative factor on Y
Confidence interval for the mean response in simple linear regression
y^0±tn−2,α/2⋅s1n+(x0−xˉ)2Sxx\hat{y}_0 \pm t_{n-2,\alpha/2} \cdot s\sqrt{\frac{1}{n} + \frac{(x_0-\bar{x})^2}{S_{xx}}}
s = residual SE, n = sample size, SxxS_{xx} = Σ(x_i-x̄)²
Prediction standard error from confidence standard error
SEpred=SE(y^0)2+s2SE_{pred} = \sqrt{SE(\hat{y}_0)^2 + s^2}
SE(ŷ₀) = CI standard error of the fit, s = residual standard error
Prediction interval for a new observation in simple linear regression
y^0±tn−2,α/2⋅s1+1n+(x0−xˉ)2Sxx\hat{y}_0 \pm t_{n-2,\alpha/2} \cdot s\sqrt{1 + \frac{1}{n} + \frac{(x_0-\bar{x})^2}{S_{xx}}}
extra +1 inside radical captures irreducible noise σ²
Variance of the fitted mean in multiple regression
Var(y^0)=σ2x0⊤(X⊤X)−1x0\text{Var}(\hat{y}_0) = \sigma^2 \mathbf{x}_0^\top (X^\top X)^{-1} \mathbf{x}_0
x0x_0 = predictor vector at new point, X = design matrix, σ² = error variance
Time Series Models 17 items
AR(1) model
Xt=ϕXt−1+εt,εt∼WN(0,σ2)X_t=\phi X_{t-1}+\varepsilon_t,\quad\varepsilon_t\sim WN(0,\sigma^2)
Stationary iff ∣ϕ∣<1|\phi|<1
MA(1) model
Xt=εt+θεt−1,εt∼WN(0,σ2)X_t=\varepsilon_t+\theta\varepsilon_{t-1},\quad\varepsilon_t\sim WN(0,\sigma^2)
Always stationary
ARMA(1,1) model
Xt=ϕXt−1+εt+θεt−1X_t=\phi X_{t-1}+\varepsilon_t+\theta\varepsilon_{t-1}
Stationary iff ∣ϕ∣<1|\phi|<1
ACF of AR(1)
ρ(h)=ϕh,h=0,1,2,…\rho(h)=\phi^h,\quad h=0,1,2,\ldots
Decays geometrically; PACF cuts off after lag 1
Ljung-Box test statistic
Q=n(n+2)∑k=1mρ^k2n−kQ=n(n+2)\sum_{k=1}^m\dfrac{\hat{\rho}_k^2}{n-k}
Tests H0:H_0: first mm autocorrelations are zero
Distributed χ2(m)\chi^2(m) under H0H_0
Random walk with drift h-step point forecast
Y^T+h=YT+hδ\hat{Y}_{T+h} = Y_T + h\delta
YTY_T = last observed value, δ\delta = drift per period, h = forecast horizon
h-step prediction interval for a time series forecast
Y^T+h±zα/2Var(eT+h)\hat{Y}_{T+h} \pm z_{\alpha/2} \sqrt{\text{Var}(e_{T+h})}
Y^T+h\hat{Y}_{T+h} = point forecast, zα/2z_{\alpha/2} = normal quantile (1.96 for 95%), Var(eT+h)\text{Var}(e_{T+h}) = h-step forecast error variance
AR(1) h-step forecast error variance
Var(eT+h)=σ2⋅1−ϕ2h1−ϕ2\text{Var}(e_{T+h}) = \sigma^2 \cdot \frac{1 - \phi^{2h}}{1 - \phi^2}
σ2\sigma^2 = white noise variance, ϕ\phi = AR(1) coefficient (∣ϕ∣<1|\phi|<1), h = forecast horizon
AR(1) long-run (unconditional) forecast variance ceiling
Var∞=σ21−ϕ2\text{Var}_\infty = \frac{\sigma^2}{1 - \phi^2}
σ2\sigma^2 = white noise variance, ϕ\phi = AR(1) coefficient with ∣ϕ∣<1|\phi|<1
Simple exponential smoothing one-step forecast
Y^t+1=αYt+(1−α)Y^t\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha)\hat{Y}_t
α ∈ (0,1) = smoothing constant, YtY_t = current observation, Ŷ_t = previous smoothed forecast
ARCH(q) conditional variance
σt2=ω+α1εt−12+⋯+αqεt−q2\sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \cdots + \alpha_q \varepsilon_{t-q}^2
ω > 0 = constant, αiα_i ≥ 0 = ARCH weights, εt−iε_{t-i} = past residuals
Long-run mean of a stationary AR(1) process
μ=c1−ϕ1\mu = \dfrac{c}{1 - \phi_1}
c = intercept, φ₁ = AR(1) coefficient with |φ₁| < 1 for stationarity
GARCH(1,1) conditional variance
σt2=ω+α1εt−12+β1σt−12\sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2
ω = constant, α₁ = ARCH term, β₁ = GARCH lagged-variance term, εt−1ε_{t-1} = prior residual
Random walk variance
Var(Yt)=tσ2\text{Var}(Y_t) = t\sigma^2 where Yt=Y0+∑i=1tεiY_t = Y_0 + \sum_{i=1}^{t}\varepsilon_i
Y0Y_0 = starting value, σ² = white noise variance, t = time index; variance grows linearly so series is non-stationary
White noise 95 percent significance band for sample ACF
ρ^k∈±1.96T\hat{\rho}_k \in \pm \dfrac{1.96}{\sqrt{T}}
T = sample size; sample autocorrelations inside this band are consistent with zero at the 5% level
Autocorrelation function at lag k
ρk=γkγ0=Cov(Yt,Yt−k)Var(Yt)\rho_k = \dfrac{\gamma_k}{\gamma_0} = \dfrac{\text{Cov}(Y_t, Y_{t-k})}{\text{Var}(Y_t)}
γkγ_k = autocovariance at lag k, γ0γ_0 = variance, ρ0ρ_0 = 1, |ρkρ_k| ≤ 1
Sample autocorrelation at lag k
ρ^k=∑t=k+1T(Yt−Yˉ)(Yt−k−Yˉ)∑t=1T(Yt−Yˉ)2\hat{\rho}_k = \dfrac{\sum_{t=k+1}^{T}(Y_t - \bar{Y})(Y_{t-k} - \bar{Y})}{\sum_{t=1}^{T}(Y_t - \bar{Y})^2}
Ȳ = sample mean, T = sample size, k = lag
Decision Trees 14 items
Gini impurity
G=∑k=1Kp^k(1−p^k)=1−∑k=1Kp^k2G = \sum_{k=1}^{K} \hat{p}_k(1-\hat{p}_k) = 1 - \sum_{k=1}^{K}\hat{p}_k^2
p^k\hat{p}_k=fraction of class kk in node
Entropy (node impurity)
H=−∑k=1Kp^klog⁡p^kH = -\sum_{k=1}^{K}\hat{p}_k\log\hat{p}_k
Bagging (bootstrap aggregation)
Train BB trees on bootstrap samples; aggregate predictions
f^(x)=1B∑b=1Bf^b(x)\hat{f}(x)=\dfrac{1}{B}\sum_{b=1}^B\hat{f}_b(x)
Reduces variance without increasing bias
Out-of-bag mean squared error
OOB MSE=1n∑i=1n(yi−y^iOOB)2\text{OOB MSE} = \frac{1}{n}\sum_{i=1}^{n} (y_i - \hat{y}_i^{\text{OOB}})^2
yiy_i = actual response, y^iOOB\hat{y}_i^{\text{OOB}} = average of trees not trained on i
Out-of-bag observation fraction
(1−1/n)n→e−1≈0.368(1 - 1/n)^n \to e^{-1} \approx 0.368
n = training rows, fraction of rows omitted from any given bootstrap sample
Boosting ensemble update rule
f^(x)←f^(x)+λf^b(x)\hat{f}(x) \leftarrow \hat{f}(x) + \lambda \hat{f}^{b}(x)
f^\hat{f} = current ensemble, f^b\hat{f}^b = new tree fit to residuals, λ\lambda = shrinkage/learning rate
Random forest default predictors per split
m=⌊p⌋m = \lfloor \sqrt{p} \rfloor (classification); m=⌊p/3⌋m = \lfloor p/3 \rfloor (regression): m = predictors sampled at each split, p = total predictors
Piecewise-constant decision tree prediction function
Y^=∑m=1Mcm⋅1{X∈Rm}\hat{Y} = \sum_{m=1}^{M} c_m \cdot \mathbf{1}\{X \in R_m\}
M = number of leaves, RmR_m = m-th rectangular region, cmc_m = constant prediction in leaf m, 1{·} = indicator function
Number of unique binary splits for an unordered categorical predictor
N=2q−1−1N = 2^{q-1} - 1
q = number of unordered levels of the categorical predictor
Cost-complexity criterion for a classification tree
Cα(T)=∑m=1∣T∣∑i∈RmL(yi,y^Rm)+α∣T∣C_\alpha(T) = \sum_{m=1}^{|T|} \sum_{i \in R_m} L(y_i, \hat{y}_{R_m}) + \alpha |T|
L = Gini or entropy impurity, |T| = number of terminal nodes, α ≥ 0 = complexity penalty
Recursive binary splitting objective for a regression tree
min⁡j,s[∑i∈R1(j,s)(yi−yˉR1)2+∑i∈R2(j,s)(yi−yˉR2)2]\min_{j,s} \big[\sum_{i \in R_1(j,s)} (y_i - \bar{y}_{R_1})^2 + \sum_{i \in R_2(j,s)} (y_i - \bar{y}_{R_2})^2\big]
j = predictor, s = cutpoint, R1R_1,R2R_2 = half-planes
Cost-complexity pruning criterion for a regression tree
Cα(T)=∑m=1∣T∣∑i:xi∈Rm(yi−y^Rm)2+α∣T∣C_\alpha(T) = \sum_{m=1}^{|T|} \sum_{i: x_i \in R_m} (y_i - \hat{y}_{R_m})^2 + \alpha |T|
|T| = number of leaves, α = complexity penalty, RmR_m = leaf region
Classification tree leaf prediction
C^Rj=arg⁡max⁡kp^Rj,k\hat{C}_{R_j} = \arg\max_k \hat{p}_{R_j, k}
RjR_j = leaf region, k = class index, p̂_{R_j,k} = training proportion of class k in leaf
Regression tree leaf prediction
y^Rj=1∣Rj∣∑i:xi∈Rjyi\hat{y}_{R_j} = \frac{1}{|R_j|} \sum_{i: x_i \in R_j} y_i
RjR_j = leaf region, |RjR_j| = number of training observations in leaf, yiy_i = response
Unsupervised Learning Techniques 16 items
PCA — proportion of variance explained
PVEk=λk∑j=1pλj\text{PVE}_k=\dfrac{\lambda_k}{\sum_{j=1}^p\lambda_j}
λk\lambda_k=kkth eigenvalue of covariance (or correlation) matrix
K-means objective function
min⁡C1,…,CK∑k=1K∑i∈Ck∥xi−xˉk∥2\min_{C_1,\ldots,C_K}\sum_{k=1}^K\sum_{i\in C_k}\|x_i-\bar{x}_k\|^2
xˉk\bar{x}_k=centroid of cluster kk
Average linkage distance between clusters
d(A,B)=1∣A∣∣B∣∑a∈A∑b∈B∥a−b∥d(A,B) = \frac{1}{|A||B|} \sum_{a \in A} \sum_{b \in B} \|a - b\|
A, B = clusters, |A|, |B| = cluster sizes, ||·|| = Euclidean distance
Complete linkage distance between clusters
d(A,B)=max⁡a∈A, b∈B∥a−b∥d(A,B) = \max_{a \in A,\, b \in B} \|a - b\|
A, B = clusters, a, b = observations in each cluster, ||·|| = Euclidean distance
Single linkage distance between clusters
d(A,B)=min⁡a∈A, b∈B∥a−b∥d(A,B) = \min_{a \in A,\, b \in B} \|a - b\|
A, B = clusters, a, b = observations in each cluster, ||·|| = Euclidean distance
K-means centroid update
μk=1∣Ck∣∑i∈Ckxi\mu_k = \frac{1}{|C_k|} \sum_{i \in C_k} x_i
μkμ_k = centroid of cluster k, CkC_k = set of points in cluster k, |CkC_k| = cluster size, xix_i = observation
First principal component as a linear combination
Z1=ϕ11X1+ϕ21X2+⋯+ϕp1Xp, ∑jϕj12=1Z_1 = \phi_{11} X_1 + \phi_{21} X_2 + \dots + \phi_{p1} X_p,\ \sum_j \phi_{j1}^2 = 1
Z₁ = PC1 score, φⱼ₁ = loading of Xⱼ on PC1, Xⱼ = centered predictor
Cumulative proportion of variance explained through component M
Cum PVEM=∑m=1Mλm/∑j=1pλj\text{Cum PVE}_M = \sum_{m=1}^{M}\lambda_m \big/ \sum_{j=1}^{p}\lambda_j
λₘ = m-th eigenvalue, M = components retained, p = total predictors
PCA loading vector as eigenvector of the covariance matrix
S ϕm=λm ϕm\mathbf{S}\,\phi_m = \lambda_m\,\phi_m
S = p×p sample covariance matrix, φₘ = unit eigenvector (loading vector for PC m), λₘ = eigenvalue = Var(Zₘ)
Principal component score for observation i on component m
zim=ϕ1mxi1+ϕ2mxi2+⋯+ϕpmxipz_{im} = \phi_{1m} x_{i1} + \phi_{2m} x_{i2} + \dots + \phi_{pm} x_{ip}
zᵢₘ = score, φⱼₘ = loading of Xⱼ on PC m, xᵢⱼ = centered value of Xⱼ for obs i
Total within-cluster sum of squares (pairwise form)
W(k)=∑c=1k12∣Cc∣∑i,j∈Cc∥xi−xj∥2W(k) = \sum_{c=1}^{k} \frac{1}{2|C_c|} \sum_{i,j \in C_c} \|x_i - x_j\|^2
k = number of clusters, CcC_c = cluster c, |CcC_c| = its size, xix_i = observation vectors; ISLR total within-cluster variation is twice this
Silhouette coefficient for an observation
s(i)=b(i)−a(i)max⁡{a(i),b(i)}s(i) = \frac{b(i) - a(i)}{\max\{a(i), b(i)\}}
a(i) = mean within-cluster distance, b(i) = mean distance to nearest other cluster; s(i) in [-1, 1]
One-standard-error rule for the gap statistic
Gap(k)≥Gap(k+1)−sk+1\text{Gap}(k) \geq \text{Gap}(k+1) - s_{k+1}
choose smallest k satisfying this; sk+1s_{k+1} = standard error of Gap(k+1) across B reference samples
Gap statistic for choosing number of clusters
Gap(k)=E∗[log⁡W∗(k)]−log⁡W(k)\text{Gap}(k) = E^{*}[\log W^{*}(k)] - \log W(k)
W(k) = observed within-cluster SS, W*(k) = SS on uniform reference data, E* = expectation over B reference samples
Principal component score
Zk=ϕ1kX1+ϕ2kX2+⋯+ϕpkXpZ_k = \phi_{1k} X_1 + \phi_{2k} X_2 + \cdots + \phi_{pk} X_p
ZkZ_k = kth component, φjkφ_{jk} = loading of variable j on PC k, XjX_j = standardized predictor
Loading vector unit-length constraint
∑j=1pϕjk2=1\sum_{j=1}^{p} \phi_{jk}^2 = 1
φjkφ_{jk} = loading of variable j on PC k, p = number of predictors
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