Associate of the Society of Actuaries (ASA/FSA) Exam SRM — Statistics for Risk Modeling Flashcards

24 sample cards from the Exam SRM — Statistics for Risk Modeling deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What link and distribution are used for Poisson regression, and what response does it model?

   Poisson regression models count data with the log link: $$\log(\mu) = X\beta, \qquad \mu = e^{X\beta} > 0.$$ The response is assumed Poisson with $E[Y]=\operatorname{Var}(Y)=\mu$.

2. What is overdispersion in a Poisson GLM, and one way to address it?

   Overdispersion occurs when $\operatorname{Var}(Y) > E[Y]$, violating the Poisson mean-variance equality. It can be addressed with a quasi-Poisson model (dispersion parameter $\phi$) or a negative binomial regression.

3. Define the deviance of a GLM and its use.

   Deviance is $D = 2(\ell_{\text{saturated}} - \ell_{\text{model}})$, twice the log-likelihood gap between the fitted model and the saturated model. It measures goodness of fit (analogous to RSS); lower deviance means better fit, and deviance differences test nested models.

4. What is the AIC for a GLM and how is it used for model comparison?

   $$\text{AIC} = -2\ell + 2k$$ where $\ell$ is the maximized log-likelihood and $k$ the number of parameters. Lower AIC indicates a better tradeoff of fit and complexity; it is used to compare (possibly non-nested) GLMs.

5. In a confusion matrix, define sensitivity, specificity, and precision.

   Sensitivity (recall) $= \frac{TP}{TP+FN}$ (true positive rate); specificity $= \frac{TN}{TN+FP}$ (true negative rate); precision $= \frac{TP}{TP+FP}$ (positive predictive value).

6. What does an ROC curve plot, and what does AUC measure?

   An ROC curve plots the true positive rate (sensitivity) against the false positive rate ($1-$specificity) across all classification thresholds. AUC (area under the curve) summarizes overall performance; $0.5$ is random guessing and $1.0$ is perfect.

7. Describe how a regression tree makes predictions and how splits are chosen.

   A regression tree partitions the predictor space into regions and predicts the mean response of the region. Splits are chosen by recursive binary splitting to minimize RSS $\sum_{j}\sum_{i\in R_j}(y_i-\hat{y}_{R_j})^{2}$ greedily at each step.

8. What impurity measures are used to grow classification trees?

   The Gini index $G = \sum_{k} \hat{p}_k(1-\hat{p}_k)$ and cross-entropy $D = -\sum_k \hat{p}_k \log \hat{p}_k$; both are small when a node is pure. Classification error rate is also defined but is less sensitive for growing trees.

9. What is cost-complexity (weakest-link) pruning of a decision tree?

   It controls tree size by minimizing $$\sum_{m}\sum_{i\in R_m}(y_i-\hat{y}_{R_m})^{2} + \alpha |T|$$ where $|T|$ is the number of terminal nodes and $\alpha \geq 0$ is a tuning parameter (chosen by CV) that trades fit against tree complexity.

10. List key advantages and disadvantages of decision trees.

    Advantages: easy to interpret/visualize, handle qualitative predictors without dummies, mirror human decisions. Disadvantages: high variance / non-robust, and generally lower predictive accuracy than ensemble methods.

11. How does bagging reduce variance, and what are out-of-bag (OOB) observations?

    Bagging fits trees on many bootstrap samples and averages their predictions (or majority votes), reducing variance. OOB observations are those not in a given bootstrap sample (~$1/3$); averaging predictions over trees where an observation was OOB gives a built-in error estimate.

12. How do random forests improve on bagging?

    At each split, a random forest considers only a random subset of $m$ predictors (often $m \approx \sqrt{p}$ for classification) instead of all $p$. This decorrelates the trees, so averaging them reduces variance more than bagging.

13. How does boosting build a predictive model, and how does it differ from bagging?

    Boosting grows trees sequentially, each fitting the residuals of the current ensemble, with contributions scaled by a shrinkage parameter $\lambda$. Unlike bagging's independent parallel trees, boosting learns slowly and dependently, using small trees and tuning the number of trees to avoid overfitting.

14. Describe how K-nearest neighbors (KNN) makes a prediction and the effect of K.

    KNN finds the $K$ training points nearest to $x_0$ and predicts by majority vote (classification) or average (regression) of their responses. Small $K$ gives a flexible, low-bias/high-variance fit; large $K$ gives a smoother, higher-bias/lower-variance fit.

15. What is Principal Components Analysis (PCA) and what does the first principal component maximize?

    PCA finds orthogonal linear combinations of standardized variables (principal components) capturing successively maximal variance. The first principal component is the direction (loading vector of unit length) along which the projected data have the largest variance.

16. In PCA, what is the proportion of variance explained (PVE) by the $m$-th component?

    $$\text{PVE}_m = \frac{\sum_{i=1}^{n} z_{im}^{2}}{\sum_{j=1}^{p}\sum_{i=1}^{n} x_{ij}^{2}}$$ the variance of the $m$-th component scores divided by total variance. A scree plot of PVE helps choose how many components to keep.

17. Why should variables be standardized before performing PCA?

    PCA is scale-sensitive: variables with larger variances would dominate the components. Standardizing each variable to mean 0 and standard deviation 1 ensures all variables contribute comparably to the analysis.

18. Describe the K-means clustering algorithm.

    Choose $K$; randomly assign points to clusters. Iterate: (1) compute each cluster's centroid (mean), (2) reassign each point to the nearest centroid. Repeat until assignments stop changing. It minimizes within-cluster sum of squares; rerun with multiple random starts since it finds a local optimum.

19. Compare K-means and hierarchical clustering.

    K-means requires pre-specifying $K$ and partitions data flatly. Hierarchical clustering builds a nested tree (dendrogram) without pre-specifying $K$; cutting the dendrogram at a chosen height yields the clusters and reveals structure at multiple scales.

20. In agglomerative hierarchical clustering, define complete, single, and average linkage.

    Linkage measures inter-cluster dissimilarity: complete linkage uses the largest pairwise distance, single linkage the smallest, and average linkage the mean of all pairwise distances between the two clusters' points.

21. Define a stationary time series (weak stationarity).

    A time series is (weakly) stationary if its mean and variance are constant over time and its autocovariance $\operatorname{Cov}(Y_t, Y_{t+h})$ depends only on the lag $h$, not on $t$.

22. What is the random walk model and is it stationary?

    $$Y_t = Y_{t-1} + \varepsilon_t$$ where $\varepsilon_t$ is white noise. It is non-stationary (its variance grows with $t$); differencing once, $Y_t - Y_{t-1} = \varepsilon_t$, produces a stationary white-noise series.

23. In time series, what are the trend, seasonal, and cyclical/irregular components of decomposition?

    Trend is the long-term increase or decrease; seasonal is a regular pattern of fixed, known period (e.g., monthly); the cyclical/irregular (remainder) component is the residual fluctuation after removing trend and seasonality.

24. What is exponential smoothing and what does its smoothing parameter control?

    Exponential smoothing forecasts with weights that decay geometrically into the past: $$\hat{Y}_{t+1} = \alpha Y_t + (1-\alpha)\hat{Y}_t, \quad 0<\alpha<1.$$ Larger $\alpha$ weights recent observations more (responsive); smaller $\alpha$ gives smoother forecasts.

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