ISYE 6525 — HIGH-DIMENSIONAL DATA ANALYTICS
210 ORIGINAL EXAM-STYLE PRACTICE QUESTIONS WITH
CORRECT ANSWERS & RATIONALES
SUMMER 2026–2027 COMPREHENSIVE STUDY SET
SECTION 1 — FUNCTIONAL DATA ANALYSIS & BASIS EXPANSIONS
Questions 1–21
1. In functional data analysis (FDA), what is the primary difference
between functional and multivariate observations?
A. Functional observations contain only categorical variables
B. Functional observations are represented as continuous or densely
sampled functions
C. Functional observations always have fewer dimensions
D. Functional observations cannot be noisy
Correct Answer: B
Rationale: FDA treats each observation as a function, such as a growth
curve or temperature trajectory, rather than merely as a finite vector.
2. What is a major purpose of representing a function using basis
functions?
A. To eliminate all measurement error
B. To transform categorical data into numerical data
,C. To represent a complex function using a weighted combination of
known functions
D. To guarantee a linear regression model
Correct Answer: C
Rationale: Basis expansion approximates a function as a weighted sum
of simpler basis functions, reducing the problem to estimating
coefficients.
3. Which expression represents a typical basis expansion?
A. 𝑥 (𝑡) = ∑𝐾𝑘=1 𝑐𝑘 𝜙𝑘 (𝑡)
B. 𝑥 (𝑡) = ∏𝐾𝑘=1 𝑐𝑘
C. 𝑥 (𝑡) = 𝑐 + 𝑡
D. 𝑥 (𝑡) = 𝜙(𝑡)/𝐾
Correct Answer: A
Rationale: A function is approximated by a linear combination of basis
functions 𝜙𝑘 (𝑡)with coefficients 𝑐𝑘 .
4. Which basis is particularly suitable for representing periodic
functions?
A. Polynomial basis
B. Fourier basis
C. Haar basis only
D. Constant basis
Correct Answer: B
,Rationale: Fourier bases consist of sine and cosine functions and are
naturally suited to periodic behavior.
5. What is a major advantage of B-splines in functional data analysis?
A. They require no tuning parameters
B. They provide flexible local representations of functions
C. They can represent only periodic functions
D. They always produce orthogonal basis functions
Correct Answer: B
Rationale: B-splines use piecewise polynomial functions and provide
local flexibility through knots.
6. In spline modeling, what is a knot?
A. A missing observation
B. A location where polynomial pieces are joined
C. A regression coefficient
D. A covariance estimate
Correct Answer: B
Rationale: Knots divide the domain into intervals where different
polynomial pieces are fitted.
7. Increasing the number of basis functions generally has what effect?
A. It reduces model flexibility
B. It increases representational flexibility
, C. It guarantees lower prediction error
D. It removes noise automatically
Correct Answer: B
Rationale: More basis functions allow a model to represent more
complex shapes, although excessive flexibility can cause overfitting.
8. What is the main purpose of a roughness penalty in functional
regression?
A. To increase dimensionality
B. To encourage excessively oscillatory functions
C. To discourage overly complex or rough fitted functions
D. To remove the response variable
Correct Answer: C
Rationale: A roughness penalty controls smoothness by penalizing
quantities such as the integrated squared second derivative.
9. A common roughness penalty for a function 𝑓is based on which
quantity?
A. ∫ 𝑓 (𝑡)𝑑𝑡
B. ∫[ 𝑓 ′′ (𝑡)]2 𝑑𝑡
C. ∑ 𝑓 (𝑡)
D. max 𝑓 (𝑡)
Correct Answer: B
210 ORIGINAL EXAM-STYLE PRACTICE QUESTIONS WITH
CORRECT ANSWERS & RATIONALES
SUMMER 2026–2027 COMPREHENSIVE STUDY SET
SECTION 1 — FUNCTIONAL DATA ANALYSIS & BASIS EXPANSIONS
Questions 1–21
1. In functional data analysis (FDA), what is the primary difference
between functional and multivariate observations?
A. Functional observations contain only categorical variables
B. Functional observations are represented as continuous or densely
sampled functions
C. Functional observations always have fewer dimensions
D. Functional observations cannot be noisy
Correct Answer: B
Rationale: FDA treats each observation as a function, such as a growth
curve or temperature trajectory, rather than merely as a finite vector.
2. What is a major purpose of representing a function using basis
functions?
A. To eliminate all measurement error
B. To transform categorical data into numerical data
,C. To represent a complex function using a weighted combination of
known functions
D. To guarantee a linear regression model
Correct Answer: C
Rationale: Basis expansion approximates a function as a weighted sum
of simpler basis functions, reducing the problem to estimating
coefficients.
3. Which expression represents a typical basis expansion?
A. 𝑥 (𝑡) = ∑𝐾𝑘=1 𝑐𝑘 𝜙𝑘 (𝑡)
B. 𝑥 (𝑡) = ∏𝐾𝑘=1 𝑐𝑘
C. 𝑥 (𝑡) = 𝑐 + 𝑡
D. 𝑥 (𝑡) = 𝜙(𝑡)/𝐾
Correct Answer: A
Rationale: A function is approximated by a linear combination of basis
functions 𝜙𝑘 (𝑡)with coefficients 𝑐𝑘 .
4. Which basis is particularly suitable for representing periodic
functions?
A. Polynomial basis
B. Fourier basis
C. Haar basis only
D. Constant basis
Correct Answer: B
,Rationale: Fourier bases consist of sine and cosine functions and are
naturally suited to periodic behavior.
5. What is a major advantage of B-splines in functional data analysis?
A. They require no tuning parameters
B. They provide flexible local representations of functions
C. They can represent only periodic functions
D. They always produce orthogonal basis functions
Correct Answer: B
Rationale: B-splines use piecewise polynomial functions and provide
local flexibility through knots.
6. In spline modeling, what is a knot?
A. A missing observation
B. A location where polynomial pieces are joined
C. A regression coefficient
D. A covariance estimate
Correct Answer: B
Rationale: Knots divide the domain into intervals where different
polynomial pieces are fitted.
7. Increasing the number of basis functions generally has what effect?
A. It reduces model flexibility
B. It increases representational flexibility
, C. It guarantees lower prediction error
D. It removes noise automatically
Correct Answer: B
Rationale: More basis functions allow a model to represent more
complex shapes, although excessive flexibility can cause overfitting.
8. What is the main purpose of a roughness penalty in functional
regression?
A. To increase dimensionality
B. To encourage excessively oscillatory functions
C. To discourage overly complex or rough fitted functions
D. To remove the response variable
Correct Answer: C
Rationale: A roughness penalty controls smoothness by penalizing
quantities such as the integrated squared second derivative.
9. A common roughness penalty for a function 𝑓is based on which
quantity?
A. ∫ 𝑓 (𝑡)𝑑𝑡
B. ∫[ 𝑓 ′′ (𝑡)]2 𝑑𝑡
C. ∑ 𝑓 (𝑡)
D. max 𝑓 (𝑡)
Correct Answer: B
