Problem Class 2

Question 1

In FDA, what does the term ”functional data” refer to?A. Data with missing values
B. Data collected from experiments
C. Data with repeated measures over a continuum
D. Categorical data

Answer 1

Answer: C. Data with repeated measures over a continuum

Question 2

What does the smoothing parameter λ in functional data analysis control?
A. The number of data points in the functional dataset
B. The degree of flexibility in the fitted curve
C. The range of values in the functional dataset
D. The significance level for hypothesis testing

Answer 2

B: The degree of flexibility in the fitted curve

Question 3

Which of the following statements about the smoothing parameter λ is true?
A. The smoothing parameter only affects the mean of the functional dataset
B. The smoothing parameter does not affect the smoothness of the curve
C. A smaller smoothing parameter leads to a smoother curve
D. A larger smoothing parameter leads to a smoother curve

Answer 3

D. A larger smoothing parameter leads to a smoother curve

Question 4

Which of the following techniques is commonly used to select the smoothing parameter λ in functional data analysis?
A. Principal component analysis (PCA)
B. Singular value decomposition (SVD)
C. Maximum likelihood estimation (MLE)
D. Generalized Cross-Validation (GCV)

Answer 4

D. Generalized Cross-Validation (GCV)

Question 5

What happens if the smoothing parameter λ is too small in functional data analysis?
A. Over-smoothing of the curve occurs
B. Under-smoothing of the curve occurs
C. No effect on the curve’s smoothness
D. The curve becomes discontinuous

Answer 5

B. Under-smoothing of the curve occurs

Question 6

How does cross-validation help in selecting the smoothing parameter λ?
A. By minimising the sum of squared errors (SSE)
B. By maximising the variance of the functional dataset
C. By maximising the likelihood function
D. By minimising the number of data points in the functional data

Answer 6

A. By minimising the sum of squared errors (SSE)

Question 7

What does the smoothing parameter λ control in Roughness Penalty Smoothing Method for functional data analysis?
A. The level of significance for hypothesis testing
B. The number of penalties applied to the model
C. The trade-off between smoothness and goodness of fit
D. The type of basis functions used in the analysis

Answer 7

C. The trade-off between smoothness and goodness of fit

Question 8

What is the purpose of the smoothing parameter λ in functional data analysis?
A. To reduce noise in the data
B. To increase the number of data points
C. To adjust the scale of the functional dataset
D. To control the degree of smoothing in the fitted curve

Answer 8

D. To control the degree of smoothing in the fitted curve

Question 9

What is the primary objective of Generalized Cross-Validation in determining the optimal smoothing parameter λ?
A. To maximize the bias of the fitted curve
B. To minimize the variance of the fitted curve
C. To select the smoothing parameter that generalizes best to unseen data
D. To estimate the number of basis functions needed for the analysis

Answer 9

C. To select the smoothing parameter that generalizes best to unseen data

Question 10

What is the matrix form of the estimated coefficient vector?
A. ˆc = (Φ⊤Φ)^−1 Φ^⊤
B. ˆc = (Φ⊤Φ)^−1 Φ^⊤ λ
C. ˆc = (Φ⊤Φ)^−1 Φ^⊤ Y
D. ˆc = (Φ⊤Φ)^⊤ Φ^⊤ Y

Answer 10

C. ˆc = (Φ⊤Φ)^−1 Φ^⊤ Y

Question 11

(11) We wish to apply the Leave One Out approach to a functional dataset of 100 samples, how many samples will be divided into the training set for each validation?
A. 100
B. 99
C. 2
D. 1

Answer 11

B. 99

Question 12

(12) Which of the following sequences does this R command ”seq (0, 1, length.out = 11)” generate?
A. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
B. 0.0000000 0.1111111 0.2222222 0.3333333 0.4444444 0.5555556 0.6666667 0.7777778 0.8888889 1.0000000
C. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
D. 0.00000000 0.09090909 0.18181818 0.27272727 0.36363636 0.45454545 0.545454550.63636364 0.72727273 0.81818182 0.90909091 1.00000000

Answer 12

A. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Question 13

hich of the following Fourier Basis expressions is correct?
A. Φ(t) = (1, sin(ωt), cos(2ωt), sin(3ωt), cos(4ωt), sin(5ωt), cos(6ωt))
B. Φ(t) = (1, sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), sin(3ωt), cos(3ωt))
C. Φ(t) = (1, sin(ωt), sin(2ωt), cos(3ωt), cos(4ωt))
D. Φ(t) = (1, sin(ωt))

Answer 13

B. Φ(t) = (1, sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), sin(3ωt), cos(3ωt))

Question 14

If we use cubic B-spline functions with 13 knots, how many basis functions do we need?
A. 17
B. 16
C. 15
D. 14

Answer 14

C. 15
Number of Basis Functions = Number of Knots + Degree − 2

Question 15

Which is the correct expression?
A. When λ → ∞, penalty increases, roughness decreases, goodness of fitting increases
B. When λ → ∞, penalty increases, roughness decreases, smoothness increases
C. When λ → 0, penalty increases, roughness decreases, smoothness increases
D. When λ → 0, penalty decreases, roughness increases, goodness of fitting decreases

Answer 15

B. When λ → ∞, penalty increases, roughness decreases, smoothness increases

Question 16

What is the primary objective of Functional Data Analysis (FDA)?
A. Analysing categorical data
B. Analysing data collected over a continuous domain
C. Studying only time-series data
D. Focusing on spatial data analysis

Answer 16

B. Analysing data collected over a continuous domain

Question 17

What is the role of basis functions in functional data analysis?
A. To transform the functional data into a matrix format
B. To model the relationship between the predictor and response variables
C. To represent the functional data in terms of a finite set of functions
D. To calculate the derivatives of the functional data

Answer 17

C. To represent the functional data in terms of a finite set of functions

Question 18

What is the primary purpose of a roughness penalty in functional data analysis?
A. To increase the complexity of the model
B. To penalize the smoothness of the fitted curve
C. To minimize the computational time required for analysis
D. To enforce linearity in the relationship between variables

Answer 18

B. To penalize the smoothness of the fitted curve