Multiple Choice Questions

Question 1

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 1

B. Analysing data collected over a continuous domain

Question 2

Which statistical technique is commonly used in Functional Data Analysis for smoothing and summarizing data?
A. Functional Principal Component Analysis (FPCA)
B. Cluster Analysis
C. Linear Regression
D. Fourier Transform

Answer 2

A. Functional Principal Component Analysis (FPCA)

Question 3

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 3

C. Data with repeated measures over a continuum

Question 4

Which of the following is a key assumption in Functional Data Analysis?
A. Independence of observations
B. Homoscedasticity
C. Time-series stationarity
D. Functional smoothness

Answer 4

D. Functional smoothness

Question 5

What role does the concept of a ”functional basis” play in Functional Data Analysis?
A. It defines the distribution of data points
B. It provides a set of functions to represent data
C. It measures the spread of data
D. It assesses outliers in the data

Answer 5

B. It provides a set of functions to represent data

Question 6

Which of the following is not a characteristic of functional data?
A. Data are measurements of smooth processes over time
B. Often have multiple measurements of the same process
C. Must be equally spaced or perfect measurements
D. Curves with similar trends

Answer 6

C. Must be equally spaced or perfect measurements

Question 7

Which of the following factors does not influence the selection of the smoothing parameter in functional data analysis?
A. Sample size
B. Noise level in the data
C. Type of basis functions used
D. Computational resources available

Answer 7

D. Computational resources available

Question 8

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 8

B. The degree of flexibility in the fitted curve

Question 9

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 9

D. A larger smoothing parameter leads to a smoother curve

Question 10

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 dataset

Answer 10

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

Question 11

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

Answer 11

D. Generalized Cross-Validation (GCV)

Question 12

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 12

B. Under-smoothing of the curve occurs

Question 13

What happens if the smoothing parameter λ is too large 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 13

A. Over-smoothing of the curve occurs

Question 14

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 14

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

Question 15

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 15

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

Question 16

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 16

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

Question 17

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 17

B. Analysing data collected over a continuous domain

Question 18

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 18

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

Question 19

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 19

B. To penalize the smoothness of the fitted curve

Question 20

What is the main objective of Functional Linear Regression Analysis (FLRA)?
A. To estimate the functional relationship between two or more functional variables
B. To model the relationship between a functional or scalar response variable and one or more functional or scalar predictors
C. To compute the mean value of a functional variable
D. To identify outliers in functional data

Answer 20

B. To model the relationship between a functional or scalar response variable and one or more functional or scalar predictors

Question 21

Which of the following best describes the difference between Traditional Linear Regression Analysis and Functional Linear Regression Analysis (FLRA)?
A. Traditional linear regression involves scalar variables, while FLRA involves functional variables
B. Traditional linear regression only considers fixed values for regression coefficients, while FLRA allows coefficients to vary as functions of time
C. Traditional linear regression only allows for one predictor variable, while FLRA can handle multiple functional predictors
D. Traditional linear regression assumes linear relationships, while FLRA assumes non-linear relationships

Answer 21

A. Traditional linear regression involves scalar variables, while FLRA involves functional variables

Question 22

When might the Permutation F -Test methodology be used in Functional Linear Regression Analysis?
A. To estimate the functional relationship between two scalar variables
B. To assess the goodness of fit of the Functional Linear Regression model
C. To evaluate the statistical significance of relationships among functional independent variables and dependent variables
D. To determine the linearity assumption of the Functional Linear Regression mode

Answer 22

C. To evaluate the statistical significance of relationships among functional independent variables and dependent variables

Question 23

Which of the following scenarios is not included in Functional Linear Regression Analysis?
A. The response variable is scalar or multivariate, and the independent variable is functional
B. The response variable is functional, and the independent variable is scalar
C. Both the response and independent variables are functional
D. Both the response and independent variables are scalars

Answer 23

D. Both the response and independent variables are scalars

Question 24

What is the main objective of Functional Principal Component Analysis (FPCA)?
A. To identify outliers in functional data
B. To estimate the relationship between functional variables and scalar outcomes
C. To reduce the dimensionality of functional data while preserving the most important variability
D. To assess the goodness of fit of functional regression models

Answer 24

C. To reduce the dimensionality of functional data while preserving the most important variability

Question 25

In Functional Principal Component Analysis (FPCA), what are the principal components (PCs) used to represent?
A. The mean function of the functional data
B. The variability within the functional data
C. The coefficients of the functional regression model
D. The distribution of the functional data

Answer 25

B. The variability within the functional data

Question 26

Which of the following best describes the interpretation of the principal component scores in Functional Principal Component Analysis?

A. They represent the coefficients of the functional regression model
B. They indicate the importance of each observation in the functional dataset
C. They provide information about the similarity between functional observations
D. They quantify the contribution of each principal component to the original functional data

Answer 26

D. They quantify the contribution of each principal component to the original functional data

Question 27

When performing Functional Principal Component Analysis, what does the scree-plot help visualize?

A. The distribution of the functional data
B. The correlation structure among principal components
C. The amount of variability explained by each principal component
D. The coefficients of the functional regression model

Answer 27

C. The amount of variability explained by each principal component

Question 28

Which of the following statements about eigenfunctions in Functional Principal Component Analysis is true?
A. Eigenfunctions represent the principal components of the functional data
B. Eigenfunctions are obtained by solving the ordinary least squares regression problem
C. Eigenfunctions represent the average function of the functional data
D. Eigenfunctions are used to estimate the coefficients of the functional regression model

Answer 28

A. Eigenfunctions represent the principal components of the functional data

Question 29

What is the primary benefit of using Functional Principal Component Analysis before conducting functional regression analysis?
A. It helps identify multicollinearity among functional predictors
B. It reduces the dimensionality of the functional predictors, making the regression model more interpretable
C. It assesses the distributional assumptions of the functional data
D. It provides information about the non-linear relationships between functional variables and scalar
outcomes

Answer 29

B. It reduces the dimensionality of the functional predictors, making the regression model more
interpretable

Question 30

Which of the following is a common application of Functional Principal Component Analysis?
A. Image classification
B. Time series forecasting
C. Dimensionality reduction in functional data
D. Text mining

Answer 30

C. Dimensionality reduction in functional data

Question 31

What is the primary objective of Functional Canonical Correlation Analysis?
A. To identify outliers in functional data
B. To summarize and reduce the dimensionality of functional data while retaining most of the variability
C. To estimate the relationship between two sets of functional variables
D. To fit a functional regression model to the data

Answer 31

C. To estimate the relationship between two sets of functional variables

Question 32

In Functional Canonical Correlation Analysis, what does the canonical correlation coefficient measure?
A. The strength of the linear relationship between two sets of functional variables
B. The variability within each set of functional variables
C. The mean function of each set of functional variables
D. The residual errors of the functional data

Answer 32

A. The strength of the linear relationship between two sets of functional variables

Question 33

Which of the following statements about canonical correlation functions in Functional Canonical Correlation Analysis is correct?
A. Canonical correlation functions represent the principal components of the functional data
B. Canonical correlation functions are obtained by fitting a regression model to the data
C. Canonical correlation functions represent the relationship between two sets of functional variables
D. Canonical correlation functions are used to estimate the coefficients of the functional regression
model

Answer 33

C. Canonical correlation functions represent the relationship between two sets of functional variables

Question 34

Which of the following is a common application of Functional Canonical Correlation Analysis?
A. Image classification
B. Text mining
C. Time series forecasting
D. Relationship modelling between functional datasets

Answer 34

D. Relationship modelling between functional datasets

Question 35

Which of the following descriptions of the Phase-plane plot is incorrect?
A. The Phase-Plane plots aim to show pairs of derivatives to explore the dynamics between phase variation and amplitude variation
B. The Phase-Plane plots require plotting the first derivative (velocity) on the horizontal axis against the second derivative (acceleration) on the vertical axis
C. Velocity and acceleration correspond to potential energy and kinetic energy respectively
D. The Phase-Plane plots show kinetic and potential energy exchange

Answer 35

C. Velocity and acceleration correspond to potential energy and kinetic energy respectively

Question 36

Which of the following is not a characteristic of the Phase-plane plot?
A. The size of the radius: the larger it is, the more energy transfer there is in the event
B. The horizontal location of the center: if it is to the right, there is net positive velocity, and if to
the left, there is net negative velocity
C. The vertical location of the center: if it is above zero, there is a net velocity increase; if below
zero, there is a velocity decrease
D. The Phase-plane plot must be a closed circle

Answer 36

D. The Phase-plane plot must be a closed circle

Question 37

Which of the following statements is incorrect regarding the difference between traditional principal
component analysis (PCA) and functional principal component analysis (FPCA)?
A. Traditional PCA is typically applied to multivariate data, where each observation is represented by a vector of variables; FPCA is specifically designed for functional data, where each observation is a curve or a function
B. In traditional PCA, each observation is represented as a vector of fixed length; in FPCA, each observation is represented as a function or a curve over a continuous domain
C. Traditional PCA deals with individual samples or cases; FPCA deals with functional data, where each observation is a function or a curve rather than a single value
D. The main objective of traditional PCA is to identify the primary modes of variation within a single variable; whereas FPCA focuses on capturing relationships between multiple variables simultaneously

Answer 37

D. The main objective of traditional PCA is to identify the primary modes of variation within a single variable; whereas FPCA focuses on capturing relationships between multiple variables simultaneously

Question 38

Which of the following R commands can generate the standard deviation of functional data?
A. mean.fd (fdobj)
B. std.fd (fdobj)
C. var.fd (fdobj1, fdobj2)
D. deriv.fd (fdobj

Answer 38

B. std.fd (fdobj)

Question 39

Which of the following models is a concurrent model?
(see sheet)

Answer 39

C

Question 40

Which of the following Phase-plane plots in Figure 3 is the correct one?
(see sheet)
A. Phase-plane plot A
B. Phase-plane plot B
C. Phase-plane plot C
D. Phase-plane plot D

Answer 40

A

Question 41

Which of the following models is not a functional linear regression model?
A. The response variable is scalar or multivariate, and the independent variable is functional: {yi
, xi(t)}, i =1, · · · , n.
B. The response variable is functional, and the independent variable is scalar or multivariate: {yi(t), xi}, i =1, · · · , n.
C. Both the response and independent variables being functional: {yi(t), xi(t)}, i = 1, · · · , n.
D. Both the response and independent variables being scalar or multivariate: {yi
, xi}, i = 1, · · · , n.

Answer 41

D. Both the response and independent variables being scalar or multivariate: {yi
, xi}, i = 1, · · · , n.

Question 42

Which sequence from the following options can generate 5 knots?
A. knots = c ( seq (0, 9, 1.5))
B. knots = c ( seq (0, 8, 2))
C. knots = c ( seq (0, 10, 2))
D. knots = c ( seq (0, 5, 1))

Answer 42

B. knots = c ( seq (0, 8, 2))

Question 43

Which sequence from the following options can generate 51 observations?
A. tobs = seq (0, 5, 0.5)
B. tobs = seq (0, 1, 0.01)
C. tobs = seq (0, 1, 0.02)
D. tobs = seq (0, 10, 0.25)

Answer 43

C. tobs = seq (0, 1, 0.02)

Question 44

Which of the following datasets is a periodic dataset?
A. Social media interactions and website traffic
B. Data on patient visits to a hospital or clinic
C. Ocean tides over a lunar month
D. Data generated from random processes, such as white noise

Answer 44

C. Ocean tides over a lunar month

Question 45

What happens if we use too many basis functions to fit a curve?
A. The model may not be able to capture the complexity of the data, resulting in a poor fit
B. It can lead to overfitting, where the model captures noise and fluctuations in the data rather than
the underlying true relationship
C. The model is too simple and fails to capture the underlying patterns in the data
D. If too many basis functions are used to fit a curve, it can lead to underfitting

Answer 45

B. It can lead to overfitting, where the model captures noise and fluctuations in the data rather than
the underlying true relationship

Question 46

Which of the following statements is incorrect regarding the trade-off between the number of basis
functions and smoothness?
A. Too many basis functions over-fit the data and reflect errors of measurement
B. Too few basis functions fail to capture interesting features of the curves
C. To control the balance between these two, we introduce the roughness penalty smoothing method
D. To ensure smoothness, we can choose any number of basis functions

Answer 46

D. To ensure smoothness, we can choose any number of basis functions

Question 47

Assuming that ”fd obj1” and ”fd obj2” are two functional objects, which of the following arithmetic
operations does not satisfy Functional Arithmetic?
A. fd obj1 ± fd obj2
B. fd obj1k
C. fd obj1 * fd obj2
D. derivative(fd obj1)

Answer 47

D. derivative(fd obj1)

Question 48

Assuming that ”fd obj1” and ”fd obj2” are two functional objects corresponding to ”Line 1” and
”Line 2” in the following Figure 4 plots, which of the following resulting plots is incorrect according
to the principles of Functional Arithmetic?
(see sheet plot)
A. Plot A
B. Plot B
C. Plot C
D. Plot D

Answer 48

D. Plot D

Question 49

Which of the following plots in Figure 5 uses the largest value of λ to fit these discrete points?
(see sheet)
A. Plot A
B. Plot B
C. Plot C
D. Plot D

Answer 49

D. Plot D

Question 50

What is the purpose of applying the VARIMAX rotation algorithm in Functional Principal Component Analysis (FPCA)?
A. To ensure orthogonality between principal components, leading to more interpretable and easily
understandable results
B. To maximize the variance explained by each principal component, leading to more effective
dimensionality reduction
C. To minimize the computational complexity of the FPCA algorithm, allowing for faster analysis
of large datasets
D. To standardize the loadings of each principal component, resulting in a more balanced representation of the original data

Answer 50

A. To ensure orthogonality between principal components, leading to more interpretable and easily
understandable results

Question 51

In functional linear regression analysis, what is the fundamental assumption about the relationship
between the response variable and the functional independent variables?
A. The relationship is linear
B. The relationship is nonlinear
C. The relationship is stochastic
D. The relationship is deterministic

Answer 51

A. The relationship is linear

Question 52

What advantage does functional linear regression analysis offer over traditional linear regression
when dealing with functional data?
A. It allows for modeling of nonlinear relationships
B. It requires fewer assumptions about the data distribution
C. It can handle predictors that are functions over a continuum
D. It is computationally simpler and faster

Answer 52

C. It can handle predictors that are functions over a continuum