Problem Class 1 Flashcards
Please summarise the characteristics of the Functional Data
The characteristics of functional data are as follows:
* Data are measurements of smooth processes over time
* Do not want to make parametric assumptions about those processes
* Often have multiple measurements of the same process
* We are interested in describing the variation of processes
* Do not need equally-spaced or perfect measurements
* Domain is usually time, but can be anything: space, energy …
* Curves with similar trends
* Viewing each replication as a single observation can make the data easier to think about
* Not easily described by a mathematical formula
In what practical scenarios would you expect a function to be best represented using the Fourier basis, and when would the spline basis be more appropriate? Provide examples to support your reasoning
The functions that we wish to model tend to fall into two main categories: periodic and non-periodic. The Fourier basis system is the usual choice for periodic functions, such as signal processing.And the spline basis system (and B-splines in particular) tends to serve well for nonperiodic functions,for example, growth curves
Describes the stages for building functional data objects.
We build functional data objects in two stages:
* First, we define a set of functional building blocks φk called basis functions.
* Then we set up a vector, matrix, or array of coefficients to define the function as a linear combina-tion of these basis functions
Write the basis function expansion in mathematical notation to represent the function x(t).And explain the meaning of each parameter.
x(t) =K∑k=1ckφk(t) = c⊤Φ(t)where φ1(t), · · · , φK (t) are basis functions. We say φ(t) is a basis system for x. And c1(t), · · · , cK (t)are coefficients to basis functions.
We observe a functional dataset at 50 equally spaced time points. Now, if we would like to use quadratic B-spline basis functions to fit these observations, please answer the following questions:
1) What is the maximum number of knots we can choose? Can we have 100 knots for this functional data set? Why?
2) If we set 16 knots, how many basis functions do we have?
Since knots are defined by dividing the observation interval into subintervals, we can select up to 50 knots.
No, we can’t select the 100 knots for this functional dataset. Since we know that each knot connects two line segments, we need to ensure that there is at least one observation in each segment. Therefore, if we set 100 knots, it becomes challenging to guarantee an observation point for each segment. Therefore, we can select up to 50 knots.
2) When we use a quadratic B-spline basis function with 16 knots, it is equivalent to degree 2 and order 2 + 1 = 3. And we have 16 - 2 = 14 interior knots, so the number of basis functions = number of interior knots + order = 14 + 3 = 17.
Discuss the differences between traditional descriptive statistical analysis and functional descriptive statistical analysis. [8 marks]
(1) [Bookwork] Some key differences between traditional descriptive statistical analysis and
functional descriptive statistical analysis include:
* Nature of Data: Traditional descriptive statistical analysis typically deals with scalar data,
where each observation is a single numerical value. In contrast, functional descriptive
statistical analysis involves data where each observation is a function or curve rather than
a single value. [2 marks]
* Representation: Traditional descriptive statistical analysis often uses summary statistics
like mean, median, and standard deviation, and graphical representations like histograms
and box plots. In contrast, functional descriptive statistical analysis utilizes techniques that
account for the entire curve or function, such as functional principal component analysis
(FPCA). [2 marks]
* Methods of Analysis: Traditional descriptive statistical analysis primarily focuses on summary statistics and measures of central tendency and variability. In contrast, functional descriptive statistical analysis emphasizes techniques that capture the underlying structure
of the entire functional data, recognizing the variability and patterns within the functions
themselves. [2 marks]
* Tools and Techniques: Traditional descriptive statistical analysis relies on standard statistical tools and methods applicable to scalar data. In contrast, functional descriptive statistical analysis requires specialized tools and techniques tailored to handle functional data,
such as functional data analysis (FDA), functional principal component analysis (FPCA),
and smoothing techniques.
Discuss the differences between traditional principal component analysis (PCA) and functional principal component analysis (FPCA) [4 marks]
- Data Type: 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. [2 marks] - Representation of Data: 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.
Generally, principal component analysis is the main method we adopt after descriptive statistics. Can you outline the steps for conducting Functional Principal Component Analysis (FPCA) and include operational formulas for each step?
Functional Principal Component Analysis (FPCA) indeed plays a crucial role in
functional data analysis because it helps to identify and display the types of variation across
a sample of functions. When we perform FPCA, we need to follow these steps:
* Data - n curves: X
∗
1
(t),X
∗
2
(t),··· ,X
∗
n
(t). [1 mark]
* Mean Curve: x(t) = 1
n ∑
n
i=1X
∗
i
(t). [1 mark]
* Centralization process: Xi(t) = X
∗
i
(t)−x(t), i = 1,2,··· ,n. [1 mark]
16
* First FPC: ξ1(t), which represents the strongest and most important mode of variation in
n curves. [1 mark]
* First FPC score: fi1 =
R
ξ1(t)Xi(t)dt. [1 mark]
* Maximize (n−1)
−1 ∑
n
i=1
f
2
i1
subject to R
ξ
2
1
(t)dt = 1. [1 mark]
* Second FPC: ξ2(t), which represents the second strongest and most important mode of
variation in n curves.
* Second FPC score: fi2 =
R
ξ2(t)Xi(t)dt. [1 mark]
* Maximize (n − 1)
−1 ∑
n
i=1
f
2
i2
subject to R
ξ
2
2
(t)dt = 1 and R
[ξ1(t)ξ2(t)]dt = 0 due to the
orthogonal properties. [1 mark]
* Similarly, we obtain the rest top K FPCs ξ1(t),ξ2(t),··· ,ξK(t), and FPCs are orthogonal
to each other. [1 mark]
* Xi(t) is projected to fi1, fi2,··· , fiK, then Xi(t) = ∑
K
k=1
fiKξk(t). [1 mark]
What is the purpose of applying the VARIMAX rotation algorithm?
The purpose of applying the VARIMAX rotation algorithm in functional principal component analysis (FPCA) is to achieve a more interpretable and simpler representation of the principal components. In FPCA, the principal components represent the main
sources of variation in the functional data. However, these components may not always be easily interpretable or distinct. The VARIMAX rotation algorithm aims to rotate the principal components so that they are more orthogonal to each other, resulting in clearer variance separation and easier interpretation. This can be particularly useful
when dealing with complex datasets where the original principal components may be
difficult to interpret.
