Problem Class 3 (T/F) Flashcards
True or False:
We build functional data objects in two stages: first, we determine whether the data is periodic or non periodic, and then we set up a vector, matrix, or array of coefficients to define the function
False (2 marks). Functional data objects are constructed in a two-step process. Initially, a set of basis functions is defined. Subsequently, a vector, matrix, or array of coefficients is established to represent the function as a linear combination of these basis functions (2 marks).
True or False:
We observe a functional dataset at 500 equally spaced time points. If we fit these observations using a sixth degree B-spline basis function with 20 knots, we would need 26 basis functions.
False (2 marks). When we use a sixth degree B-spline basis function with 20 knots, now order is 6 +
1 = 7. And we have 20 - 2 = 18 interior knots, so the number of basis functions = number of interior
knots + order = 18 + 7 = 25. Therefore, we need 25 basis functions instead of 26 (2 marks).
True or False:
When we choose enough basis functions, the goodness of fitting is very high, while the smoothness performance is poor.
True (2 marks). This statement is true because when you choose a large number of basis functions
for fitting a dataset, the model becomes very flexible and can closely fit the data points, resulting in a high goodness of fit. However, when using a large number of basis functions, the resulting curves or functions may lead to over fitting and may exhibit poor smoothing performance (2 marks).
True or False:
We wish to apply the Leave One Out approach, a form of Generalized Cross-Validation, to a functional dataset containing 50 samples. This means that one sample will be allocated to the training set, while the remaining 49 samples will comprise the test set for each validation iteration.
False (2 marks). In the context of Leave One Out Cross-Validation, for each iteration, one sample
is withheld from the dataset and used as the test set, while the remaining 49 samples are used for
training. Therefore, 49 samples will be allocated to the training set, and one sample will comprise
the test set for each validation iteration (2 marks).
True or False:
For the ’Medfly’ functional dataset, we observed 50 flies and recorded the number of eggs for each fly over a span of 25 days. After visualization by functional data analysis, there are 25 curves in total.
False (2 marks). In the ’Medfly’ functional dataset, the number of eggs for each of the 50 flies is recorded over a span of 25 days. Therefore, after visualization by functional data analysis, there would be a curve for each fly, resulting in a total of 50 curves, not 25. Each curve represents the egg-laying pattern of an individual fly over the 25-day period (2 marks).
True or False:
In Functional Data Analysis (FDA), the initial crucial step is to smooth the dataset using functional smoothing techniques before conducting descriptive and exploratory statistical analyses.
True. In FDA, it is often essential to smooth the dataset using functional smoothing techniques as an initial step before conducting descriptive and exploratory statistical analyses. Smoothing helps to reduce noise and variability in the data, making subsequent analyses more reliable and interpretable.
True or False:
In Functional Principal Component Analysis (FPCA), we can always obtain any arbitrary number of the principal components (or harmonics).
False. In FPCA, the number of principal components (or harmonics) is usually limited by the number of observations or the dimensionality of the dataset. We can obtain the number of principal components (or harmonics) by plotting a Scree-plot.
True or False:
In Functional Linear Regression Analysis (FLRA), if we build a functional linear model, for example, yi(t) = β0(t) +from j = 1 to n ∑xij(t)βj(t) +εi(t). Its intercept function β0(t) and regression coefficient function βj(t) (for j = 1,2,…,n) are two functional curves instead of two single values.
True.
In FLRA, the intercept function, denoted as β0(t), and the regression coefficient functions, denoted as βj(t) for j = 1,2,…,n, are indeed functional curves rather than single values. This is because FLRA deals with functional data, where each observation is a function defined over time. Therefore, the intercept and regression coefficients are also functions of time t rather than fixed values.
True or False:
When we use B-splines to smooth the dataset and are interested in exploring the information of the 4th derivatives of functional data, we can choose 2 order B-splines.
False. We know a useful rule: the order of the spline basis should be at least two higher than the highest order derivative to be used. Therefore, if we are interested in 4th derivatives, the order of the spline should be at least 6 (i.e., 4 + 2).
