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Applied Functional Data Analysis: Methods and Case Studies
This book contains the ideas of functional data analysis by a number of case studies. The case studies are accessible to research workers in a wide range of disciplines. Every reader should gain not only a specific understanding of the methods of functional data analysis, but more importantly a general insight into the underlying patterns of thought. There is an associated web site with MATLABr and S?PLUSr implementations of the methods discussed.
203 pages, Paperback
First published January 1, 2002
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October 21, 2021
Speaking of time-series data, people commonly relate to a sequence of observations taken at equal intervals across the time dimension. However, despite being a solid factor indicating the shape pattern, to my knowledge, the time continuum primarily plays a non-essential role as a fixed frame capturing observation interdependency. There are indeed sophisticated algorithms, say, RNN, that can embed time recursively. But surprisingly, since more than two decades ago, Ramsay & Silverman have already written about a new research branch that defined time as a feature dimension, glued all the observations to a single entity and called it Functional Data Analysis (FDA).
As in the name, functional data view data in the form of a function (or curve) y=f(t) that smoothly spreads across a continuum t, be it time, probability density, or even cyclical shape (draw it with a single stroke) of an object. The method brings a new perspective to data treatment: 30 years of daily temperature now becomes 30 data samples, and so do your handwriting or 2D images of an object.
To transform discrete observations into a continuous curve, we fit a function. FDA provides the basis, a standard set of function families(BSplines, Fourier, Polynomial), that we can use as building blocks to construct almost any underlying function of interest. A valuable by-product of fitting the curve is the smoothing effect that can help remove unwanted noise at the point data collected.
Once obtaining the new representation, there are two main usages. First, suppose you are interested in the curve shape or its rate of change to gain further insight. In that case, the first order (velocity), second-order (acceleration) or higher-order derivative (if applicable) can be applied and studied. Second, some popular ML methods are extended to this new data type if you are an ML practitioner. The book gave an equal share to the two topics, but the differential equations are more complicated, and therefore I solely focus on the latter.
Functional PCA decomposes all your curves to a linear combination of the same set of curves (PC) that only differs in coefficients (PC score). Not only do we know PC captures variation in data, but since it's a curve, we can pinpoint where the variation stems from when plotting together with the mean curve. Moreover, PC score turns data to conventional tabular type, which any prediction algorithm can apply on top of, for example, functional linear regression.
Time-series data is typically annoying for trace comparison, as their general shapes are rarely well-aligned in the time dimension. The literature denotes this "lagging effect" as phase variation, while the difference in value is as magnitude variation. As the curve takes the form of y=f(t), warping function h(t) is introduced to map two traces to the same "time frame" before feeding into the curve y = f(h[t]), therefore removing phase variation. Given the monotonicity and one-to-one mapping global constraint, I view it as DTW extends to a continuous domain.
All in all, the book is a good reference for new beginners, but could be more greatly effective when accompanied with the following materials:
FDA python package: https://fda.readthedocs.io/en/latest/
Original code of the book: http://psych.mcgill.ca/misc/fda/
As in the name, functional data view data in the form of a function (or curve) y=f(t) that smoothly spreads across a continuum t, be it time, probability density, or even cyclical shape (draw it with a single stroke) of an object. The method brings a new perspective to data treatment: 30 years of daily temperature now becomes 30 data samples, and so do your handwriting or 2D images of an object.
To transform discrete observations into a continuous curve, we fit a function. FDA provides the basis, a standard set of function families(BSplines, Fourier, Polynomial), that we can use as building blocks to construct almost any underlying function of interest. A valuable by-product of fitting the curve is the smoothing effect that can help remove unwanted noise at the point data collected.
Once obtaining the new representation, there are two main usages. First, suppose you are interested in the curve shape or its rate of change to gain further insight. In that case, the first order (velocity), second-order (acceleration) or higher-order derivative (if applicable) can be applied and studied. Second, some popular ML methods are extended to this new data type if you are an ML practitioner. The book gave an equal share to the two topics, but the differential equations are more complicated, and therefore I solely focus on the latter.
Functional PCA decomposes all your curves to a linear combination of the same set of curves (PC) that only differs in coefficients (PC score). Not only do we know PC captures variation in data, but since it's a curve, we can pinpoint where the variation stems from when plotting together with the mean curve. Moreover, PC score turns data to conventional tabular type, which any prediction algorithm can apply on top of, for example, functional linear regression.
Time-series data is typically annoying for trace comparison, as their general shapes are rarely well-aligned in the time dimension. The literature denotes this "lagging effect" as phase variation, while the difference in value is as magnitude variation. As the curve takes the form of y=f(t), warping function h(t) is introduced to map two traces to the same "time frame" before feeding into the curve y = f(h[t]), therefore removing phase variation. Given the monotonicity and one-to-one mapping global constraint, I view it as DTW extends to a continuous domain.
All in all, the book is a good reference for new beginners, but could be more greatly effective when accompanied with the following materials:
FDA python package: https://fda.readthedocs.io/en/latest/
Original code of the book: http://psych.mcgill.ca/misc/fda/
Displaying 1 of 1 review