""" Introduction ============ In this section, we will briefly explain what is :term:`functional data analysis (FDA) `, and we will introduce scikit-fda, a library that provides FDA tools to staticians, engineers or machine learning practicioners in the Python scientific ecosystem. .. Disable isort isort:skip_file """ # Author: Carlos Ramos Carreño # License: MIT # %% # What is functional data analysis? # --------------------------------- # # Traditional multivariate statistics focus on the simultaneous analysis of # a finite number of variables. In this setting, we have several observations, # each of them consisting on a vector of measured values. The variables, or # coordinates of this vector, could be correlated, but otherwise they can be # arbitrarily ordered inside the observed vector, provided that the order is # the same for each observation. Usually these observations are considered # to be instances of a random vector, and a big part of the analysis is # centered in finding the distribution associated with it. # # In contrast, in functional data analysis each observation is a function of # one or several variables, such as curves or surfaces. These functions are # usually continuous and often they are smooth, and derivatives can be # computed. The number of variables of these objects is then infinite, as each # evaluation of the function at one point could be considered as one variable. # Moreover, now it is not possible to reorder the variables of the # observations without altering substantially its structure. If the functions # are continuous, nearby variables are highly correlated, a characteristic # that makes some classical multivariate methods unsuitable to work with this # data. # # In this setting, observations can also be considered to be instances # of a "functional random variable", usually called a stochastic process or # a random field. However, some of the concepts that proved very useful to # analyze multivariate data, such as density functions, are not applicable # to :term:`functional data`, while new tools, such as taking derivatives, # become available. # # As such, functional data can benefit of a separate analysis from # multivariate statistics, but also adapting and extending multivariate # techniques when possible. # %% # What is scikit-fda? # ------------------- # # scikit-fda is a Python library containing classes and functions that allow # you to perform functional data analysis tasks. Using it you can: # # - Represent functions as Python objects, both in a discretized fashion # and as a basis expansion. # - Apply preprocessing methods to functional data, including smoothing, # registration and dimensionality reduction. # - Perform a complete exploratory analysis of the data, summarizing its # main properties, detecting possible outliers and visualizing the data # in several ways. # - Apply statistical inference tools developed for functional data, such # as functional ANOVA. # - Perform usual machine learning tasks, such as classification, # regression or clustering, using functional observations. # - Combine the tools offered by scikit-fda with other tools of the Python # scientific ecosystem, such as those provided by the popular machine # learning library `scikit-learn `_. # %% # Anatomy of a function # --------------------- # # We would like to briefly remind the reader the basic concepts that are # employed to talk about functions. Functions in math are a relation between # two sets, the :term:`domain` and the :term:`codomain` in which each element # of the :term:`domain` is restricted to be related to exactly one element of # the :term:`codomain`. The intuition behind this is that a function # represents some type of deterministic process, that takes elements of the # :term:`domain` as inputs and produces elements of the :term:`codomain` as # outputs. # # In :term:`FDA`, the inputs or parameters of a function are assumed to be # continuous parameters, and so are the outputs, or values of the function. # Thus, it is usual to restrict our functional observations to be functions # :math:`\{x_i: \mathcal{T} \subseteq \mathbb{R}^p \to \mathbb{R}^q\}_{i=1}^N`. # In this case both the domain and codomain are (subsets of) vector spaces of # real numbers, and one could talk of the dimension of each of them as a # vector space (in this case the domain dimension is :math:`p` and the # codomain dimension is :math:`q`). # # The most common case of functional observation, and the one that has # received more attention in the functional data literature, is the case of # functions # :math:`\{x_i: \mathcal{T} \subseteq \mathbb{R} \to \mathbb{R}\}_{i=1}^N` # (curves or trajectories). # %% # As an example, the following code shows the Berkeley Growth dataset, one # of the classical datasets used in :term:`FDA`. The curves are heights of # 93 boys and girls measured at several points since their birth to # their 18th birthday. Here the domain :math:`\mathcal{T}` is the interval # :math:`[0, 18]` and both the domain and codomain have a dimension of one. import matplotlib.pyplot as plt import skfda X, y = skfda.datasets.fetch_growth(return_X_y=True) X.plot() plt.show() # %% # Functions where the domain dimension is greater than one ( # such as surfaces or higher dimensional objects) are referred to as functions # of several variables. Functions where the codomain dimension is greater than # one are called vector-valued functions. # %% # As an example we show another popular dataset: Canadian Weather. Here each # observation correspond to data taken from a different weather station in # Canada. For each day of the year we have two values: the average temperature # at that day among several years and the average precipitation among the same # years. Thus, here the domain :math:`\mathcal{T}` is the interval # :math:`[0, 365)`, the domain dimension is one and the codomain dimension # is two. We can see that by default each coordinate of the values of the # function is plotted as a separate coordinate function. X, y = skfda.datasets.fetch_weather(return_X_y=True) X.plot() plt.show()