Data Visualization

Functional Boxplot: Visualizing Distributions of Functional Data

Functional boxplot thus utilizes techniques from other traditional statistical graphics to interpret functional data. Functional data can be seen as curves or surfaces. They extend the statistical application of boxplots to data where each observation has the form of a function, such as a curve or image. Functional boxplots are of help when exploring and analyzing functional data in terms of identifying centrality, variability, and outliers.

Understand Functional Data

Functional data analysis treats data that includes one observation as a function defined on some continuum, such as time or space. Typical examples of functional data are: 

  • Curves: Growth functions, stock prices, or temperature fluctuations on a normal day.
  • Surfaces: Images, spatio-temporal information, or sensor readings over an area.
  • Probability distributions: Uncertainties.

Functional data thus might not be amenable to traditional statistical application as developed for techniques for scalar data generally.

Definition of Functional Boxplot

Functional boxplot is an exploratory tool visualizing functional data. It adapts the concepts of the traditional box plot to represent a visual summary of distribution. Some key features of the functional boxplot are: 

  • Data Ordering: Functional data (curves or surfaces) is ordered from the center outwards using a notion of band depth modified as band depth.
  • Median Curve: The median curve represents the most central observation and is thereby a robust statistic for centrality.
  • The 50% Central Region: This region is defined by the deepest or most central observations, which represents 50% of the data, and gives a robust measure for spread.
  • Alienation Detection: Identification of outliers relies on an empirical rule that states 1.5 times the size of the portion of the 50% central region.

Construction Steps for a Functional Boxplot

The construction of a functional boxplot has several steps. 

  1. Ordering Data: Unlike traditional boxplots, where data points are ordered from small to large observations, one orders functional data from the center outward using band depth or modified band depth. Band depth measures the centrality of a curve by counting how many other curves stay within its ‘band’.
  2. Identifying the Median Curve: The median curve represents the most central observation in the data set. It is a robust measure of centrality; it is less sensitive to outliers than the mean. 
  3. Determining the 50% Central Region: This region is defined by the band delimited by the 50% of the deepest, or more central observations. It is therefore an indication similar to the interquartile range (IQR) in a classical boxplot, giving an indication of the spread of the central 50% of the curves. 
  4. Outlier Detection: Outliers can be detected using the 1.5 ct. of the 50% central region empirical rule, similar to the 1.5 IQR rule for classical boxplots. All observations outside the fence are flagged for investigation as potential outliers.

Expanding the Functional Boxplots

More so, enhanced functional boxplots can be given by using more included central regions, like the 25% and 75% central regions. Such a representation would be a greater view of the distribution of the functional data.

Surface Boxplots

A volume-based surface band depth is determined to arrange the sampling surfaces so that a three-dimensional surface boxplot can be made. Surface boxplot is a natural extension of the functional boxplot to R3.

Implementation

Functional boxplots can be easily implemented using several different software packages. For example, the

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fda

package in R provides functions for creating functional boxplots. The

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statsmodels

library in Python also offers functionality for plotting functional boxplots.

In R, the

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fbplot

function can be used to produce functional boxplots or enhanced functional boxplots.

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library(fda)

# Produce functional boxplots of the given functional data

fbplot(data)

 

Benefits of Functional Boxplots

Some of the benefits functional boxplots have for the visualization of functional data are:

  1. Visual Summary: They provide a visual summary of the distribution of functional data as far as central tendencies, outliers, and variability.
  2. Outlying: Outlier identification is systemically done in the functional data.
  3. Robustness: Use robust statistics such as median and band-depth insensitive to outlier effects.
  4. Extension of Classical Boxplots: Have extended the normal understanding of boxplots to the functional data domain.

Functional Boxplots Limitations

In functional boxplots, some limitations are involved in the following:

  • Complexity: It becomes complex sometimes for interpretation as compared to classical boxplots, especially by some people to whom functional data analysis is alien.
  • Computation Cost: Band depth measurement is often computationally very intensive, especially with large data sets.
  • Subjectivity: The 1.5 times the 50 % central region rule to get the outlier detection mainly depends on the choice of band depth.

Applications of Functional Boxplots

Various fields where functional data is analyzed have applications for functional boxplots:

  • Environmental Science: Spatially-temporally minded analysis of data, such as temperature or pollution levels.
  • Finance: Visualization of stock prices or interest rate curves over time.
  • Biomedical Engineering: Analysis of medical images and physiological signals.
  • Sports science: Analysis of athlete performance data such as running speed or heart rate.

Important Consideration:

  • Depth Calculation: The choice of inherent depth measure is relevant. Research applicable methods for your data. 
  • Outlier Rule: Adapt the 1.5 * “IQR” rule for functional data. 
  • Libraries: Learn the functional data analysis special libraries for more robust and efficient implementations. 
  • Customization: Add customization to the plot. You can add titles and labels to the axes. 

This serves as a basis for implementing functional box plots in R or Python. Make sure to check the documentation of specific libraries for detailed usage and available options.

Conclusion

This information pertains to the functional boxplot, which is a very effective visualization device for functional data analysis. It is an extension of the traditional boxplot to more complex data forms, including curves and surfaces. The functional boxplot portrays the distribution of functional data in a robust and very informative way elucidating centrality, variation, and outliers. Several advantages include summarizing complex data and outlier detection; limitations range from computation to subjective interpretations. In functional data analysis applications, the functional boxplot is a great asset that enables researchers and analysts to discover new insights deeply into complex data structures.