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HistogramTools for Efficient Data Analysis and Distribution Representation in Large Data Sets

Shubham Malhotra

TL;DR

Various methods of histogram bin manipulation, distance measures, quantile approximation, and error estimation in cumulative distribution functions (CDFs) derived from histograms are presented.

Abstract

Histograms provide a powerful means of summarizing large data sets by representing their distribution in a compact, binned form. The HistogramTools R package enhances R built-in histogram functionality, offering advanced methods for manipulating and analyzing histograms, especially in large-scale data environments. Key features include the ability to serialize histograms using Protocol Buffers for distributed computing tasks, tools for merging and modifying histograms, and techniques for measuring and visualizing information loss in histogram representations. The package is particularly suited for environments utilizing MapReduce, where efficient storage and data sharing are critical. This paper presents various methods of histogram bin manipulation, distance measures, quantile approximation, and error estimation in cumulative distribution functions (CDFs) derived from histograms. Visualization techniques and efficient storage representations are also discussed alongside applications for large data processing and distributed computing tasks.

HistogramTools for Efficient Data Analysis and Distribution Representation in Large Data Sets

TL;DR

Various methods of histogram bin manipulation, distance measures, quantile approximation, and error estimation in cumulative distribution functions (CDFs) derived from histograms are presented.

Abstract

Histograms provide a powerful means of summarizing large data sets by representing their distribution in a compact, binned form. The HistogramTools R package enhances R built-in histogram functionality, offering advanced methods for manipulating and analyzing histograms, especially in large-scale data environments. Key features include the ability to serialize histograms using Protocol Buffers for distributed computing tasks, tools for merging and modifying histograms, and techniques for measuring and visualizing information loss in histogram representations. The package is particularly suited for environments utilizing MapReduce, where efficient storage and data sharing are critical. This paper presents various methods of histogram bin manipulation, distance measures, quantile approximation, and error estimation in cumulative distribution functions (CDFs) derived from histograms. Visualization techniques and efficient storage representations are also discussed alongside applications for large data processing and distributed computing tasks.

Paper Structure

This paper contains 15 sections, 3 theorems, 18 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Information Loss Due to Binning
  4. First Moment
  5. Case Analysis
  6. Second Moment
  7. Polynomial Histograms
  8. Empirical Validation
  9. Quantiles and Cumulative Distribution Functions
  10. Applications
  11. Efficiently Binning Large Data Sets Using MapReduce
  12. Approaches to Histogram Generation with MapReduce
  13. Application in R and Other Languages
  14. Visual Illustration
  15. Conclusion

Key Result

Theorem 1

$F_1$ and $F_2$ minimize $F_{-}(x)$ and maximize $F(x)$, for $x \le \mu$ and $x \ge \mu$, respectively.

Figures (6)

  • Figure 1: An example histogram (left) with its CDF representation and a yellow area of uncertainty showing where the true empirical cdf of the unbinned data must lie (right).
  • Figure 2: Distributions that minimize $F_{h-}$ and maximize $F_{h+}$. The left column has $\mu_2$ unknown ($F_2$ and $F_1$). The middle column has variance increasing from 0 to $\sigma_*^2$ ($F_3$). The right column has variance increasing from $\sigma_*^2$ to the maximum ($F_4$).
  • Figure 3: Lower and upper bounds $F_{h-}$ and $F_{h+}$, for $\mu$ unknown and small, middle and large variances.
  • Figure 4: Yellow areas of uncertainty for where the ecdf of the unbinned data must lie given a histogram bin bisected in two (left) or a histogram annotated with the mean of values in that bin (right).
  • Figure 5: The Information gained from storing the mean in 24 integer buckets of log file sizes across 315 storage users.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof