Overview

This project is an implementation of the streaming, one-pass histograms described in Ben-Haim's Streaming Parallel Decision Trees. Inspired by Tyree's Parallel Boosted Regression Trees, the histograms are extended so that they may track multiple values.

The histograms act as an approximation of the underlying dataset. They can be used for learning, visualization, discretization, or analysis. The histograms may be built independently and merged, making them convenient for parallel and distributed algorithms.

While the core of this library is implemented in Java, it includes a full featured Clojure wrapper. This readme focuses on the Clojure interface, but Java developers can find documented methods in com.bigml.histogram.Histogram.

Installation

histogram is available as a Maven artifact from Clojars.

For Leiningen:

For Maven:

<repository>
  <id>clojars.org</id>
  <url>http://clojars.org/repo</url>
</repository>
<dependency>
  <groupId>bigml</groupId>
  <artifactId>histogram</artifactId>
  <version>4.1.2</version>
</dependency>

Basics

In the following examples we use Incanter to generate data and for charting.

The simplest way to use a histogram is to create one and then insert! points. In the example below, ex/normal-data refers to a sequence of 200K samples from a normal distribution (mean 0, variance 1).

user> (ns examples
        (:use [bigml.histogram.core])
        (:require (bigml.histogram.test [examples :as ex])))
examples> (def hist (reduce insert! (create) ex/normal-data))

You can use the sum fn to find the approximate number of points less than a given threshold:

examples> (sum hist 0)
99814.63248

The density fn gives us an estimate of the point density at the given location:

examples> (density hist 0)
80936.98291

The uniform fn returns a list of points that separate the distribution into equal population areas. Here's an example that produces quartiles:

examples> (uniform hist 4)
(-0.66904 0.00229 0.67605)

Arbritrary percentiles can be found using percentiles:

examples> (percentiles hist 0.5 0.95 0.99)
{0.5 0.00229, 0.95 1.63853, 0.99 2.31390}

We can plot the sums and density estimates as functions. The red line represents the sum, the blue line represents the density. If we normalized the values (dividing by 200K), these lines approximate the cumulative distribution function and the probability distribution function for the normal distribution.

examples> (ex/sum-density-chart hist) ;; also see (ex/cdf-pdf-chart hist)

The histogram approximates distributions using a constant number of bins. This bin limit is a parameter when creating a histogram (:bins, defaults to 64). A bin contains a :count of the points within the bin along with the :mean for the values in the bin. The edges of the bin aren't captured. Instead the histogram assumes that points of a bin are distributed with half the points less than the bin mean and half greater. This explains the fractional sum in the example below:

examples> (def hist (-> (create :bins 3)
                        (insert! 1)
                        (insert! 2)
                        (insert! 3)))
examples> (bins hist)
({:mean 1.0, :count 1} {:mean 2.0, :count 1} {:mean 3.0, :count 1})
examples> (sum hist 2)
1.5

As mentioned earlier, the bin limit constrains the number of unique bins a histogram can use to capture a distribution. The histogram above was created with a limit of just three bins. When we add a fourth unique value it will create a fourth bin and then merge the nearest two.

examples> (bins (insert! hist 0.5))
({:mean 0.75, :count 2} {:mean 2.0, :count 1} {:mean 3.0, :count 1})

A larger bin limit means a higher quality picture of the distribution, but it also means a larger memory footprint. In the chart below, the red line represents a histogram with 8 bins and the blue line represents 64 bins.

examples> (ex/multi-pdf-chart
           [(reduce insert! (create :bins 8) ex/mixed-normal-data)
            (reduce insert! (create :bins 64) ex/mixed-normal-data)])

Another option when creating a histogram is to use gap weighting. When :gap-weighted? is true, the histogram is encouraged to spend more of its bins capturing the densest areas of the distribution. For the normal distribution that means better resolution near the mean and less resolution near the tails. The chart below shows a histogram without gap weighting in blue and with gap weighting in red. Near the center of the distribution, red uses more bins and better captures the gaussian distribution's true curve.

examples> (ex/multi-pdf-chart
           [(reduce insert! (create :bins 8 :gap-weighted? true)
                    ex/normal-data)
            (reduce insert! (create :bins 8 :gap-weighted? false)
                    ex/normal-data)])

Merging

A strength of the histograms is their ability to merge with one another. Histograms can be built on separate data streams and then combined to give a better overall picture.

In this example, the blue line shows a density distribution from a histogram after merging 300 noisy histograms. The red shows one of the original histograms:

examples> (let [samples (partition 1000 ex/mixed-normal-data)
                hists (map #(reduce insert! (create) %) samples)
                merged (reduce merge! (create) (take 300 hists))]
            (ex/multi-pdf-chart [(first hists) merged]))

Targets

While a simple histogram is nice for capturing the distribution of a single variable, it's often important to capture the correlation between variables. To that end, the histograms can track a second variable called the target.

The target may be either numeric or categorical. The insert! fn is overloaded to accept either type of target. Each histogram bin will contain information summarizing the target. For numeric targets the sum and sum-of-squares are tracked. For categoricals, a map of counts is maintained.

examples> (-> (create)
              (insert! 1 9)
              (insert! 2 8)
              (insert! 3 7)
              (insert! 3 6)
              (bins))
({:target {:sum 9.0, :sum-squares 81.0, :missing-count 0.0},
  :mean 1.0,
  :count 1}
 {:target {:sum 8.0, :sum-squares 64.0, :missing-count 0.0},
  :mean 2.0,
  :count 1}
 {:target {:sum 13.0, :sum-squares 85.0, :missing-count 0.0},
  :mean 3.0,
  :count 2})
examples> (-> (create)
              (insert! 1 :a)
              (insert! 2 :b)
              (insert! 3 :c)
              (insert! 3 :d)
              (bins))
({:target {:counts {:a 1.0}, :missing-count 0.0},
  :mean 1.0,
  :count 1}
 {:target {:counts {:b 1.0}, :missing-count 0.0},
  :mean 2.0,
  :count 1}
 {:target {:counts {:d 1.0, :c 1.0}, :missing-count 0.0},
  :mean 3.0,
  :count 2})

Mixing target types isn't allowed:

examples> (-> (create)
              (insert! 1 :a)
              (insert! 2 999))
Can't mix insert types
  [Thrown class com.bigml.histogram.MixedInsertException]

insert-numeric! and insert-categorical! allow target types to be set explicitly:

examples> (-> (create)
              (insert-categorical! 1 1)
              (insert-categorical! 1 2)
              (bins))
({:target {:counts {2 1.0, 1 1.0}, :missing-count 0.0}, :mean 1.0, :count 2})

The extended-sum fn works similarly to sum, but returns a result that includes the target information:

examples> (-> (create)
              (insert! 1 :a)
              (insert! 2 :b)
              (insert! 3 :c)
              (extended-sum 2))
{:sum 1.5, :target {:counts {:c 0.0, :b 0.5, :a 1.0}, :missing-count 0.0}}

The average-target fn returns the average target value given a point. To illustrate, the following histogram captures a dataset where the input field is a sample from the normal distribution while the target value is the sine of the input. The density is in red and the average target value is in blue:

examples> (def make-y (fn [x] (Math/sin x)))
examples> (def hist (let [target-data (map (fn [x] [x (make-y x)])
                                           ex/normal-data)]
                      (reduce (fn [h [x y]] (insert! h x y))
                              (create)
                              target-data)))
examples> (ex/pdf-target-chart hist)

Continuing with the same histogram, we can see that average-target produces values close to original target:

examples> (def view-target (fn [x] {:actual (make-y x)
                                    :approx (:sum (average-target hist x))}))
examples> (view-target 0)
{:actual 0.0, :approx -0.00051}
examples>  (view-target (/ Math/PI 2))
{:actual 1.0, :approx 0.9968169965429206}
examples> (view-target Math/PI)
{:actual 0.0, :approx 0.00463}

Missing Values

Information about missing values is captured whenever the input field or the target is nil. The missing-bin fn retrieves information summarizing the instances with a missing input. For a basic histogram, that is simply the count:

examples> (-> (create)
              (insert! nil)
              (insert! 7)
              (insert! nil)
              (missing-bin))
{:count 2}

For a histogram with a target, the missing-bin includes target information:

examples> (-> (create)
              (insert! nil :a)
              (insert! 7 :b)
              (insert! nil :c)
              (missing-bin))
{:target {:counts {:a 1.0, :c 1.0}, :missing-count 0.0}, :count 2}

Targets can also be missing, in which case the target missing-count is incremented:

examples> (-> (create)
              (insert! nil :a)
              (insert! 7 :b)
              (insert! nil nil)
              (missing-bin))
{:target {:counts {:a 1.0}, :missing-count 1.0}, :count 2}

Array-backed Categorical Targets

By default a histogram with categorical targets stores the category counts as Java HashMaps. Building and merging HashMaps can be expensive. Alternatively the category counts can be backed by an array. This can give better performance but requires the set of possible categories to be declared when the histogram is created. To do this, set the :categories parameter:

examples> (def categories (map (partial str "c") (range 50)))
examples> (def data (vec (repeatedly 100000
                                     #(vector (rand) (str "c" (rand-int 50))))))
examples> (doseq [hist [(create) (create :categories categories)]]
            (time (reduce (fn [h [x y]] (insert! h x y))
                          hist
                          data)))
"Elapsed time: 1295.402 msecs"
"Elapsed time: 516.72 msecs"

Group Targets

Group targets allow the histogram to track multiple targets at the same time. Each bin contains a sequence of target information. Optionally, the target types in the group can be declared when creating the histogram. Declaring the types on creation allows the targets to be missing in the first insert:

examples> (-> (create :group-types [:categorical :numeric])
              (insert! 1 [:a nil])
              (insert! 2 [:b 8])
              (insert! 3 [:c 7])
              (insert! 1 [:d 6])
              (bins))
({:target
  ({:counts {:d 1.0, :a 1.0}, :missing-count 0.0}
   {:sum 6.0, :sum-squares 36.0, :missing-count 1.0}),
  :mean 1.0,
  :count 2}
 {:target
  ({:counts {:b 1.0}, :missing-count 0.0}
   {:sum 8.0, :sum-squares 64.0, :missing-count 0.0}),
  :mean 2.0,
  :count 1}
 {:target
  ({:counts {:c 1.0}, :missing-count 0.0}
   {:sum 7.0, :sum-squares 49.0, :missing-count 0.0}),
  :mean 3.0,
  :count 1})

Rendering

There are multiple ways to render the charts, see examples.clj. An example of rendering a single function, namely cumulative probability:

examples> (def hist (reduce hst/insert! (hst/create) [1 1 2 3 4 4 4 5]))
examples> (let [{:keys [min max]} (hst/bounds hist)]
            (core/view (charts/function-plot (hst/cdf hist) min max)))

(core and charts are Incanter namespaces.)

To render multiple functions on the same chart, you would use add-function with the result of function-plot:

examples> (core/view (-> (charts/function-plot (hst/cdf hist) min max :legend true)
                         (charts/add-function (hst/pdf hist) min max)))

Performance-related concerns

Freezing a Histogram

While the ability to adapt to non-stationary data streams is a strength of the histograms, it is also computationally expensive. If your data stream is stationary, you can increase the histogram's performance by setting the :freeze parameter. After the number of inserts into the histogram have exceeded the :freeze parameter, the histogram bins are locked into place. As the bin means no longer shift, inserts become computationally cheap. However the quality of the histogram can suffer if the :freeze parameter is too small.

examples> (time (reduce insert! (create) ex/normal-data))
"Elapsed time: 333.5 msecs"
examples> (time (reduce insert! (create :freeze 1024) ex/normal-data))
"Elapsed time: 166.9 msecs"

Performance

There are two implementations of bin reservoirs (which support the insert! and merge! functions). Either of the two implementations, :tree and :array, can be explicitly selected with the :reservoir parameter. The :tree option is useful for histograms with many bins as the insert time scales at O(log n) with respect to the # of bins. The :array option is good for small number of bins since inserts are O(n) but there's a smaller overhead. If :reservoir is left unspecified then :array is used for histograms with <= 256 bins and :tree is used for anything larger.

examples> (time (reduce insert! (create :bins 16 :reservoir :tree)
                        ex/normal-data))
"Elapsed time: 554.478 msecs"
examples> (time (reduce insert! (create :bins 16 :reservoir :array)
                        ex/normal-data))
"Elapsed time: 183.532 msecs"

Insert times using reservoir defaults:

![timing chart] (https://docs.google.com/spreadsheet/oimg?key=0Ah2oAcudnjP4dG1CLUluRS1rcHVqU05DQ2Z4UVZnbmc&oid=2&zx=mppmmoe214jm)

License

Copyright (C) 2013 BigML Inc.

Distributed under the Apache License, Version 2.0.