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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 to 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, convenient for parallel and distributed algorithms.

Building

  1. Make sure you have Java 1.6 or newer
  2. Install leiningen
  3. Checkout the histogram project with git
  4. Run lein jar

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 100K samples from a normal distribution (mean 0, variance 1).

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

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

examples> (hst/sum hist 0)
50044.02331806754

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

examples> (hst/density hist 0)
39687.562791977114

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

examples> (hst/uniform hist 4)
(-0.6723425970050285 -0.0011145378611749357 0.6713314937601746)

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 100K), these lines approximate the cumulative distribution function and the probability distribution function for the normal distribution.

examples> (ex/sum-density-chart hist)

Histogram from normal distribution

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 are distributed evenly with half the points less than the mean and half greater. This explains the fraction sum in the example below:

examples> (def hist (-> (hst/create :bins 3)
                        (hst/insert! 1)
                        (hst/insert! 2)
                        (hst/insert! 3)))
examples> (hst/bins hist)
({:mean 1.0, :count 1} {:mean 2.0, :count 1} {:mean 3.0, :count 1})
examples> (hst/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> (hst/bins (hst/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 16 bins and the blue line represents 64 bins.

examples> (ex/multi-density-chart
           [(reduce hst/insert! (hst/create :bins 16) ex/normal-data)
            (reduce hst/insert! (hst/create :bins 64) ex/normal-data)])

64 and 32 bins histograms

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 five bins in roughly the same space that blue uses three.

examples> (ex/multi-density-chart
           [(reduce hst/insert! (hst/create :bins 16 :gap-weighted? true)
                    ex/normal-data)
            (reduce hst/insert! (hst/create :bins 16 :gap-weighted? false)
                    ex/normal-data)])

Gap weighting vs. No gap weighting

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.

examples> (let [samples (partition 1000 ex/normal-data)
                hist1 (reduce hst/insert! (hst/create :bins 16) (first samples))
                hist2 (reduce hst/insert! (hst/create :bins 16) (second samples))
                merged (-> (hst/create :bins 16)
                           (hst/merge! hist1)
                           (hst/merge! hist2))]
            (ex/multi-density-chart [hist1 hist2 merged]))

Merged histograms

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 numerics the targets sums are tracked. For categoricals a map of counts is maintained.

examples> (-> (hst/create)
              (hst/insert! 1 9)
              (hst/insert! 2 8)
              (hst/insert! 3 7)
              (hst/insert! 3 6)
              (hst/bins))
({:target {:sum 9.0, :missing-count 0.0}, :mean 1.0, :count 1}
 {:target {:sum 8.0, :missing-count 0.0}, :mean 2.0, :count 1}
 {:target {:sum 13.0, :missing-count 0.0}, :mean 3.0, :count 2})
examples> (-> (hst/create)
              (hst/insert! 1 :a)
              (hst/insert! 2 :b)
              (hst/insert! 3 :c)
              (hst/insert! 3 :d)
              (hst/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> (-> (hst/create)
              (hst/insert! 1 :a)
              (hst/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> (-> (hst/create)
              (hst/insert-categorical! 1 1)
              (hst/insert-categorical! 1 2)
              (hst/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> (-> (hst/create)
              (hst/insert! 1 :a)
              (hst/insert! 2 :b)
              (hst/insert! 3 :c)
              (hst/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 (but scaled and shifted to make plotting easier). The density is in red and the average target value is in blue:

examples> (def make-y (fn [x] (+ 10000 (* 10000 (Math/sin x)))))
examples> (def hist (let [target-data (map (fn [x] [x (make-y x)])
                                           ex/normal-data)]
                      (reduce (fn [h [x y]] (hst/insert! h x y))
                              (hst/create)
                              target-data)))
examples> (ex/density-target-chart hist)

Numeric target

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 (hst/average-target hist x)}))
examples> (view-target 0)
{:actual 10000.0, :approx {:sum 9617.150788081583, :missing-count 0.0}}
examples> (view-target (/ Math/PI 2))
{:actual 20000.0, :approx {:sum 19967.590011881348, :missing-count 0.0}}
examples> (view-target Math/PI)
{:actual 10000.000000000002, :approx {:sum 9823.774137889975, :missing-count 0.0}}

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

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

examples> (-> (hst/create)
              (hst/insert! nil :a)
              (hst/insert! 7 :b)
              (hst/insert! nil :c)
              (hst/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> (-> (hst/create)
              (hst/insert! nil :a)
              (hst/insert! 7 :b)
              (hst/insert! nil nil)
              (hst/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 [(hst/create) (hst/create :categories categories)]]
            (time (reduce (fn [h [x y]] (hst/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> (-> (hst/create :group-types [:categorical :numeric])
              (hst/insert! 1 [:a nil])
              (hst/insert! 2 [:b 8])
              (hst/insert! 3 [:c 7])
              (hst/insert! 1 [:d 6])
              (hst/bins))
({:target ({:counts {:a 1.0, :d 1.0}, :missing-count 0.0}
           {:sum 6.0, :missing-count 1.0}),
  :mean 1.0, :count 2}
 {:target ({:counts {:b 1.0}, :missing-count 0.0}
           {:sum 8.0, :missing-count 0.0}),
  :mean 2.0, :count 1}
 {:target ({:counts {:c 1.0}, :missing-count 0.0}
           {:sum 7.0, :missing-count 0.0}),
  :mean 3.0, :count 1})

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 hst/insert! (hst/create) ex/normal-data))
"Elapsed time: 391.857 msecs"
examples> (time (reduce hst/insert! (hst/create :freeze 1024) ex/normal-data))
"Elapsed time: 99.92 msecs"

Performance

Insert time scales log(n) with respect to the number of bins in the histogram.

timing chart

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