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Histograms

Snippet from Wikipedia: Histogram

In statistics, a histogram is a graphical representation of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson. A histogram is a representation of tabulated frequencies, shown as adjacent rectangles or squares (in some situations), erected over discrete intervals (bins), with an area proportional to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. A histogram may also be normalized displaying relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size.

To construct a histogram representing a density distribution of some variable one should follow these 2 steps: construct a histogram object using the H1D class and then fill it.

Here is an simple example of how to build a histogram with 100 bins between 0 and 5:

from jhplot import *
h1=H1D("Test histogram",100,0,5)

Use the method “fill” to fill this histogram.

Plotting a histogram

This is an example using the Jython code and the jhplot package (it is rather trivial to rewrite it in Java):

Code example

  Download for this example is disabled for non-members
 1: from java.util import Random
 2: from jhplot  import *
 3:
 4: c1 = HPlot("Canvas")
 5: c1.setGTitle("A histogram")
 6: c1.visible()
 7: c1.setAutoRange()
 8:
 9: h1 = H1D("1D histogram",100, -2, 2)
10: rand = Random()
11: for i in range(100):
12:     h1.fill(rand.nextGaussian())
13: c1.draw(h1)
14: c1.drawStatBox(h1)

The output with a statistical summary is plotted as well (the method drawStatBox()). By default, the plot shows statistical uncertainties in each bin.

h1d.jpg

In the above example, “100” is the number of bins between -2 and 2, thus all bins are of the same size. You can get some information about histograms as:

  • getBinSize() Get bin width in case of fixed-size bins
  • getDensity() Get a density distribution dividing each bin of the histogram by the bin width and the total number of heights for all bins.
  • getProbability() Return a probability distribution derived from a histogram.
  • integral(i1,i2) Integrate a histogram between two bin indices

See more details in the SCaVis H1D API

You can get a complete different look and feel using various attributes of the H1D class or HPlot.

You can use different canvases also (see Section graphics). For example, replacing HPlot with HPlotter leads to the image shown below:

 HPlotter example

Statistics

You can get detailed statistics on a given histogram using the method getStat(). It returns a map (for JAVA) or Python dictionary (for Jython) where each statistical characteristics can be accessed using a key, such as mean, RMS, variance, error on the mean at.

stat=h1.getStat()  # get PYTHON dictionary with statistics
for key in  stat: 
     print key , 't', stat[key]

This will print the following values:

overflowBin          4.0
error                  0.0824244396904
underflowBin          4.0
rms                  0.793596244181
variance             0.625028519761
allEntries          100.0
maxBinHeight           5.0
minBinHeight           0.0
mean                  -0.0690396916107
entries           92.0
underflowHeight       4.0
stddev                   0.790587452317
overflowHeight           4.0

Learn more in Sect. statistics.

Histogram input/output

As any object in ScaVis, you can serialize histogram into a file and then read it back. Read IO section. Here we show a simple example how to write a histogram into a human-readable text file (jdat) and then read it back.

Code example

  Download for this example is disabled for non-members
 1: # Input-Output.Write/read histogram to  a text file
 2: from jhplot  import *
 3: from jhplot.io import *
 4: from java.util import Random
 5:
 6: h1=H1D('Simple1',20, -2.0, 2.0)
 7: r=Random()
 8: for i in range(1000):
 9:       h1.fill(r.nextGaussian())
10: hb = HBook("output.jdat","w") # HBook object
11: hb.write("data",h1)
12: hb.close()
13:
14: print "Reading the histogram"
15: hb = HBook("output.jdat")  # read HBook object
16: print hb.getKeys()         # print all the keys
17: h2=hb.get("data")
18: c1 = HPlot("Canvas",600,400)
19: c1.setGTitle("Histograms from a file");
20: c1.visible(1)
21: c1.setAutoRange()
22: c1.draw(h2)

This example also shows a canvas with the histogram from a file. Open the output.jdat file and study. Most tags are optional. The histogram entries are stored between “data” tag.

Histogram conversions

Histograms can be generated from F1D and F2D functions as explained converting_functions_to_histograms. The opposite is not true.

Histograms can be converted to the P1D or P2D data points as explained in data_structures.

Histograms in 2D

Build a histogram in 2 dimensions using the Java class H2D. This is an example using the JHPLOT package (here we are using again Jython syntax, instead of Java):

Code example

  Download for this example is disabled for non-members
 1: from java.util import Random
 2: from jhplot  import *
 3:
 4: # build a standard 3D canvas
 5: c1 = HPlot3D("Canvas")
 6: c1.setGTitle("Global title")
 7: c1.setNameX("X")
 8: c1.setNameY("Y")
 9: c1.visible(1)
10:
11: h1 = H2D("2D histogram",10,-3.0, 3.0,10,-3.0, 3.0)
12: rand = Random();
13: for i in range(200):
14:     h1.fill(rand.nextGaussian(),rand.nextGaussian())
15: c1.draw(h1)

The output with statistical summary is shown here. By default, the plot shows statistical uncertainties in each bin.

 Histogram in 2D

Histograms in 3D

Similarly, histograms can be defined in 3D using the class H3D

Histograms with variable bin size

One can use also variable-size bins as:

h1 = H1D("Variable-size bins",[-2,-1,0,2,10])

where the list used in the H1D constructor specifies edges of the bins. Similarly, one can define H2D and H3D histogram by passing 2 lists (one for X, one for Y) or 3 lists (X,Y,Z).

Histogram operations

The histogram classes support many mathematical operations (division, subtraction, multiplication, scaling, shifting, smoothing etc). Histogram arithmetic can be done with the method “oper(h,”New Title”,”operation”)”, where “h” is an object represented a histogram which is used to subtract, divide, multiply and add. All these operations should be defined by a string operation as ”-, /, *, +”, and the histograms must have the same binning. It should also be noted that all such operations take into account propagation of statistical errors for each bin assuming that histograms do not correlate.

from java.util import Random
from jhplot  import *
h1 = H1D("First",10, -2.0, 2.0)
h2 = H1D("Second",10, -2.0, 2.0)
r = Random()
for i in range(5000):  h1.fill(r.nextGaussian())
for i in range(5000):  h2.fill(r.nextGaussian())
h3=h1.oper(h2,"subtract","-")
h4=h1.oper(h2,"add","+")
h5=h1.oper(h2,"multiply","*")
h6=h1.oper(h2,"divide","/")

A histogram can be scaled by a constant using the method “operScale(title,scaleFactor)”

Jython scripting using histograms

Instead of calling Java classes using the Jython (or Python) language, one can use the native Jython classes based on the Java classes. In this case, many Java methods can conveniently be overloaded. For example, histograms can be added, subtracted, divided and multiplied using the conventional arithmetic operators ”+,-,/,*”. To be able to use Python-derived classes for the histogram objects, import the class “shplot” (“scripting” HPlot package). The histogram classes have the same names, but they start from a lower case. Let us give an example:

Code example

  Download for this example is disabled for non-members
 1: import shplot
 2: from java.awt import Color
 3: from java.util import Random
 4: from shplot import *
 5: print shplot.__doc__
 6:
 7: c1=hplot("scripting",1,2) # build a canvas with 2 plot regions
 8: c1.visible()
 9: c1.auto()
10: h1=h1d("h1",40,-3,3) # define histograms
11: h2=h1d("h2",40,-3,3)
12: h3=h1d("h3",40,-3,3)
13: r = Random()
14:
15: for i in range(2000):              # fill histograms
16:       h1.fill(r.nextGaussian())
17:       h2.fill(0.6*r.nextGaussian()+2)
18:       if i<1000:  h3.fill(0.5*r.nextGaussian()+1.0)
19:
20: h1.setColor(Color.red)
21: c1.draw(h1)
22:
23: h12=h1+h2           # add 2 histograms
24: h12.setFill(1)
25: h12.setFillColor(Color(20,30,20))
26: h12.setColor(Color.blue)
27: c1.draw(h12)
28:
29: h13=h12+h3             # sum 2 histogram and draw
30: h13.setFill(1)
31: h13.setFillColor(Color(50,90,20))
32: c1.draw(h13)
33:
34: scaled=h1*2.5           # scale a histogram by 2.5
35: scaled.setColor(Color.green)
36: c1.draw(scaled)
37:
38: c1.cd(1,2)               # create a new plotting region
39: c1.auto()
40: h13=h1+h3                # draw the sum of 2 histograms
41: h13.setColor(Color.blue)
42: c1.draw(h13)
43:
44: h113=h1-h3                # subtract 2 histograms
45: h113.setFill(1)
46: h113.setColor(Color(10,200,100))
47: h113.setFillColor(Color(20,200,90))
48: c1.draw(h113)

The resulting figure is shown here

Histogram scripting using shplot package

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A complete description of how to use Java, Jython and SCaVis for scientific analysis is described in the book Scientific data analysis using Jython and Java published by Springer Verlag, London, 2010 (by S.V.Chekanov)

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