Gaussian Analysis

The graph of a Gaussian Analysis is a symmetric "bell curve" the shape of which depends on its Mean and Variance:

These functions are named after Carl Friedrich Gauss (1777-1855) who applied them to areas such as:

astronomy (e.g., describing conic sections in space)
physics (e.g., Gauss’s Law which relates electric charge to the resulting electric field)

Gaussian Analysis has a number of Machine Learning applications, including:

in statistics to describe normal distributions
in signal processing to define Gaussian filters
in image processing where two-dimensional Gaussians are used for Gaussian blurs
classification

Python Example

To download the code below, click here.

"""
normal_gaussian_distribution.py
displays a normal (gaussian) distribution
"""

# Import needed libraries.
import numpy as np
import matplotlib.pyplot as plotlib

# Define parameters.
mean = 0
standard_deviation = .1
number_of_bars = 30
number_of_values = 200

# Get normally distributed random values for input to the histogram.
values = np.random.normal(mean, standard_deviation, number_of_values)
print("Values:")
print(values)

# Plot the histogram bars.
y_axis_counts, x_axis_bins, ignored = plotlib.hist(values, number_of_bars, density=True)
print("Y axis counts: ")
print(y_axis_counts)
print("X axis bins: ")
print(x_axis_bins)

# Define the predictive gaussian curve equation.
curve_equation = 1/(standard_deviation * np.sqrt(2 * np.pi)) * \
                 np.exp(-(x_axis_bins - mean)**2 / (2 * standard_deviation**2))

# Plot the curve.
plotlib.plot(x_axis_bins, curve_equation, linewidth=2, color='r')

# Display the plot.
plotlib.xlabel('X')
plotlib.ylabel('Y')
plotlib.show()

The output is shown below:

Values:
[ 0.10474705 -0.30987932 -0.0377035  -0.12976679  0.03038214  0.02884987
  0.06735474  0.02050336  0.10587408  0.0791276   0.03052759  0.21124234
  0.12064312  0.10327687  0.14145668 -0.09847649  0.09387778 -0.11535636
  0.07428888  0.09126191  0.02139157  0.12499734  0.08815726  0.04669107
  0.11204913 -0.00150977 -0.17304188 -0.05482239 -0.12933726 -0.17945018
 -0.09856835 -0.062865    0.06568228 -0.01367314 -0.16178722  0.20493866
  0.14503332 -0.14496856  0.03617333 -0.18822113 -0.06582592 -0.17168639
 -0.0391193  -0.01552944  0.10939517  0.08769728  0.07535222  0.11443495
  0.04564379 -0.19244263 -0.10538806  0.01349989 -0.07703975 -0.17040789
 -0.31217889  0.06574343  0.00646494  0.2068757   0.2335022  -0.01168252
  0.05205067 -0.02333681 -0.03391463  0.02957848 -0.03660433 -0.05374718
 -0.15526708  0.05358534  0.00534359 -0.03919433 -0.12330592 -0.16946084
  0.00989148  0.04596242 -0.05949043  0.01114637  0.03046224  0.17880669
  0.05638037  0.11397048 -0.03805841  0.07311613  0.15061437  0.08709982
  0.08485248  0.0721531   0.00141072  0.08259899 -0.11273119 -0.1687727
 -0.05965978 -0.12921086 -0.13001872 -0.0482734  -0.11063765 -0.07264496
  0.14480709 -0.01091887 -0.06735846  0.0097877  -0.09959458 -0.00281473
  0.11152834  0.07634176 -0.06880425 -0.06444    -0.08786654  0.27057934
  0.00130303  0.1040938  -0.04027242 -0.09381493  0.16435642 -0.06778927
 -0.06378324  0.02909034 -0.08235546  0.00622552  0.16772477 -0.01604748
 -0.05742619  0.07295698 -0.16177074  0.15175463  0.0111284   0.01002375
  0.1121463   0.04910416  0.03420192 -0.23352333  0.02109191  0.00578421
  0.18632515 -0.04966773  0.051996   -0.04810638 -0.05649257 -0.01946372
 -0.06099288 -0.16894184 -0.19070416 -0.06760437  0.03209812 -0.10352239
 -0.06656983 -0.05270806 -0.0106319  -0.05207719  0.07823332  0.13054697
 -0.1449979  -0.03860409 -0.10919588 -0.03190828  0.047055    0.10942066
 -0.07273786  0.19440813  0.0173874   0.04903521 -0.02009606  0.0572564
  0.13853029  0.09609025  0.00606739  0.10157311  0.09431825  0.02695916
 -0.09729319  0.07581314  0.04807245 -0.05871703 -0.13058004 -0.17450026
  0.21163954  0.06790068 -0.06012134  0.02367296  0.09584136  0.27604811
 -0.14405045 -0.09499542 -0.00078646 -0.13857509  0.05205091  0.05981628
  0.06830568  0.17722134  0.01685421 -0.18746567  0.01508161  0.10049187
 -0.06885735  0.08311225 -0.01847542 -0.07042699  0.18187776  0.17013984
  0.13434615 -0.0960767 ]
Y axis counts:
[0.51000719 0.         0.         0.         0.2550036  0.
 1.27501798 2.29503237 1.27501798 1.53002158 2.55003597 1.53002158
 4.84506834 3.57005035 1.78502518 2.80503956 4.08005755 3.06004316
 3.57005035 3.57005035 2.80503956 3.31504676 1.27501798 1.27501798
 1.02001439 1.02001439 1.02001439 0.2550036  0.         0.51000719]
X axis bins:
[-0.31217889 -0.29257132 -0.27296376 -0.25335619 -0.23374862 -0.21414106
 -0.19453349 -0.17492592 -0.15531836 -0.13571079 -0.11610322 -0.09649566
 -0.07688809 -0.05728052 -0.03767296 -0.01806539  0.00154218  0.02114974
  0.04075731  0.06036488  0.07997244  0.09958001  0.11918758  0.13879514
  0.15840271  0.17801028  0.19761784  0.21722541  0.23683298  0.25644054
  0.27604811]

Gaussian Analysis

Python Example

References