Polynomial Regression in IronPython QuickStart Sample

Illustrates how to fit data to polynomials using the PolynomialRegressionModel class in IronPython.
```Python
import numerics

from Extreme.Mathematics import *
from Extreme.Statistics import *

# Illustrates the use of the PolynomialRegressionModel class
# to perform polynomial regression.

# Polynomial regression can be performed using 
# the PolynomialRegressionModel class.
#
# This QuickStart sample uses data from the National Institute
# for Standards and Technology's Statistical Reference Datasets
# library at http:#www.itl.nist.gov/div898/strd/.

# Note that, due to round-off error, the results here will not be exactly
# the same as the NIST results, which were calculated using 500 digits
# of precision!

# We use the 'Pontius' dataset, which contains measurement data
# from the calibration of load cells. The independent variable is the load.
# The dependent variable is the deflection.
deflectionData = Vector([ \
    .11019, .21956, .32949, .43899, .54803, .65694, \
    .76562, .87487, .98292, 1.09146, 1.20001, 1.30822, \
    1.41599, 1.52399, 1.63194, 1.73947, 1.84646, 1.95392, \
    2.06128, 2.16844, .11052, .22018, .32939, .43886, \
    .54798, .65739, .76596, .87474, .98300, 1.09150, \
    1.20004, 1.30818, 1.41613, 1.52408, 1.63159, 1.73965, \
    1.84696, 1.95445, 2.06177, 2.16829 ])
loadData = Vector([ \
    150000, 300000, 450000, 600000, 750000, 900000, \
    1050000, 1200000, 1350000, 1500000, 1650000, 1800000, \
    1950000, 2100000, 2250000, 2400000, 2550000, 2700000, \
    2850000, 3000000, 150000, 300000, 450000, 600000, \
    750000, 900000, 1050000, 1200000, 1350000, 1500000, \
    1650000, 1800000, 1950000, 2100000, 2250000, 2400000, \
    2550000, 2700000, 2850000, 3000000 ])

deflection = NumericalVariable("deflection", deflectionData)
load = NumericalVariable("load", loadData)

# Now create the regression model. We supply the dependent and independent
# variable, and the degree of the polynomial:
model = PolynomialRegressionModel(deflection, load, 2)

# The Compute method performs the actual regression analysis.
model.Compute()

# The Parameters collection contains information about the regression 
# parameters.
print "Variable                  Value   Std.Error   t-stat  p-Value"
for parameter in model.Parameters:
    # Parameter objects have the following properties:
    print "{0:19}{1:12.4e}{2:12.2e}{3:9.2f}{4:9.4f}".format( # Name, usually the name of the variable:
		parameter.Name, # Estimated value of the parameter:
		parameter.Value, # Standard error:
		parameter.StandardError, # The value of the t statistic for the hypothesis that the parameter
		# is zero.
		parameter.Statistic, # Probability corresponding to the t statistic.
		parameter.PValue)
print 

# In addition to these properties, Parameter objects have a GetConfidenceInterval
# method that returns a confidence interval at a specified confidence level.
# Notice that individual parameters can be accessed using their numeric index.
# Parameter 0 is the intercept, if it was included.
confidenceInterval = model.Parameters[0].GetConfidenceInterval(0.95)
print "95% confidence interval for constant term: {0:.4e} - {1:.4e}".format(
    confidenceInterval.LowerBound, confidenceInterval.UpperBound)
print 
			
# There is also a wealth of information about the analysis available
# through various properties of the LinearRegressionModel object:
print "Residual standard error: {0:.3e}".format(model.StandardError)
print "R-Squared:               {0:.4f}".format(model.RSquared)
print "Adjusted R-Squared:      {0:.4f}".format(model.AdjustedRSquared)
print "F-statistic:             {0:.4f}".format(model.FStatistic)
print "Corresponding p-value:   {0:.5e}".format(model.PValue)
print 

# Much of this data can be summarized in the form of an ANOVA table:
print model.AnovaTable.ToString()
```