When curve fitting with Python's statsmodels' OLS linear regression, how do I choose the constant in the formula?

Question

• I'd like to fit linear regression models of different degrees to a data set and choose the best fitting one based on adjusted r^2 .
• Based on other answers , I'm using the OLS formula "y ~ 1 + " + " + ".join("I(x**{})".format(i) for i in range(1, degree+1)) ,
• I don't have enough statistics knowledge to understand: is the 1 + constant needed and, if so, what should the constant value be?

import numpy
import pandas
import matplotlib
import matplotlib.offsetbox
import statsmodels.tools
import statsmodels.formula.api

data = numpy.array([
  [1999, 197.0],
  [2000, 196.5],
  [2001, 194.3],
  [2002, 193.7],
  [2003, 192.0],
  [2004, 189.2],
  [2005, 189.3],
  [2006, 187.6],
  [2007, 186.9],
  [2008, 186.0],
  [2009, 185.0],
  [2010, 186.2],
  [2011, 185.1],
  [2012, 185.6],
  [2013, 185.0],
  [2014, 185.6],
  [2015, 185.4],
  [2016, 185.1],
  [2017, 183.9],
])

df = pandas.DataFrame(data, columns=["Year", "CrudeRate"])

cause = "Malignant neoplasms"
x = df["Year"].values
y = df["CrudeRate"].values
degree = 2
predict_future_years = 5

# https://stackoverflow.com/a/34617603/4135310
olsdata = {"x": x, "y": y}
formula = "y ~ 1 + " + " + ".join("I(x**{})".format(i) for i in range(1, degree+1))
model = statsmodels.formula.api.ols(formula, olsdata).fit()

print(model.summary())

ax = df.plot("Year", "CrudeRate", kind="scatter", grid=True, title="Deaths from {}".format(cause))

# https://stackoverflow.com/a/37294651/4135310
func = numpy.poly1d(model.params.values[::-1])
matplotlib.pyplot.plot(df["Year"], func(df["Year"]))

predicted = func(df.Year.values[-1] + predict_future_years)
print("Predicted in {} years: {}".format(predict_future_years, predicted))

ax.add_artist(matplotlib.offsetbox.AnchoredText("$\\barR^2$ = {:0.2f}".format(model.rsquared_adj), loc="upper center"))
ax.add_artist(matplotlib.offsetbox.AnchoredText("Predicted in +{} = {:0.2f}".format(predict_future_years, predicted), loc="upper right"))

ax.xaxis.set_major_formatter(matplotlib.ticker.FormatStrFormatter("%d"))
fig = matplotlib.pyplot.gcf()
fig.autofmt_xdate(bottom=0.2, rotation=30, ha="right", which="both")
matplotlib.pyplot.tight_layout()
cleaned_title = cause.replace(" ", "_").replace("(", "").replace(")", "")
#matplotlib.pyplot.savefig("{}_{}.png".format(cleaned_title, degree), dpi=100)
matplotlib.pyplot.show()

Answer 1

Based on comments from @ALollz, when using Patsy notation (eg statsmodels.formula.api.ols("y ~ x") ), you don't need to include 1 + , as the constant is added by default to the model, although this does not specify that your model has a constant that takes on the value of 1. Instead, it specifies that you have a constant whose magnitude will be given by the intercept coefficient. This is the constant determined by OLS, so it's the one you want.