[FIXED] how to constrain a (cubic) regression model to pass through certain points?

Issue

Consider the following example:

import statsmodels.formula.api as smf
import random
import pandas as pd

df = pd.DataFrame({'y' : [x**2 + random.gauss(2)   for x in range(10)],
             'x' : [x for x in range(10)]})



model = smf.ols(data = df, formula = 'y ~ x + I(x**2) +  I(x**3)').fit()

df['pred'] = model.predict(df)

df.set_index('x').plot()

As you can see, I fit a cubic model to my data and the fit is overall pretty good. However, I would like to constrain my cubic model to have the following values at two specific x points:

• f(0) = 10
• f(8) = 60

How can I do that in statsmodels or sklearn? Thanks!

Solution

You can use fit_constrained using glm.

import random

import pandas as pd
import statsmodels.formula.api as smf

df = pd.DataFrame(
    {
        "y" : [x ** 2 + random.gauss(2, 1) for x in range(10)],
        "x" : [x for x in range(10)],
    }
)

untrained_glm = smf.glm("y ~ x + I(x ** 2) + I(x ** 3)", df)

trained_glm = untrained_glm.fit_constrained(
    ([[1, 0, 0, 0], [1, 8, 64, 512]], [8, 60])
)

df["pred"] = trained_glm.predict(df)

Result:

>>> df
           y  x       pred
0   0.191139  0   8.000000
1   3.225092  1   6.110541
2   5.353590  2   7.008272
3   9.367904  3  10.498092
4  16.512384  4  16.384900
5  28.742154  5  24.473595
6  36.584476  6  34.569078
7  51.006869  7  46.476246
8  66.839006  8  60.000000
9  82.163031  9  74.945239

---

(Edit to add explanation of the constraints)

Suppose the model is y = a + b * x + c * (x ** 2) + d * (x ** 3) + e, where e is the error term, a is the intercept and b, c, d are coefficients for other degrees.

The fitted model f will satisfy f(0) = 8 if and only if 8 = a_est + b_est * 0 + c_est * 0 + d_est * 0 and will satisfy f(8) = 60 if and only if 60 = a_est + b_est * 8 + c_est * (8 ** 2) + d_est * (8 ** 3).

So, I am adding the following constraints to the model:

1 * a + 0 * b + 0 * c + 0 * d = 8
1 * a + 8 * b + 64 * c + 512 * d = 60

Answered By - yarnabrina