Why does the output of scikit PolynomialFeatures' degree always have one?

Question

I don't understand why the output of scikit's PolynomialFeatuers' degree always has a one.

Example for degree=2 and [a, b], output is

[1, a, b, a^2, b^2, ab]

I don't know why it always has a 1.

What you have above is: [a^0 * b^0, a^1 * b^0, a^0 * b^1, a^2 * b^0, a^0 * b^2, a^1 * b^1] — sshashank124, Commented Jan 11, 2020 at 8:40
that 1 is for scale shifts detection. Intuitionally, take that as allowing the math to include outer dimensional forces, which doesn't relate with those variables a or b, or their compositions, which thus are constant. — Vicrobot, Commented Jan 11, 2020 at 8:41

gunes · Accepted Answer · 2020-01-11 17:27:03Z

A few things to add:

An $n$-th degree univariate polynomial is of the form $\sum_{i=0}^n a_ix^i$, which includes the bias term (i.e. $1=x^0$), even if it can be zero.
sklearn has the option to omit the bias term via include_bias option. When set to False, you won't see any $1$'s.
As commented by @sshashank124 you'll have each term's exponent $\leq$ n, if you have $k$ features (i.e. $x_1\dots x_k$), the terms will be of the form: $\prod_{i=1}^k x_i^{n_i}$, where $\sum_{i=1}^k n_i\leq n$.
This sometimes produces lots of features, specifically: ${{n+k}\choose {n}}$. You may also want to use interaction_only parameter to only get the interactions (i.e. terms with each feature degree having at most $1$).

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