## Cannot understand with sklearn's PolynomialFeatures

fit_transform sklearn

sklearn polynomial regression

sklearn binning

sklearn pipeline

sklearn polynomialfeatures pipeline

sklearn feature cross

feature combination sklearn

Need help in sklearn's Polynomial Features. It works quite well with one feature but whenever I add multiple features, it also outputs some values in the array besides the values raised to the power of the degrees. For ex: For this array,

X=np.array([[230.1,37.8,69.2]])

when I try to

X_poly=poly.fit_transform(X)

It outputs

[[ 1.00000000e+00 2.30100000e+02 3.78000000e+01 6.92000000e+01 5.29460100e+04 8.69778000e+03 1.59229200e+04 1.42884000e+03 2.61576000e+03 4.78864000e+03]]

Here, what is `8.69778000e+03,1.59229200e+04,2.61576000e+03`

?

If you have features `[a, b, c]`

the default polynomial features(in `sklearn`

the degree is 2) should be `[1, a, b, c, a^2, b^2, c^2, ab, bc, ca]`

.

`2.61576000e+03`

is `37.8x62.2=2615,76`

(`2615,76 = 2.61576000 x 10^3`

)

In a simple way with the `PolynomialFeatures`

you can create new features. There is a good reference here. Of course there are and disadvantages("Overfitting") of using `PolynomialFeatures`

(see here).

**Edit:**
We have to be careful when using the polynomial features. The formula for calculating the number of the polynomial features is `N(n,d)=C(n+d,d)`

where `n`

is the number of the features, `d`

is the degree of the polynomial, `C`

is binomial coefficient(combination). In our case the number is `C(3+2,2)=5!/(5-2)!2!=10`

but when the number of features or the degree is height the polynomial features becomes too many. For example:

N(100,2)=5151 N(100,5)=96560646

So in this case you may need to apply **regularization** to penalize some of the weights. It is quite possible that the algorithm will start to suffer from curse of dimensionality (here is also a very nice discussion).

**Frequently Asked Questions,** If your problem raises an exception that you do not understand (even after Outside of neural networks, GPUs don't play a large role in machine learning today, When using anaconda with MKL and trying to build sklearn, I get Intel MKL FATAL ERROR: Cannot load libmkl_avx2.so or libmkl_def.so. Can anyone explain what is happening / how to built?

PolynomialFeatures generates a new matrix with all polynomial combinations of features with given degree.

Like [a] will be converted into [1,a,a^2] for degree 2.

You can visualize input being transformed into matrix generated by PolynomialFeatures.

from sklearn.preprocessing import PolynomialFeatures a = np.array([1,2,3,4,5]) a = a[:,np.newaxis] poly = PolynomialFeatures(degree=2) a_poly = poly.fit_transform(a) print(a_poly)

Output:

[[ 1. 1. 1.] [ 1. 2. 4.] [ 1. 3. 9.] [ 1. 4. 16.] [ 1. 5. 25.]]

You can see matrix generated in form of [1,a,a^2]

To observe polynomial features on scatter plot, let's use number 1-100.

import numpy as np from sklearn.preprocessing import StandardScaler from sklearn.preprocessing import PolynomialFeatures #Making 1-100 numbers a = np.arange(1,100,1) a = a[:,np.newaxis] #Scaling data with 0 mean and 1 standard Deviation, so it can be observed easily scaler = StandardScaler() a = scaler.fit_transform(a) #Applying PolynomialFeatures poly = PolynomialFeatures(degree=2) a_poly = poly.fit_transform(a) #Flattening Polynomial feature matrix (Creating 1D array), so it can be plotted. a_poly = a_poly.flatten() #Creating array of size a_poly with number series. (For plotting) xarr = np.arange(1,a_poly.size+1,1) #Plotting plt.scatter(xarr,a_poly) plt.title("Degree 2 Polynomial") plt.show()

Output:

Changing degree=3 ,we get:

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You have 3-dimensional data and the following code generates all poly features of degree 2:

X=np.array([[230.1,37.8,69.2]]) from sklearn.preprocessing import PolynomialFeatures poly = PolynomialFeatures() X_poly=poly.fit_transform(X) X_poly #array([[ 1.00000000e+00, 2.30100000e+02, 3.78000000e+01, # 6.92000000e+01, 5.29460100e+04, 8.69778000e+03, # 1.59229200e+04, 1.42884000e+03, 2.61576000e+03, # 4.78864000e+03]])

This can also be generated with the following code:

a, b, c = 230.1, 37.8, 69.2 # 3-dimensional data np.array([[1,a,b,c,a**2,a*b,c*a,b**2,b*c,c**2]]) # all possible degree-2 polynomial features # array([[ 1.00000000e+00, 2.30100000e+02, 3.78000000e+01, 6.92000000e+01, 5.29460100e+04, 8.69778000e+03, 1.59229200e+04, 1.42884000e+03, 2.61576000e+03, 4.78864000e+03]])

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##### Comments

- Why does it gives ab,bc,ca?
- @TechieBoy101: It's polynomial features, not monomial features. There's nothing restricting it to only one variable at a time.
- @TechieBoy101, The default
`PolynomialFeatures`

in`sklearn`

includes all polynomial combinations. You can add`interaction_only=True`

to exclude the powers like`a^2, b^2, c^2`

. Of course you can exclude the interaction if your model performs better - the`PolynomialFeatures`

are a simple way to derive new features (in some artificial manner).