what is polynomial regression?

polynomial linear regression is a type of regression analysis in which the relationship between the independent variable and the dependent variable is modeled as an n-th degree polynomial function. polynomial regression allows for a more complex relationship between the variables to be captured beyond the linear relationship in simple linear regression and multiple linear regression.

why polynomial regression?

in machine learning (ml) and data science, choosing between a linear regression or polynomial regression depends upon the characteristics of the dataset. a non-linear dataset can't be fitted with a linear regression. if we apply linear regression to a nonlinear dataset, it will not be able to capture the non-linear patterns in the data.

look at the below diagram to understand why we need polynomial regression for non-linear data.

simple linear regression vs. polynomial regression

the above diagram shows the simple linear model hardly fits the data points whereas the polynomial model fits most of the data points.

equation of polynomial regression model

in machine learning, the general formula for polynomial regression of degree ${n}$ is as follows −

$$\mathrm{y \: = \: w_{0} \: + \:w_1 x \:+ \: w_2 x^2 \: + \: w_{3} x^3 \: + \: \dotso\: + \: w_n x^n \: + \: \epsilon}$$

where

${y}$ is the dependent variable (output).
${x}$ is the independent variable (input).
${w_{0}, w_{1}, w_{2}, \dots, w_{n}}$ are the coefficients (parameters) of the model.
${n}$ is the degree of the polynomial (the highest power of ${x}$).
${\epsilon}$ is the error term or residual, representing the difference between the observed value and the model's prediction.

for a quadratic (second-degree) polynomial regression, the formula would be:

$$\mathrm{y \:=\: w_{0} \:+\: w_{1} x \:+\: w_{2} x^2 \:+\: \epsilon}$$

this would fit a parabolic curve to the data points.

how does polynomial regression work?

in machine learning, the polynomial regression actually works in a similar way as linear regression works. it is modeled as multiple linear regression. the input feature is transformed into polynomial features of higher degrees (${x^2, x^3, ..., x^n}$). these features are now treated as separate independent variables as in multiple linear regression. now, a multiple linear regressor is trained on these transformed polynomial features.

the polynomial regression is a special case of multiple linear regression but there is a difference that multiple linear regression assumes linearity of input features. here, in polynomial regression, the transformed polynomial features are dependent on the original input feature.

implementation of polynomial regression using python

let's implement polynomial regression using python. we will use a well known machine learning python library, scikit-learn for building a regression model.

step 1: data preparation

in machine learning model building, the data preparation is very important step. let's prepare our data first. we will be using a dataset named ice_cream_selling_data.csv. it contains 49 data examples. it has an input feature/ independent variable (temperature (c)) and target feature/ dependent variable (ice cream sales (units)).

the following table represents the data in ice_cream_selling_data.csv file.

ice_cream_selling_data.csv

temperature (c)	ice cream sales (units)
-4.662262677	41.84298632
-4.316559447	34.66111954
-4.213984765	39.38300088
-3.949661089	37.53984488
-3.578553716	32.28453119
-3.455711698	30.00113848
-3.108440121	22.63540128
-3.081303324	25.36502221
-2.672460827	19.22697005
-2.652286793	20.27967918
-2.651498033	13.2758285
-2.288263998	18.12399121
-2.11186969	11.21829447
-1.818937609	10.01286785
-1.66034773	12.61518115
-1.326378983	10.95773134
-1.173123268	6.68912264
-0.773330043	9.392968661
-0.673752802	5.210162615
-0.149634867	4.673642541
-0.036156498	0.328625517
-0.033895286	0.897603187
0.008607699	3.165600008
0.149244574	1.931416029
0.688780908	2.576782245
0.693598873	4.625689458
0.874905029	0.789973651
1.024180814	2.313806358
1.240711619	1.292360811
1.359812674	0.953115312
1.740000012	3.782570136
1.850551926	4.857987801
1.999310369	8.943823209
2.075100597	8.170734936
2.31859124	7.412094028
2.471945997	10.33663062
2.784836463	15.99661997
2.831760211	12.56823739
2.959932091	21.34291574
3.020874314	20.11441346
3.211366144	22.8394055
3.270044068	16.98327874
3.316072519	25.14208223
3.335932412	26.10474041
3.610778478	28.91218793
3.704057438	17.84395652
4.130867961	34.53074274
4.133533788	27.69838335
4.899031514	41.51482194

note − create a csv file with the above data and save it as ice_cream_selling_data.csv.

import python libraries and packages for data preparation

let's first import libraries and packages required in the data preparation step. we use python pandas for reading csv files. we use numpy to convert the pandas data frame to numpy array. input and output features are numpy arrays. we use preprocessing package from the scikit-learn library for preprocessing related tasks such as transforming input feature to polynomial features.

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.preprocessing import polynomialfeatures

load the dataset

load the ice_cream_selling_data.csv as a pandas dataframe. learn more about data loading here.

data = pd.read_csv('/ice_cream_selling_data.csv')
data.head()

output

	temperature (c)	ice cream sales (units)
0	-4.662263	41.842986
1	-4.316559	34.661120
2	-4.213985	39.383001
3	-3.949661	37.539845
4	-3.578554	32.284531

let's create independent variable (${x}$) and the dependent variable (${y}$).

x = data.iloc[:, 0].values.reshape(-1, 1)
y = data.iloc[:, 1].values

visualize the original datapoints

let's visualize the original data points to get some insight.

# visualize the original data points
plt.scatter(x, y, color="green")
plt.title("original data")
plt.xlabel("temperature (c)")
plt.ylabel("ice cream sales (units)")
plt.show()

output

the above graph shows a parabolic curve (polynomial with degree 2) that will fit the datapoints.

so the relationship between the dependent variable ("ice cream sales (units)") and independent variable ("temperature (c)") can be modeled using polynomial regression of degree 2.

create a polynomial features object

now, let's create a polynomial feature object with degree 2. we will use polynomialfeatures class from sklearn.preprocessing module to create the feature object.

degree = 2  # degree of the polynomial
poly_features = polynomialfeatures(degree=degree)

let's now transform the input data to include polynomial features

x_poly = poly_features.fit_transform(x)

here ${x\_poly}$ is transformed polynomial features of original input features (${x}$). the transformed data is of (49, 3) shape.

step 2: model training

we have created polynomial features. now, let's build out the model. we use linearregression class from sklearn.linear_model module. as we already discussed, polynomial regression is a special type of linear regression.

let's create a linear regression object lr_model and train (fit) the model with data.

from sklearn.linear_model import linearregression
lr_model = linearregression()
#now, fit the model (linear regression object) on the data
lr_model.fit(x_poly, y)

so far, we have trained our regression model lr_model

step 3: model prediction and testing

now, we can use our model to predict the output. before going to predict for new data, let's predict for the existing data.

# generate predictions
y_pred = lr_model.predict(x_poly)

df = pd.dataframe({'actual values':y, 'predicted values':y_pred})
print(df)

output

    actual values  predicted values
0       41.842986         46.564507
1       34.661120         40.600548
2       39.383001         38.915089
3       37.539845         34.749272
4       32.284531         29.331940
5       30.001138         27.649735
6       22.635401         23.192862
7       25.365022         22.863178
8       19.226970         18.222266
9       20.279679         18.009098
10      13.275828         18.000794
11      18.123991         14.418541
12      11.218294         12.853070
13      10.012868         10.504868
14      12.615181          9.364587
15      10.957731          7.264266
16       6.689123          6.437055
17       9.392969          4.683654
18       5.210163          4.337906
19       4.673643          3.116139
20       0.328626          2.983983
21       0.897603          2.981829
22       3.165600          2.944811
23       1.931416          2.869446
24       2.576782          3.251711
25       4.625689          3.259923
26       0.789974          3.630683
27       2.313806          4.026226
28       1.292361          4.744891
29       0.953115          5.213321
30       3.782570          7.055902
31       4.857988          7.690948
32       8.943823          8.616039
33       8.170735          9.118494
34       7.412094         10.874961
35      10.336631         12.092557
36      15.996620         14.843721
37      12.568237         15.287199
38      21.342916         16.539614
39      20.114413         17.156188
40      22.839406         19.171090
41      16.983279         19.818497
42      25.142082         20.335157
43      26.104740         20.560474
44      28.912188         23.826884
45      17.843957         24.998282
46      34.530743         30.764287
47      27.698383         30.802396
48      41.514822         42.821195

you can compare the predicted values with actual values.

step 4: evaluating model performance

to evaluate the model performance, the best metric is the r-squared score (coefficient of determination). it measures the proportion of the variance in the dependent variable that is predictable from the independent variables.

from sklearn.metrics import r2_score

# get the predicted values for test dat
y_pred = lr_model.predict(x_poly)
r2 = r2_score(y, y_pred)
print(r2)

outout

0.9321137090423877

the r2_score is the most common metric used to evaluate a regression model. the high score indicates a better fit of the model with data. 1 represent perfect fit and 0 represents no relation between the predicted values and actual values.

result explanation − you can examine the above metrics. our model shows an r-squared score of around 0.932, which means that approximately 93% of data points are scattered around the fitted regression curve. another interpretation is that 93% of the variation in the output variables is explained by the input variables.

step 5: visualize the polynomial regression results

let's visualize the regression results for better understanding. we use the pyplot module from the matplotlib library to plot the graph.

import matplotlib.pyplot as plt

# visualize the polynomial regression results
plt.scatter(x, y, color="green")
plt.plot(x, y_pred, color='red', label=f'polynomial regression (degree={degree})')
plt.xlabel("temperature (c)")
plt.ylabel("ice cream sales (units)")
plt.legend()
plt.title('polynomial regression')
plt.show()

output

the above graph shows that the polynomial regression with degree 2 fits well with the original data. the polynomial curve (parabola), in red color, represents the best-fit regression curve. this regression curve is used to predict the value. the graph also shows that the predicted values are close to the actual values.

step 5: model prediction for new data

up to now, we have predicted the values in the dataset. let's use our regression model to predict new, unseen data.

let's take the temperature (c) as 1.9929c and predict the units of ice cream sales.

# predict a new value
x_new = np.array([[1.9929]])  # example value to predict
x_new_poly = poly_features.transform(x_new)
y_new_pred = lr_model.predict(x_new_poly)
print(y_new_pred)

output

[8.57450466]

the above result shows that the predicted value of ice cream sales is 8.57450466.

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