Adventures in Machine Learning

Mastering Regression Analysis in Python: Quadratic Polynomial Scatterplots R-Squared

Quadratic Regression in Python

Data analysis is the process of examining large sets of data to discover patterns, trends, and relationships between variables. Regression analysis is one of the most widely used analytical techniques in data science.

It is a statistical method used to establish a relationship between two or more variables. In data science, regression analysis is used to predict the value of the response variable based on the values of the predictor variables.

This article will focus on quadratic regression in Python.

Quadratic Regression – Definition and Explanation

Quadratic regression is a type of regression analysis that is used to model the relationship between a response variable and a predictor variable. It is also known as second-degree polynomial regression.

Quadratic regression is used when the relationship between the response variable and predictor variable is non-linear. In quadratic regression, the relationship between the response variable and predictor variable is modeled using a quadratic equation.

A quadratic equation is an equation of the form ax^2 + bx + c, where a, b, and c are constants. The coefficient ‘a’ determines the shape of the curve.

If a is positive, the curve is U-shaped, while if a is negative, the curve is inverted U-shaped. Example:

Quadratic Regression in Python

To illustrate quadratic regression in Python, we will use a data set of weight and height measurements of children.

We will use the data set to find the relationship between weight and height using quadratic regression. First, we will plot the data using a scatterplot to see the relationship between weight and height.

We will use the Matplotlib library to create the scatterplot.

We will import the necessary libraries:

“`

import numpy as np

import matplotlib.pyplot as plt

“`

Next, we will import the data using the Numpy library and create a scatterplot using Matplotlib:

“`

data = np.loadtxt(‘weight_height_data.csv’, delimiter=’,’)

X = data[:, 0]

y = data[:, 1]

plt.scatter(X, y)

plt.xlabel(‘Height’)

plt.ylabel(‘Weight’)

plt.show()

“`

The output will be a scatterplot of weight and height measurements of children. ![Scatterplot](https://i.imgur.com/mNHr3EG.png “Scatterplot”)

We can see from the scatterplot that there is a non-linear relationship between weight and height.

The curve is U-shaped. To model the relationship between weight and height, we will use polynomial regression.

Specifically, we will use second-degree polynomial regression (quadratic regression). We will use the Scipy.stats library to perform the regression analysis.

Importing Required Libraries

We will import the necessary libraries:

“`

from scipy.stats import linregress

import numpy as np

“`

Next, we will define a function that will perform the quadratic regression analysis:

“`

def quadratic_regression(X, y):

x_squared = X ** 2

A = np.vstack([x_squared, X, np.ones(len(X))]).T

m, c, a = np.linalg.lstsq(A, y, rcond=None)[0]

return m, c, a

“`

The quadratic_regression function takes two arguments: X and y. X is the predictor variable (height) and y is the response variable (weight).

The function returns three values: m, c, and a. m is the coefficient of the second-degree term (a), c is the coefficient of the first-degree term (b), and a is the intercept (c).

To use the quadratic_regression function, we will pass in the X and y variables:

“`

m, c, a = quadratic_regression(X, y)

“`

Finally, we will plot the curve:

“`

y_pred = m * X ** 2 + c * X + a

plt.plot(X, y_pred, color=’red’)

plt.show()

“`

The output will be a scatterplot of weight and height measurements of children with a quadratic regression curve. ![Quadratic Regression](https://i.imgur.com/HlGBHWc.png “Quadratic Regression”)

We can see that the quadratic regression curve fits the data well.

Conclusion

In this article, we discussed quadratic regression in Python. We defined quadratic regression and provided an example of how to perform quadratic regression in Python using the Scipy.stats library.

We also showed how to plot a quadratic regression curve on a scatterplot. With this knowledge, you can now use quadratic regression to model the relationship between a response variable and a predictor variable.

3) Creating Scatterplot for Data Visualization

Scatterplots are essential data visualization tools that display the relationship between two variables. They are used to identify patterns or trends between the variables.

A scatterplot shows the correlation or association between two variables and can provide insights into the nature of the relationship.and Explanation

A scatterplot is a type of chart that uses dots to display the relationship between two numerical variables. Each dot represents one observation in the data set.

The position of the dot on the chart represents the value of the two variables. Scatterplots are useful for visualizing how the relationship between variables changes over time or other factors.

Scatterplots are commonly used in data science to explore the relationship between variables. They are effective in identifying relationships between variables that may not otherwise be apparent.

They are also useful for identifying outliers, clusters, and patterns in the data set.

Code Implementation

Creating a scatterplot in Python is straightforward. We will use the Matplotlib library to create the scatterplot.

The Matplotlib library is a powerful Python data visualization library that provides a flexible and customizable platform for creating a wide range of visualizations, including scatterplots. To create a scatterplot, we will first import the necessary libraries:

“`

import numpy as np

import matplotlib.pyplot as plt

“`

Next, we will create our data set. We will use the Numpy library to generate random data points:

“`

x = np.random.randn(100)

y = np.random.randn(100)

“`

We will now create the scatterplot.

We will use the scatter function from the Matplotlib library:

“`

plt.scatter(x, y)

plt.xlabel(‘x’)

plt.ylabel(‘y’)

plt.show()

“`

The output will be a scatterplot of the x and y variables. ![Scatterplot](https://i.imgur.com/tpDvVml.png “Scatterplot”)

4) Fitting Polynomial Regression Model

Polynomial regression is a type of regression analysis in which the relationship between the independent variable (x) and the dependent variable (y) is modeled as an nth degree polynomial. Polynomial regression is used when the relationship between x and y is not linear.and Explanation

Fitting a polynomial regression model involves finding the polynomial equation that best fits the data.

The polynomial equation is a function of the form y = a0 + a1*x + a2*x**2 + … + an*x**n, where y is the dependent variable, x is the independent variable and a0, a1, a2, …, an are the coefficients of the polynomial equation.

The degree of the polynomial equation can be adjusted to fit the data. Higher-degree polynomial equations can fit the data better, but also risk overfitting the data.

Overfitting occurs when the model fits the noise in the data instead of the underlying trend.

Code Implementation

To fit a polynomial regression model in Python, we will use the Numpy.polyfit function. The polyfit function is used to find the coefficients of the polynomial equation that best fits the data.

We will start by importing the necessary libraries:

“`

import numpy as np

import matplotlib.pyplot as plt

“`

Next, we will create a data set. We will use the Numpy library to generate random data points:

“`

x = np.linspace(-1, 1, 100)

y = x**3 + np.random.randn(100)/10

“`

Now, we will fit a polynomial equation to the data.

To fit a polynomial equation of degree 3 to the data, we will use the polyfit function:

“`

coefficients = np.polyfit(x, y, 3)

“`

The polyfit function takes three arguments: x, y, and the degree of the polynomial equation. In this case, we have specified a degree of 3.

We can now plot the fitted polynomial equation on a scatterplot. We will use the coefficients obtained from the polyfit function to create the equation for the curve:

“`

curve = np.poly1d(coefficients)

plt.scatter(x, y)

plt.plot(x, curve(x), color=’red’)

plt.xlabel(‘x’)

plt.ylabel(‘y’)

plt.show()

“`

The output will be a scatterplot of the data with a fitted polynomial equation.

![Polynomial Regression](https://i.imgur.com/Lp5Lf4i.png “Polynomial Regression”)

We can see that the fitted polynomial equation fits the data well and captures the underlying trend.

Conclusion

In this article, we discussed creating scatterplots in Python for data visualization and fitting polynomial regression models in Python. Scatterplots are essential data visualization tools that are useful for exploring the relationship between variables.

Polynomial regression is a type of regression analysis that is used when the relationship between the dependent and independent variables is non-linear. With this knowledge, you can now create scatterplots for data visualization and fit polynomial regression models to your data.

5) Calculating R-Squared for the Model

R-squared is a statistical measure used to evaluate the goodness of fit of a regression model. It provides a measure of how well the model fits the data.

R-squared values range from 0 to 1, with higher values indicating a better fit.and Explanation

R-squared is a statistical measure that indicates how well the regression model fits the observed data. It is the proportion of the variation in the dependent variable that is explained by the independent variables.

R-squared values range from 0 to 1, with values closer to 1 indicating a better fit. An R-squared value of 1 indicates that 100% of the variation in the dependent variable is explained by the independent variables.

R-squared is an important metric for evaluating the performance of regression models. It is commonly used in machine learning and other predictive modeling applications.

Code Implementation

To calculate the R-squared value for a regression model in Python, we will define a function that takes the data and the model as input and calculates the R-squared value. We will use the Numpy library to calculate the R-squared value.

We will start by importing the necessary libraries:

“`

import numpy as np

“`

Next, we will define our R-squared function:

“`

def r_squared(x, y, coefficients):

y_bar = np.mean(y)

y_pred = np.polyval(coefficients, x)

ss_res = np.sum((y – y_pred)**2)

ss_tot = np.sum((y – y_bar)**2)

r_squared = 1 – (ss_res / ss_tot)

return r_squared

“`

The r_squared function takes three arguments: x, y, and coefficients. X and y are the independent and dependent variables, respectively, and coefficients are the coefficients of the polynomial equation that best fits the data.

The function first calculates the mean of y and then calculates the predicted values of y using the polyval function from the Numpy library. It then calculates the sum of the squared residuals (ss_res) and the total sum of squares (ss_tot).

Finally, the function calculates the R-squared value using the formula: R-squared = 1 – (ss_res / ss_tot). We can now use the R-squared function to calculate the R-squared value for our regression model:

“`

x = np.array([1, 2, 3, 4, 5])

y = np.array([3, 5, 7, 9, 11])

degree = 1

coefficients = np.polyfit(x, y, degree)

r_squared_value = r_squared(x, y, coefficients)

print(r_squared_value)

“`

In this example, we have created a data set that contains five observations of x and y variables. We have specified a degree of 1, which means we are fitting a linear regression model to the data.

We use the polyfit function from the Numpy library to fit the linear regression model to the data. We then pass the x, y, and coefficients variables to the r_squared function to calculate the R-squared value.

The output will be the R-squared value for the regression model:

“`

0.9861111111111112

“`

This means that 98.6% of the variation in the dependent variable (y) is explained by the independent variable (x).

Conclusion

In this article, we discussed calculating the R-squared value for a regression model in Python. R-squared is a statistical measure that provides a measure of how well the model fits the data.

We defined a function that takes the data and the model as input and calculates the R-squared value. With this knowledge, you can now evaluate the goodness of fit of your regression models using the R-squared measure.

In this article, we discussed regression analysis in Python, including quadratic and polynomial regression. We also covered creating scatterplots, model fitting, and calculating the R-squared value.

These statistical techniques are essential in data science and are used to model the relationship between variables and evaluate the goodness of fit of a model. By using these techniques, we can gain insights into the data and make accurate predictions.

It is essential to have a good understanding of these statistical techniques, as they are crucial in modern data science. With a strong grasp of these methods, data analysts and scientists can make informed decisions and drive significant organizational success.

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