Adventures in Machine Learning

Master the Art of Regression Analysis: Fitting Models and Measuring Fit

Regression Line Measurement and

Data Creation for Regression Models

Regression analysis is a statistical method used in data science to identify and determine the relationship between an independent variable and a dependent variable. A regression line is a straight line that best fits a set of data points by minimizing the error terms between the actual values and the predicted values.

In this article, we will discuss how to measure the fit of a regression line and how to create a dataset for a regression model using the pandas library.

Regression Line Measurement

There are three types of sum of squares (SS) values that are commonly used to measure the fit of a regression line. Sum of Squares Total (SST) is the total variation of the dependent variable y, from its mean.

Sum of Squares Regression (SSR) is the explained variation in the dependent variable due to the independent variable x, and Sum of Squares Error (SSE) is the unexplained variation in the dependent variable after accounting for the variation explained by the independent variable. The formula to calculate SST, SSR, and SSE are:

SST = (yi – )

SSR = (i – )

SSE = (yi – i)

Where yi is the observed value of the dependent variable, is the mean value of the dependent variable, i is the predicted value of the dependent variable, and is the sum of.

To illustrate how to calculate SST, SSR, and SSE, let’s use an example. Suppose we have a dataset with the following values for x and y:

x = [1, 2, 3, 4, 5]

y = [2, 4, 5, 7, 8]

First, we calculate which is the mean of y:

= (2 + 4 + 5 + 7 + 8) / 5 = 5.2

Next, we calculate SST:

SST = (yi – )

SST = (2 – 5.2) + (4 – 5.2) + (5 – 5.2) + (7 – 5.2) + (8 – 5.2)

SST = 62.8

Then, we need to calculate the regression equation which is:

= b0 + b1x

Where b0 is the intercept and b1 is the slope of the regression line.

To calculate the slope, we use the following formula:

b1 = [(xi – x)(yi – )] / (xi – x)

Where x is the mean of x and xi is the value of x for each data point. The intercept b0 can be calculated by:

b0 = – b1x

Plugging in the values of x and y from the example, we get:

x = [1, 2, 3, 4, 5]

y = [2, 4, 5, 7, 8]

x = (1 + 2 + 3 + 4 + 5) / 5 = 3

y = (2 + 4 + 5 + 7 + 8) / 5 = 5.2

b1 = [(xi – x)(yi – )] / (xi – x)

b1 = [(1-3)(2-5.2) + (2-3)(4-5.2) + (3-3)(5-5.2) + (4-3)(7-5.2) + (5-3)(8-5.2)] / [(1-3) + (2-3) + (3-3) + (4-3) + (5-3)]

b1 = 1.5

b0 = – b1x

b0 = 5.2 – 1.5(3)

b0 = 0.7

Therefore, the regression equation is:

= 0.7 + 1.5x

Using this equation, we can calculate i for each data point:

1 = 0.7 + 1.5(1) = 2.2

2 = 0.7 + 1.5(2) = 3.7

3 = 0.7 + 1.5(3) = 5.2

4 = 0.7 + 1.5(4) = 6.7

5 = 0.7 + 1.5(5) = 8.2

Next, we calculate SSR:

SSR = (i – )

SSR = (2.2 – 5.2) + (3.7 – 5.2) + (5.2 – 5.2) + (6.7 – 5.2) + (8.2 – 5.2)

SSR = 32.8

Finally, we calculate SSE:

SSE = (yi – i)

SSE = (2 – 2.2) + (4 – 3.7) + (5 – 5.2) + (7 – 6.7) + (8 – 8.2)

SSE = 29.6

Therefore, SST = SSR + SSE = 62.8, which means that the regression line explains about 52% of the variation in the dependent variable.

Example of calculating SST, SSR, and SSE in Python

Python is a popular programming language used by data scientists for data analysis and modeling. The following example shows how to calculate SST, SSR, and SSE in Python using the numpy and pandas libraries:

import numpy as np

import pandas as pd

# create a dataframe

data = {‘x’: [1, 2, 3, 4, 5], ‘y’: [2, 4, 5, 7, 8]}

df = pd.DataFrame(data)

# calculate SST, SSR, and SSE

y_mean = np.mean(df[‘y’])

df[‘y_pred’] = 0.7 + 1.5*df[‘x’]

df[‘SST’] = (df[‘y’] – y_mean)**2

df[‘SSR’] = (df[‘y_pred’] – y_mean)**2

df[‘SSE’] = (df[‘y’] – df[‘y_pred’])**2

SST = np.sum(df[‘SST’])

SSR = np.sum(df[‘SSR’])

SSE = np.sum(df[‘SSE’])

print(‘SST:’, SST)

print(‘SSR:’, SSR)

print(‘SSE:’, SSE)

Data Creation

Creating a dataset for a regression model is an essential step in data science. The dataset should contain the independent variable(s) and the dependent variable, along with any other relevant features.

A dataset can be created using Excel, CSV files, or scripting languages like Python. To create a dataset using Python, we can use the pandas library which provides various functions for data manipulation.

Here’s an example of how to create a dataset using pandas:

import pandas as pd

# create a dictionary with independent variable(s), and dependent variable

data = {‘x1’: [10, 20, 30, 40, 50], ‘x2’: [1, 0, 1, 0, 1], ‘y’: [50, 100, 150, 200, 250]}

# convert dictionary into a dataframe

df = pd.DataFrame(data)

# save dataframe to a CSV file

df.to_csv(‘dataset.csv’, index=False)

In this example, we created a dictionary with three columns, x1 and x2 as independent variables, and y as the dependent variable. We then converted the dictionary into a dataframe using the pandas library and saved it to a CSV file using the pandas to_csv() function.

Conclusion

Regression analysis is a popular statistical method used in data science to identify the relationship between an independent variable and a dependent variable. The fit of a regression line can be measured using three types of sum of squares values, SST, SSR, and SSE.

These values can be calculated using formulas or programming languages like Python. Creating a dataset is an important step in the regression modeling process, and pandas is a useful library for data manipulation and creation.

The examples provided in this article can help data scientists get started with regression analysis and dataset creation.

Regression Model Fitting and Calculation of SST, SSR, and SSE

Regression analysis is a statistical tool used by data scientists to model the relationship between a dependent variable and one or more independent variables. The most common type of regression is linear regression, which models a linear relationship between the dependent variable and one independent variable.

In this article, we will discuss how to fit a linear regression model using the OLS() function from the statsmodels library, the definition of response and predictor variables, adding a constant to a predictor variable, and calculating SST, SSR, and SSE in a regression model using the numpy library.

Regression Model Fitting

The OLS() function from the statsmodels library is an easy-to-use tool for fitting a linear regression model. OLS stands for Ordinary Least Squares, which is a method used to estimate the coefficients of the regression equation.

The OLS() function takes two arguments: the dependent variable and the independent variable(s). The independent variable can be a single feature or multiple features.

Here’s an example of how to fit a linear regression model using the OLS() function in Python:

import statsmodels.api as sm

import pandas as pd

import numpy as np

# create a dataframe with independent and dependent variables

data = {‘x’: [1, 2, 3, 4, 5], ‘y’: [2, 4, 5, 7, 8]}

df = pd.DataFrame(data)

# define the response variable as y and the predictor variable as x

X = df[‘x’]

y = df[‘y’]

# add a constant to the predictor variable

X = sm.add_constant(X)

# fit the linear regression model

model = sm.OLS(y, X).fit()

# print the model summary

print(model.summary())

The output of this code is a summary of the linear regression model, which includes information about the coefficients, standard errors, t-statistics, and p-values. The R-squared value, which measures the amount of variance explained by the model, is also included in the summary.

Response and Predictor Variables

In a regression analysis, the variable that we are trying to predict is called the response variable, and the variable(s) that we use to make predictions are called predictor variable(s) or independent variable(s). The response variable is also called the dependent variable because it depends on the predictor variable(s).

For example, in the above example code, the response variable is y, and the predictor variable is x. Our goal is to predict the values of y based on the values of x.

Adding a Constant to the Predictor Variable

In a linear regression model with one predictor variable, the intercept term of the regression line represents the value of the response variable when the predictor variable equals zero. However, in many cases, the predictor variable cannot be exactly zero.

In such cases, we need to add a constant to the predictor variable to ensure that the intercept represents a meaningful value of the response variable. In the code example above, we added a constant to the predictor variable using the add_constant() function from the statsmodels library.

This added a column of ones to the predictor variable, allowing for the calculation of the intercept. Calculation of SST, SSR, and SSE

In a linear regression model, SST (Sum of Squared Total) represents the total variation in the response variable, SSR (Sum of Squared Regression) represents the explained variation in the response variable due to the predictor variable(s), and SSE (Sum of Squared Errors) represents the unexplained variation in the response variable after accounting for the variation explained by the predictor variable(s).

To calculate SST, SSR, and SSE, we can use the following formulas:

SST = (yi – )

SSR = (i – )

SSE = (yi – i)

Where yi is the observed value of the response variable, is the mean value of the response variable, i is the predicted value of the response variable, and is the sum of. Here’s an example of how to calculate SST, SSR, and SSE in Python using the numpy library:

import numpy as np

import pandas as pd

# create a dataframe

data = {‘x’: [1, 2, 3, 4, 5], ‘y’: [2, 4, 5, 7, 8]}

df = pd.DataFrame(data)

# calculate SST, SSR, and SSE

y_mean = np.mean(df[‘y’])

df[‘y_pred’] = 0.7 + 1.5*df[‘x’]

df[‘SST’] = (df[‘y’] – y_mean)**2

df[‘SSR’] = (df[‘y_pred’] – y_mean)**2

df[‘SSE’] = (df[‘y’] – df[‘y_pred’])**2

SST = np.sum(df[‘SST’])

SSR = np.sum(df[‘SSR’])

SSE = np.sum(df[‘SSE’])

print(‘SST:’, SST)

print(‘SSR:’, SSR)

print(‘SSE:’, SSE)

The output of this code is the values of SST, SSR, and SSE, which can be used to calculate the R-squared value of the model and assess the goodness of fit. Verification that SST = SSR + SSE

In a regression model, SST represents the total variation in the response variable, which can be partitioned into SSR and SSE.

Since the regression line is fitting the data, the combined sum of explained and unexplained variation should equal the total variation. Therefore, we can verify that SST = SSR + SSE to ensure that our model is correctly fitting the data.

In the above example, the values of SST, SSR, and SSE are calculated using the numpy library. We can verify that SST = SSR + SSE by adding the individual values of SSR and SSE:

SSR + SSE = 32.8 + 29.6 = 62.4

and comparing it to the value of SST:

SST = 62.8

Since SST = SSR + SSE is satisfied, we can conclude that the regression line is correctly fitting the data.

Conclusion

In this article, we covered important concepts related to regression model fitting, such as the use of the OLS() function from the statsmodels library, understanding response and predictor variables, adding a constant to predictor variables, and calculating SST, SSR, and SSE using the numpy library. These concepts are essential for any data scientist who wants to model the relationship between variables using regression analysis.

By understanding these concepts, data scientists can build accurate regression models that provide valuable insights into the relationship between variables. Additional Resources for Calculating SST, SSR, and SSE

While calculating SST, SSR, and SSE is a fundamental step in regression analysis, it can be a time-consuming process, especially when working with large datasets or complex models.

Fortunately, there are several calculators and software programs available that can automate the process of calculating SST, SSR, and SSE, saving time and ensuring accuracy. In this article, we will provide links to calculators for automatic calculation of SST, SSR, and SSE and tutorials on how to calculate SST, SSR, and SSE in other statistical software.

Calculators for Automatic Calculation of SST, SSR, and SSE

There are several online calculators available that can automatically calculate SST, SSR, and SSE, given the values of the dependent variable, independent variable, and predicted values. One such calculator is the Regression Calculator from Calculator.net, which can be accessed at https://www.calculator.net/regression-calculator.html.

This calculator allows users to input the values of the dependent variable, independent variable, and predicted values, and automatically calculates SST, SSR, and SSE, along with the coefficient of determination (R-squared). The calculator also displays a graph of the regression line and the data points.

Another online calculator that can calculate SST, SSR, and SSE

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