harshamth522.sites.umassd.edu

“Linear Regression Analysis and Plot: % OBESE vs. % DIABETIC”

1. Import necessary libraries: The code begins by importing the required libraries:

– `statsmodels.api` for performing linear regression analysis.

– `matplotlib.pyplot` for creating plots.

– `numpy` to create a range of X values for the model line.

2. Fit the linear regression model:

– It assumes that you already have `X` and `y` defined, where `X` is the independent variable (in this case, “% OBESE”) with a constant term added, and `y` is the dependent variable (“% DIABETIC”).

– It fits a linear regression model (`sm.OLS`) using the `X` and `y` data.

3. Get the coefficients of the model:

– The code retrieves the intercept and slope coefficients from the fitted linear regression model.

4. Create a range of X values for the model line:

– It generates a range of X values (`x_range`) using `np.linspace` that spans the range of the original “% OBESE” values.

5. Calculate predicted Y values:

– The code calculates the predicted Y values (`y_pred`) based on the linear regression model by applying the intercept and slope to the `x_range`.

6. Create a scatter plot of the data points:

– It creates a scatter plot (`plt.scatter`) of the original data points, where “% OBESE” is on the x-axis and “% DIABETIC” is on the y-axis. This visually represents the data.

7. Plot the regression line:

– It overlays a red regression line (`plt.plot`) on the scatter plot, using the calculated `y_pred` values. This represents the linear regression model’s predictions.

8. Add labels and a legend:

– The code adds labels to the x-axis and y-axis to provide context for the plot.

– It includes a legend to distinguish between the data points and the regression line.

9. Show the plot:

– Finally, the code uses `plt.show()` to display the generated plot, allowing you to visualize the data points and the fitted regression line.

A regression plot is a visual representation that displays the relationship between two variables, often in the context of a linear regression model. It includes a scatterplot of data points and a fitted regression line that represents the relationship between the variables. This plot helps assess the linearity of the relationship, evaluate how well the model fits the data, and visualize predictions.

 

 

How to create a Regression plot?

 

To create a regression plot, you can use various programming languages and libraries, such as Python with matplotlib and seaborn or R with ggplot2. Here are the general steps to create a regression plot:

 

1. Import Libraries: Start by importing the necessary libraries for data visualization and regression analysis. In Python, we might use matplotlib and seaborn, while in R, we use ggplot2.
2. Load or Generate Data: Load your dataset or generate data that you want to analyze using regression.
3. Fit a Regression Model: Depending on your data and research question, fit an appropriate regression model. For a simple linear regression plot, you would fit a linear regression model. For more complex relationships, you might use polynomial regression or other regression techniques.
4. Create Scatterplot: Create a scatterplot of your data with the independent variable (X-axis) on one axis and the dependent variable (Y-axis) on the other axis. This step helps you visualize the data distribution.
5. Overlay Regression Line: Overlay the regression line on the scatterplot. This line represents the relationship between the variables as determined by the regression model.
6. Optional Enhancements: You can enhance the plot by adding confidence intervals, prediction intervals, labels, titles, or other relevant information to make the plot more informative.

In summary, creating a regression plot involves plotting your data, fitting a regression model, and displaying the relationship between variables for visualization and analysis.

K-fold cross-validation is a technique where your dataset is split into K equal parts. Your model is trained and tested K times, with each part taking a turn as the test set. This helps evaluate your model’s performance more reliably and ensures it generalizes well to unseen data.

 

The results from each round of testing are averaged to provide an overall assessment of your model’s effectiveness, making K-fold cross-validation a valuable tool for robustly evaluating machine learning models. It involves dividing your dataset into K equally sized subsets or “folds.” The model is trained and evaluated K times, with each fold serving as the test set once while the others are used for training. This technique provides a more comprehensive and reliable evaluation of how well the model will perform on new, unseen data by reducing the impact of random data splits and variance in performance estimates. The results from each fold are averaged to obtain an overall performance assessment. K-fold cross-validation is a valuable tool for model evaluation and selection in machine learning.

 

Distribution plots are visual representations used in data analysis to show how data points are spread across a dataset. They help you understand the shape, central values, and patterns in your data. Common types include histograms, which show data frequencies, and density plots, which provide smoothed representations of data distribution. Other plots like box plots and violin plots reveal data quartiles and outliers, while Q-Q plots compare data distribution to theoretical models. These plots are essential for exploring and understanding your data’s characteristics.

Observations:

1. Data Collection: Collect data on crab sizes before and after molting.
2.  Kurtosis Assessment: Recognize that both groups have high kurtosis, indicating non-normal data distributions.
3. Hypothesis Testing Challenge: Due to data non-normality, traditional t-tests aren’t suitable. An alternative approach is needed.
4. Monte Carlo Test:

Data Pooling: Combine data from both groups into one dataset.

Random Sampling: Randomly split the combined dataset into two groups numerous times (e.g., 10 million).

Calculate Mean Differences: For each split, compute the mean difference between the two groups.

Distribution of Mean Differences: Aggregate mean differences to create a distribution representing what’s expected under the null hypothesis (no real difference).

Compare Observed Difference: Compare the observed mean difference in the actual data to the distribution of permuted mean differences.

Calculate p-value: The p-value is the proportion of permuted mean differences as extreme as or more extreme than the observed mean difference. A low p-value suggests the observed difference is likely not due to chance, supporting null hypothesis rejection.

This Monte Carlo permutation test is a robust way to assess the significance of the observed mean difference while accommodating non-normal data. If the calculated p-value is below your chosen significance level (e.g., 0.05), it implies a significant difference between premolt and postmolt crab sizes.

Overfitting:

Overfitting is when a model fits the training data too closely, capturing noise and leading to excellent training performance but poor generalization to new data. It happens when the model is overly complex or flexible. To counter overfitting, one can simplify the model, use regularization, cross-validation, more data, early stopping, or ensemble methods to ensure better predictive performance on unseen data.

Simple Linear Regression Model:

A fundamental statistical technique called simple linear regression is used to examine and measure the relationship between two variables. With this method, our main goal is to comprehend how changes in one variable—the independent variable—are related to changes in another—the dependent variable. Ten essential ideas about simple linear regression are as follows:

Basic Concept: To model this relationship, Simple Linear Regression looks at the linear relationship between two variables.

Two variables are: It involves an independent predictor variable and a dependent response variable as its two main variables.

The approach makes the assumption that the correlation between these two variables can be described by a straight line equation.

Finding the line that fits the data the best and reduces the variation between the observed and anticipated values is the best-fit line.

Simple linear regression, which uses the independent variable to predict the value of the dependent variable, is frequently used for prediction.

Strength and Direction: By quantifying how much the dependent variable varies for every unit change in the independent variable, it expresses the relationship’s strength and direction.

Intercept and Slope: The fitted line contains two parameters: an intercept (the value at zero for the independent variable) and a slope (the rate of change).

The least squares method, which minimizes the sum of the squared differences between observed and predicted values, is commonly used to define the line.

Applications: For analyzing and making predictions based on data, simple linear regression is widely utilized in a variety of domains, including economics, biology, and social sciences.

The assumption of a linear relationship, which may not always hold true in real-world situations, is a limitation. More complicated regression procedures may be required when the connection is nonlinear.

In conclusion, Simple Linear Regression is a fundamental statistical technique that offers insightful information about the relationship between two variables, making it a useful tool for data analysis and forecasting across a variety of fields.

I have observed that the correlation between the two datasets there are no correlation between them, but while merging the two datasets we can see that there will be a correlation between them.

I have done this while merging the two datasets by using the same columns in the datasets.

We can see the pair plot clearly that this is a left skewed

 

 

By removing the duplicate columns and merging with the FIPS Columns.

 

I have merged that the diabetics and obesity data sheets as seen in the below

By using the drop function, we can remove the columns which has come the same.

On the next post I will make the obesity and inactivity sheets merge.

 

I have observed the CDC Dataset contains 3 sheets of Excel files. Diabetics, Obesity, & Inactivity.  By observing the data set the FIPS is common to all Excel sheets. With the help of FIPS, we can merge all the data sets and take the correlation value. By individual Datasets, we can observe the individually there is no correlation between them. we can check the FIPS in Diabetics corelation value is -0.083521

 

 

By Observing the pair plot we can see the left side skewed  for diabetics

 

P-value: P value or probability value tells how likely data is occurred null hypothesis.  The null hypothesis is statement that assume there is no relation between the variables.

 

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