Multiple Linear Regression Analysis

Project Description

This project implements multiple linear regression analysis to predict voltage output based on input voltage settings and frequency parameters. The analysis uses real-world data from an Excel file to build a predictive model.

Theory

Multiple linear regression is a statistical technique used to model the relationship between a dependent variable (target) and multiple independent variables (features). It extends simple linear regression by allowing multiple predictors to explain the variance in the outcome variable.

Mathematical Formula

The general form of multiple linear regression is:

Y = β₀ + β₁X₁ + β₂X₂ + ... + βₙXₙ + ε

Where:

• Y = Dependent variable (voltage_out)
• β₀ = Intercept term
• β₁, β₂, ..., βₙ = Regression coefficients for each feature
• X₁, X₂, ..., Xₙ = Independent variables (voltage_set, frequency)
• ε = Error term

In This Project

For this voltage analysis:

voltage_out = β₀ + β₁(voltage_set) + β₂(frequency) + ε

Features

• Data Loading: Reads data from Excel files (long_format_data.xlsx)
• Statistical Analysis: Uses statsmodels for comprehensive regression analysis
• Model Summary: Provides detailed statistical output including:
  • R-squared values
  • P-values for significance testing
  • Confidence intervals
  • Residual analysis

Requirements

This project requires Python 3.8+ and the packages listed in requirements.txt.

Setup and Installation

1. Create Virtual Environment

# Create virtual environment
python -m venv venv

# Activate virtual environment
# On Windows:
venv\Scripts\activate
# On macOS/Linux:
source venv/bin/activate

2. Install Dependencies

# Install all required packages from requirements.txt
pip install -r requirements.txt

3. Verify Installation

# Check installed packages
pip list

How to Use

1. Data Preparation

Ensure your Excel file (long_format_data.xlsx) contains the following columns:

• voltage_set: Input voltage settings
• frequency: Frequency parameters
• voltage_out: Output voltage measurements (target variable)

2. Running the Analysis

Execute the main script to train the regression model:

# Make sure virtual environment is activated
python main.py

3. Extract Model Coefficients

After running main.py, note down the coefficients from the output:

• Intercept (const): The β₀ value
• voltage_set coefficient: The β₁ value
• frequency coefficient: The β₂ value

4. Use Coefficients for Predictions

Update the coefficients in cal.py with your actual model results, then run:

python cal.py

5. Understanding the Output

The script will display:

• Data count verification: Shows the number of observations for each variable
• Regression summary: Comprehensive statistical analysis including:
  • R-squared: Proportion of variance explained by the model
  • P-values: Statistical significance of each coefficient
  • Coefficients: The β values in the regression equation
  • Confidence intervals: Range of likely values for coefficients

Data Structure

The project expects data in long format where each row represents one observation with:

• One voltage setting value
• One frequency value
• One corresponding voltage output value

Model Interpretation

• Intercept (β₀): Expected voltage output when all predictors are zero
• voltage_set coefficient (β₁): Change in voltage output per unit change in voltage setting
• frequency coefficient (β₂): Change in voltage output per unit change in frequency
• R-squared: Percentage of variance in voltage output explained by the model
• P-values < 0.05: Indicate statistically significant predictors

File Structure

├── main.py                    # Main regression analysis script
├── cal.py                     # Calculation script using model coefficients
├── requirements.txt           # Package dependencies
├── long_format_data.xlsx      # Input training data file
├── voltage_output_data.xlsx   # Alternative data source
├── venv/                      # Virtual environment (after setup)
└── README.md                  # This documentation

Complete Workflow

Step 1: Environment Setup

# Clone or download the project
# Navigate to project directory
cd multi_regression_formula

# Create and activate virtual environment
python -m venv venv
venv\Scripts\activate  # Windows
# source venv/bin/activate  # macOS/Linux

# Install dependencies
pip install -r requirements.txt

Step 2: Train the Model

# Run the regression analysis
python main.py

Step 3: Extract Coefficients

From the output, copy the coefficient values:

==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         74.9390     X.XXX      X.XXX      0.XXX      XX.XXX      XX.XXX
voltage_set    1.5959     X.XXX      X.XXX      0.XXX       X.XXX       X.XXX
frequency    -69.4982     X.XXX      X.XXX      0.XXX       X.XXX       X.XXX
==============================================================================

Step 4: Update cal.py

Edit cal.py and update the coefficients:

# Update these values with your actual model results
intercept = 74.9390           # const value from model output
coeff_set_voltage = 1.5959    # voltage_set coefficient
coeff_frequency = -69.49817   # frequency coefficient

Step 5: Make Predictions

# Run calculations with your trained model
python cal.py

Example Output

count 40, 40, 40
                            OLS Regression Results                            
==============================================================================
Dep. Variable:            voltage_out   R-squared:                       0.XXX
Model:                            OLS   Adj. R-squared:                  0.XXX
Method:                 Least Squares   F-statistic:                     XX.XX
Date:                Mon, 22 Aug 2025   Prob (F-statistic):            X.XXe-XX
Time:                        XX:XX:XX   Log-Likelihood:                 -XX.XX
No. Observations:                  40   AIC:                             XX.XX
Df Residuals:                      37   BIC:                             XX.XX
Df Model:                           2                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         XX.XXXX      X.XXX      X.XXX      0.XXX      XX.XXX      XX.XXX
voltage_set    X.XXXX      X.XXX      X.XXX      0.XXX       X.XXX       X.XXX
frequency      X.XXXX      X.XXX      X.XXX      0.XXX       X.XXX       X.XXX
==============================================================================

Applications

This regression model can be used for:

• Predicting voltage output for new voltage/frequency combinations
• Understanding the relationship between input parameters and output
• Quality control in voltage regulation systems
• Optimization of voltage settings for desired outputs
• Reverse calculation: Finding required input voltage for desired output (as shown in cal.py)

Using cal.py for Predictions

The cal.py script provides a practical implementation for:

Forward Prediction

Calculate output voltage given input parameters:

output = intercept + (coeff_voltage * voltage_set) + (coeff_frequency * frequency)

Reverse Calculation

Find required input voltage for desired output:

required_voltage = (desired_output - intercept - (coeff_frequency * frequency)) / coeff_voltage

Example Usage

python cal.py
# Output:
# Frequency: 2.0 MHz, Desired Output Voltage: 100 V, Required Set Voltage: 73.25
# Frequency: 1.9 MHz, Desired Output Voltage: 100 V, Required Set Voltage: 79.18

Notes

• Ensure data quality before analysis (check for outliers, missing values)
• Validate model assumptions (linearity, normality of residuals, homoscedasticity)
• Consider model validation techniques (cross-validation, train/test split) for production use