Regression Overview and Implementation

Regression Overview

• Regression is a statistical method used to predict the relationship between several independent variables and one dependent variable.
• It's a supervised learning technique that aims to find the best-fitting curve (linear or non-linear) for a dependent variable within a multi-dimensional space.
• The quality of fit is measured by the coefficient of correlation (r) and the R² value.
• R² represents the variance explained by the curve, and r is the square root of the explained variance.

Learning Objectives

• Understand the concept of regression
• Learn how to implement regression in Excel.
• Improve regression model prediction accuracy.
• Define and discuss logistic regression.
• Evaluate the pros and cons of the regression method.
• Practice performing regression in Excel.

What is Regression?

• Regression is a well-known statistical technique to predict the relationship between several independent and one dependent variable.
• It's a supervised machine learning approach.
• A model can be created by determining an equation.
• The equation often uses one or more predictor variables (independent variables) and a single target variable or dependent variable.
• The equation describes how the dependent variable is expected to change in response to changes in the independent variables.

How Much to Produce?

• Example scenario: A pizza business needs to determine daily dough production based on weather patterns and sales history.
• Various factors influence sales.
• The scenario involves collecting data (temperature and sales) over the summer to create a model.
• The goal is to create a model that can predict how much dough is needed based on the temperature and historical sales.

Key Steps for Regression

• List all accessible attributes.
• Select the target dependent variable .
• Graph/visually review relationships between variables.
• Create an equation to predict the target variable using other attributes.

Case Study: Data Driven Prediction

• Nate Silver, a data-driven political forecaster, used big data and advanced analytics to predict election outcomes, with successful predictions in Presidential and Senate elections.
• His methodology focuses on developing hypotheses, gathering data, analyzing it with sophisticated models and algorithms to produce insightful results.

Correlations and Relationships

• Categorize variables having relationships (correlated) and unrelated ones.
• Correlation measures the strength of a relationship.
• The correlation coefficient (r) ranges from -1 to +1.
• A correlation of 0 indicates no relationship; +1 means a perfect positive relationship, and -1 means a perfect negative relationship.

Visual Look at Relationships

• Scatter plots visualize the relationship between two variables.
• Each point on the plot represents a data point.
• Scatter plots help to visually identify trends or patterns in the data.
• Scatterplots help to determine if the correlation is linear and the strength of the relationship between variables.
• Scatterplots show linear and non-linear relationships.

Regression Exercise

• The regression equation is represented as a straightforward equation (y = β0 + β1x + ε).
•  y is the variable being predicted, which is also referred to as the dependent variable.
• x is the independent or predictor variable
• The regression model can include many predictor variables (x1, x2,…).
• The regression equation has one and only one dependent variable (y).

House Data

• Example scenario: Predicting house prices based on size (predictor) using a scatter plot. A positive correlation exists between house price and area. This is not a perfect correlation.

Correlation and Regression

• Example: A correlation of 0.891 between house size and price suggests a strong positive relationship.
• An R² value of 0.794 suggests the model explains approximately 79.4% of the variance in house prices.
• Regression equations can be formed based on the calculated coefficients.
• Example equation: House Price ($) = 139.48 * Size(sqft) – 54191

House Data (Correlation and Regression)

• Adding other variables like the number of rooms leads to a stronger predictive model.
• The variables in this data are positively and strongly correlated (e.g., 0.984 correlation coefficient, 0.968 R²).
• The added variable increases the predictive power of the model.

Predict the House Price

• Generate a predictive equation using the calculated regression coefficients.
• Example: House Price ($) = 65.6 * Size(sqft) + 23613 * Rooms + 12924

Non-Linear Regression Exercise

• The scenario illustrates the situation where a non-linear relationship exists between variables (e.g., temperature related electricity consumption data).
• A linear model fits poorly.
• Applying a quadratic term (i.e., Temp²) provides a strong linear relationship which significantly improves the accuracy.
• Using a quadratic term (Temp²) provides a significant improvement and results in a high R² value (near 1) for the model which indicates a strong correlation.

Predict Energy Consumption

• Calculate the energy consumption given temperature levels using the updated regression equation.
• The new equation has a strong positive correlation with the target variable, with R² approaching 1.
• Example: "Energy Consumption = 15.87 * Temp² -1911 * Temp + 67245".

Logistic Regression

• Regression models typically work with continuous numerical data.
• Logistic regression uses binary (yes/no) values for the target variable.

Logistic Regression (cont'd)

• Logistic Regression models use probability scores.
• A logit (natural logarithm of the odds) function is used.
• The model provides a continuous criterion.
• This type of regression is used to predict the binary outcome of an event from a combination of independent variables.

Advantages of Regression Models

• Easy to comprehend and use.
• Based on established statistical principles.
• Simple equations.
• Useful for predicting outcomes of other modeling techniques.
• Enables inclusion of multiple variables in the model.
• Easy to implement.

Disadvantages of Regression Models

• Sensitive to data quality (missing data, errors, non-normal distributions).
• Affected by high correlations between predictors.
• Overly sensitive to adding too many independent variables (collinearity).
• Data must be numerical and doesn't easily accommodate categorical variables.
• Cannot automatically handle non-linear relationships.

Which Technique to Use?

• Choose regression for continuous target variables.
• Choose classification for discrete target variables (e.g., yes/no, categories).

In Class Exercise

• Create a regression model to predict Test 2 scores from Test 1 score
• Predict a student's Test 2 score who scored 46 on Test 1.
• Identify the dependent variable (Test 2 score).
• Identify input variables (Test 1 score).

This quiz explores the fundamentals of regression analysis, focusing on predicting relationships between variables. You'll learn about implementing regression in Excel and improving model prediction accuracy. Key concepts like logistic regression and the evaluation of regression methods will also be covered.