Linear Regression Algorithm

Category: Data Mining Algorithms Last Updated: October 26, 2023

The Linear Regression algorithm in SQL Server Analysis Services (SSAS) is used for data mining to build a predictive model. It identifies the relationship between a dependent variable and one or more independent variables, assuming a linear correlation. This algorithm is particularly useful for forecasting and understanding how changes in predictor variables affect an outcome.

Overview

Linear regression models a relationship between a dependent variable and one or more explanatory variables by fitting a linear equation to the observed data. The equation takes the form:

Y = b0 + b1*X1 + b2*X2 + ... + bn*Xn + e

Where:

Y is the dependent variable (the target you want to predict).
X1, X2, ..., Xn are the independent variables (predictors).
b0, b1, b2, ..., bn are the coefficients calculated by the algorithm.
e is the error term.

The algorithm aims to minimize the sum of the squared differences between the observed and predicted values of the dependent variable.

When to Use Linear Regression

When you need to predict a continuous numerical value (e.g., sales, price, temperature).
When you suspect a linear relationship between the target variable and its predictors.
To understand the strength and direction of the relationship between variables.
For anomaly detection by identifying data points that deviate significantly from the predicted line.

Key Concepts and Terminology

Dependent Variable: The variable you are trying to predict. It must be a continuous numerical attribute.
Independent Variables (Predictors): The variables used to predict the dependent variable. These can be continuous or discrete.
Coefficients: The weights assigned to each independent variable in the regression equation, indicating their impact on the dependent variable.
Intercept (b0): The predicted value of the dependent variable when all independent variables are zero.
R-squared: A statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model. A higher R-squared value indicates a better fit.

Parameters

The Linear Regression algorithm in SSAS has several configurable parameters:

Parameter	Description	Default Value	Allowed Values
`ORDER`	Specifies the order of the polynomial to use for regression. For standard linear regression, this should be 1.	1	Integer >= 1
`MAX_INPUT_ATTRIBUTES`	The maximum number of input attributes that can be used in the model.	100	Integer >= 0
`MAX_OUTPUT_ATTRIBUTES`	The maximum number of output attributes that can be used in the model.	100	Integer >= 0
`COMPUTE_PROBABILITY`	When set to TRUE, the algorithm computes the probability for each prediction.	FALSE	TRUE, FALSE
`ENABLE_HIERARCHY_VOTING`	Enables or disables hierarchy voting. Not typically relevant for linear regression.	FALSE	TRUE, FALSE

Example Usage

Imagine you want to predict the price of a house based on its size (square footage) and the number of bedrooms. The Linear Regression algorithm can help model this relationship.

Scenario: Predicting House Prices

Dependent Variable: Price (continuous numerical)

Independent Variables:

SquareFootage (continuous numerical)
NumberOfBedrooms (discrete numerical)

After training the model with historical house sales data, the algorithm might produce a model with an equation like:

Price = 50000 + 150 * SquareFootage + 10000 * NumberOfBedrooms

This equation suggests that for every additional square foot, the price increases by $150, and each additional bedroom adds $10,000 to the price, assuming the base intercept of $50,000.

DMX Query Example (Conceptual)

A Data Mining eXpressions (DMX) query to predict a house price:

Predicting a house price using DMX
DMX
SELECT
    [House Price Regressor].[Price] AS PredictedPrice,
    [House Price Regressor].Query([SquareFootage], [NumberOfBedrooms]) AS PredictionDetails
FROM
    [House Price Model]
PREDICTION JOIN
    (SELECT 1500 AS [SquareFootage], 3 AS [NumberOfBedrooms]) AS InputData
ON
    [House Price Model].[SquareFootage] = InputData.[SquareFootage]
    AND [House Price Model].[NumberOfBedrooms] = InputData.[NumberOfBedrooms]

Note: This is a simplified DMX example. Actual syntax might vary based on model structure.

Pros and Cons

Pros:

Simplicity: Easy to understand and interpret.
Efficiency: Computationally inexpensive to train.
Interpretability: Coefficients provide clear insights into variable impact.
Foundation: Forms the basis for more complex models.

Cons:

Linearity Assumption: Assumes a linear relationship, which may not hold true for all data.
Sensitivity to Outliers: Can be heavily influenced by extreme values.
Multicollinearity: Performance can degrade if independent variables are highly correlated.
Limited to Continuous Output: Cannot directly predict categorical outcomes.

Related Algorithms

Logistic Regression (for binary classification)
Decision Trees (for both classification and regression)
Linear Regression (used here)