Linear Regression Algorithm

Algorithm Summary

Purpose: Predicts a continuous value based on a linear relationship with input attributes.
Type: Regression Algorithm.
Use Cases: Forecasting sales, predicting stock prices, estimating project completion time, etc.
Key Concepts: Regression Equation, Coefficients, Intercept, R-squared.

Overview

The Linear Regression algorithm in SQL Server Analysis Services (SSAS) is a powerful tool for modeling the relationship between a dependent continuous variable and one or more independent variables. It assumes a linear relationship and finds the best-fitting line (or hyperplane in multiple dimensions) to represent this relationship.

How it Works

The algorithm uses the method of least squares to determine the coefficients of the regression equation. For a single predictor variable X and a target variable Y, the equation is:

Y = b₀ + b₁ * X + e

Where:

Y is the dependent variable.
X is the independent variable.
b₀ is the intercept (the value of Y when X is 0).
b₁ is the slope or coefficient of X (the change in Y for a unit change in X).
e is the error term, representing the unexplained variance.

When multiple independent variables are involved (X1, X2, ..., Xn), the equation becomes:

Y = b₀ + b₁*X₁ + b₂*X₂ + ... + b_n*X_n + e

Parameters

The Linear Regression algorithm in SSAS offers the following configurable parameters:

MAX_OUTPUT_ATTRIBUTES: Specifies the maximum number of output attributes that the algorithm can generate.
MISSING_VALUE_TREATMENT: Defines how missing values are handled (e.g., REPLACE_WITH_MEAN, LIST).
SPLIT_PROBABILITY: Used for determining when to split a node in the decision tree representation (if applicable during modeling).
CALCULATE_PERMUTATION_IMPORTANCE: Determines whether to calculate feature importance using permutation importance.

Usage in SSAS

To use the Linear Regression algorithm in SSAS:

Create a new Mining Structure in SQL Server Data Tools (SSDT) or SQL Server Management Studio (SSMS).
Select "Linear Regression" as the algorithm type.
Define your mining columns:
- Select a Predictable column (continuous).
- Select one or more Input columns (can be numeric or categorical). SSAS will automatically handle the encoding of categorical variables.
Configure algorithm parameters as needed.
Process the mining structure and model.

Example DMX Query (Predicting Sales)

Assume you have a model named [Sales_LinearRegression_Model] and you want to predict sales based on advertising spend and seasonality.

SELECT
    Predict([Sales]),
    PredictProbability([Sales]) AS SalesProbability
FROM
    [Sales_LinearRegression_Model]
PREDICTION JOIN
    
    (SELECT 'Spring' AS [Seasonality], 1500 AS [AdvertisingSpend]) AS T
ON T.[Seasonality] = T.[Seasonality]
AND T.[AdvertisingSpend] = T.[AdvertisingSpend];

Advantages

Simple to understand and interpret.
Computationally efficient, especially for large datasets.
Provides clear insights into the linear relationship between variables.
Can identify which input features have the most significant impact.

Disadvantages

Assumes a linear relationship, which may not always hold true.
Sensitive to outliers.
May not perform well if there are complex non-linear interactions between variables.

SQL Server Analysis Services Documentation