Linear Regression Algorithm
Table of Contents
The Linear Regression algorithm in SQL Server Analysis Services (SSAS) is used to predict a continuous numerical value based on a set of independent variables. It's a fundamental technique in predictive analytics, widely used for forecasting and understanding relationships between variables.
Introduction
Linear regression models the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. The goal is to find the coefficients of this linear equation that best explain the variation in the dependent variable.
In SSAS, the Linear Regression algorithm can be used to:
- Forecast sales based on advertising spend.
- Predict housing prices based on features like size, location, and number of bedrooms.
- Estimate customer lifetime value based on demographics and purchase history.
- Analyze the impact of various factors on a continuous outcome.
How the Algorithm Works
The algorithm works by finding a linear combination of input attributes that best predicts the output attribute. The core of the algorithm involves calculating the coefficients for the linear equation:
Y = b0 + b1*X1 + b2*X2 + ... + bn*Xn
Where:
Yis the predicted dependent variable.X1, X2, ..., Xnare the independent variables.b1, b2, ..., bnare the coefficients calculated by the algorithm.b0is the intercept (the value of Y when all Xs are zero).
SSAS uses a sophisticated method to find these coefficients, typically employing techniques like Ordinary Least Squares (OLS) or variations thereof, to minimize the sum of the squared differences between the observed and predicted values.
Algorithm Parameters
The Linear Regression algorithm in SSAS offers several parameters to control its behavior and performance:
| Parameter | Description | Default |
|---|---|---|
MAX_GRADIENT_HEIGHT |
Specifies the maximum allowed gradient height for the model. Affects model complexity and training time. | 100000 |
MAX_RESPONSE_ பணிகள் |
Sets the maximum number of weights per attribute that the algorithm can consider. | 10 |
PRIZING_RATE |
A regularization parameter to prevent overfitting by penalizing large coefficients. | 0.0001 |
SAMPLE_SIZE |
Controls the proportion of the training data used to build the model. A larger sample size can improve accuracy but increase training time. | 1.0 (100%) |
REGULARIZATION_COEFFICIENT |
Another regularization parameter, similar to PRIZING_RATE, influencing the model's complexity. |
0.01 |
Using Parameters
These parameters can be set when creating a mining model using DMX (Data Mining Extensions) or through the SQL Server Data Tools (SSDT) interface.
ALTER MINING MODEL [MyLinearRegressionModel] WITH PARAMETERS (
MAX_RESPONSE_ பணிகள் = 20,
PRIZING_RATE = 0.001
)
Mining Model Content
The content of a Linear Regression mining model reveals the discovered relationships and coefficients. Key components include:
- Coefficients: The numerical weights assigned to each independent variable.
- Intercept: The constant term in the linear equation.
- Attribute Importance: Measures how much each attribute contributes to the prediction.
- R-squared Value: Indicates the proportion of the variance in the dependent variable that is predictable from the independent variables.
You can query the mining model content using DMX:
SELECT
ATTRIBUTENAME,
RELEVANCE,
VALUETYPE
FROM
[MyLinearRegressionModel].VARIABLES
WHERE
(PATH IS NULL OR PATH = 'Value')
Usage Examples
Here are common scenarios where Linear Regression is applied:
Forecasting Sales
Predicting future sales figures based on historical sales data, marketing expenditure, seasonality, and economic indicators.
Price Prediction
Estimating the selling price of a product or service by considering factors like features, target audience, competitor pricing, and market demand.
Risk Assessment
Assessing the likelihood of an event (e.g., loan default, customer churn) by analyzing contributing factors and their linear impact on the probability.