The unified model is a mathematical framework showing that many regression algorithms can be expressed in a single, general form.
Mathematical Formulation
Components
- : Basis functions (typically Gaussian/RBF)
- : Weight vectors for local linear models
- : Bias/offset terms
- : Number of local models
- : Parameters of basis functions (e.g., mean, variance)
Deconstructing the Model
For any given , the model calculates which expert is most relevant and gives its output more weight in the final sum.
Prediction from Expert This is the simple linear model ( ). Each of the “experts” is a linear model with its own weight vector and bias .
Weight for Expert This is the gating function . It calculates a value based on the input . Typically, these gating functions are designed to be high for some regions of the input space and low for others.
Relating Models to the Unified Model
1. Simple Linear Regression
- Setting: , (constant everywhere)
- Result:
- Interpretation: One global linear model, no local weighting
2. RBF (Radial Basis Function) Networks
- Setting: (zero weight vectors), keep only
- Result:
- Interpretation: Weighted sum of constants, each Gaussian holds one value
3. Weighted Regression (WR)
- Setting: , but varies with input
- Result:
- Interpretation: Single linear model weighted by input-dependent function
4. Locally Weighted Regression (LWR)
- Setting: Full unified model with all parameters
- Result:
- Interpretation: Multiple local linear models, each weighted by Gaussian
Key Relationships:
- RBF ⊂ LWR: RBF is special case of LWR with
- Linear Regression ⊂ LWR: Linear regression is LWR with ,
- LWR = Most General: Can represent all others by parameter choices
Hierarchy of complexity
Linear Regression → RBF → LWR