Unified Model

The unified model is a mathematical framework showing that many regression algorithms can be expressed in a single, general form.

Mathematical Formulation

$$t(x)=\sum _{e=1}^E{\varphi }_e(x,{\theta }_e)(w_e^{T}x+b_e)$$

Components

• ${\varphi }_e(x,{\theta }_e)$: Basis functions (typically Gaussian/RBF)
• $w_e^T$: Weight vectors for local linear models
• $b_e$: Bias/offset terms
• $E$: Number of local models
• ${\theta }_e$: Parameters of basis functions (e.g., mean, variance)

Deconstructing the Model

For any given $x$ , the model calculates which expert is most relevant and gives its output more weight in the final sum.

$$t(x)=\sum _{e=1}^E\text{Weight for Expert}(e)\times \text{Prediction from Expert}(e)$$

Prediction from Expert This is the simple linear model ( $w_e^{T}x+b_e$ ). Each of the $E$ “experts” is a linear model with its own weight vector $w_e$ and bias $b_e$.

Weight for Expert This is the gating function ${\varphi }_e\left(x,{\theta }_e\right)$. It calculates a value based on the input $x$. Typically, these gating functions are designed to be high for some regions of the input space and low for others.