A weighted sum multiplies each input by a corresponding coefficient, then adds the products. For inputs x₁ and x₂ with weights w₁ and w₂, the result is w₁x₁ + w₂x₂. Changing a weight changes that input’s contribution. section 8.2.2
In a perceptron, the weighted sum represents the combined strength of the inputs. An activation function uses that value to produce an output, which can feed another neuron. A bias is a separate constant term; a simple threshold example produces zero below the threshold and one above it. section 8.2.2 section 8.2.2
A weighted sum also creates new variables in principal component analysis (PCA). PCA combines original variables into principal components, choosing the first to capture as much variation in the data as possible. Later components are orthogonal to earlier ones. section 3.5
(3.1)
new_variable = a*area – b*bedrooms + c*distance – d*schools
The home-variable equation illustrates a new variable formed by adding and subtracting weighted inputs. Its coefficients are illustrative, not fitted values. section 3.5
