LINEAR REGRESSION

• We can simpify the previous equation by imposing a central difference type formulation of the problem, as shown below: The x-axis is relabeled in terms of equispaced sample indices, and centered about zero.
  
  
• This simplifies the calculation to:
• This equation is the form we desire, and is extremely efficient to compute. The denominator can be precomputed, and the integer multiplications are easily implemented even in fixed-point DSPs.
  
  
• Obviously, this approach can be extended to higher order derivatives. However, historically, second derivatives in speech recognition have been computed by applying two first-order derivatives in succession.
  
  
• Further, the order of regression used, N, is most commonly set to 2, which means a five-frame sequence of features is required to compute the first-order derivative.