Estimating the Derivative

As no direct measurements of the continuous derivative exist, it must be estimated from the samples. This is done using a simple difference form:

$$S_t'(\ensuremath{\mathbf{x}}) = \frac{S_t(\ensuremath{\mathb... ...suremath{\mathbf{d}})} {2 \Vert \ensuremath{\mathbf{d}}\Vert}.$$ (40)

The direction of the derivative is specified by the direction of the difference vector $\ensuremath{\mathbf{d}}$.

It is assumed that this approximation ( $S_t'(\ensuremath{\mathbf{x}}) \approx \frac{\partial S_t(\ensuremath{\mathbf{x}})}{\partial x}$) is sufficiently accurate so that estimation described in equation 36 is not significantly biased.