Numerical Smoothing and Differentiation - Convolution and Correlation

Convolution and Correlation

Convolution affects the correlation between errors in the data. The effect of convolution can be expressed as a linear transformation.

By the law of error propagation, the variance-covariance matrix of the data, A will be transformed into B according to

To see how this applies in practice, consider the effect of a 5-point quadratic smoothing function for the first five calculated points, Y₃ - Y₇.

If one assumes that the data points have equal variance and that there is no correlation between them, A will be an identity matrix multiplied by a constant, σ2, the variance at each point. In that case, . The correlation coefficients, between calculated points i and j will be obtained by vector multiplication of rows i and j of C.

ρ₃₄ = (12×−3 + 17×12 + 12×17 + −3×12) / 352 = 0.56

ρ₃₅ = (-3×17 + 12×12 −3×17) / 352 = 0.034

ρ₃₆ = (-3×12 −3×12) / 352 = -0.058

ρ₃₇ = (-3×−3) / 352 = 0.0073

ρ₃₈ = 0

Thus, the calculated values are correlated even when the observed values are not correlated. The same pattern applies to the rest of the calculated points. The correlation extends over m-1 calculated points at a time.