Statistical Methods
Trace a convolution before interpreting a filtered series
Define the task
A convolution combines an input sequence with a specified kernel. Record the sample interval and the kernel order, then distinguish the output index from the original time index. A filter can shift a peak or include values outside the observed span depending on the selected output mode and boundary convention.
Carry out the check
For full discrete convolution, each input value contributes a shifted, scaled copy of the kernel. Add overlapping contributions. An input of length n and kernel of length m produces n+m−1 values. Full, same and valid output modes retain different parts of that result. A normalized smoothing kernel preserves a constant interior signal, but boundary padding can still change edge values.
Worked hypothetical example
For hypothetical input (1,2,3) and kernel (1,−1), the full convolution is (1,1,1,−3). The last value includes the assumed zero beyond the input and is not evidence of an observed drop of three. Write out each contributing product before interpreting an edge. Convolution also differs from correlation when the kernel is asymmetric, so retain the function name and mode in the analysis record.
Method checked 14 November 2026 against SciPy documentation: convolve.