FormalDescriptions.md

Mathematical descriptions

A collection of detailed mathematical descriptions.

Index

• spikes_flagRaise
• spikes_flagSpektrumBased

spikes_flagRaise

The value x_{k} of a time series x with associated timestamps t_i, is flagged a rise, if:

1. There is any value x_{s}, preceeding x_{k} within raise_window range, so that:
   • M = |x_k - x_s | > thresh > 0
2. The weighted average \mu^* of the values, preceeding x_{k} within average_window range indicates, that x_{k} doesnt return from an outliererish value course, meaning that:
   • x_k > \mu^* + ( M / mean_raise_factor )
3. Additionally, if min_slope is not None, x_{k} is checked for being sufficiently divergent from its very predecessor x_{k-1}, meaning that, it is additionally checked if:
   • x_k - x_{k-1} > min_slope
   • t_k - t_{k-1} > min_slope_weight*intended_freq

The weighted average \mu^* was calculated with weights w_{i}, defined by:

• w_{i} = (t_i - t_{i-1}) / intended_freq, if (t_i - t_{i-1}) < intended_freq and w_i =1 otherwise.

The value x_{k} of a time series x_t with timestamps t_i is considered a spikes, if:

spikes_flagSpektrumBased

1. The quotient to its preceding data point exceeds a certain bound:
   • |\frac{x_k}{x_{k-1}}| > 1 + raise_factor, or
   • |\frac{x_k}{x_{k-1}}| < 1 - raise_factor
2. The quotient of the second derivative x'', at the preceding and subsequent timestamps is close enough to 1:
   • |\frac{x''_{k-1}}{x''_{k+1}} | > 1 - deriv_factor, and
   • |\frac{x''_{k-1}}{x''_{k+1}} | < 1 + deriv_factor
3. The dataset X = x_i, ..., x_{k-1}, x_{k+1}, ..., x_j, with |t_{k-1} - t_i| = |t_j - t_{k+1}| = noise_window fulfills the following condition: noise_func(X) < noise_thresh