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Conceived and designed the experiments: RS FG. Performed the experiments: RS FG. Analyzed the data: FG. Contributed reagents/materials/analysis tools: RS. Wrote the paper: RS FG.

The authors have declared that no competing interests exist.

Staircase-like structures in the log-log correlation plot of a time series indicate patterns against a noisy background, even under condition of strong jitter. We analyze the method for different jitter-noise-combinations, using quantitative criteria to measure the achievement by the method. A phase diagram shows the remarkable potential of this method even under very unfavorable conditions of noise and jitter. Moreover, we provide a novel and compact analytical derivation of the upper and lower bounds on the number of steps observable in the ideal noiseless case, as a function of pattern length and embedding dimension. The quantitative measure developed combined with the ideal bounds can serve as guiding lines for determining potential periodicity in noisy data.

The detection of patterns against a noisy signal background is a particularly important task for engineering and neuroscience

Given a time series

Given a time series generated from a noise-free pattern of length

For

For

These results extend and detail insights from previous approaches

By searching for steps, we can not only pin down data that are likely to contain patterns. With the help of the table presented in

Lower bound

To what extent is the method reliable? In realistic time series, especially in neuroscience, a regular signal will be contaminated by jitter and noise. Jitter is commonly defined as the addition of an amount of signed (or unsigned) noise to the signal. Under its influence, a period-three signal of interspike intervals (ISIs)

Upon the addition of jitter and noise, the steps gradually smear out and finally may no longer be visible. An example of a log-log plot displaying a step-like behavior is shown in

a)

In the log-log plot, jitter predominantly smoothens out the steps, whereas noise decreases the heights as well as the slopes of the stairs. We assess the ability of our method to highlight regular patterns of length

For the first criterion, we verified whether the predicted decrease of the number of steps as a function of

For the second criterion, in order to quantify the visibility of exactly three derivative peaks at

For all criteria assessments (a), (b) and (c), we approximated the derivative values as difference quotients between two consecutive data points, for which

By dividing through the observed maximal measure, the three measures were normalized and a contour-plot with suitable contours was drawn.

Fulfillment of the criteria is expressed by three degrees: Region I: excellent, region II: fair, region III: ambiguous. a) Measure for the decrease in steps with

For a proof of (2)–(5), we decompose the graph of componentwise distances

Decomposition of potential distances in the maximum norm for odd and for even pattern lengths

To summarize, we emphasize the remarkable performance of the method under very noisy conditions. As a general advice (generally true for time series analysis!) we propose not to rely on one single criterion, but to combine all aspects to obtain a coherent picture. The reader may thus derive an overall goodness-of-method measure by adding the measures obtained from the different criteria. This might help to

All computations were performed in a C++ and