Zero-crossing statistics for non-Markovian time series

The problem

A bridge subjected to irregular loads will gradually be damaged. If it is not maintained, disastrous consequences will follow. It’s important, then, to know when maintenance is needed.  If we know the mean number of loads within a given time interval as well as the spread around the mean number of loads, then we can identify when to work on the bridge. The loads can be viewed as zero-crossings of a stochastic process.

To understand what we mean by zero-crossings, consider a coin tossing game between two players, Sofia and Peter.  When both Sofia and Peter have equal scores we say that a zero-crossing has occurred. Zero-crossings play an important role in many areas, for example, rapid DNA sequencing, edge detection in image analysis, speech recognition and machine learning. To better understand these applications we need to know how to calculate the statistics associated with the zero-crossing.

The solution

When studying zero-crossings, ‘moments’ are the quantities of statistical importance. They are embedded in the probability distribution P_m(n). This distribution gives the probability that m number of zero-crossings has occurred up to the n:th time step. The mean number of crossings is where P_m(n) is peaked, the first moment. The shape of P_m(n) such as the width of the peak, its skewness and tails are described by the higher moments. Together they constitute the pieces in the puzzle to find P_m(n). For a Gaussian and stationary process the first moment is known due to Rice. However, this is the only piece that exists even though zero-crossings has received great interest of both mathematicians and physicists over decades.

Combined with Rice’s result and a generalization of the independent interval approximation we are able to calculate the moments for a Gaussian and stationary process in discrete time. Our result is compared to simulations with excellent agreement for processes with a short memory.

Read the original paper

Markus Nyberg, Ludvig Lizana, and Tobias Ambjörnsson. Zero-crossing statistics for non-Markovian time series. Phys. Rev. E 97, 032114