Algorithm for error-free determination of the variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data

Abstract

The article presents an algorithm for fast and error-free determination of statistics such as the arithmetic mean and variance of all contiguous subsequences and fixed-length contiguous subsequences for a sequence of industrial measurement data. Additionally, it shows that both floating-point and integer representation can be used to perform this kind of statistical calculations. The author proves a theorem on the number of bits of precision that an arithmetic type must have to guarantee error-free determination of the arithmetic mean and variance. The article also presents the extension of Welford’s formula for determining variance for the sliding window method—determining the variance of fixed-length contiguous subsequences. The section dedicated to implementation tests shows the running times of individual algorithms depending on the arithmetic type used. The research shows that the use of integers in calculations makes the determination of the aforementioned statistics much faster.

Application of python libraries for variance, normal distribution and Weibull distribution analysis in diagnosing and operating production systems

“…In practice, it means that for a small number of 10 4 measurements and value 𝑤 = 100 it is necessary to perform 10 10 operations. Therefore, to determine the entire set of variances, we will use the approach presented in Lemma 1 [17]. It will allow to reduce the computational complexity of the problem under consideration to the level of 𝑂(𝑛 2 /𝑤), which will significantly improve the efficiency of calculations.…”

“…This formula will be used in the initial stage of calculations -until the sequence s reaches the size 2𝛿 + 1. In the next stage, the window method presented in [17]…”

“…The above-mentioned relationships allow for the definition of a computationally effective algorithm, which was presented in [17].…”

“…This example shows how complicated the situation is when there is no information about the potential location of an anomaly. The publication [17] gives some intuitions about trying to solve this problem from the algorithmic point of view, but it can be said that this is only the discovery of the tip of the iceberg. It should be emphasized that the level of complexity of such problems increases significantly when data matrices appear instead of simple sequences/vectors.…”

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