# Difficulty with small samples! Don’t be superstitious about data and machine learning. Without data, you have to rely on basic scientific principles.

Design

I am convinced that the reliability theory brings new hope for solving the three major reliability engineering problems of mathematical expression of reliability, the difficulty of small samples and the mystery of the product theorem!

The story also starts from the Second World War.

During the Second World War, Germany took the lead in the germination of reliability engineering ideas. In 1943, during the development of the V1 rocket, the German rocket expert Robert Lusser proposed that the reliability of a system composed of N components is equal to the product of the respective reliability of the N components.

This calculation method uses the product theorem in probability theory. Robert Lusser calculated the reliability of the V1 rocket navigation device to be 0.75, which is the first time in history that people have expressed reliability in a quantitative way.

The calculation method proposed by Robert Lusser was later developed into the reliability calculation formula of the series system in the reliability theory. It is precisely because of his pioneering contribution in the field of reliability that people call him the “father of reliability.”

In the 1950s, reliability engineering developed vigorously in the United States. Facing the problem of frequent failures of military electronic equipment and extremely high maintenance costs, the US Department of Defense established the “Advisory Group on Reliability of Electronic Equipment, AGREE” in 1952 to implement comprehensive reliability development. plan.

In 1957, the advisory group issued the famous AGREE report. The report comprehensively explained the basic methods of reliability engineering from the perspectives of design, testing, production, and use, and became a foundational document for reliability development. Its release also marked that reliability has become an independent discipline.

Since then, the United States has formulated a series of reliability-related military standards and development specifications based on this report, which has greatly improved the reliability of US military equipment, and the basic theory of reliability has achieved comprehensive development.

In 1954, Japan began to import reliability technology and economic management technology from the United States. In the 1960s, Japan successively established the National Quality Committee and the Electronic Components Reliability Center to apply the reliability research results of the United States in the aviation, aerospace, and military industries to the civilian industry, especially the civilian electronics industry.

This series of measures has greatly improved the quality of Japanese civilian electronic products, creating a large number of Japanese national brands such as Toyota, Hitachi, and NSK, which have won a good quality reputation in the world.

Although reliability engineering has gradually matured after more than half a century of development, the basic theory of reliability engineering still has obvious shortcomings, which are mainly reflected in the definition and mathematical expression of reliability.

The AGREE report pointed out: Reliability refers to the ability of a product to complete a specified function under specified conditions and within a specified time. It can be seen that reliability is a kind of ability of a product, and this ability is closely related to the specified conditions, specified time and specified functions of the product.

The prescribed function refers to the specific content of the product serving human beings and is the basis of the product’s reliability capability. The prescribed time refers to the length of time that the product serves humans, and it needs to be determined according to the nature of human practical activities. The prescribed conditions refer to the various effects that products endure in the process of serving humans, including external environments such as temperature, humidity, and vibration, as well as internal loads such as electrical stress and mechanical stress, and interaction factors such as maintenance methods.

Reliability assumes the responsibility of quantifying product reliability capabilities. It is defined as the probability that a product completes a specified function under specified conditions and within a specified time.

Mathematically, it can be expressed as R(t)=Pr(T>t), R(t) is the reliability at time t, also known as the “reliability function”, t is the prescribed time, and T is the time before product failure, Pr{.} is a measure of probability.

For more than half a century, this definition and mathematical expression of reliability has been widely used in textbooks, monographs and industry standards. However, numerous scientific researches and engineering practices have increasingly proved that the mathematical expression of reliability is unreasonable and unsuitable. The team of Professor Rui of Beijing University of Aeronautics and Astronautics summarized it into three major problems.
The question of reliability function

The definition of reliability clearly points out that reliability is closely related to the “three regulations”, but why is there only one element of “specified time” in the reliability function? Why didn’t “prescribed conditions” and “prescribed functions” enter it? Obviously, such a reliability function fails to achieve a scientific expression of the definition of reliability.

Difficulties with small sample problems

Under this reliability function, it is necessary to collect the time data before the product failure, and calculate the reliability by the method of probability statistics. This implies a prerequisite for probability statistical methods-the “law of large numbers”. The law of large numbers states that when the sample size approaches infinity, the frequency approaches probability.

However, it is often difficult to obtain enough failure time data in actual engineering. “Small samples” are the norm and reality of reliability engineering practice. In other words, such a reliability function does not meet the actual requirements of a large number of projects.

The mystery of the product theorem

The calculation of reliability still continues Lusser’s law, which is actually caused by the product theorem of probability. Under this rule, if 30 components with a reliability of 0.9 are connected in series to form a system, the reliability of the system will quickly decay to 0.04!

At the beginning, Lusser warned that NASA’s lunar exploration project would definitely fail. However, it turns out that even for a series system, its reliability is not that low. Therefore, the product theorem under the probability measure is not suitable for reliability engineering.

In response to the above problems, I am convinced that the reliability theory has achieved three breakthroughs:

The first is to put forward the three reliability scientific principles of margin reliability, degeneration eternity and uncertainty, and establish discipline equations, margin equations, degeneration equations, and measurement equations from this, and organically integrate the “prescribed conditions” and “prescribed functions”. Incorporating into the reliability function realizes the mathematical expression based on scientific principles and answers the question of the reliability function.

The second is to introduce the “uncertainty theory” created by Professor Liu Baoding of Tsinghua University, an axiomatic mathematical system, into reliability measurement, and use the uncertainty measurement to solve the reliability evaluation problem in the small sample scenario;

The third is based on the product axiom of “measure the smaller” in the uncertainty theory, to solve the embarrassing situation of the probability measure product theorem in the practice of reliability engineering.