CC BY
28
Mathematical
Structures and Modeling 2014. N. 4(32). PP. 49-52
UDC 519.2:681.2
HOW TO GAUGE UNKNOWN UNKNOWNS: A POSSIBLE THEORETICAL EXPLANATION OF THE USUAL SAFETY
FACTOR OF 2
J. Lorkowski
Ph.D. Student, e-mail: lorkowski@computer.org V. Kreinovich
Ph.D. (Math.), Professor, e-mail: vladik@utep.edu
University of Texas at El Paso, El Paso, TX 79968, USA
Abstract. To gauge the accuracy of a measuring instrument, engineers analyze possible factors contributing to the instrument's inaccuracy. In addition to known factors, however, there are usually unknown factors which also contribute to the instrument's inaccuracy. To properly gauge the instrument's accuracy — and thus, to make sure that we do not compromise our safety by underestimating the inaccuracy - we need to also take these "unknown unknowns" into account. In practice, this is usually done by multiplying the original estimate for inaccuracy by a "safety" factor of 2. In this paper, we provide a possible theoretical explanation for this empirical factor.
Keywords: safety factor, measurement accuracy, Laplace principle of indifference.
1. Safety Factor: Description of the Problem
What is a safety factor? When engineers design an accurate measuring instrument, they try their best to make it as accurate as possible. For this purpose, they apply known techniques to eliminate (or at least to drastically decrease) all major sources of error. For example:
thermal noise can be drastically decreased by cooling,
the effect of the outside electromagnetic fields can be drastically decreased
by barriers made of conductive or magnetic materials,
the effect of vibration can be decreased by an appropriate suspension, etc.
Once the largest sources of inaccuracy are (largely) eliminated, we need to deal with the next largest sources, etc. No matter how many sources of inaccuracy we deal with, there is always the next one. As a result, we can never fully eliminate all possible sources of inaccuracy.
50 J. Lorkowski, V. Kreinovich. How to Gauge Unknown Unknowns...
At each stage of the instrument design, we usually understand reasonably well what is the main source of the remaining inaccuracy, and how to gauge the corresponding inaccuracy. As a result, we have a good estimate for the largest possible value A of the inaccuracy caused by this source.
This value A, however, does not provide a full description of the measurement inaccuracy. Indeed, in addition to the "known unknown" (the known main source of remaining inaccuracy), there are also "unknown unknowns" — smaller additional factors that also contribute to measurement inaccuracy.
To take these "unknown unknowns" into account, practitioners usually multiply A by a larger-than-one factor. This factor is known as the safety factor.
A related idea is used when we design engineering objects such as roads, bridges, houses, etc. For example, when we design a bridge, we want to make sure that its deformation caused by different loads does not exceed the limit after which catastrophic changes may occur. For this purpose, we first estimate the deformation caused by the known factors, and then — to be on the safe side -multiply this deformation by a safety factor.
Safety factor of 2 is most frequently used. In civil engineering, a usual recommendation is to use the safety factor of 2; see, e.g., [6].
This value has been in used for more than 150 years: according to [5], the earliest recorded use of this value can traced to a book published in 1858 [3].
This value is also widely used beyond standard civil engineering projects: e.g., it was used in the design of Buran, a successful Soviet-made fully automatic pilotless Space Shuttle [2].
Comment. It should be mentioned that in situations requiring extreme caution — e.g., in piloted space flights - usually, a larger value of safety factor is used, to provide additional safety.
Open problem. The fact that the safety factor of 2 has been in use for more than 150 years shows that this value is reasonable. However, there seems to be no convincing explanation of why this particular value is empirically reasonable.
In this paper, we provide a possible theoretical explanation for this value.
2. Safety Factor of 2: Our Explanation
Analysis of the problem. We know that in addition to the largest inaccuracy of size A, there is also next largest inaccuracy of size A1 < A. Once we take that inaccuracy into account, then we will need to take into account the next inaccuracy, of size A2 < Ai, etc. The final inaccuracy can be estimated as the sum of all these inaccuracies, i.e., as the sum A + A1 + A2 + ..., where
... < Afc+i < Ak < ... < A2 < Ai < A. To estimate this sum, it is therefore necessary to estimate all the sizes A1, A2,
Let us estimate these sizes one by one.
Mathematical Structures and Modeling. 2014. N.4(32)
51
Estimating A^ The only information that we have about the value Ai is that it is larger than 0 and smaller than A. In other words, in principle, A1 can take any value from the interval (0, A). We have no information about the probabilities of different values from this interval.
Since we have no reason to think that some specific value from this interval is more probable and some other specific value is less probable, it is reasonable to assume that all the values from this interval are equally probable. This argument — known as Laplace principle of indifference — is widely used in statistics applications; see, e.g., [1,4]
In precise terms, this means that we assume that the distribution of possible values A1 on the interval (0, A) is uniform. For the uniform distribution on an interval, the expected value is the interval's midpoint. Thus, the expected value of Ai is equal to A/2.
This expected value is what we will use as an estimate for A1.
Estimating A2, etc. Once we produced an estimate A1 = A2, a next step is to estimate the next inaccuracy component A2. The only information that we have about the value A2 is that it is larger than 0 and smaller than A1. In other words, in principle, A2 can take any value from the interval (0, A1). We have no information about the probability of different values from this interval. Thus, similarly to the previous section, it is reasonable to assume that A2 is uniformly distributed on the interval (0, A1). In this case, the expected value of A2 is equal to A1/2. This expected value A2 = A1/2 is what we will use as an estimate for A2.
Similarly, we conclude that a reasonable estimate for A3 < A2 is A3 = A2/2, and, in general, that for every k, a reasonable estimate for Ak+1 < Ak is Ak+1 = = Afc/2.
Adding up all these estimates lead to the desired explanation. From A1 = A/2 and Ak+1 = Ak/2, we can conclude, by induction, that Ak = 2-k■ A. Substituting these estimates into the formula for the overall inaccuracy 5 = A+A1+A2+..., we conclude that
5 = A + 2-1 ■ A + 2-2 ■ A + ... = (1 + 2-1 + 2-2 + ...) ■ A.
The sum of the geometric progression 1 + 2-1 + 2-2 + ... is 2, so we get 5 = 2A.
This is exactly the formula that we tried to explain — with a safety factor of two.
Acknowledgments
This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721.
References
1. Jaynes E.T. Probability Theory: The Logic of Science. Cambridge University Press,
Cambridge, Massachisetts, 2003.
52 J. Lorkowski, V. Kreinovich. How to Gauge Unknown Unknowns.
2. Maslova N.B. Private communication.
3. Rankine W.J.M. A Manual on Applied Mechanics. Glasgow, Scotland: Richard Griffin and Company, 1858. P. 274. URL: https://archive.org/stream/ manualappmecha00rankrich#page/n8/mode/1up
4. Sheskin D.J. Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC Press, Boca Raton, Florida, 2011.
5. Smith R.A., Hillmansen S. A brief historical overview of the fatigue of railway axles // Proceeding of the Institute of Mechanical Engineering. 2004. V. 218. P. 267-277.
6. Ullman D. Mechanical Design Process. New York: McGraw-Hill, 2009. P. 314-317.
КАК ОЦЕНИТЬ «НЕИЗВЕСТНЫЕ НЕИЗВЕСТНЫЕ»: ВОЗМОЖНОЕ ТЕОРЕТИЧЕСКОЕ ОБЪЯСНЕНИЕ ИСПОЛЬЗОВАНИЯ КОЭФФИЦИЕНТА
«НАДЁЖНОСТИ» 2
Дж. Лорковский
аспирант, e-mail: lorkowski@computer.org В. Крейнович
к.ф.-м.н., профессор, e-mail: vladik@utep.edu
Техасский университет в Эль Пасо, США
Аннотация. Чтобы оценить точность измерительного прибора, инженеры анализируют возможные факторы, способные вызвать погрешность. Однако, в добавок к известным факторам, обычно существуют неизвестные факторы, которые также способны вызвать погрешность. Чтобы правильно оценить точность прибора и тем самым убедиться, что мы не рискуем надёжностью, недооценивая погрешность, мы должны также принимать во внимание эти «неизвестные неизвестные». На практике для этого изначальную оценку погрешности обычно умножают на коэффициент «надёжности» 2. В данной статье мы представляем возможное теоретическое объяснение для этого эмпирического коэффициента.
Ключевые слова: коэффициент надёжности, точность измерения, принцип безразличия Лапласа.
