Extreme Events: Examining the “Tails” of a Distribution

Although our engineering training treats all physics as deterministic, we also know that random variation is a normal part of nature. Strength of parts and loads on parts var. Unusually low strengths and unusually high loads do occur , for example a flood or a hurricane in the case of a building or a bridge, or a slug of liquid refrigerant in the case of a compressor. Accidents can occur when extreme events happen. Failure of a part occurs when the load on the part is greater than the strength. Extreme events happen much more frequently than predicted by theories based on the normal distribution. Statisticians describe extreme value distributions as “heavy tailed” as a result. In this paper, models of extreme values are discussed for both load and strength. Modeling examples are given for loads, strength of materials, applications to predicting time to failure and maintenance intervals. Extreme values are a part of our normal engineering lives. Introduction Most engineering problems are not, by their nature, completely deterministic. While deterministic physics may govern simple electrical circuits via Ohm’s law, neither the applied voltage nor the resistance is completely deterministic. Even the most basic electrical circuit, such as a light bulb, is subject to variation. Small differences in material properties and manufacturing affect the level of resistance of the light, even when the circuit is new. As the circuit ages, corrosion, wear and strain further affect the resistance. Material properties and age affect the voltage delivered by a battery powering the circuit. The result is that a nominally deterministic problem has many features of a problem with random variations. Human factors are another source of seemingly “random” variations. ASHRAE standard 55 (2010) , attempts to define the thermal environmental parameters that lead to comfort for human occupants. This problem is full of variation. First, in the same room environment, all occupants will have different levels of clothing, and will have different metabolic rates. Second, experiments (Fanger, 1972) have shown that people in the same environment, with the same clothing, at nominally the same metabolic rate, still do not respond identically to the question “are you too hot or too cold?” In a building, there are always multiple spaces (or zones), and each space is not identical, so there is further variation in the comfort of occupants. To overcome the problem of variation, engineers use “factors of safety” or other constants in expressions from “experience”. The notion is that the deterministic expressions are used for design, but then an added “margin” is given to account for the unknown variations in the load and strength of the structure. Since failure occurs when load is greater than strength, and the levels of load and strength are not truly deterministic, the question becomes what is the probability that load is greater than strength? In this question, the mean load and strength are not as important as the extreme values of load and strength. Unfortunately, classes in basic statistics focus on statistics for the mean and predicting the main effects for various factors. The distributions learned in basic statistics, such as the Gaussian or Normal distribution, that are valuable for predicting main effects are not suitable for predicting extreme values (see O‘Connor, 2002). Consider the thermal comfort problem of an entire building with many spaces. For simplicity, we will ignore the human factors and state that all people will react identically to the environment. Further, we will ignore radiation effects from the walls and through the windows. The “strength” variable in this example is temperature (to include radiation, one might use an operative temperature). The temperature in each zone will be slightly different and we will model it as random. The use of zoning, personal control, or other control strategies will certainly effect the standard deviation of the temperature, but will not affect the basic fact of variation all sensors and systems will have variation. The “load” variable is the combination of clothing and metabolic rate of the occupants that determines if they are comfortable. Consider the case of “too cold”: a person will be too cold if the combination of “clothing and metabolism” is too small for the 1 Is this probability acceptable? Although the consequences of failure will not be discussed here, its importance cannot be overemphasized. The level of analysis and/or the factor of safety used in design must be much larger for events that endanger life when compared to events that might make us uncomfortable. given temperature. The statistical problem is to determine how often someone is too cold in the building. It is important to predict the extremes of the distribution: how many people are wearing very light clothing, and how many rooms are much colder than average. Consider a second problem: is the strength of a beam is sufficient to hold a given load when both the beam strength and the load are subject to variation. Consider the charts in Figure 1. In Figure 1a, the load is much less than the strength, or using the thermal comfort problem, all people are dressed so that they will not be too cold (ignore the “too hot” problem). In Figure 1a, the probability of the beam breaking is the small gray shaded area where the two distributions intersect. Here, load is greater than strength, even though the average load is much smaller than strength. In the case of Figure 1a, there is a very, very small probability of failure. In Figure 1b, the strength is not sufficiently larger than the load and some fraction of time the load is larger than the strength and failure will occur. In Figure 1a and 1b, the naïve assumption of normal distributions was assumed for both load and strength. For simplicity, the standard deviation is assumed unity, but this assumption can be relaxed without any change in the conclusions. The normal distribution has the property that the tail of the distribution is very light that is a very small fraction of the population lies outside 3 standard deviations from the mean. Further, the normal distribution is symmetric, so the probability of an event a certain distance greater than the mean is equal to the probability of an event the same distance less than the mean. In Figure 1c, the same mean and standard deviation is assumed for both the load and strength, but for this Figure non-normal distributions are used. These distributions have the property that they are skewed rather than symmetric. The load distribution is assumed to be positively skewed. This occurs physically because many loads cannot be negative while in practice there is often little reason why the maximum load is limited, thus the distribution must be right skewed. The strength distribution shown is left-skewed, indicating that some items have a much lower strength than others. This is often the case because flaws will limit the strength of the item, where a completely unflawed (e.g. single crystal) item will have a maximum possible strength. Further, weakness might occur due to the natural effects of aging, which tend to create an upper limit for the strength value but no lower limit. The failure rate of the items in Figure 1c is seen to be larger than in Figure 1b. This is the case even though the mean and standard deviations are identical! The distributions in Figure 1c have the property that they have “heavy” tails compared with the normal distribution, which is a larger fraction of the population lies to the extremes relative to the normal distribution. This is quantified in Figure 1d, where the probability of an event larger than a given value is shown for the distributions in Figure 1b and 1c.