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2.1.1: Statistical Estimation and Simulation

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    Random Models and Phenomena

    In science and engineering environments, we often encounter experiments whose outcome cannot be determined with certainty in practice and is better described as random. An example of such a random experiment is a coin flip. The outcome of flipping a (fair) coin is either heads (H) or tails (T), with each outcome having equal probability. Here, we define the probability of a given outcome as the frequency of its occurrence if the experiment is repeated a large number of times. \({ }^{1}{ }^{-}\)In other words, when we say that there is equal probability of heads and tails, we mean that there would be an equal number of heads and tails if a fair coin is flipped a large (technically infinite) number of times. In addition, we expect the outcome of a random experiment to be unpredictable in some sense; a coin that consistently produces a sequence HTHTHTHT or HHHTTTHHHTTT can be hardly called random. In the case of a coin flip, we may associate this notion of unpredictability or randomness with the inability to predict with certainty the outcome of the next flip by knowing the outcomes of all preceding flips. In other words, the outcome of any given flip is independent of or unrelated to the outcome of other flips.

    While the event of heads or tails is random, the distribution of the outcome over a large number of repeated experiments (i.e. the probability density) is determined by non-random parameters. In the case of a coin flip, the sole parameter that dictates the probability density is the probability of heads, which is \(1 / 2\) for a fair coin; for a non-fair coin, the probability of heads is (by definition) different from \(1 / 2\) but is still some fixed number between 0 and 1 .

    Now let us briefly consider why the outcome of each coin flip may be considered random. The outcome of each flip is actually governed by a deterministic process. In fact, given a full description of the experiment - the mass and moment of inertia of the coin, initial launch velocity, initial angular momentum, elasticity of the landing surface, density of the air, etc - we can, in principle, predict the outcome of our coin flip by solving a set of deterministic governing equations - Euler’s equations for rigid body dynamics, the Navier-Stokes equations for aerodynamics, etc. However, even for something as simple as flipping a coin, the number of variables that describe the state of the system is very large. Moreover, the equations that relate the state to the final outcome (i.e.

    \({ }^{1}\) We adhere to the frequentistic view of probability throughout this unit. We note that the Baysian view is an alternative, popular interpretation of probability. heads or tails) are complicated and the outcome is very sensitive to the conditions that govern the experiments. This renders detailed prediction very difficult, but also suggests that a random model - which considers just the outcome and not the myriad "uncontrolled" ways in which we can observe the outcome - may suffice.

    Although we will use a coin flip to illustrate various concepts of probability throughout this unit due to its simplicity and our familiarity with the process, we note that random experiments are ubiquitous in science and engineering. For example, in studying gas dynamics, the motion of the individual molecules is best described using probability distributions. Recalling that 1 mole of gas contains approximately \(6 \times 10^{23}\) particles, we can easily see that deterministic characterization of their motion is impractical. Thus, scientists describe their motion in probabilistic terms; in fact, the macroscale velocity and temperature are parameters that describe the probability distribution of the particle motion, just as the fairness of a given coin may be characterized by the probability of a head. In another instance, an engineer studying the effect of gust on an airplane may use probability distributions to describe the change in the velocity field affected by the gust. Again, even though the air motion is well-described by the Navier-Stokes equations, the highly sensitive nature of turbulence flows renders deterministic prediction of the gust behavior impractical. More importantly, as the engineer is most likely not interested in the detailed mechanics that governs the formation and propagation of the gust and is only interested in its effect on the airplane (e.g., stresses), the gust velocity is best described in terms of a probability distribution.

    Statistical Estimation of Parameters/Properties of Probability Distributions

    Statistical estimation is a process through which we deduce parameters that characterize the behavior of a random experiment based on a sample - a set of typically large but in any event finite number of outcomes of repeated random experiments. \({ }^{2}{ }^{2}\) In most cases, we postulate a probability distribution - based on some plausible assumptions or based on some descriptive observations such as crude histogram - with several parameters; we then wish to estimate these parameters. Alternatively, we may wish to deduce certain properties - for example, the mean - of the distribution; these properties may not completely characterize the distribution, but may suffice for our predictive purposes. (In other cases, we may need to estimate the full distribution through an empirical cumulative distribution function; We shall not consider this more advanced case in this text.) In Chapter \(\underline{9}\), we will introduce a variety of useful distributions, more precisely parametrized discrete probability mass functions and continuous probability densities, as well as various properties and techniques which facilitate the interpretation of these distributions.

    Let us illustrate the statistical estimation process in the context of a coin flip. We can flip a coin (say) 100 times, record each observed outcome, and take the mean of the sample - the fraction which are heads - to estimate the probability of heads. We expect from our frequentist interpretation that the sample mean will well approximate the probability of heads. Note that, we can only estimate - rather than evaluate - the probability of heads because evaluating the probability of heads would require, by definition, an infinite number of experiments. We expect that we can estimate the probability of heads - the sole parameter dictating the distribution of our outcome - with more confidence as the sample size increases. For instance, if we wish to verify the fairness of a given coin, our intuition tells us that we are more likely to deduce its fairness (i.e. the probability of heads equal to \(0.5\) ) correctly if we perform 10 flips than 3 flips. The probability of landing HHH using a fair coin in three flips - from which we might incorrectly conclude the coin as unfair - is \(1 / 8\), which is not so unlikely, but that of landing HHHHHHHHHH in 10 flips is less than 1 in 1000 trials, which is very unlikely.

    \({ }^{2}\) We will provide a precise mathematical definition of sample in Chapter \(10 .\) In Chapters 10 and 11 , we will introduce a mathematical framework that not only allows us to estimate the parameters that characterize a random experiment but also quantify the confidence we should have in such characterization; the latter, in turn, allows us to make claims - such as the fairness of a coin - with a given level of confidence. We consider two ubiquitous cases: a Bernoulli discrete mass density (relevant to our coin flipping model, for example) in Chapter 10 ; and the normal density in Chapter \(11 .\)

    Of course, the ultimate goal of estimation is inference - prediction. In some cases the parameters or quantities estimated, or corresponding "hypothesis tests," in fact suffice for our purposes. In other cases, we can apply the rules of probability to make additional conclusions about how a system will behave and in particular the probability of certain events. In some sense, we go from a finite sample of a population to a probability "law" and then back to inferences about particular events relevant to finite samples from the population.

    Monte Carlo Simulation

    So far, we have argued that a probability distribution may be effectively used to characterize the outcome of experiments whose deterministic characterization is impractical due to a large number of variables governing its state and/or complicated functional dependencies of the outcome on the state. Another instance in which a probabilistic description is favored over a deterministic description is when their use is computationally advantageous even if the problem is deterministic.

    One example of such a problem is determination of the area (or volume) of a region whose boundary is described implicitly. For example, what is the area of a unit-radius circle? Of course, we know the answer is \(\pi\), but how might we compute the area if we did not know that \(A=\pi r^{2}\) ? One way to compute the area may be to tessellate (or discretize) the region into small pieces and employ the deterministic integration techniques discussed in Chapter 7 . However, application of the deterministic techniques becomes increasingly difficult as the region of interest becomes more complex. For instance, tessellating a volume intersected by multiple spheres is not a trivial task. More generally, deterministic techniques can be increasingly inefficient as the dimension of the integration domain increases.

    Monte Carlo methods are better suited for integrating over such a complicated region. Broadly, Monte Carlo methods are a class of computational techniques based on synthetically generating random variables to deduce the implication of the probability distribution. Let us illustrate the idea more precisely for the area determination problem. We first note that if our region of interest is immersed in a unit square, then the area of the region is equal to the probability of a point drawn randomly from the unit square residing in the region. Thus, if we assign a value of 0 (tail) and 1 (head) to the event of drawing a point outside and inside of the region, respectively, approximating the area is equivalent to estimating the probability we land inside (a head). Effectively, we have turned our area determination problem into an statistical estimation problem; the problem is now no different from the coin flip experiment, except the outcome of each "flip" is determined by performing a (simple) check that determines if the point drawn is inside or outside of the region. In other words, we synthetically generate a random variable (by performing the in/out check on uniformly drawn samples) and deduce the implication on the distribution (in this case the area, which is the mean of the distribution). We will study Monte-Carlo-based area integration techniques in details in Chapter 12 .

    There are several advantages to using Monte Carlo methods compared to deterministic integration approaches. First, Monte Carlo methods are simple to implement: in our case, we do not need to know the domain, we only need to know whether we are in the domain. Second, Monte Carlo methods do not rely on smoothness for convergence - if we think of our integrand as 0 and 1 (depending on outside or inside), our problem here is quite non-smooth. Third, although Monte Carlo methods do not converge particularly quickly, the convergence rate does not degrade in higher dimensions - for example, if we wished to estimate the volume of a region in a threedimensional space. Fourth, Monte Carlo methods provide a result, along with a simple built-in error estimator, "gradually" - useful, if not particularly accurate, answers are obtained early on in the process and hence inexpensively and quickly. Note for relatively smooth problems in smooth domains Monte Carlo techniques are not a particularly good idea. Different methods work better in different contexts.

    Monte Carlo methods - and the idea of synthetically generating a distribution to deduce its implication - apply to a wide range of engineering problems. One such example is failure analysis. In the example of an airplane flying through a gust, we might be interested in the stress on the spar and wish to verify that the maximum stress anywhere in the spar does not exceed the yield strength of the material - and certainly not the fracture strength so that we may prevent a catastrophic failure. Directly drawing from the distribution of the gust-induced stress would be impractical; the process entails subjecting the wing to various gust and directly measuring the stress at various points. A more practical approach is to instead model the gust as random variables (based on empirical data), propagate its effect through an aeroelastic model of the wing, and synthetically generate the random distribution of the stress. To estimate the properties of the distribution such as the mean stress or the probability of the maximum stress exceeding the yield stress - we simply need to use a large enough set of realizations of our synthetically generated distribution. We will study the use of Monte Carlo methods for failure analysis in Chapter 14 .

    Let us conclude this chapter with a practical example of area determination problem in which the use of Monte Carlo methods may be advantageous.

    This page titled 2.1.1: Statistical Estimation and Simulation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Masayuki Yano, James Douglass Penn, George Konidaris, & Anthony T Patera (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.