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Inbunden Engelska, Spara som favorit. Skickas inom vardagar. Integrals are one of the primary computational tools in mathematics, and hence are of great importance in just about any numerate discipline, including statistics, mathematical finance, computer science and engineering. Although it is occasionally possible to compute integrals exactly this is typically not the case. In these situations it becomes necessary to approximate integrals. This book covers all the most useful approximation techniques so far discovered; the first time that all such techniques have been included in a single book and at a level accessible for students.

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If we evaluate the function at x2, we over estimate the area. It's not surprising in a way as the rectangles which are too large compensate for the rectangles which are too small. And in fact, we will soon give the proof that summing them up and averaging their areas actually converges to the integral "area" as the number of samples used in the calculation increases. This idea is illustrated in the following figure. The function was evaluated in four different locations.

The result of the function as these four values of x randomly chosen, are then multiplied by b-a , summed up and averaged we divide the sum by 4. The result can be considered as an approximation of the actual integral. Of course, as usual with Monte Carlo methods, this approximation converges to the integral result as the number of rectangles or samples used increases. Where N here, is the number of samples used in this approximation.

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Monte Carlo integration

This equation is called a basic Monte Carlo estimator. The random point in the interval [a,b] can easily be obtained by multiplying the result of a random generator producing uniformly distributed numbers in the interval [0,1] with b-a :. Since the random numbers are produced with equiprobability each number is produced with the same probability than the others , we just divide 1 by the total number of outcomes as in the case of a dice.

However in this example, the function is continuous as opposed to discrete , so we divide 1 by the interval [a,b].


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The law of large numbers which we talked in lesson 16, tells us that as N approaches infinity, our Monte Carlo approximation converges in probabiliy to the right answer the probabiliy is 1. Now, as mentioned above, the formula we used for the Monte Carlo estimator is basic.

[R] two-dimensional integration?

Because it only works if the PDF of the random variable X is uniform. The more generic formula is then:. This is the more generalized form of the Monte Carlo estimator, and the one you should remember if there's only one equation to remember from the last two chapters, it is the one. As with the basic Monte Carlo estimator, to be sure that this formula is valid, we need to check that this estimator has the correct expected value. Let's check:. Take the time to understand these equations. As we just said, this is the most important result of everything we have studied so far, and is the backbone of almost every algorithm we are going to study in the next lessons.

If you don't understand this algorithm, you won't understand monte carlo ray tracing.


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With the rendering equation this is probably the second most important equation. Now, you will ask why would I ever want to draw samples from any other distribution than a uniform distribution? And that would be a very good question. The weak answer is "because maybe you can only use a given random generator to produce samples and that this generator has a non-uniform PDF".

But you will also see that this result will become handy when we will study variance reduction in the next chapter.

So keep reading and you will soon understand why this result is important! As you can see, a Monte Carlo estimation is nothing else than a sample mean, only, we substitute the population for a real-value arbitrary function. For this reason, Monte Carlo estimations and sample means share the same properties:.

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Some additional remarks can be made. One of the most popular of these techniques is important sampling.

Generalization to Arbitrary PDF

First some notation. We make a slight change in our notation for expectation to emphasize the corressponding density. The main idea here is to take samples from a different distribution so as to reduce variance of the estimator. Consider the following motivating example from MMT page However, because this is a toy example let us cheat and use R.

Now, let us use Monte Carlo with this expectation!

Generalization to Arbitrary PDF

Introduction Expectation! Importance Sampling. Introduction Last lecture we saw how to generate random samples from probability distributions.