Monte Carlo Integration Pattern Recognition Cosc 750a Direct

Monte Carlo Integrationpattern Recognition Cosc 750a Direct Applicat

Monte Carlo Integration pattern Recognition (COSC 750) A direct application for random number generation is in evaluating an integral. This approach is known as Monte Carlo Integration.

1. a) Evaluate the following integrals by the Monte Carlo method

b) Compute the area under the function given above by integration

c) Compare the areas obtained by method a) and b). If there is a discrepancy, explain the results.

2. Evaluate the following integral by Monte Carlo Integration

(a) For µ=0, σ=1

(b) For µ=10, σ=4

dx e^(-x²)

Paper For Above instruction

Monte Carlo integration is a computational technique that employs random sampling to numerically estimate integrals, particularly valuable in high-dimensional or otherwise challenging scenarios for traditional numerical methods. This method relies on generating random points within a specified domain and using statistical approaches to approximate the value of the integral. The application of Monte Carlo methods to pattern recognition illustrates its versatility and importance across various scientific and engineering disciplines, including machine learning, image analysis, and signal processing.

In this paper, we focus on the application of Monte Carlo integration techniques to evaluate definite integrals. The core concept involves generating random samples uniformly within a domain and computing the average value of the function at these samples. Multiplying this average by the measure of the domain yields an estimate of the integral. This technique is especially useful when the integral's domain is irregular or high-dimensional, where traditional analytical and numerical methods may falter.

Part 1: Evaluating integrals via Monte Carlo method

The first part of the problem involves evaluating specific integrals, computing their areas under the curve, and comparing these estimates with analytical solutions obtained through traditional integration methods.

Consider the example integral of a function, say, \(f(x) = e^{-x^2}\), over the domain \([-∞, ∞]\). For Monte Carlo estimation, the integral is approximated by simulating a large number of random points uniformly in a finite domain that captures most of the function's significant values, typically around \([-k,k]\) for some large k. The integral estimate is then the average of the function values at these points, multiplied by the total domain length.

Method (a): Monte Carlo evaluation

Assuming we evaluate the integral \(I = \int_{a}^{b} e^{-x^2} dx\) over finite bounds, say, \([-L, L]\), where \(L\) is sufficiently large, we generate N uniformly distributed random points within \([-L, L]\). For each point \(x_i\), compute \(f(x_i) = e^{-x_i^2}\), then estimate the integral as:

\[ I_{MC} = \frac{2L}{N} \sum_{i=1}^N e^{-x_i^2} \]

where \(\frac{2L}{N}\) is the measure of the sampled domain.

Method (b): Analytical computation

The integral of \(e^{-x^2}\) over the entire real line is well-known:

\[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \]

For finite bounds, the integral can be expressed with the error function:

\[ \int_{a}^{b} e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \left[ \operatorname{erf}(b) - \operatorname{erf}(a) \right] \]

Comparison and discussion

By comparing the results from Monte Carlo estimation and analytical solutions, we can analyze the accuracy of the Monte Carlo method. Discrepancies arise due to the stochastic nature of sampling, the finite number of samples, and the choice of domain bounds. Increasing the number of samples N reduces the variance, thus improving the estimate’s accuracy, while selecting sufficiently large bounds minimizes truncation errors. The convergence of Monte Carlo estimates is probabilistic and follows the law of large numbers.

Part 2: Monte Carlo evaluation for different parameters

The second part involves evaluating the integral of \(e^{-x^2}\) over the entire real line for different parameters \(\mu\) and \(\sigma\), interpreted as a Gaussian distribution:

• (a) \( \mu=0, \sigma=1 \)
• (b) \( \mu=10, \sigma=4 \)

Given the parameters, and recognizing that the integrand resembles a Gaussian with these parameters, the integral to estimate is:

\[ I = \int_{-\infty}^{\infty} e^{-\frac{(x - \mu)^2}{2\sigma^2}} dx \]

which analytically equates to:

\[ I = \sigma \sqrt{2\pi} \]

Applying Monte Carlo integration, the approach involves generating random samples from the specified Gaussian distributions, then averaging the function values at these samples, scaled by the domain's measure, similar to previous steps. Alternatively, because the Gaussian's integral over the entire real line has a known closed form, it serves as a benchmark for Monte Carlo estimates.

Discussion on results

Comparing Monte Carlo estimates for the Gaussian integrals with the theoretical values demonstrates the method's robustness and limitations. For \(\mu=0, \sigma=1\), the known value is \(\sqrt{2\pi}\). For \(\mu=10, \sigma=4\), it is \(4\sqrt{2\pi}\). As the samples increase, the Monte Carlo estimates should converge towards these known values, illustrating the method's consistency. Variability in estimates depends on the number of samples and the distribution's variance.

Conclusion

Monte Carlo integration proves to be an effective computational tool for approximating integrals, especially where traditional analytical methods are infeasible. Its application to integrals involving the Gaussian distribution highlights both its strengths and limitations, emphasizing the importance of sample size, choice of domain, and variance reduction techniques. As computational power advances, Monte Carlo methods continue to expand their utility in scientific research, pattern recognition, financial modeling, and beyond.

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