Project 6 - Calculating π and using building models for data

Transcription

1 Project 6 - Calculating π and using building models for data MTH337 - Fall 2018 Due date: November 29, 2018 Instructor - Rahul Kashyap Part 1 - Calculating the value of π The mathematical constant π is quite well-known and heavily used. This number also has a long history going 4000 years back in time. Some information is available here. Calculating the value of π is a hard thing to do. For one, it is irrational. Further, it is believed the digits might be randomly distributed. But, it still remains an important number because of its heavy use in the sciences. A lot of theories/models would not work without the use of π. Even if they could, π occurs very naturally in their interpretation. For instance, consider simple harmonic motion. A simple sinusoidal motion from a linear force turns out to be a 1D projection of circular motion in 2D, hence we find a π in the solutions. We will try to calculate the value of π using a few different methods. The first method is using definite integrals. The integral x2 dx = π 2 (1) 1. Evaluate this integral using the trapezoidal rule. Then calculate the value of π. 2. This integral can also be calculated by using a Monte-Carlo method. Remember, a Monte-Carlo method relies on randomly sampling and then averaging to estimate the right answer. In other words, if you want to buy a product on Amazon, you would not read only one review. You would read a bunch of reviews of all types and then form an opinion and make a decision. Use the random number generation and Monte-Carlo integration method we discussed in class to calculate this integral. 3. Which of the two calculations do you think is better? You would evaluate this using a variety of criteria like speed, convergence, precision etc. Keep in mind that with the Monte-Carlo methods, you will get a slightly different answer every time. This is due to the random sampling. 1

2 4. Another way to calculate π is to consider a square of side 1. In this square, a quadrant of a circle with radius 1 can be inscribed. Generate pairs of random numbers on the square. Each pair of random numbers can be considered to be the (x, y) coordinates of that point on the square (Python generates random numbers in the interval [0,1)). Now, check which of these points fall within the circle. The ratio of the points within the circle to the total generated points must be the same as the ratio of the areas of the quarter circle to the square. From this ratio, calculate π. Try out different number of points. Try few hundreds few thousands, tens of thousands etc. 5. Calculate π in two other ways of your own choice. The Wikipedia page on π has some examples. You can use other examples if you wish. But for each example you use, I want to see proper background and explanation. If I see a method implemented but not explained convincingly, I will not consider it. Don t forget to cite your sources and remember, no copy pasting. Also, avoid mistakes like using the value of π somewhere in the calculation implicitly. Some of these mistakes might not be very obvious sometimes. Bonus Quantify the error from your calculations more rigorously. For instance, calculate the relative error in π given by (π numerical π actual )/π actual for the different integration algorithms and check how it scales with the number of iterations. By scaling, I mean, in what specific manner does the error depend on the number of iterations? These arguments must be quantitative i.e. the error goes as square root of iterations or inverse of square root of iterations etc. It is obvious that increasing iterations will cause error to go down in a general manner. The question is exactly how in a mathematical sense. Part 2 - Estimating parameters of simple harmonic oscillations If you remember the simple harmonic oscillator we saw previously in class, its equation is given as d 2 x dt 2 = ω 2 x. (2) For the initial conditions dx/dt = Aω at t = 0, we get the exact solution for the equations as x = A sin (ωt) (3) 2

3 The solution describes sinusoidal oscillations of a particle with time. The frequency is given by ω and the size of the oscillations is given by A. In other words, ω relates to how quickly the object goes from one end to another and A relates to how large the swing is on each side. In the data file provided to you, I have given you measured data on a simple harmonic oscillator i.e. there is column for time t and there is a column for the position x. What we will do now, is figure what the amplitude and the frequency are by fitting a curve of the form x = β 0 sin (β 1 t). (4) The parameters β 0 and β 1 are the best estimates of the values of A and ω respectively. For this part, we will use the Particle Swarm Optimization (PSO) algorithm. PSO is a meta-heuristic algorithm based on Monte-Carlo type sampling but with intelligent behavior instead of functioning entirely randomly [1]. It mimics a flock of birds searching for food. So, imagine a flock of birds searching for the best solution in your parameter space, i.e you are scanning a space of β 0 and β 1 for the best (lowest) corresponding error. They find this best solution by a combination of random sampling, personal memories and communication with each other and keeping track of how the entire flock/swarm is doing. A combination of these methods typically lets these birds approach the best set of parameters on the error landscape. The Wikipedia article on PSO is a very nice and probably the simplest introduction to the algorithm that I have seen. This page has more information on swarming and swarm intelligence in general. 1. Write an error function for the simple harmonic oscillator(very similar to the error function for the free-fall data we wrote in class). The function would take in the name of the data file, called sinedata.txt, and read in the values of t and x into two arrays. It must also accept a set of values for beta (given as a 1D array). Using these values of β j, it must be able to compute the error S between the model and the data as S = N (β 0 sin (β 1 t i ) x i ) 2, (5) i where t i and x i are the ith value of t and x in the columns. 2. You would have noticed that conventional least-squares using derivatives here will probably not work. If one writes down the minimization equations based on the derivaties, the equations look like 3

4 S β 0 = S β 1 = N 2(β 0 sin (β 1 t i ) x i ) sin (β 1 t i ) (6) i N 2(β 0 sin (β 1 t i ) x i )β 0 t i cos (β 1 t i ) (7) i It should be clear to see that S/ β j = 0 is not easy to solve. Therefore, we will side-step the derivative based methods and opt for PSO which is a Monte-Carlo based intelligent algorithm. 3. You can import the data into your notebook outside of the error function, plot it and convince yourself that the data is indeed sinusoidal. Estimate what the values of A and ω can be from the plot. It will be helpful for setting bounds of the parameter space when you use PSO i.e. you should figure out the ranges in which you expect reasonable values of β 0 and β The PSO code is given to you. The code takes in a function and its parameters (file name and other necessary arguments if needed). There are also a bunch of the parameters of the algorithm itself you can set by hand. The default ones should work in this case, but feel free to play around with them. Into the PSO solver, pass in your error function for the simple harmonic oscillator and its parameters. You can set some parameters for the PSO algorithm. You can start with adjusting numparticles and iterations which control the number of particles you have searching the parameter space for the best parameter values and the number of iterations they perform before declaring a final value respectively. You will also need to specify the upper and lower bound arrays for the search space. Use the estimated values from the previous part to set suitable bounds. You can set fairly large bounds, PSO is quite good at finding global minima given enough number of particles and iterations. 5. Finally, report your values of β 0 and β 1 and show that the fit works well by overlaying the data points with a curve generated using your parameter values. Hint: You don t need to change most of the PSO parameters. You should be able to tweak the total number of particles and the number of iterations to get it to converge. Bonus 1. Benchmark the performance of PSO as the number of particles and the number of iterations are varied. Which one do you think is PSO more sensitive to? You can also check how the error behaves with the iterations. 4

5 If you need to, modify the PSO function. I have also included another data file for the temperature variation on the surface of Mars. It requires a model with a higher dimensional parameter space. Hence, it should offer a bigger challenge for PSO. The data file for Mars has two columns. The first is time and the second is temperature. 2. The Python function I have written for error is actually quite inefficient. Can you identify why that is and fix it? You will need to change the PSO function format a little bit, but it should speed up your code substantially. Avoid relying on global variables here. Tips for this Project You have done about a third of this project in class already. For part 2, the PSO code is provided and you don t need to write much code on your own. So I will be focusing a lot on your background research, your understanding and your analysis of results you obtain. I want to know that you understand the concepts behind what we are doing - both for the actual problem as well the techniques. For part 2, you don t have to dig too deep into the simple harmonic oscillator related background if you don t want to, but I do expect that you know the standard least-squares, rationale behind rejecting it, advantages of AI based methods etc. When you do part 1, and even in general, make sure your references are included and cited. Code must also be easy to read. I want to remind you again - absolutely no copying and pasting any code or content from anywhere. You can use the code from class notes, but that s it. If you like the idea of swarm intelligence, you should read the novel Prey by Michael Crichton. Michael Crichton was a well-known science fiction author. He passed away recently, but many of this books have become cult classics, for instance, Jurassic Park, The Lost World, Timeline, Congo, Disclosure etc. and a lot of these novels have been adopted into movies. He writes frenetically paced novels and they are quite good reads. The science is typically quite accurate and where it does turn into fiction, it still seems quite believable. For instance, in Jurassic Park, he uses the concept of Chaos theory to make the argument that the park was doomed to begin with. There is also a healthy dose of genetic engineering he uses to bring the dinosaurs to life. Strangely, he never believed in global warming and he even wrote a book called State of Fear in which he portrays global warming as a money-making farce furthered by vested interests. In Prey, he writes about a swarm of nanobots gone rogue. You should try it out if you like the swarming stuff we do here. Abebooks has cheap used versions of the book (around 3-4 dollars including shipping). Best of luck! 5

6 References [1] J. Kennedy and R. Eberhart, Proceedings of the IEEE International Conference on Neural Networks (1995). 6