Algebraic Topology: A First Course This is a good book if you have some prior knowledge in the subject. So if you have already read a bit about the subject and want to learn more, buy it. The author appears to know how to position material to make it interesting to on board a reader but it can a bit long winded and abstract towards the end. A Concise Course in Algebraic Topology Like the above, only a basic introduction to algebraic topology is needed to get started with this book. Check for the version or edition of the book, and buy the latest one. The book has some typos which will be corrected in the latest version. The homework problems in the book can get very demanding. Algebraic Topology See this book as a go-between undergraduate and graduate levels for the subject. The book starts off well by giving an overview and introduction to topology but gets complex towards the later chapters. Algebraic Topology 1st Edition Think carefully before you buy this book, becaus...
Posts
This is a write up to describe an algorithm described in an ancient Indian manuscript. Its called the Bhakshali manuscript. The manuscript describes some mathematical assertions, methods and algorithms that has been dated to several thousands years ago. A really cool algorithm described in that manuscript is an approximation for finding the square root of a number. What I liked about this algorithm is that its handy. You could quickly approximate the square root of a real number with just some basic division and addition. This is how the algorithm works: If 'X' is the number you want to find a square root of, find the nearest whole number 'N' that approximates it. So if X = 23.2 then N = 5. Find the difference between X and N*N. Call it D. In this case it works out to -1.8. This should be too tedious to work out either. Now comes the magical part, divide this difference (D) by 2*N. So that's -1.8/10. Again, this shouldn't be that difficult to do in you...
Q. A jar has a single cell of a bacteria. After every fixed interval of time the bacteria either splits into two with probability 2/5, does nothing with probability 2/5 or dies out with probability 1/5. What is the probability that the bacteria would continue to make a large family tree over an extended period of time? A. The situation can be described by the following visual. Assume that the required probability is 'p'. The term 1 - p would represent the probability that the ecosystem eventually dies out. Each of the above scenarios contributes a quantum of probability towards the ecosystem eventually dying out. Lets start off by represent 1 - p as 'x'. The probability that the bacteria die out is The total of each of the above must add up to the probability that the bacteria eventually die out, which is 'x'. So you can phrase the problem recursively as This simplifies to which is a quadratic equation, yielding a solution as This...
Often times a lot of people working with data are trying to create an index of some sort. Something that captures a set of key business metrics. If you are a site (or an app) you want to create some sort of an engagement index, which if trending up implies good things are happening, bad if it is trending down. The creators of such metrics (think analysts) tend to prefer a weighted arithmetic mean of the influencing factors. If the influencing factors are f1,f2, f3 (say) with weights w1, w2, w3 then the index would be computed as However, what does not get factored in are the final consumers of the index (think product managers) and there could be many. They will invariably try to check it with something else they have handy. For example, if clicks on a site went up 20% the index may be up by just 5% (say) or vice-versa. If resources are being allocated based on the movement of such an index, it will invariably lead to contention on what is the right weighting to be given to each f...
Q: You have a set of thirty six cards. The cards are six in color ( six each) and each color is numbered from 1 to 6. You draw two cards at random. What is probability that they are of a different color and have a different number? A: The first card can be drawn at random. It does not matter what its color or number is. To compute the probability that the second card is different in color and number from the first, it helps to visualize the situation in a simple way as shown below. In the figure above, assume the green dot represents the card that was picked. The marked out cards represent the cards that should not be picked to get a different color and number. Also, the act of picking a card bought down the pool of cards from 36 to 35. The remaining unmarked space represents the available set of cards to pick from. This can be computed easily as This yields an overall probability of If you are interested in learning the art of probability, some of the best book...
Python is a great programming language. It's primary merits are readability and the numerous packages that are available online. However, being an interpreted language the speed of execution is always an issue. Here is a simple example of a piece of code written in Python that tries to add two numbers from a grid. #!/usr/bin/python def overlap_pp(x,y): count = 0 for i in range(x): for j in range(y): count += i + j return count for _ in range(1000): q = overlap_pp(500,500) If you run the above script (saved as n.py) on the command line terminal with the time command you should see some numbers like the below time ./n.py real 0m22.379s user 0m22.363s sys 0m0.407s The process running on one processor took about 22 seconds to complete the run. Now lets do something seemingly magical. Lets add a decorator. Decorators are python speak for a mechanism to do wrapper functions. The decorator we are going to add is '@jit'. The code looks l...
This write up is about an estimator. A statistical estimator, is used when you already have a model in mind, data at hand and want to estimate some parameters needed for the model. For example, you want to predict how many runs a batter would make in a game given recent history \(x_1,x_2,x_3,\ldots\) . If we assume (we are making a choice of a model now) that the scores come from a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) then the probability density function for a given value \(x\) is The likelihood that a series of points \(x_1,x_2,x_3,\ldots\) come from such a distribution can be expressed as Next is basic calculus. Take the logarithm on both sides, set the partial derivative w.r.t. \(\mu\) to zero yields (excluding the algebra) To verify, you also need to check the second derivative's sign to see if its negative to ensure that it is indeed a maxima you have found. So a simple estimator would be to use the average ...
I chanced on this excellent puzzle on the net that tends to reveal a cognitive bias in our heads against Bayesian reasoning. The puzzle statement is quite simple You are given four cards. Each card has a letter on one side and number on the other side. You are told the statement "If there is a D on one side, there is a 5 on the other side". Which two cards would you flip over to validate the statement? The original article is here , think hard before you click through for an answer :)
Follow @ProbabilityPuz Q: You are about to play a game wherein you flip a biased coin. The coin falls heads with probability \(p\) and tails with \(1 - p\) where \(p \le \frac{1}{2}\). You are forced to play by selecting heads so the game is biased against you. For every toss you make, your opponent gets to toss too. The winner of this game is the one who wins the toss the most. You, however get to choose the number of rounds that get played. Can you ever hope to win? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: At a first look, it might appear that the odds are stacked against you as you are forced to play by choosing heads. You would think that your chances or winning decrease as you play more and more. But, surprisingly there is a way to choose the optimal number of tosses (remember, you get to choose the number of times this game is played). To see how, lets crank out some numbers. If you get to toss the coin \(n\) times, then t...
The following are some of the best books to own to learn Monte Carlo methods for sampling and estimation problems Monte Carlo Methods in Statistics (Springer) This is a good book which discusses both Bayesian methods from a practical point of view as well as theoretical point of view with integrations. The explanations given are also fairly comprehensive. There are also a fair amount of examples in this text. Overall, this is an excellent book to own if you want to understand Monte Carlo sampling methods and algorithms at an intermediate to graduate level. Explorations in Monte Carlo Methods (Undergraduate Texts in Mathematics) This is good book to own to get you started on Monte Carlo methods. It starts with fairly simple and basic examples and illustrations. The mathematics used is also fairly basic. Buy this if you are at an undergraduate level and want to get into using Monte Carlo methods but have only a basic knowledge of statistics and probability. Monte Carlo Methods in...
Follow @ProbabilityPuz Q: You are conducting a survey and want to ask an embarrassing yes/no question to subjects. The subjects wouldn't answer that embarrassing question honestly unless they are guaranteed complete anonymity. How would you conduct the survey? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: One way to do this is to assign a fair coin to the subject and ask them to toss it in private. If it came out heads then answer the question truthfully else toss the coin a second time and record the result (heads = yes, tails = no). With some simple algebra you can estimate the proportion of users who have answered the question with a yes. Assume total population surveyed is \(X\). Let \(Y\) subjects have answered with a "yes". Let \(p\) be the sort after proportion. The tree diagram below shows the user flow. The total expected number of "yes" responses can be estimated as $$ \frac{pX}{2} +...
Follow @ProbabilityPuz A class of analysis that has piqued the interest of mathematicians across millennia are Diophantine equations . Diophantine equations are polynomials with multiple variables and seek integer solutions. A special case of Diophantine equations is the Pell's equation . The name is a bit of a misnomer as Euler mistakenly attributed it to the mathematician John Pell. The problem seeks integer solutions to the polynomial $$ x^{2} - Dy^{2} = 1 $$ Several ancient mathematicians have attempted to study and find generic solutions to Pell's equation. The best known algorithm is the Chakravala algorithm discovered by Bhaskara circa 1114 AD. Bhaskara implicitly credits Brahmagupta (circa 598 AD) for it initial discovery, though some credit it to Jayadeva too. Several Sanskrit words used to describe the algorithm appear to have changed in the 500 years between the two implying other contributors. The Chakravala technique is simple and implementing it ...
Follow @ProbabilityPuz All too often, when we deal with data the outcome needed is a strategy or an algorithm itself. To arrive at that strategy we may have historic data or some model on how entities in system respond to various situations. In this write up, I'll go over the method of reinforcement learning. The general idea behind reinforcement learning is to come up with a strategy to maximize some measurable goal. For example, if you are modelling a robot that learns to navigate around obstacles, you want the learning process to come back with a strategy that minimizes collisions (say) with other entities in the environment. Pattern Recognition and Machine Learning (Information Science and Statistics) For the sake of simplicity, lets assume the following scenario. A robot is placed (at random) on flat plank of wood which has some sticky glue in the center. To its left there is a hole which damages the robot a bit and to its right is a reward which is its dest...