1 + 2 + 3 + 4 + 5 + …. = -1/12

sum of all natural numbersWait a minute, what? That looks very very wrong. You must be wondering if the title is some horrendous typo, right? Well, let me assure you that it’s absolutely not! The sum of all natural numbers is equal to -1/12. This blog post is not just about mathematical trickery either. The equation in the title is actually a very important result used in theoretical physics, particularly in string theory. Now how can that be possible? Are physicists really that bad at mathematics? That can’t be it! What is the proof behind this? Do we ever encounter it in real life?   Continue reading “1 + 2 + 3 + 4 + 5 + …. = -1/12”

What Are Confidence Intervals?

mainConfidence interval is a concept in statistics that is used extensively in many diverse areas like physics, chemistry, computer vision, machine learning, genetics, etc. This concept is so fundamental that any modern science would eventually end up using it. Let’s say you have collected some data and you want to understand the behavior of that data. For example, you can say that the data is centered around some value or that the data is distributed with a certain amount of variance. This is very common in many fields where you have estimate the underlying parameters that govern the data distribution. When you estimate a statistical parameter from some data, you can’t be certain about its true value. If you have a lot of high-quality data, then you’re more confident that your estimate is near its true value. But if you don’t have a lot of data, or if it’s of poor quality, then you don’t have much confidence in it. So how do we deal with these situations? Can we measure this uncertainty?   Continue reading “What Are Confidence Intervals?”

What Are P-Values?

mainLet’s say you are a part of the sub-atomic physics team and you are working on discovering an important effect. The thing about sub-atomic physics is that nothing is certain and you cannot say something has happened with 100% certainty. The best we can do is to say that we are x-percent sure that something interesting happened. One fine day, you see some pattern in your data which looks pretty much like what that effect would look like. Now the problem is, your experiment produced data with a lot of noise. People are therefore skeptical of you, and think that the supposed “effect” you claimed to see might just have been a funny pattern in some random noise. How would you convince them that it’s not? Before that, how do you convince yourself that it’s not just noise? A good strategy for arguing your point would be to say, “Alright listen, suppose you’re right, and the patterns in my data really are in fact just from random noise, then how would you explain the fact that random noise very rarely produces patterns like this?”. Pretty good strategy right? Now how do we formulate this mathematically?   Continue reading “What Are P-Values?”

What Is Random Walk?

1 mainConsider the following situation. We have a drunkard who is clinging to a lamppost, and now he decides to start walking. He is in the middle of the street and the road runs from east to west. In his inebriated state, he is as likely to take a step towards the east as he is towards the west. It just means that there is a 50% chance that he will go in either direction. From each new position, he is again as likely to go east or west. Each of his steps are of the same length but in random direction. After having taken ‘n’ number of steps, he is to be found standing at some position on the street. This is what a random walk is. We can plot the position against the number of steps taken for any particular random walk. Now the question is, can we model his movement so that we can predict where he will be after taking ‘n’ steps?   Continue reading “What Is Random Walk?”

What Is A Galois Field?

mainA Galois Field is actually a corn field owned by any person named Galois! Too obvious? Alright, that was a joke. Anyway, the word ‘field’ is being used in the mathematical context here. Évariste Galois was a mathematical prodigy who laid strong foundations for abstract algebra. His collected works contain important ideas that have had far-reaching consequences for nearly all branches of mathematics, thus etching his name in mathematics forever. Unfortunately, he died at a tender age of 20. The work he did as a teenager is now being used by mathematicians around the world for their doctorate studies and related research work. Prodigy indeed! Just so we are clear, this blog post has nothing to do with corn fields. Well then, what else can it be about?   Continue reading “What Is A Galois Field?”

Transcendental Functions

mainTranscendentalism refers to the philosophical movement that developed in the 1800s. It taught people to believe in the inherent goodness of people and nature. It also said that religious organizations and political parties corrupt the purity of an individual. So obviously, any ceremonies or functions attended by transcendentalists should be called transcendental functions. Right? No, not really! That has actually nothing to do with what we will be discussing here. When we hear the term ‘function’ used in a scientific context, we immediately jump to mathematics. If you are somewhat familiar with mathematics, you know what a function is. If not, no worries; we will discuss it further soon. But for now, all we need to know is that real functions are divided into two groups: algebraic functions and transcendental functions. Wait, what? Aren’t all functions and equations ‘algebraic’? Well, not exactly. As it turns out, transcendentalism exists in mathematics!   Continue reading “Transcendental Functions”

Asymmetric Dominance Effect

mainLet’s consider a situation. There is a company with two products, A and B. Both these products have their own merits and demerits. Product A has relatively less features, but it’s price is low. Product B, on the other hand, has more features but it’s more expensive. Consumers tend to pick both these products depending on their needs. Now the company introduces a third product, C. The asymmetric dominance theory says that you can affect the consumer behavior using this third product. You can make the consumers shift towards product A or product B by designing product C in different ways. Now how is that possible? How can we change consumer preference between A and B without even modifying these products?   Continue reading “Asymmetric Dominance Effect”

Derandomization Of RANSAC

mainLet’s say you are a clothes designer and you want to design a pair of jeans. Since you are new to all this, you go out and collect a bunch of measurements from people to see how to design your jeans as far as sizing is concerned. One aspect of this project would be to see how the height of a person relates to the size of the jeans you are designing. From the measurements you took from those people, you notice a certain pattern that relates height of a person to the overall size of the jeans. Now you generalize this pattern and say that for a given height, a particular size is recommended. To deduce the pattern, you just took a bunch of points and drew a line through them so that it is close to all those points. Pretty simple right! What if there are a few points that are way off from all the other points? Would you consider them while deducing your pattern? You will probably discard them because they are outliers. This was a small sample set, so you could notice these outliers manually. What if there were a million points?   Continue reading “Derandomization Of RANSAC”

What Is The Poincaré Conjecture?

colored shapeBefore we start, let me put something out there. Poincaré Conjecture is one of the seven millennium problems established by the Clay Mathematics Institute. Those problems are worth a million dollars each! The Poincaré conjecture depends on the mind-numbing problem of understanding the shapes of spaces. This field of study is referred to as “topology” by mathematicians. Topology is an important field within mathematics concerned with the study of shapes, spaces and surfaces. We interact with these in our everyday lives. We see surfaces breaking and shapes getting deformed all the time. Ever wondered if there is any law governing these deformations? Or are these deformations just too random to be studied? Does Poincaré Conjecture have any real world applications?   Continue reading “What Is The Poincaré Conjecture?”

Expectation Maximization

3d surfaceProbabilistic models are commonly used to model various forms of data, including physical, biological, seismic, etc. Much of their popularity can be attributed to the existence of efficient and robust procedures for learning parameters from observations. Often, however, the only data available for training a probabilistic model are incomplete. Missing values can occur which will not be sufficient to get the model. For example, in medical diagnosis, patient histories generally include results from a limited battery of tests. In gene expression clustering, incomplete data arise from the intentional omission of gene-to-cluster assignments in the probabilistic model. If we use regular techniques to estimate the underlying model, then we will get a wrong estimate. What do we do in these situations?   Continue reading “Expectation Maximization”