RK's Musings

A Certain Ambiguity : Book Summary

For the last one month I was completely occupied with work . Weekdays and Weekends flew away before I realized that it has been quite sometime since I had read any book. This weekend , being a long weekend , decided to spend my time reading a book on ambiguity. I stumbled on to a reference to this book on some blog and was intending to read at some point in time. A nice holiday break and festive mood around was good enough to motivate me to read this book, which was touted as a mathematical novel.

There are 2 stories which are intertwined in this novel. One about Ravi Kapoor, who narrates his learnings from a class on Infinity at Stanford. The second story is about circumstances in the life of Ravi's grandpa, Vijay Sahni who gets imprisoned for a brief period in his life. The narration of these two stories can be visualized by the following image:

The upper box represents the story of Vijay Sahni. The lower box represents Ravi's story in the novel.

The conversations between Vijay Sahni and Judge Taylor brings out Euclidean and Non Euclidean Geometry

The five postulates of Euclid based on which the entire "ELEMENTS" is built are :
Postulates :
1. It is possible to draw a straight line from any point to any point
2. It is possible to produce a finite straight line continuously in a straight line
3. It is possible to describe any circle with center and radius
4. All right angles equal one another
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

The fifth postulate controversial nature lead to the development of Non-Euclidean Geometry

The lower box is Ravi's story where his stanford professor Nico, uses a perfect blend of examples and historical narrative and leads to the Continuum problem

Continuum Problem: Is there a set whose cardinality is greater than the cardinality of natural numbers, but less than the cardinality of the real
numbers?

What is the connection between the two stories ? Why does the book develop these themes in the first place ? What set theory and geometry have in common ? A reader would definitely be delighted from the way connections are made in the book using two stories.

Here is a snippet from one of the conversations between Vijay Sahni and the Judge
This freedom has me in awe. It is unbounded, and every single one of us possesses it. We are free to believe or not to believe; we may create mathematics or build homes or write poetry or do nothing at all; we can marry and raise a family or stay in bachelorhood; we can quest for new adventure or find comfort in the familiar; we can seek meaning or we can doubt that it is possible to find meaning. Every path is there to be taken or ignored, and none is ordained. We are given no certainties, yet we are given the capacity to feel certainty. There is no absolute meaning to latch onto, yet transcendence is within our grasp. We are free to chart our course, free to pursue our passions, and free to create the axioms of our lives. And it is in this glorious freedom that I find grace. This freedom, then, is my proof of His existence.

Here is a list of mathematicians who figure in the book as the stories progress

Zeno popularly known for his Paradox

Italian thinker Giordano Bruno
Galileo

Bhaskara (1114-1185)
Pythagoras (570-490 bc)

Cantor
Baruch (Benedict) Spinoza (1632-1677)
Euclid

David Hilbert (1862-1943),

Girolamo Saccheri (1667-1733):

Farkas (Wolfgang) Bolyai (1775-1856)
Farkas Bolyai's son Janos (Johann) Bolyai

Nikolay Ivanovich Lobachevsky (1792-1856)

Johann Carl Friedrich Gauss (1777-1855)

Bernhard Riemann (1826-1866)

Overall this book reminds me of "The Lady Tasting Tea" which gives a good historical narrative of people contributing to the field of statistics.

I guess the take away from the book, besides getting a thrilling ride on the basis of mathematics is this gem of a statement:

It is true that absolute certainty may lie outside our reach, but we live for that magic moment of discovery when we are attuned to this sense of order and connectedness. And the existence of this order and connectedness is a leap of faith.

Visualizing Abacus

This is what an 8 year slog with Abacus/ Soroban (as called in Japan) can enable one to do .

Look at the video and watch how one can visualize huge mental arithmetic problem..It was a jaw dropping experience to watch kids do this!!

 

Its more like V

Smiles & Smirks are what one would get to see about Vols in US Markets. However a quick crunching of NIFTY options show that implied volatility is more V Shaped. Infact Puts are tilted towards right and Calls towards left. Here is an illustration between moneyness and implied vol.

Something to think about :How do you model this quasi V shaped market phenomenon ? and more importantly How do you make money ?

Gaussian Copula : Formula that killed Wall St.

Came across a wired article which says the following copula formula by David X.Li was widely used for securitization models. Results are known to everyone now!

 

Link :Wired

ArcSin Density

Learning about probability distribution is pretty boring unless one knows how to relate these to at least some real life application. Well, that can be said about almost anything learnt in life .However in the case of probability distributions, it is strikingly more true.

There is a fair chance that a lot of people know about normal,log normal, student t,weibull,poisson,Cauchy, exponential, chi-square ,brownian, gbm, levy, etc etc.. Somehow I have never found merely learning about these distributions interesting. There are all sorts of distributions which have various parameters. Sometimes I feel I should least know half a dozen real life apps for each of these distributions.

It is always an aha moment for me when I see a application of these distributions in some practical way. One of the aha moments I had recently, was with the arc sine density function. Most of the people working in finance know about brownian motion becoz finance literature is filled with BM constructs. I remember reading about Zero Hitting time of a brownian motion which follows arc sine density function. I conveniently forgot about it. Honestly, who the hell cares whether it follows arc sine or arc cos density function. This was until a few days back when I came across a powerful way of looking at a residual trading strategy where author looks at a spread trade and uses Zero hitting time of a brownian motion to discard trading in that spread!! NOW THAT's WHAT I LIKE. A true application of something from theory.

Well , the reasoning is elegant. The trader looks at a spread and compares the zero crossing frequency of the spread. If the spread is to be traded, it has to mean revert which means it has to cross 0 often, 0 often meaning, probability of hitting 0 given it hit 0 at some point in its history should be high. AND, brownian motion doesnt let that happen. One might think.....Its a random walk so the path hits 0 pretty regularly!! But that's not the case. In any case, the brownian motion can thus be used to check the non stationarity of the zero hitting time.

ArcSin Density

I don't think I will ever forget this arc sine density function again becoz a viewpoint from a practical application is worth more than 1000 hours of studying about it.

Factor analysis needs a priori theory

While hacking some trading techniques, I came across an author trying to link Factor analysis and Arbitrage Pricing Theory(Ross). I was wondering the use for it . My thinking was a classic case of an empiricist. Today I came across a reference to a old paper which says the link provided has some meaning to it. It is always better to have some theoretical model before pca/factor analysis/ eigen value decomposition.

Just because there is computational power and software doesn't mean that your hypotheses(read crap, most of the times) are true.

In the words of the J. Scott Armstrong

"The cost of doing factor analytic studies has dropped substantially in recent years. In contrast with earlier time,. it is now much easier to perform the factor analysis than to decide what you want to factor analyze. It is not clear that the resulting proliferation of the literature will lead us to the development of better theories. Factor analysis may provide a means of evaluating theory or of suggesting revisions in theory. This requires, however, that the theory be explicitly specified prior to the analysis of the data. Otherwise, there will be insufficient criteria for the evaluation of the results.
If principal components is used for generating hypotheses without an explicit a priori analysis. the world will soon be overrun by hypotheses."

Link to the paper : Tom Swift and His Electric Factor Analysis Machine

Abacus

 

Picked up this book from B&N today . Reader friendly 150 page book that I managed to read over the subway ride back.

Well, I was just curious to find out what makes the system so addictive . In the past, I had witnessed a couple of abacus competitions and was amazed by the speed with which calculations are done. There are atleast 2 clubs that I am aware of in Hyd(India). The expert in one of the clubs was so proficient that he could add about 15 numbers in air with just a mental image of abacus. He was able to add all those numbers in less than 3-4 sec.

Anyways, the book is definitely a good way to begin understanding the basics and the last chapter of the book is fascinating account of the possibilities of computations with abacus !

Hiatus - Fourier week

My laptop crashed 10 days back and I had to make do with out one till the new one arrives. After a looooong time, staying completely away from system allowed me some time to think on some aspects which I had not focused on . Managed to read some terrific books on Fourier series in the last 10 days. Fourier series is something that I learnt in undergrad and conveniently forgot it as I never came across a practical application(may be bhag daud ki zindagi mein, never took time to understand i guess), until I came across Vol Modeling.

There is a vast amount of literature where FFT (Fast Fourier transformation) is used for option valuation and the math is beautiful . A combination of sines and cosines can do wonders. For those who would want to get down to basics and see what a fourier transform looks like, where one can use it and how is it related to math finance, here is a list of books that I recommend(purely from my own experience after making false starts on a set of literature) :

This book is a gem . It is like HeadFirst Series for Fourier. If you have not read any of HeadFirst Series books, then you are missing something !. Ironically HeadFirst Books were first suggested to me by a High school teacher who was kicked about the way things are communicated.Head First books are books on Technology and the way High Schoole teacher applied Head First principles for her curriculum is amazing, when I first heard about it.

Title is misleading. This book is more than a pedestrian introduction to Fourier. I guess some knowledge of fourier would be good before you embark on this book

One should read the chapter on fourier from this book to enjoy the great way the author talks about Fourier. Gilbert Strang is very popular for his ways of teaching as he makes any subject intuitive and easily understandable. This book has to be read after you think that you know pretty well some stuff on Fourier. What this book will tell you is that, your knowledge is no good for real life apps. It proceeds to show where one can exactly use it

After reading the above 3 books ,you would be in a better shape to code/program/understand FFTs use in Option valuation. Once you take an effort in understanding the basics, rest of math and application is BEAUTIFUL.

FFT eFFecTs :)

 

Convergence

Any freshman will know that integral of (1/x) between 0 and 1 doesn't converge. But integral of (1/x^2) between 0 and 1 does converge.But WHY ? Questions often , even as simple as this , are not given time and the outcome is - we spend most of our lives on answering somebody else's questions rather than posing questions.

A particularly nice way of looking at this convergence cropped up when Anupam gave this insight, while discussing Fourier Transforms. This is a popular technique , a 10,000 ft view of this technique is this : It reduces the tails so that you can calculate the integral easily.That is exactly the reason why (1/x^2) converges ...we are reducing the tail area! ..Wow!! never ever thought about it in this way!!

Sometimes I have this feeling that I should spend my entire life in math...But I guess one has to afterall, reconcile to "Sex and Cash Theory "

Can numbers be illegal ?

Can a number be illegal ?? Well, I came to know about it very recently , that in fact, a few numbers are illegal...!!!

Via  Illegal prime - Wikipedia:

An illegal prime is a prime number that represents information forbidden to possess or distribute, because when interpreted a particular way, it describes a computer program which bypasses copyright protection schemes. Distribution of that program in the United States is illegal under the Digital Millennium Copyright Act. Illegal primes are a subset of illegal numbers.

Explore a dozen :) for 1/e % success !

An article about  "optimal stopping time" , in a completely different context.

Link : Mathematics, marriage and finding somewhere to eat

Random Numbers Patent

We all know that the random number generator that comes along in excel is not actually random. If you sample 1000 x coordinates and 1000 y coordinates and draw a scatter plot, you can easily see large gaps where sample points are not picked.
Look at the picture below - The first one is from excel function rand()

If one is using random numbers a lot in one's analysis, one has to ignore rand() and go for something else..One of the popular methods is LCG, a pseudo random number generator. If one looks at the second graph, it seems to be doing a better job than excel.

However the best approach comes from Quasi random numbers (QSR)where the space gets evenly filled out.The third graph illustrates.

One caveat in using QSR...Columbia university has a patent on developing the most efficient 50 or greater dimensional random number. So, watch out when you use a 50 dim rand number algo that you might develop for your usage...You might end up in a litigation :)..So as long as it is not super efficient, you are fine..:) strange are the ways of life in which patents rule

Best of the 20th Century Algorithms

Via SIAMS News:Top 10 Algorithms of the 20th Century

• 1946: John von Neumann, Stan Ulam, and Nick Metropolis, all at the Los Alamos Scientific Laboratory, cook up the Metropolis algorithm, also known as the Monte Carlo method.
• 1947: George Dantzig, at the RAND Corporation, creates the simplex method for linear programming.
• 1950: Magnus Hestenes, Eduard Stiefel, and Cornelius Lanczos, all from the Institute for Numerical Analysis at the National Bureau of Standards, initiate the development of Krylov subspace iteration methods.
• 1951: Alston Householder of Oak Ridge National Laboratory formalizes the decompositional approach to matrix computations.
• 1957: John Backus leads a team at IBM in developing the Fortran optimizing compiler.
• 1959–61: J.G.F. Francis of Ferranti Ltd., London, finds a stable method for computing eigenvalues, known as the QR algorithm.
• 1962: Tony Hoare of Elliott Brothers, Ltd., London, presents Quicksort.
• 1965: James Cooley of the IBM T.J. Watson Research Center and John Tukey of Princeton University and AT&T Bell Laboratories unveil the fast Fourier transform.
• 1977: Helaman Ferguson and Rodney Forcade of Brigham Young University advance an integer relation detection algorithm.
• 1987: Leslie Greengard and Vladimir Rokhlin of Yale University invent the fast multipole algorithm.

Tidbit : Google uses a flavor of QR Algorithm in its algorithmic attack on the search problem

Moore Method

A nice method for the math instructors -
Link :  WHAT IS THE MOORE METHOD?

Neat and elegant method

Crout LU Decomposition

This is a neat and an elegant LU Decomposition method which can be coded with  both partial or full pivoting. Came across this algo in Numerical Recipes and loved it from the word go.

Nature's Numbers : Book Summary

Ian Stewart's books have really caught my attention nowadays. After reading his book, " Letters to a Young Mathematician", I gulped "Nature's Numbers". The book is all about seeing nature from a mathematician's POV.

The book starts off with an introduction of nature and patterns. Patterns of form and Patterns of movement are so widespread in the nature that it is difficult not to observe them. Ripples in the pond, Stripes on Zebra tiger, movement of horses, elephants, mammals all happen based on a pattern. Patterns posses beauty as well as utility. If one were to study the patterns, it would be the best thing to leverage those patterns in our daily life applications. When such a complicated creature like Nature can work with a pattern, it is certain that Pattern would work for a less complex purpose driven application.

Patterns are basically numerical patterns, geometric patterns, and movement (translation, rotation, reflection) patterns. A mathematician's instinct is to structure the process of understanding by seeking generalities that cut across various sub divisions.A lot of physics proceeded with out the any major advances in the mathematical world. For 200 years, calculus was in a different position. It was being used with great success in Physics But the mathematicians were really concerned about what it really meant. Thus there is a fundamental difference in the way of thinking of a mathematician. They tend to ask WHY rather than HOW. HOW related questions are left to domain experts, be it physicists , chemists, scientists etc Mathematicians concentrate on WHY and that opens a whole set of areas for people to work on HOWs. For example snail develops a spiral shell, mathematician will be interested in the ways a spiral is formed whereas how the snail makes the shell is matter of genetics / chemistry.

Is Mathematics about numbers, real , complex , functions, transformations, proofs, theorems etc..No , Math is about story telling. If you can take a natural phenomenon / application and can a tell an effective story using some tools, that is what Math is all about

There is a chapter titled "From Violins to Videos", which is a beautiful summarization of the events starting from the purposeless study of 1d strings on a violin to a very practical device tv. A lot of physicists and mathematicians played a role in cracking the 1d wave equation of a violin string. jean le rond d'alemert , euler, bernoulli all were instrumental in bringing about the solution for 1d waves. This was extended to the vibrations of the surface of the drum which is a 2d. Finally it showed up in the areas of Electricity and Magnetism. Michael Faraday and subsequently Maxwell came up with electromagnetic forces which was a giant leap in the advancement of scientific understanding.  Visible electromagnetic waves with different frequencies produce different colors.

The book also deals with the pattern of movement. One complete chapter is dedicated to gait analysis where trot, pace, bound, walk, rotary, gallop, traverse gallop and canter is analyzed.

Chapter 8 titled - "Do Dice Play God ?" is my favorite chapter of the book.  It starts off by introducing a concept called phase space which is nothing but a solution space that is obtained based on the initial conditions. The chapter's main theme is that random movements at the microscopic level can result in deterministic movements at the macroscopic level. Also simple cause results in complex effects. One superb example that's given to justify the theme is the half life period. One can never say that at an instant a particular atom will disintegrate or not, but one can always calculate half life period of an elements. So, one knows the half life of an elements, with out knowing which half will disintegrate..Its like the famous ad saying, I know my marketing ad budget gets me results but I do not know which half.

The last chapter is collection of 3 case studies - One , water from a tap , Two , a simulated artificial ecology example, and the final one is that of petals in various flowers. Each of the case studies is a gem that goes on tell that mathematical complexity results in simple patterns and it is well worth understanding mathematical complexity , for it is such study that creates a better understanding of nature's patterns.

I am certain to read this book again at a later point of time