Oct 20

People need real knowledge

People need real knowledge

Traffic analysis

The number of visits to my website has exceeded 206,000. This number depends on what counts as a visit. An external counter, visible to everyone, writes cookies to the reader's computer and counts many visits from one reader as one. The number of individual readers has reached 23,000. The external counter does not give any more statistics. I will give all the numbers from the internal counter, which is visible only to the site owner.

I have a high percentage of complex content. After reading one post, the reader finds that the answer he is looking for depends on the preliminary material. He starts digging it and then has to go deeper and deeper. Hence the number 206,000, that is, one reader visits the site on average 9 times on different days. Sometimes a visitor from one post goes to another by link on the same day. Hence another figure: 310,000 readings.

I originally wrote simple things about basic statistics. Then I began to write accompanying materials for each advanced course that I taught at Kazakh-British Technical University (KBTU). The shift in the number and level of readership shows that people need deep knowledge, not bait for one-day moths.

For example, my simple post on basic statistics was read 2,300 times. In comparison, the more complex post on the Cobb-Douglas function has been read 7,100 times. This function is widely used in economics to model consumer preferences (utility function) and producer capabilities (production function). In all textbooks it is taught using two-dimensional graphs, as P. Samuelson proposed 85 years ago. In fact, two-dimensional graphs are obtained by projection of a three-dimensional graph, which I show, making everything clear and obvious.

The answer to one of the University of London (UoL) exam problems attracted 14,300 readers. It is so complicated that I split the answer into two parts, and there are links to additional material. On the UoL exam, students have to solve this problem in 20-30 minutes, which even I would not be able to do.

Why my site is unique

My site is unique in several ways. Firstly, I tell the truth about the AP Statistics books. This is a basic statistics course for those who need to interpret tables, graphs and simple statistics. If you have a head on your shoulders, and not a Google search engine, all you need to do is read a small book and look at the solutions. I praise one such book in my reviews. You don't need to attend a two-semester course and read an 800-page book. Moreover, one doesn't need 140 high-quality color photographs that have nothing to do with science and double the price of a book.

Many AP Statistics consumers (that's right, consumers, not students) believe that learning should be fun. Such people are attracted by a book with anecdotes that have no relation to statistics or the life of scientists. In the West, everyone depends on each other, and therefore all the reviews are written in a superlative degree and streamlined. Thank God, I do not depend on the Western labor market, and therefore I tell the truth. Part of my criticism, including the statistics textbook selected for the program "100 Textbooks" of the Ministry of Education and Science of Kazakhstan (MES), is on Facebook.

Secondly, I have the world's only online, free, complete matrix algebra tutorial with all the proofs. Free courses on Udemy, Coursera and edX are not far from AP Statistics in terms of level. Courses at MIT and Khan Academy are also simpler than mine, but have the advantage of being given in video format.

The third distinctive feature is that I help UoL students. It is a huge organization spanning 17 universities and colleges in the UK and with many branches in other parts of the world. The Economics program was developed by the London School of Economics (LSE), one of the world's leading universities.

The problem with LSE courses is that they are very difficult. After the exams, LSE puts out short recommendations on the Internet for solving problems like: here you need to use such and such a theory and such and such an idea. Complete solutions are not given for two reasons: they do not want to help future examinees and sometimes their problems or solutions contain errors (who does not make errors?). But they also delete short recommendations after a year. My site is the only place in the world where there are complete solutions to the most difficult problems of the last few years. It is not for nothing that the solution to one problem noted above attracted 14,000 visits.

Fourthly, my site is unique in terms of the variety of material: statistics, econometrics, algebra, optimization, and finance.

The average number of visits is about 100 per day. When it's time for students to take exams, it jumps to 1-2 thousand. The total amount of materials created in 5 years is equivalent to 5 textbooks. It takes from 2 hours to one day to create one post, depending on the level. After I published this analysis of the site traffic on Facebook, my colleague Nurlan Abiev decided to write posts for the site. I pay for the domain myself, $186 per year. It would be nice to make the site accessible to students and schoolchildren of Kazakhstan, but I don't have time to translate from English.

Once I was looking at the requirements of the MES for approval of electronic textbooks. They want several copies of printouts of all (!) materials and a solid payment for the examination of the site. As a result, all my efforts to create and maintain the site so far have been a personal initiative that does not have any support from the MES and its Committee on Science.

Sep 17

Geometry behind optimization

Geometry behind optimization

Optimization is an animal of many faces. More precisely, this post is about what is called the First Order Conditions (FOC's) and Second Order Conditions (SOC's). Make sure to go through this visual introduction to derivatives, then everything will be easy.

Case of a minimum

Draw a simple function that has a clear minimum. For example, just use a parabola with branches looking upward. Below it draw the graph of its derivative. Satisfy yourself that the derivative at the minimum point is zero. Then look at the behavior of the second derivative at the minimum point. Establish its sign; that will be the second order condition.

Conditions sufficient for a minimum point

Video 1. Conditions sufficient for a minimum point


Case of a maximum

Here you turn the previous discussion upside down. Draw a simple function that has a clear maximum. For example, just use a parabola with branches looking downward. Below it draw the graph of its derivative. Satisfy yourself that the derivative at the maximum point is zero. Then look at the behavior of the second derivative at the maximum point. Establish its sign; that will be the second order condition.

Conditions sufficient for a maximum point

Video 2. Conditions sufficient for a maximum point

When first and second order conditions don't work

The example in case is the cubic function. The derivative at zero is zero. However, the origin is neither maximum nor minimum. This is an inflection point (concavity changes to convexity). The general rule that arises from the analysis of the Taylor decomposition is this. Look for the first derivative in the decomposition that is not zero. If the order of the derivative is even, then the sign of the derivative determines whether it's a maximum or a minimum. If the order of the derivative is odd, then it's an inflection point, regardless of the sign of the derivative.

Inflection point

Video 3. Inflection point



A point where the first derivative vanishes is called a critical point. Suppose x_0 is a critical point.

Case 1. If the second derivative at x_0 is positive, you have a minimum because locally the function is a parabola with branches extended upward.

Case 2. If the second derivative at x_0 is negative, you have a maximum because locally the function is a parabola with branches hanging downward.

Case 3. If the second derivative at x_0 is zero, you are out of luck. Look for the first derivative f^{(n)} that is different from zero at x_0. Your course of action is explained by the fact that locally the function is

f(x)\approx f(x_0)+\frac{f^{(n)}}{n!}(x-x_0)^n.

If n is even, it is a minimum in case f^{(n)}(x_0)>0 and a maximum in case f^{(n)}(x_0)<0. If n is odd, it's neither, and the investigation is over.