Author Archives: gmgolem

Cryptographic roots in Blockchain technology

The Blockchain is expected to be the revolutionary technology to take the centre stage in our society where the traditional ledger system once dominates, from bitcoin that emerged in the finance sector to fields where transactions are dependent on authenticity, be it a paper document from bank, an import / export data exchange, or even documents in judicial systems, it is important to understand the principles of its fundamental roots in cryptography.

For example, to ensure the rightful spending of currency in bitcoins, there are a lot of technology being in place on the virtual money market. One being the Elliptic Curve Cryptography that is based on mathematics to ensure the identity of parties involved in any bitcoin transactions.

blockchain1

Advertisements

Plotting gamma distribution in TI Nspire

The gamma function is not a built-in function in the TI Nspire. Nevertheless this function can easily be defined and used for visualising the gamma probability distribution function.

gdf2

gdf1

Poisson prediction for soccer match odds

It is the World Cup season. There are many prediction models, and one of the most widely used statistical technique is the Poisson distribution:

soccer-poisson1

The historic match results are available in the public domain, for example, at http://eloratings.net/. The data are analysed to obtain an index referred to attack strength or goal expectancy. This can further elaborates into more complex data like home team and away team expectancy.

soccer-poisson2

By using a matrix of score scales, usually from 0 to 9, all possible outcomes under 10 goals per team are defined with the respective probability. An example matrix with scores from 0 to 2 will look like one below:

soccer-poisson3

This is the basics of predicting soccer in a quantitative way using the Poisson distribution. There are of course many other prediction methods and models, including organic method like asking the famous Paul the octopus 🙂

51% Blockchain attack

One of the predicted, and proved feasible, attack to the blockchain technology is the 51% attack. There are several forms of this attack including weakness in the blockchain algorithm itself that allowed easier than usual forking of chain to dishonestly win the popular vote mechanism. In other cases, the perpetrator has to amass more hashing power than all other nodes in the blockchain combined to corrupt the distributed ledger, and hence the name of “51%”.

An interesting mathematical study on the effectiveness of this attack is based on the regularized incomplete beta function to calculate the probability of success of a perpetrator.

incompletebeta

Looking for insights from Fitbit data with R

With a month’s of Fitbit data, it’s about time to harvest for some insights from this technology packed wristband.

r-fitbit4


library(lubridate)
library(dplyr)
fitbitdata = read.csv('fitbit.csv')
fitbitdata % mutate(dow = wday(Date))
fitbitdata$dowlabel <- factor(fitbitdata$dow,levels=1:7,labels=c("Mon","Tue","Wed","Thu","Fri","Sat","Sun"),ordered=TRUE)
fitbitdata$scyl <- as.factor(as.integer(fitbitdata$distance)/max(fitbitdata$distance))
head(fitbitdata)
c1 <- rainbow(7)
c2 <- rainbow(7, alpha=0.4)
c3 <- rainbow(7, v=0.8)
boxplot(fitbitdata$steps~fitbitdata$dowlabel, col=c2, medcol=c3, whiskcol=c1, staplecol=c3, boxcol=c3, outcol=c3, pch=23, cex=2, alpha=fitbitdata$scyl)

r-fitbit3

Number of steps and distance traveled data per day is collected from the Fitbit’s phone app, converted into CSV format, and then uploaded to R for data analysis. With a few lines of R code to draw a box-plot for day of week analysis, this data set with a third part statistical package will fill the gap until the Fitbit App offers something more sophisticated .

 

 

RStudio Server and Apache

rstudio2To install R and RStudio Server on Ubuntu:

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys E298A3A825C0D65DFD57CBB651716619E084DAB9
sudo add-apt-repository 'deb [arch=amd64,i386] https://cran.rstudio.com/bin/linux/ubuntu xenial/'
sudo apt-get update
sudo apt-get install r-base
sudo -i R
wget https://download2.rstudio.org/rstudio-server-1.1.453-amd64.deb
sudo gdebi rstudio-server-1.1.453-amd64.deb

Configure Apache 2.4 to proxy RStudio, install required modules.

sudo apt-get install libapache2-mod-proxy-html
sudo apt-get install libxml2-dev

Edit configuration file 000-default.conf to add the followings. Assuming RStudio runs on default port 8787 and preferred path is /rstudio:

<Proxy *>
	Allow from localhost
< /Proxy *>

RewriteEngine on
RewriteCond %{HTTP:Upgrade} =websocket
RewriteRule /rstudio/(.*) ws://localhost:8787/$1  [P,L]
RewriteCond %{HTTP:Upgrade} !=websocket
RewriteRule /rstudio/(.*) http://localhost:8787/$1 [P,L]
ProxyPass /rstudio/ http://localhost:8787/
ProxyPassReverse /rstudio/ http://localhost:8787/
ProxyRequests Off

Finally restart apache.


sudo a2enmod proxy && sudo a2enmod proxy_http && sudo a2enmod proxy_wstunnel && sudo service apache2 restart

Exploring optimization problems in Excel

Excel is able to solve optimization problems. Two commonly available tools are the build-in Solver tool and the Excel plugin for Microsoft Solver Foundation (MSF). The former is not installed by default but can be easily enabled through the Excel Options menu. The latter is a separate download available from Microsoft.

For a simple comparison of the performance of the two, the non-linear data fitting example from the MSF is used as benchmark.
msf-1

MSF provided additional menu pane within Excel for complex optimization operations.
msf-2

Optimization results and log of this benchmark run of a non-linear data fitting sample from the MSF, based on an NIST sample.
msf-10
msf-3

Goals setting screen.
msf-5

Model Display.
msf-6

On the other hand, the built-in Solver offered a simpler interface but still provide detailed reports, including answer, sensitivity, and limits reports in separate spreadsheets.
msf-7

msf-8

The built-in Excel Solver offered easy to use interface, while the Microsoft Solver Foundation is more capable for complex problems and modelling.

The NelderMead solver is selected in this benchmark by the MSF. Check out this previous installment for details of running Nelder-Mead on TI Nspire. The same data set is performed on the Nspire using Nelder-Mead to obtain the following results.
msf-9
msf-11