File:2014.12.26rubleDollar.png

Price of 100 ruddian rubles, measured in the USA cents; data for the end of year 2014 and the approximations with elementary functions.

The green thick curve represents the experimental data $y=g(x)=\mathrm{Measured}(x)$ by https://www.mataf.net/en/currency/converter-USD-RUB ; coordinate $x$ has sense of time, measured in days since the date of beginning of the project, 2014,10,27.

For each datum stored for specified year,month,day, the time $x$ is evaluated as

$x = \rm daju24(year,month, day)- daju24(2014,10, 27)$

with function daju24 defined below in C++: \begin{verbatim} int daju24(int Y,int M, int D){ int a, y, m; a = (14-M)/12; y = Y + 4800 - a; m = M + 12*a - 3; return D + (153*m+2)/5 + 365*y + y/4 - y/100 + y/400 - 32045 - 2400000; }

These data are stored as array $\{x_n,g_n\}, \{n,1,M\}$, where $M$ is total number of experimental data. Approximating functions are specified below:

$ \begin{array}{c|l|r|r} \rm Label &~ ~ ~ ~ ~ f(x) & D ~ ~ & Q ~ ~\\ \hline \rm Linear &227.323 - 0.583872 x &\! 10.52733 &\! 13.15973\\ %{227.32289289657714` - 0.5838719175561293` x, 10.52725166548383`, 13.15967254961913`} \rm Qu\! & 233.214 - 0.908933 x - 0.00361841 x^2 & 3.94609& 5.84960\\ %3.946087936982961`, 5.849600340588587 \rm Ellipse &~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1.19332 \sqrt{(99.8879 - x) (388.557 + x)} & 3.72305 & 5.74477\\ \rm Ed & -96.8595 +  1.46555 \sqrt{(127.305 - x) (401.761 + x)} & 3.74359 & 5.71801\\ \rm Cu & 234.636 - 0.905604 x - 0.00456087 x^2 - 6.98113\!\times\!10^{-6}~ x^3 & 3.77297 & 5.72430\\ \rm Bell & 291.207 / \cosh(0.715005 + 0.00630878 x)& 4.91540& 6.96990\\ \rm Gauss & 290.656 \exp\Big(-0.0000168642 (116.58 + x)^2\Big) & 4.45347&6.43202\\ \rm Dex &\!\! 100/\Big(0.337261 \exp(0.000020293 x) +  0.0881639 \exp(0.0189706 x)\Big) \!& 3.79927& 5.72350\\ \hline \end{array} $

The last two columns of the table above characterise the precision of each approximation:

$\displaystyle D= \frac{1}{M} \sum_{n=1}^{M} |f(x_n)-g_n|$

$\displaystyle Q= \sqrt{\frac{1}{M} \sum_{n=1}^{M} (f(x_n)-g_n)^2}$

These quantities refer to date 2014.12.26; $M=209$.

Refrerences
http://mizugadro.mydns.jp/PAPERS/2015ruble.pdf D.Kouznetsov. Fitting of economical data with elementary functions: rouble versus dollar in 2014.