File:2014rubleDollar3param.png

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Price of 100 Russian roubles, evaluated in the USA cents, versus time.

Time $x$ is measured in days since the beginning of the project, 2014.10.27.

Pink dots represent the experimental data by https://www.mataf.net/en/currency/converter-USD-RUB

The thick smooth lines are formed by the four sets of approximations. Each set is formed with specific four-parametric function $f$, and values of parameters are chosen to minimise the mean square deviation

$\displaystyle Q=\sqrt{ \sum_{n=1}^m \big(F_n-f(X_n)\big)^2 }$

for $M-50\le m\le M$, where $M$ is total number of experimental dots available for the day of plotting. For each $m$,the single curve is drawn, emulating situation when the data with number larger $m$ are not available. With growth of $m$, the evolution of the approximation with time can be traced. The thickness of the resulting line qualifies the instability of the approximation.

In the way mentioned above, the following four functions $f$ are considered and plotted:

$f(x)=\mathrm{Bell}(x)=a/\cosh(b(x+c))$

$f(x)=\mathrm{Gauss}(x)=a \exp(b x+c x^2))$

$f(x)=\mathrm{Quadratic}(x)=a + b x +c x^2$

$f(x)=\mathrm{Ellipse}(x)=a \sqrt{(b-x)(c+x)}$

For these functions, the mean square deviations is of order of 10.

The input data are stored as array of pairs $X_n$, $F_n$

References

The image is used as Fig.2 in article

http://mizugadro.mydns.jp/PAPERS/2015ruble.pdf D.Kouznetsov. Currency band and the approximations: Fitting of rouble with 3-parametric functions. 2015, under consideration.

Mathematica generator of lines


g0 = Import["~/Q/RUBLE/BUL/TRY01/ddat.txt", "Table"];
T0[i_] := Extract[Extract[g0, i], 1];
G0[i_] := Extract[Extract[g0, i], 2]; M0 = Length[g0]

lp = ListPlot[g0, PlotRange -> All, PlotStyle -> {RGBColor[1, 0, 1], PointSize[.04]}]

For[i = M0 - 52, i < M0, i++; Print[i];
 g = Table[{T0[j], G0[j]}, {j, 1, i}]; M = Length[g];
 T[i_] := Extract[Extract[g, i], 1];
 G[i_] := Extract[Extract[g, i], 2];
 F[x_] = a + b x + c x^2; sub = FindFit[g, F[x], {a, b, c}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[b, sub], ReplaceAll[c, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo31[i] =
    Plot[f[x], {x, -260, 201}, PlotStyle -> {RGBColor[1, 0, 0]},
     PlotRange -> All]; Show[lp, plo31[i]]}]
 ]

For[i = M0 - 52, i < M0, i++; Print[i];
 g = Table[{T0[j], G0[j]}, {j, 1, i}]; M = Length[g];
 T[i_] := Extract[Extract[g, i], 1];
 G[i_] := Extract[Extract[g, i], 2];
 F[x_] = 300 a Exp[-.1 b x - .01 c x^2];
 sub = FindFit[g, F[x], {a, b, c}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[b, sub], ReplaceAll[c, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo32[i] =
    Plot[f[x], {x, -261, 261}, PlotStyle -> {RGBColor[0, 1, 0]},
     PlotRange -> All]; Show[lp, plo32[i]]}]
 ]

Show[lp, Table[plo31[n], {n, M0 - 51, M0}],
 Table[plo32[n], {n, M0 - 51, M0}],
 PlotRange -> {{-261, 201}, {-2, 310}}, AspectRatio -> Automatic,
 GridLines -> Automatic]

For[i = M0 - 52, i < M0, i++; Print[i];
 g = Table[{T0[j], G0[j]}, {j, 1, i}]; M = Length[g];
 T[i_] := Extract[Extract[g, i], 1];
 G[i_] := Extract[Extract[g, i], 2];
 F[x_] = 300 a Sqrt[(100 + b - x) (400 + c + x)];
 sub = FindFit[g, F[x], {a, b, c}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[b, sub], ReplaceAll[c, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo33[i] =
    Plot[f[x], {x, -261, 261}, PlotRange -> All,
     PlotStyle -> RGBColor[0, 0, 0]]; Show[lp, plo33[i]]}]
 ]

Show[lp, Table[plo31[n], {n, M0 - 51, M0}],
 Table[plo32[n], {n, M0 - 51, M0}], Table[plo33[n], {n, M0 - 51, M0}],
  PlotRange -> {{-261, 201}, {-2, 310}}, AspectRatio -> Automatic,
 GridLines -> Automatic]

For[i = M0 - 52, i < M0, i++; Print[i];
 g = Table[{T0[j], G0[j]}, {j, 1, i}]; M = Length[g];
 T[i_] := Extract[Extract[g, i], 1];
 G[i_] := Extract[Extract[g, i], 2];
 F[x_] = 300 a /Cosh[.01 b (100 + c + x)];
 sub = FindFit[g, F[x], {a, b, c}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[b, sub], ReplaceAll[c, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo34[i] = Plot[f[x], {x, -261, 261}, PlotRange -> All,
     PlotStyle -> RGBColor[0, 0, 1]]; Show[lp, plo34[i]]}]
 ]

p3 = Show[lp, Table[plo31[n], {n, M0 - 51, M0}],
  Table[plo32[n], {n, M0 - 51, M0}],
  Table[plo33[n], {n, M0 - 51, M0}],
  Table[plo34[n], {n, M0 - 51, M0}],
  PlotRange -> {{-251, 251}, {-2, 310}}, AspectRatio -> Automatic,
  GridLines -> {{-250, -200, -150, -100, -50, 50, 100, 150, 200,
     250}, {50, 100, 150, 200, 250, 300}}]

Export["p34.pdf", p3]

C++ generator of dates


#include<stdio.h>
#include<math.h>

void ju24da(int Mjd, int *Year, int *Month, int *Day) { int J, C, Y, M;
    J = Mjd + 2400000 + 68569;
    C = 4 * J / 146097;
    J = J - (146097 * C + 3) / 4;
    Y = 4000 * (J + 1) / 1461001;
    J = J - 1461 * Y / 4 + 31;
    M = 80 * J / 2447;
    *Day = J - 2447 * M / 80;
    J = M / 11;
    *Month = M + 2 - (12 * J);
    *Year = 100 * (C - 49) + Y + J;
// http://www.leapsecond.com/tools/gpsdate.c
}

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;
}

int main(){int j, k, n, y,m,d;
k=daju24(2014,10,27);
ju24da(k,&y,&m,&d); printf("%4d %4d %2d %2d\n",k,y,m,d);

n=-2300,ju24da(k+n,&y,&m,&d); printf("%4d %4d %2d %2d\n",n,y,m,d);
n=-2200,ju24da(k+n,&y,&m,&d); printf("%4d %4d %2d %2d\n",n,y,m,d);
n=-2100,ju24da(k+n,&y,&m,&d); printf("%4d %4d %2d %2d\n",n,y,m,d);
n=-2000,ju24da(k+n,&y,&m,&d); printf("%4d %4d %2d %2d\n",n,y,m,d);
ju24da(k-226,&y,&m,&d); printf("%4d %4d %2d %2d\n",-226,y,m,d);

for(n=-300;n<600;n+=50){
        ju24da(k+n,&y,&m,&d);
        j=daju24(y,m,d);
        printf("%4d %4d %2d %2d %4d\n",n,y,m,d,j-k); }
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 368pt
\paperheight 242pt
\topmargin -66pt
\oddsidemargin -70pt
\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\thispagestyle{empty}
\parindent 0pt
\begin{document}
\begin{picture}(400,200)
\put(0,0){\includegraphics{p34}}
\put(6,19){\sx{1.}{\rot{90}{\bf 2014.02.19}\ero}}
\put(42,19){\sx{1.}{\rot{90}{\bf 2014.04.10}\ero}}
\put(77,19){\sx{1.}{\rot{90}{\bf 2014.05.30}\ero}}
\put(114,19){\sx{1.}{\rot{90}{\bf 2014.07.19}\ero}} %
\put(151,19){\sx{1.}{\rot{90}{\bf 2014.09.07}\ero}} %
\put(185,19){\sx{1.}{\rot{90}{\bf 2014.10.27}\ero}} %
%\put(220,19){\sx{1.}{\rot{90}{\bf 2014.12.16}\ero}} %
%\put(256,19){\sx{1.}{\rot{90}{\bf 2015.02.23}\ero}}
%\put(292,19){\sx{1.}{\rot{90}{\bf 2015.04.13}\ero}}
%\put(328,19){\sx{1.}{\rot{90}{\bf 2015.05.15}\ero}}
%\put(364,19){\sx{1.}{\rot{90}{\bf 2015.07.04}\ero}}
\put(168,238){\sx{1.1}{$y$}}
%\put(324,163){\sx{1.1}{\rot{-4}$f_{23}(x)$\ero}}
%\put(324,150){\sx{1.1}{\rot{-6}$f_{22}(x)$\ero}}
\put(302,114){\sx{1.1}{\rot{-28}$y=\mathrm{Bell}(x)$\ero}}
\put(299,77){\sx{1.1}{\rot{-35}$y\!=\!\mathrm{Gauss}(x)$\ero}}
\put(277,86){\sx{1.1}{\rot{-56}$y\!=\!\mathrm{Quadratic}(x)$\ero}}
\put(230,95){\sx{1.1}{\rot{-80}$y\!=\!\mathrm{Ellipse}(x)$\ero}}
%\put(190,105){\sx{1.2}{experimental}}
%\put(204,92){\sx{1.2}{data}}
\put(355,2){\sx{1.1}{$x$}}
\end{picture}
\end{document}

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