File:2014rubleDollar2param.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

10 red lines represent the linear approximations,

$fx)=L(x)=a+b x$

Parameters $a$ and $b$ are evaluated, minimising the mean square deviation

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

for $M-10\le m \le M$; in such a way, for function $f=L$, 10 curves are plotted, and these curve form the red strip.

The same is plotted also $f=J$ and for $f=K$, where

$K(x)=a \exp(-b x)$

$J(x)=a / (1+b x)$

References

The image is used as Fig.1 in the 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.

Input file


-249 279.71
-248 281.64
-247 281.64
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-245 281.56
-244 280.07
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-240 277.92
-239 277.92
-238 273.87
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-236 277.34
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-232 274.70
-231 274.96
-230 274.41
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-187 280.15
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-180 280.92
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 -76 278.19
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 -71 276.80
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 -67 274.97
 -66 277.41
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 -61 276.51
 -60 278.26
 -59 272.08
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 -57 269.77
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 -53 271.69
 -52 270.57
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 -50 270.67
 -49 270.45
 -48 269.93
 -47 269.59
 -46 268.02
 -45 266.44
 -44 264.65
 -43 264.65
 -42 264.59
 -41 261.08
 -40 260.85
 -39 260.19
 -38 259.89
 -37 260.18
 -36 260.18
 -35 260.20
 -34 258.19
 -33 259.19
 -32 261.74
 -31 259.78
 -30 255.41
 -29 255.41
 -28 255.47
 -27 253.60
 -26 251.98
 -25 252.71
 -24 252.71
 -23 252.71
 -22 250.12
 -21 250.16
 -20 250.70
 -19 250.70
 -18 250.70
 -17 250.70
 -16 250.70
 -15 250.70
 -14 250.70
 -13 250.70
 -12 250.70
 -11 250.70
 -10 250.70
  -9 245.59
  -8 245.59
  -7 244.29
  -6 244.08
  -5 243.90
  -4 240.73
  -3 238.51
  -2 238.69
  -1 238.69
   0 238.18
   1 235.07
   2 233.92
   3 230.48
   4 236.57
   5 232.41
   6 232.41
   7 230.63
   8 229.33
   9 224.56
  10 220.42
  11 208.61
  12 214.13
  13 214.54
  14 218.75
  15 216.71
  16 215.48
  17 216.06
  18 210.84
  19 211.21
  20 211.68
  21 211.00
  22 214.45
  23 212.92
  24 214.27
  25 219.87
  26 218.47
  27 218.47
  28 225.32
  29 220.79
  30 214.30
  31 209.51
  32 202.55
  33 199.15
  34 199.15
  35 191.10
  36 195.33
  37 182.72
  38 189.91
  39 186.14
  40 189.83
  41 189.83
  42 187.19
  43 184.48
  44 184.11
  45 181.95
  46 174.51
  47 171.89
  48 171.89
  49 171.08
  50 152.41
  51 146.39
  52 165.41
  53 165.88
  54 169.84
  55 169.84
  56 176.53
  57 182.76
  58 183.13
  59 190.59
  60 195.90
  61 195.89
  62 195.89
  63 195.89
  64 173.54
  65 173.57
  66 165.16
  67 163.92
  68 170.27
  69 170.27
  70 171.73
  71 159.74
  72 157.38
  73 163.57

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 ; sub = FindFit[g, F[x], {a, b}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[100 + b, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo21[i] =
    Plot[f[x], {x, -261, 261}, PlotStyle -> {RGBColor[1, 0, 0]},
     PlotRange -> All]; Show[lp, plo21[i]]}]
 ]

For[i = M0 - 11, 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_] = Exp[a + b x] ; sub = FindFit[g, F[x], {a, b}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[100 + b, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo22[i] =
    Plot[f[x], {x, -261, 261}, PlotStyle -> {RGBColor[0, 1, 0]},
     PlotRange -> All]; Show[lp, plo22[i]]}]
 ]

For[i = M0 - 11, 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/(100 + .001 b x) ; sub = FindFit[g, F[x], {a, b}, x];
 Print[{f[x_] = ReplaceAll[F[x], sub], ReplaceAll[a, sub],
   ReplaceAll[100 + b, sub],
   Sum[Abs[f[T[i]] - G[i]], {i, 1, M}]/M,
   Sqrt[Sum[(f[T[i]] - G[i])^2, {i, 1, M}]/M],
   plo23[i] =
    Plot[f[x], {x, -261, 261}, PlotStyle -> {RGBColor[0, 0, 1]},
     PlotRange -> All]; Show[lp, plo23[i]]}]
 ]

p23 = Show[lp, Table[plo21[n], {n, M0 - 10, M0}],
  Table[plo22[n], {n, M0 - 10, M0}],
  Table[plo23[n], {n, M0 - 10, M0}],
  PlotRange -> {{-251, 251}, {0, 310}}, AspectRatio -> Automatic,
  GridLines -> {{-250, -200, -150, -100, -50, 50, 100, 150, 200,
     250}, {50, 100, 150, 200, 250, 300}}]

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 \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray}}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

%\renewcommand{\baselinestretch}{1}
%\setcounter{page}{1}
%\setlength{\textheight}{21.6cm}
%\setlength{\textwidth}{14cm}
%\setlength{\oddsidemargin}{1cm}
%\setlength{\evensidemargin}{1cm}
%\pagestyle{myheadings}
\thispagestyle{empty}
\parindent 0pt
\begin{document}

%\begin{figure}
%\begin{center}
\begin{picture}(400,200)
\put(0,0){\includegraphics{p23}}
\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(329,161){\sx{1.1}{\rot{-4}$J(x)$\ero}}
\put(328,142){\sx{1.1}{\rot{-11}$K(x)$\ero}}
\put(328,120){\sx{1.1}{\rot{-16}$L(x)$\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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