memory.tex

Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round]
2 \path[use as bounding box] (0,-.6) rectangle (14,4.8);
3 \draw[black!13,line width=.25pt] (0.8,0.65) -- (0.8,3.85);
4 \draw[] (0.8,0.59) -- (0.8,0.71);
5 \node[font=\small,align=center,font=\scriptsize] at (0.8,0.39) {0};
6 \draw[black!13,line width=.25pt] (2.925,0.65) -- (2.925,3.85);
7 \draw[] (2.925,0.59) -- (2.925,0.71);
8 \node[font=\small,align=center,font=\scriptsize] at (2.925,0.39) {1};
9 \draw[black!13,line width=.25pt] (5.05,0.65) -- (5.05,3.85);
10 \draw[] (5.05,0.59) -- (5.05,0.71);
11 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.39) {2};
12 \draw[black!13,line width=.25pt] (7.175,0.65) -- (7.175,3.85);
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14 \node[font=\small,align=center,font=\scriptsize] at (7.175,0.39) {3};
15 \draw[black!13,line width=.25pt] (9.3,0.65) -- (9.3,3.85);
16 \draw[] (9.3,0.59) -- (9.3,0.71);
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20 \draw[black!13,line width=.25pt] (0.8,2.25) -- (9.3,2.25);
21 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,2.25) {0.5};
22 \draw[black!13,line width=.25pt] (0.8,3.85) -- (9.3,3.85);
23 \node[font=\small,align=center,anchor=east,font=\scriptsize] at (0.64,3.85) {1};
24 \draw[->,line width=.6pt] (0.8,0.65) -- (9.42,0.65);
25 \draw[->,line width=.6pt] (0.8,0.65) -- (0.8,3.97);
26 \node[font=\small,align=center,font=\scriptsize] at (5.05,0.1) {time $t$};
27 \node[font=\small,align=center,anchor=west,font=\scriptsize] at (0.8,4.25) {retained state $x(t)$};
28 \draw[line width=1.1pt] plot coordinates {(0.8,3.85) (0.885,3.726986) (0.97,3.61343) (1.055,3.508605) (1.14,3.411838) (1.225,3.322512) (1.31,3.240053) (1.395,3.163935) (1.48,3.093668) (1.565,3.028804) (1.65,2.968926) (1.735,2.913653) (1.82,2.862629) (1.905,2.815527) (1.99,2.772048) (2.075,2.731911) (2.16,2.69486) (2.245,2.660657) (2.33,2.629084) (2.415,2.599939) (2.5,2.573034) (2.585,2.548198) (2.67,2.525272) (2.755,2.504108) (2.84,2.484571) (2.925,2.466536) (3.01,2.449888) (3.095,2.43452) (3.18,2.420334) (3.265,2.407238) (3.35,2.395149) (3.435,2.383989) (3.52,2.373688) (3.605,2.364178) (3.69,2.3554) (3.775,2.347296) (3.86,2.339816) (3.945,2.33291) (4.03,2.326536) (4.115,2.320651) (4.2,2.31522) (4.285,2.310205) (4.37,2.305576) (4.455,2.301303) (4.54,2.297359) (4.625,2.293718) (4.71,2.290357) (4.795,2.287254) (4.88,2.28439) (4.965,2.281746) (5.05,2.279305) (5.135,2.277052) (5.22,2.274972) (5.305,2.273052) (5.39,2.27128) (5.475,2.269644) (5.56,2.268133) (5.645,2.266739) (5.73,2.265452) (5.815,2.264264) (5.9,2.263168) (5.985,2.262155) (6.07,2.261221) (6.155,2.260358) (6.24,2.259562) (6.325,2.258827) (6.41,2.258148) (6.495,2.257521) (6.58,2.256943) (6.665,2.256409) (6.75,2.255917) (6.835,2.255462) (6.92,2.255042) (7.005,2.254654) (7.09,2.254296) (7.175,2.253966) (7.26,2.253661) (7.345,2.25338) (7.43,2.25312) (7.515,2.25288) (7.6,2.252658) (7.685,2.252454) (7.77,2.252265) (7.855,2.252091) (7.94,2.25193) (8.025,2.251782) (8.11,2.251645) (8.195,2.251519) (8.28,2.251402) (8.365,2.251294) (8.45,2.251195) (8.535,2.251103) (8.62,2.251018) (8.705,2.25094) (8.79,2.250867) (8.875,2.250801) (8.96,2.250739) (9.045,2.250682) (9.13,2.25063) (9.215,2.250581) (9.3,2.250537)};
29 \draw[dashed,line width=.8pt] plot coordinates {(0.8,3.85) (0.885,3.724526) (0.97,3.603972) (1.055,3.488145) (1.14,3.37686) (1.225,3.269938) (1.31,3.167209) (1.395,3.068508) (1.48,2.973677) (1.565,2.882564) (1.65,2.795024) (1.735,2.710917) (1.82,2.630107) (1.905,2.552466) (1.99,2.477869) (2.075,2.406197) (2.16,2.337336) (2.245,2.271174) (2.33,2.207607) (2.415,2.146533) (2.5,2.087853) (2.585,2.031474) (2.67,1.977305) (2.755,1.925261) (2.84,1.875257) (2.925,1.827214) (3.01,1.781055) (3.095,1.736706) (3.18,1.694095) (3.265,1.653156) (3.35,1.613821) (3.435,1.576029) (3.52,1.539719) (3.605,1.504833) (3.69,1.471314) (3.775,1.43911) (3.86,1.408169) (3.945,1.378441) (4.03,1.349878) (4.115,1.322435) (4.2,1.296069) (4.285,1.270736) (4.37,1.246397) (4.455,1.223012) (4.54,1.200544) (4.625,1.178956) (4.71,1.158216) (4.795,1.138288) (4.88,1.119142) (4.965,1.100747) (5.05,1.083073) (5.135,1.066092) (5.22,1.049777) (5.305,1.034101) (5.39,1.01904) (5.475,1.00457) (5.56,0.990667) (5.645,0.977309) (5.73,0.964475) (5.815,0.952145) (5.9,0.940297) (5.985,0.928915) (6.07,0.917978) (6.155,0.907471) (6.24,0.897375) (6.325,0.887675) (6.41,0.878356) (6.495,0.869402) (6.58,0.860799) (6.665,0.852534) (6.75,0.844592) (6.835,0.836962) (6.92,0.829631) (7.005,0.822588) (7.09,0.815821) (7.175,0.809319) (7.26,0.803072) (7.345,0.79707) (7.43,0.791303) (7.515,0.785762) (7.6,0.780439) (7.685,0.775324) (7.77,0.77041) (7.855,0.765689) (7.94,0.761153) (8.025,0.756794) (8.11,0.752607) (8.195,0.748584) (8.28,0.744718) (8.365,0.741004) (8.45,0.737436) (8.535,0.734008) (8.62,0.730714) (8.705,0.727549) (8.79,0.724508) (8.875,0.721586) (8.96,0.71878) (9.045,0.716083) (9.13,0.713492) (9.215,0.711002) (9.3,0.70861)};
30 \node[font=\small,align=center] at (11.7,3.75) {exact: $\dfrac{1+e^{-2t}}2$};
31 \node[font=\small,align=center] at (11.7,2.7) {naive deletion: $e^{-t}$};
32 \node[font=\small,align=center,font=\small] at (11.7,1.25) {$\displaystyle K(t)=e^{-t}$\\returns hidden excursions.};
33 \end{tikzpicture}

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