From fd921a1c05c98dda4975254ed961556c98a25aca Mon Sep 17 00:00:00 2001 From: me Date: Wed, 8 Apr 2026 22:01:01 +0300 Subject: [PATCH] update --- notes/120326.pdf | Bin 0 -> 17792 bytes notes/120326.typ | 8 + rgr1/archive/report.pdf | 16989 ++++++++++++++++++++++++++ rgr1/archive/report.typ | 571 + rgr1/assets/ecdf-X1.png | Bin 0 -> 71879 bytes rgr1/assets/ecdf-X2.png | Bin 0 -> 72659 bytes rgr1/assets/ecdf-X3.png | Bin 0 -> 75465 bytes rgr1/assets/ecdf-X4.png | Bin 0 -> 72722 bytes rgr1/assets/histogram-X1.png | Bin 0 -> 104941 bytes rgr1/assets/histogram-X2.png | Bin 0 -> 93388 bytes rgr1/assets/histogram-X3.png | Bin 0 -> 112889 bytes rgr1/assets/histogram-X4.png | Bin 0 -> 105076 bytes rgr1/assets/variation-series-X1.png | Bin 0 -> 72240 bytes rgr1/assets/variation-series-X2.png | Bin 0 -> 72304 bytes rgr1/assets/variation-series-X3.png | Bin 0 -> 77516 bytes rgr1/assets/variation-series-X4.png | Bin 0 -> 72235 bytes rgr1/data.csv | 201 + rgr1/main.py | 380 + rgr1/task.pdf | Bin 0 -> 225917 bytes rgr1/theory/theory.pdf | Bin 0 -> 23820 bytes rgr1/theory/theory.typ | 37 + task2.ipynb | 287 + variant_5.csv | 100 + 23 files changed, 18573 insertions(+) create mode 100644 notes/120326.pdf create mode 100644 notes/120326.typ create mode 100644 rgr1/archive/report.pdf create mode 100644 rgr1/archive/report.typ create mode 100644 rgr1/assets/ecdf-X1.png create mode 100644 rgr1/assets/ecdf-X2.png create mode 100644 rgr1/assets/ecdf-X3.png create mode 100644 rgr1/assets/ecdf-X4.png create mode 100644 rgr1/assets/histogram-X1.png create mode 100644 rgr1/assets/histogram-X2.png create mode 100644 rgr1/assets/histogram-X3.png create mode 100644 rgr1/assets/histogram-X4.png create mode 100644 rgr1/assets/variation-series-X1.png create mode 100644 rgr1/assets/variation-series-X2.png create mode 100644 rgr1/assets/variation-series-X3.png create mode 100644 rgr1/assets/variation-series-X4.png create mode 100644 rgr1/data.csv create mode 100644 rgr1/main.py create mode 100644 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53 0 R + /K [69 0 R 64 0 R 59 0 R 55 0 R] +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /TR + /P 54 0 R + /K [58 0 R 57 0 R 56 0 R] +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /TD + /P 55 0 R + /A [<< + /O /Table + /Headers [(U10x2y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [22] + /Pg 1024 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /TD + /P 55 0 R + /A [<< + /O /Table + /Headers [(U10x1y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [21] + /Pg 1024 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /TD + /P 55 0 R + /A [<< + /O /Table + /Headers [(U10x0y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [20] + /Pg 1024 0 R +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /TR + /P 54 0 R + /K [62 0 R 61 0 R 60 0 R] +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /TD + /P 59 0 R + /A [<< + /O /Table + /Headers [(U10x2y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [19] + /Pg 1024 0 R +>> +endobj + 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obj +<< + /Type /StructElem + /S /Strong + /P 80 0 R + /K [0] + /Pg 1024 0 R +>> +endobj + +82 0 obj +<< + /Type /StructElem + /S /Caption + /P 53 0 R + /K [84 0 R 83 0 R] +>> +endobj + +83 0 obj +<< + /Type /StructElem + /S /Formula + /P 82 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [24 25] + /Pg 1024 0 R +>> +endobj + +84 0 obj +<< + /Type /StructElem + /S /Span + /P 82 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [23] + /Pg 1024 0 R +>> +endobj + +85 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [86 0 R] +>> +endobj + +86 0 obj +<< + /Type /StructElem + /S /Em + /P 85 0 R + /K [58] + /Pg 1022 0 R +>> +endobj + +87 0 obj +<< + /Type /StructElem + /S /H2 + /P 49 0 R + /T + /K [51 89 0 R 53 88 0 R 57] + /Pg 1022 0 R +>> +endobj + +88 0 obj +<< + /Type /StructElem + /S /Formula + /P 87 0 R + /K [54 55 56] + /Pg 1022 0 R +>> +endobj + +89 0 obj +<< + /Type /StructElem + /S /Formula + /P 87 0 R + /K [52] + /Pg 1022 0 R +>> +endobj + +90 0 obj +<< + /Type 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R + /T + /K [43] + /Pg 1022 0 R +>> +endobj + +100 0 obj +<< + /Type /StructElem + /S /H1 + /P 49 0 R + /T + /K [40 101 0 R] + /Pg 1022 0 R +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /Formula + /P 100 0 R + /K [41 42] + /Pg 1022 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [26 105 0 R 28 29 104 0 R 31 103 0 R 37 38 39] + /Pg 1022 0 R +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /Formula + /P 102 0 R + /K [32 33 34 35 36] + /Pg 1022 0 R +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /Em + /P 102 0 R + /K [30] + /Pg 1022 0 R +>> +endobj + +105 0 obj +<< + /Type /StructElem + /S /Em + /P 102 0 R + /K [27] + /Pg 1022 0 R +>> +endobj + +106 0 obj +<< + /Type /StructElem + /S /L + /P 49 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [115 0 R 111 0 R 107 0 R] +>> +endobj + +107 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [110 0 R 108 0 R] +>> +endobj + +108 0 obj +<< + /Type /StructElem + /S /LBody + /P 107 0 R + /K [109 0 R 25] + /Pg 1022 0 R +>> +endobj + +109 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [23 24] + /Pg 1022 0 R +>> +endobj + +110 0 obj +<< + /Type /StructElem + /S /Lbl + /P 107 0 R + /K [22] + /Pg 1022 0 R +>> +endobj + +111 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [114 0 R 112 0 R] +>> +endobj + +112 0 obj +<< + /Type /StructElem + /S /LBody + /P 111 0 R + /K [113 0 R 21] + /Pg 1022 0 R +>> +endobj + +113 0 obj +<< + /Type /StructElem + /S /Formula + /P 112 0 R + /K [19 20] + /Pg 1022 0 R +>> +endobj + +114 0 obj +<< + /Type /StructElem + /S /Lbl + /P 111 0 R + /K [18] + /Pg 1022 0 R +>> +endobj + +115 0 obj +<< + /Type /StructElem + /S /LI + /P 106 0 R + /K [118 0 R 116 0 R] +>> +endobj + +116 0 obj +<< + /Type /StructElem + /S /LBody + /P 115 0 R + /K [117 0 R 17] + /Pg 1022 0 R +>> +endobj + +117 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [15 16] + /Pg 1022 0 R +>> +endobj + +118 0 obj +<< + /Type /StructElem + /S 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+ /Pg 1020 0 R +>> +endobj + +128 0 obj +<< + /Type /StructElem + /S /H2 + /P 49 0 R + /T + /K [189] + /Pg 1020 0 R +>> +endobj + +129 0 obj +<< + /Type /StructElem + /S /Table + /P 49 0 R + /A [<< + /O /Layout + /BorderColor [0 0 0] + /BorderThickness 1 + >>] + /K [150 0 R 144 0 R 130 0 R] +>> +endobj + +130 0 obj +<< + /Type /StructElem + /S /TBody + /P 129 0 R + /K [140 0 R 135 0 R 131 0 R] +>> +endobj + +131 0 obj +<< + /Type /StructElem + /S /TR + /P 130 0 R + /K [133 0 R 132 0 R] +>> +endobj + +132 0 obj +<< + /Type /StructElem + /S /TD + /P 131 0 R + /A [<< + /O /Table + /Headers [(U9x1y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [182] + /Pg 1020 0 R +>> +endobj + +133 0 obj +<< + /Type /StructElem + /S /TD + /P 131 0 R + /A [<< + /O /Table + /Headers [(U9x0y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [179 134 0 R] + /Pg 1020 0 R +>> +endobj + +134 0 obj +<< + /Type /StructElem + /S /Formula + /P 133 0 R + /K [180 181] + /Pg 1020 0 R +>> +endobj + 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87 88 89 90 91 92] + /Pg 1020 0 R +>> +endobj + +163 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [169 0 R 167 0 R 166 0 R 164 0 R 47] + /Pg 1020 0 R +>> +endobj + +164 0 obj +<< + /Type /StructElem + /S /Formula + /P 163 0 R + /K [165 0 R] +>> +endobj + +165 0 obj +<< + /Type /StructElem + /S /Strong + /P 164 0 R + /K [45 46] + /Pg 1020 0 R +>> +endobj + +166 0 obj +<< + /Type /StructElem + /S /Strong + /P 163 0 R + /K [44] + /Pg 1020 0 R +>> +endobj + +167 0 obj +<< + /Type /StructElem + /S /Formula + /P 163 0 R + /K [168 0 R] +>> +endobj + +168 0 obj +<< + /Type /StructElem + /S /Strong + /P 167 0 R + /K [43] + /Pg 1020 0 R +>> +endobj + +169 0 obj +<< + /Type /StructElem + /S /Strong + /P 163 0 R + /K [42] + /Pg 1020 0 R +>> +endobj + +170 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [29 172 0 R 33 171 0 R 41] + /Pg 1020 0 R +>> +endobj + +171 0 obj +<< + /Type /StructElem + /S /Formula + /P 170 0 R + /K [34 35 36 37 38 39 40] + /Pg 1020 0 R +>> +endobj + +172 0 obj +<< + /Type /StructElem + /S /Formula + /P 170 0 R + /K [30 31 32] + /Pg 1020 0 R +>> +endobj + +173 0 obj +<< + /Type /StructElem + /S /H2 + /P 49 0 R + /T + /K [27 28] + /Pg 1020 0 R +>> +endobj + +174 0 obj +<< + /Type /StructElem + /S /Table + /P 49 0 R + /A [<< + /O /Layout + /BorderColor [0 0 0] + /BorderThickness 1 + >>] + /K [204 0 R 191 0 R 175 0 R] +>> +endobj + +175 0 obj +<< + /Type /StructElem + /S /TBody + /P 174 0 R + /K [186 0 R 181 0 R 176 0 R] +>> +endobj + +176 0 obj +<< + /Type /StructElem + /S /TR + /P 175 0 R + /K [180 0 R 179 0 R 178 0 R 177 0 R] +>> +endobj + +177 0 obj +<< + /Type /StructElem + /S /TD + /P 176 0 R + /A [<< + /O /Table + /Headers [(U8x3y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [18] + /Pg 1020 0 R +>> +endobj + +178 0 obj +<< + /Type /StructElem + /S /TD + /P 176 0 R + /A [<< + /O /Table + /Headers [(U8x2y0)] + >> << + /O /Layout + /BorderStyle /Solid + >>] + /K [17] + /Pg 1020 0 R +>> +endobj + +179 0 obj +<< + /Type 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1018 0 R +>> +endobj + +212 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [169] + /Pg 1018 0 R +>> +endobj + +213 0 obj +<< + /Type /StructElem + /S /H2 + /P 49 0 R + /T + /K [166 214 0 R] + /Pg 1018 0 R +>> +endobj + +214 0 obj +<< + /Type /StructElem + /S /Formula + /P 213 0 R + /K [167 168] + /Pg 1018 0 R +>> +endobj + +215 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [149 217 0 R 158 216 0 R 165] + /Pg 1018 0 R +>> +endobj + +216 0 obj +<< + /Type /StructElem + /S /Formula + /P 215 0 R + /K [159 160 161 162 163 164] + /Pg 1018 0 R +>> +endobj + +217 0 obj +<< + /Type /StructElem + /S /Formula + /P 215 0 R + /K [150 151 152 153 154 155 156 157] + /Pg 1018 0 R +>> +endobj + +218 0 obj +<< + /Type /StructElem + /S /H1 + /P 49 0 R + /T + /K [148] + /Pg 1018 0 R +>> +endobj + +219 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [220 0 R] +>> +endobj + +220 0 obj +<< + /Type /StructElem + /S /Em + /P 219 0 R + /K [145 146 147] + /Pg 1018 0 R +>> +endobj + +221 0 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222 223 224 225 226] + /Pg 1016 0 R +>> +endobj + +317 0 obj +<< + /Type /StructElem + /S /Formula + /P 315 0 R + /K [201 202 203 204] + /Pg 1016 0 R +>> +endobj + +318 0 obj +<< + /Type /StructElem + /S /Lbl + /P 314 0 R + /K [199] + /Pg 1016 0 R +>> +endobj + +319 0 obj +<< + /Type /StructElem + /S /LI + /P 313 0 R + /K [324 0 R 320 0 R] +>> +endobj + +320 0 obj +<< + /Type /StructElem + /S /LBody + /P 319 0 R + /K [161 323 0 R 168 322 0 R 187 321 0 R 198] + /Pg 1016 0 R +>> +endobj + +321 0 obj +<< + /Type /StructElem + /S /Formula + /P 320 0 R + /K [188 189 190 191 192 193 194 195 196 197] + /Pg 1016 0 R +>> +endobj + +322 0 obj +<< + /Type /StructElem + /S /Formula + /P 320 0 R + /K [169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186] + /Pg 1016 0 R +>> +endobj + +323 0 obj +<< + /Type /StructElem + /S /Formula + /P 320 0 R + /K [162 163 164 165 166 167] + /Pg 1016 0 R +>> +endobj + +324 0 obj +<< + /Type /StructElem + /S /Lbl + /P 319 0 R + /K [160] + /Pg 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112 113 114 115 116 117 118 119 120 121 122 123 124] + /Pg 1016 0 R +>> +endobj + +333 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [334 0 R] +>> +endobj + +334 0 obj +<< + /Type /StructElem + /S /Strong + /P 333 0 R + /K [104] + /Pg 1016 0 R +>> +endobj + +335 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [94 336 0 R 103] + /Pg 1016 0 R +>> +endobj + +336 0 obj +<< + /Type /StructElem + /S /Formula + /P 335 0 R + /K [95 96 97 98 99 100 101 102] + /Pg 1016 0 R +>> +endobj + +337 0 obj +<< + /Type /StructElem + /S /H1 + /P 49 0 R + /T + /K [86 338 0 R] + /Pg 1016 0 R +>> +endobj + +338 0 obj +<< + /Type /StructElem + /S /Formula + /P 337 0 R + /K [87 88 89 90 91 92 93] + /Pg 1016 0 R +>> +endobj + +339 0 obj +<< + /Type /StructElem + /S /P + /P 49 0 R + /K [340 0 R] +>> +endobj + +340 0 obj +<< + /Type /StructElem + /S /Em + /P 339 0 R + /K [82 83 84 85] + /Pg 1016 0 R +>> +endobj + +341 0 obj +<< + /Type /StructElem + /S /Table + /P 49 0 R + /A [<< + /O /Layout + 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).XuIRXqpwEgݩ}cotwT=g#ֳDy-wFĨ +}aWc;`pSwF?Z)w8cs7ESFC,2ӟW;RlNQǡW5.;liah!ZHݑ71RS \T%(HH* HCzbS32lص R3)SPVZnj6׶۫|؛wxo"*it]ȴzf:HCdwG'-R.W)Eح5ml.qK Yº#yy7X?uۨ&F4؛ _o.&I- L;G.f~Q6(y4~4d@^EaFZxCpAsd?>lq(tin#FF& hgn 9Y,Hͺ5ߦ;ʝF/26K)P9h5飃Pϑd@$\ple.s'K^CCkDlDΙ:KȴJP"R7՞m5{]0>YZ@R-v&+l,a4`=eT|AHy/W +#ܑ`[W!Zw{*A2j 8yqll0=T 1{IJ#LUTB9fn8N& +K%:%"L*`,3#K6U'D6{HO9-sJ4i}0K1rs54%(Tr)"sHw؋XvyJX^ (~Wr"7;|kk`VeK&V}?LލK^a#942/_ɂ?'߮(S4m {3%9̢{&f{Ѩ -~ua1RLF<Sf0o/NV >7~čY3/y$d;&rJ1MԂ eO]~a-Eo| _^/R%Ls<2ܤɇB~u_k|6/BUda;[x'vܴ LM[J8&!:69s Fa:ĴWGpڊ@>ƥ{Bo~fönv遾^k7 o߽>_ONhڈ_^qi\o)[{y\Ͷ)ǎ߲ Kw'Q[:loSl&mݭ@wow_τ;{y~I*/7/io7Œ܄:&ݓ'M1mXrΪnww5vpg nni;mp{f~fa^f}r/rrcq> +endobj + +917 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /FUYYLL+NewCMMath-Bold + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 919 0 R + /DW 0 + /W [0 0 280 1 1 947 2 4 647 5 5 708 6 6 686] +>> +endobj + +918 0 obj +<< + /Length 9 + /Filter /FlateDecode +>> +stream +x +endstream +endobj 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a/rgr1/archive/report.typ b/rgr1/archive/report.typ new file mode 100644 index 0000000..81ad23a --- /dev/null +++ b/rgr1/archive/report.typ @@ -0,0 +1,571 @@ +#set document(title: "РГР №1") +#set page( + paper: "a4", + margin: (top: 2.5cm, bottom: 2.5cm, left: 3cm, right: 2cm), + numbering: "1", +) +#set text(font: "New Computer Modern", size: 14pt, lang: "ru") +// #set heading(numbering: "1.") +#set par(justify: true, leading: 0.75em) +#show math.equation: set text(size: 14pt) +#show link: underline + +#set page(footer: context { + if counter(page).get().first() > 1 [ + #align(center)[ + #counter(page).display("1") + ] + ] + if counter(page).get().first() == 1 [ + #align(center)[ + Санкт-Петербург \ 2026 + ] + ] +}) + +#set page(header: context { + if counter(page).get().first() == 1 [ + #align(center)[ + Санкт-Петербургский национальный исследовательский университет информационных технологий, механики и оптики + ] + ] +}) + +#show raw.where(block: false): box.with( + fill: luma(240), + inset: (x: 3pt, y: 0pt), + outset: (y: 3pt), + radius: 2pt, +) + +#show raw.where(block: true): block.with( + fill: luma(240), + inset: 10pt, + radius: 4pt, +) + + +#for _ in range(5) { linebreak() } + +#align(center)[Лабораторная работа №4. Postgres] + +#for _ in range(15) { linebreak() } + +#align(right)[Выполнил:] +#align(right)[Дощенников Никита] +#align(right)[Группа: К3221] +#align(right)[Проверила:] +#align(right)[Александра Валерьевна Купера] + +#pagebreak() + +#outline(title: "Содержание", indent: 1.5em) + +#pagebreak() + += Первичное описание выборки + +В данном разделе выполняется первичный анализ столбцов $X_1$, $X_2$, $X_3$ +из предоставленного CSV-файла объёмом $n = 200$ наблюдений. + +== Вариационный ряд + +Вариационный ряд --- упорядоченная по возрастанию последовательность наблюдений $x_1 lt.eq x_2 lt.eq dots lt.eq x_n$. + +#align(center)[ + #figure( + image("assets/variation-series-X1.png"), + caption: [Вариационный ряд столбца $X_1$] + ) +] + +#align(center)[ + #figure( + image("assets/variation-series-X2.png"), + caption: [Вариационный ряд столбца $X_2$] + ) +] + +#align(center)[ + #figure( + image("assets/variation-series-X3.png"), + caption: [Вариационный ряд столбца $X_3$] + ) +] + +#align(center)[ + #figure( + image("assets/variation-series-X4.png"), + caption: [Вариационный ряд столбца $X_4$] + ) +] + +== Эмпирическая функция распределения + +Эмпирическая функция распределения определяется как + +$ +F_n (x) = frac(nu_n (x), n), +$ + +где $nu_n (x)$ --- число наблюдений, строго меньших $x$. График $F_n (x)$ является ступенчатой функцией: в каждой точке $x_i$ происходит скачок на $1/n$. + +#align(center)[ + #figure( + image("assets/ecdf-X1.png"), + caption: [Эмпирическая функция распределения $F_n (x)$ для ряда $X_1$] + ) +] + +#align(center)[ + #figure( + image("assets/ecdf-X2.png"), + caption: [Эмпирическая функция распределения $F_n (x)$ для ряда $X_2$] + ) +] + +#align(center)[ + #figure( + image("assets/ecdf-X3.png"), + caption: [Эмпирическая функция распределения $F_n (x)$ для ряда $X_3$] + ) +] + +#align(center)[ + #figure( + image("assets/ecdf-X4.png"), + caption: [Эмпирическая функция распределения $F_n (x)$ для ряда $X_4$] + ) +] + +== Гистограммы + +Для выбора числа интервалов использовались три правила: + +- *Правило Скотта:* $h = 3.5 hat(sigma) n^(-1/3)$, откуда $k = ceil.l (x_max - x_min) / h ceil.r$. +- *Правило Фридмана–Диакониса:* $h = 2 dot "IQR" dot n^(-1/3), space "IQR" = q_0.75 - q_0.25$. +- *Правило Стерджеса:* $k = 1 + floor.l log_2 n floor.r$. + +Результаты для $n = 200$: + +#figure( + table( + columns: 4, + align: horizon + center, + table.header([*Правило*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [Скотт], [10], [6], [11], + [Фридман–Диакон.], [12], [6], [17], + [Стерджес], [8], [8], [8], + ), + caption: [Число интервалов $k$ по разным правилам], +) + +В дальнейшем используется правило Скотта как наиболее устойчивое к форме +распределения. + +#figure( + rect(width: 100%, height: 7cm, fill: luma(240), stroke: luma(180))[ + #align(center + horizon)[_[Вставить гистограммы X1 (три правила рядом)]_] + ], + caption: [Гистограммы $X_1$ по трём правилам], +) + +#figure( + rect(width: 100%, height: 7cm, fill: luma(240), stroke: luma(180))[ + #align(center + horizon)[_[Вставить гистограммы X2 (три правила рядом)]_] + ], + caption: [Гистограммы $X_2$ по трём правилам], +) + +#figure( + rect(width: 100%, height: 7cm, fill: luma(240), stroke: luma(180))[ + #align(center + horizon)[_[Вставить гистограммы X3 (три правила рядом)]_] + ], + caption: [Гистограммы $X_3$ по трём правилам], +) + +// ─── 1.4 ───────────────────────────────────────────────────────────────────── +== Числовые характеристики + +Выборочное среднее: +$ +macron(x) = frac(1,n) sum_(i=1)^n x_i. +$ + +Смещённая и несмещённая дисперсии: +$ +S^2 = frac(1,n) sum_(i=1)^n (x_i - macron(x))^2, quad +hat(sigma)^2 = frac(1,n-1) sum_(i=1)^n (x_i - macron(x))^2. +$ + +#figure( + table( + columns: (2.2fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header( + [*Характеристика*], [*$X_1$*], [*$X_2$*], [*$X_3$*] + ), + [$macron(x)$ — выборочное среднее], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$S^2$ — смещённая дисперсия], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(sigma)^2$ — несмещённая дисперсия], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$S$ — смещённое ст. откл.], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(sigma)$ — несмещённое ст. откл.],[\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$m_e$ — медиана], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$q_{0.25}$ — первый квартиль], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$q_{0.75}$ — третий квартиль], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$"IQR" = q_{0.75} - q_{0.25}$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$x_min$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$x_max$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Числовые характеристики выборок], +) + +// ─── 1.5 ───────────────────────────────────────────────────────────────────── +== Описание формы распределений + +*Столбец $X_1$.* +_(Заполнить после анализа: симметрия/асимметрия, наличие выбросов, +естественные границы значений, характер хвостов.)_ + +*Столбец $X_2$.* +_(Заполнить после анализа.)_ + +*Столбец $X_3$.* +_(Заполнить после анализа.)_ + +// ───────────────────────────────────────────────────────────────────────────── += Гипотезы о виде закона распределения +// ───────────────────────────────────────────────────────────────────────────── + +На основании гистограмм, ЭФР и числовых характеристик для каждого столбца +выдвигается гипотеза о законе распределения из списка: +нормальное $N(a, sigma^2)$, равномерное $U(a, b)$, +экспоненциальное со сдвигом $"Exp"_(lambda, c)$. + +*Столбец $X_1$ — предполагаемый закон: \_\_\_\_\_\_\_\_\_.* + +_(Обоснование: форма гистограммы, симметрия, соотношение среднего и медианы, +характер ЭФР и т.д. — 2–6 предложений.)_ + +*Столбец $X_2$ — предполагаемый закон: \_\_\_\_\_\_\_\_\_.* + +_(Обоснование.)_ + +*Столбец $X_3$ — предполагаемый закон: \_\_\_\_\_\_\_\_\_.* + +_(Обоснование.)_ + +// ───────────────────────────────────────────────────────────────────────────── += Оценивание параметров +// ───────────────────────────────────────────────────────────────────────────── + +// ─── 3.1 ───────────────────────────────────────────────────────────────────── +== Метод моментов + +Идея метода: приравнять теоретические моменты $E X$ и $D X$ к выборочным +оценкам и решить систему уравнений. + +=== Нормальное распределение $N(a, sigma^2)$ + +Теория: $E X = a$, $D X = sigma^2$. Оценки: +$ +hat(a) = macron(x), quad hat(sigma)^2 = S^2. +$ + +=== Равномерное распределение $U(a, b)$ + +Теория: $E X = (a+b)/2$, $D X = (b-a)^2 / 12$. Оценки: +$ +hat(a) = macron(x) - sqrt(3 S^2), quad hat(b) = macron(x) + sqrt(3 S^2). +$ + +=== Экспоненциальное распределение со сдвигом $"Exp"_(lambda,c)$ + +Плотность: $f(x) = lambda e^{-lambda(x-c)}$, $x gt.eq c$, $lambda > 0$. + +Теория: $E X = c + 1/lambda$, $D X = 1/lambda^2$. Оценки: +$ +hat(lambda) = 1/S, quad hat(c) = macron(x) - 1/hat(lambda). +$ + +=== Результаты метода моментов + +#figure( + table( + columns: (2fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header([*Параметр*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [Параметр 1 (\_\_\_)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [Параметр 2 (\_\_\_)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Оценки параметров методом моментов], +) + +// ─── 3.2 ───────────────────────────────────────────────────────────────────── +== Метод максимального правдоподобия + +Функция правдоподобия и логарифм правдоподобия: +$ +L(theta) = product_(i=1)^n f(x_i | theta), quad +ell(theta) = sum_(i=1)^n ln f(x_i | theta). +$ + +Оценка ММП: $hat(theta) = arg max_theta ell(theta)$. + +=== Нормальное распределение + +$ +ell(a, sigma) = -n ln sigma - frac(1, 2 sigma^2) sum_(i=1)^n (x_i - a)^2 + "const". +$ + +Из условий первого порядка: +$ +hat(a)_("МП") = macron(x), quad hat(sigma)^2_("МП") = S^2 = frac(1,n) sum_(i=1)^n (x_i - macron(x))^2. +$ + +=== Равномерное распределение + +Плотность $f(x) = 1/(b-a)$ при $x in [a,b]$. Правдоподобие максимизируется при +наименьшем возможном $(b-a)$, откуда: +$ +hat(a)_("МП") = min{x_1, dots, x_n}, quad hat(b)_("МП") = max{x_1, dots, x_n}. +$ + +=== Экспоненциальное распределение со сдвигом + +$ +ell(lambda, c) = n ln lambda - lambda sum_(i=1)^n (x_i - c). +$ + +Так как $ell$ возрастает по $c$ (при $c lt.eq x_min$): +$ +hat(c)_("МП") = min{x_1, dots, x_n}, quad hat(lambda)_("МП") = frac(1, macron(x) - hat(c)). +$ + +=== Результаты ММП + +#figure( + table( + columns: (2fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header([*Параметр*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [Параметр 1 (\_\_\_)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [Параметр 2 (\_\_\_)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Оценки параметров методом максимального правдоподобия], +) + +// ─── 3.3 ───────────────────────────────────────────────────────────────────── +== Сравнение оценок двух методов + +#figure( + table( + columns: (2fr, 1fr, 1fr, 1fr, 1fr, 1fr, 1fr), + align: center, + table.header( + [], table.cell(colspan: 2)[$X_1$], table.cell(colspan: 2)[$X_2$], table.cell(colspan: 2)[$X_3$], + [*Параметр*], [*ММ*], [*МП*], [*ММ*], [*МП*], [*ММ*], [*МП*], + ), + [Параметр 1], [\_\_], [\_\_], [\_\_], [\_\_], [\_\_], [\_\_], + [Параметр 2], [\_\_], [\_\_], [\_\_], [\_\_], [\_\_], [\_\_], + ), + caption: [Сравнение оценок методом моментов и ММП], +) + +_(Комментарий: для нормального распределения оценки совпадают. Для равномерного +и экспоненциального оценки различаются — пояснить почему, 2–4 предложения.)_ + +// ───────────────────────────────────────────────────────────────────────────── += Оценивание вероятности $P(X > x_0)$ +// ───────────────────────────────────────────────────────────────────────────── + +В качестве порога выбирается $x_0 = macron(x) + hat(sigma)$. + +*Эмпирическая оценка:* +$ +hat(p)_("эмп") = frac(1,n) sum_(i=1)^n bold(1)[x_i > x_0]. +$ + +*Параметрическая оценка* — подстановка найденных оценок в теоретическую формулу: + +- Для $N(a, sigma^2)$: $quad P(X > x_0) = 1 - Phi\( (x_0 - hat(a)) / hat(sigma) \)$. +- Для $U(a,b)$: $quad P(X > x_0) = (hat(b) - x_0) / (hat(b) - hat(a))$ при $x_0 in [hat(a), hat(b)]$. +- Для $"Exp"_(lambda,c)$: $quad P(X > x_0) = e^{-hat(lambda)(x_0 - hat(c))}$. + +#figure( + table( + columns: (2.5fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header([*Оценка*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [$x_0 = macron(x) + hat(sigma)$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(p)_"эмп"$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(p)_"пар"$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [Расхождение $|hat(p)_"эмп" - hat(p)_"пар"|$], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Сравнение эмпирической и параметрической оценок вероятности], +) + +_(Комментарий к расхождению — 2–3 предложения.)_ + +// ───────────────────────────────────────────────────────────────────────────── += Оценка моментов по сгруппированной выборке +// ───────────────────────────────────────────────────────────────────────────── + +Пусть гистограмма содержит $m$ интервалов, в $k$-й интервал попало $n_k$ +наблюдений, $hat(X)_k$ — середина $k$-го интервала. Тогда: + +$ +hat(X)_g = frac(1,n) sum_(k=1)^m n_k hat(X)_k, quad +hat(sigma)^2_g = frac(1,n-1) sum_(k=1)^m n_k (hat(X)_k - hat(X)_g)^2. +$ + +#figure( + table( + columns: (2.5fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header([*Характеристика*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [$hat(X)_g$ (по сгруппированной)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$macron(x)$ (по исходным)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(sigma)^2_g$ (по сгруппированной)],[\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [$hat(sigma)^2$ (по исходным)], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Сравнение оценок по сгруппированной и исходной выборкам], +) + +_(Комментарий: потеря точности при группировке обусловлена заменой каждого +наблюдения серединой интервала — 2–3 предложения.)_ + +// ───────────────────────────────────────────────────────────────────────────── += Доверительные интервалы +// ───────────────────────────────────────────────────────────────────────────── + +Уровень доверия $1 - alpha = 0.95$, то есть $alpha = 0.05$. + +== Асимптотический доверительный интервал для $E X$ + +По центральной предельной теореме для всех трёх столбцов: + +$ +E X in + lr(( macron(x) - z_{1-alpha/2} frac(hat(sigma), sqrt(n)), space + macron(x) + z_{1-alpha/2} frac(hat(sigma), sqrt(n)) )), +$ + +где $z_{0.975} approx 1.960$. + +#figure( + table( + columns: (2fr, 1fr, 1fr, 1fr), + align: (left, center, center, center), + table.header([*Граница*], [*$X_1$*], [*$X_2$*], [*$X_3$*]), + [Нижняя граница], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [Верхняя граница], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + [Ширина интервала], [\_\_\_\_], [\_\_\_\_], [\_\_\_\_], + ), + caption: [Асимптотические ДИ для $E X$ на уровне $0.95$], +) + +== Точные доверительные интервалы для нормального столбца + +Для столбца $X\_\_$ (отнесённого к $N(mu, sigma^2)$): + +*ДИ для $mu$ при неизвестной $sigma^2$* (распределение Стьюдента): +$ +mu in lr(( + macron(x) - t_{1-alpha/2,, n-1} frac(hat(sigma), sqrt(n)), + space + macron(x) + t_{1-alpha/2,, n-1} frac(hat(sigma), sqrt(n)) +)), +$ +где $t_{0.975, 199} approx \_\_\_\_$. + +*ДИ для $sigma^2$* (распределение $chi^2$): +$ +sigma^2 in lr(( + frac((n-1) hat(sigma)^2, chi^2_{1-alpha/2,, n-1}), + space + frac((n-1) hat(sigma)^2, chi^2_{alpha/2,, n-1}) +)). +$ + +#figure( + table( + columns: (2.5fr, 1fr), + align: (left, center), + table.header([*Интервал*], [*Значение*]), + [ДИ для $mu$ (асимптотический)], [(\_\_\_\_, \_\_\_\_)], + [ДИ для $mu$ (точный, $t$-распределение)], [(\_\_\_\_, \_\_\_\_)], + [ДИ для $sigma^2$], [(\_\_\_\_, \_\_\_\_)], + ), + caption: [Точные ДИ для параметров нормального распределения ($X\_\_$)], +) + +// ─── 6.3 ───────────────────────────────────────────────────────────────────── +== Интерпретация доверительных интервалов + +Доверительный интервал с уровнем $1 - alpha = 0.95$ означает следующее: +если бы мы многократно повторяли эксперимент и строили интервал по каждой +новой выборке, то примерно 95% таких интервалов покрывали бы истинное значение +параметра. Это *не* означает, что истинный параметр попадает в данный конкретный +интервал с вероятностью 95% — параметр либо лежит в интервале, либо нет. +Вероятностный смысл относится к процедуре построения, а не к конкретному результату. + +Чем уже интервал, тем точнее оценка. Ширина убывает как $O(1/sqrt(n))$, поэтому +для вдвое более узкого интервала требуется вчетверо большая выборка. + +// ───────────────────────────────────────────────────────────────────────────── += Итоговый вывод +// ───────────────────────────────────────────────────────────────────────────── + +_(Заполнить после получения числовых результатов — 5–12 строк.)_ + +По результатам анализа установлено следующее соответствие столбцов и законов +распределения: + +- $X_1$ — \_\_\_\_\_\_\_\_ с параметрами \_\_\_\_\_\_\_\_; +- $X_2$ — \_\_\_\_\_\_\_\_ с параметрами \_\_\_\_\_\_\_\_; +- $X_3$ — \_\_\_\_\_\_\_\_ с параметрами \_\_\_\_\_\_\_\_. + +Оценки методом моментов и методом максимального правдоподобия _(совпали / +незначительно расходятся)_. Доверительные интервалы для среднего имеют ширину +порядка \_\_\_\_, что свидетельствует о _(высокой / умеренной)_ точности оценок +при $n = 200$. Асимптотический и точный интервалы для нормального столбца +практически совпадают, что подтверждает применимость ЦПТ при данном объёме +выборки. + +// ───────────────────────────────────────────────────────────────────────────── += Бонус: анализ столбца $X_4$ +// ───────────────────────────────────────────────────────────────────────────── + +== Первичное описание + +#figure( + rect(width: 100%, height: 5cm, fill: luma(240), stroke: luma(180))[ + #align(center + horizon)[_[Вставить гистограмму и ЭФР для X4]_] + ], + caption: [Гистограмма и ЭФР столбца $X_4$], +) + +_(Описание формы: признаки бимодальности, неоднородности, наличие двух мод.)_ + +== Кластеризация ($k$-средних, $k=2$) + +_(Описание метода и результат разбиения на два кластера.)_ + +#figure( + table( + columns: (2fr, 1fr, 1fr), + align: (left, center, center), + table.header([*Характеристика*], [*Кластер 1*], [*Кластер 2*]), + [Объём $n_j$], [\_\_\_\_], [\_\_\_\_], + [Среднее $macron(x)_j$], [\_\_\_\_], [\_\_\_\_], + [Ст. откл. $hat(sigma)_j$], [\_\_\_\_], [\_\_\_\_], + [Мин / Макс], [\_\_\_\_ / \_\_\_\_], [\_\_\_\_ / \_\_\_\_], + ), + caption: [Характеристики кластеров столбца $X_4$], +) + +«Общее среднее» $macron(x)$ плохо описывает смесь двух режимов, поскольку +является взвешенным средним двух различных распределений и может не +соответствовать ни одному из них. 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