Finish writing

Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
2020-11-12 18:24:15 +01:00
parent 291912c344
commit a295b83c4c
18 changed files with 1474 additions and 31 deletions
--- a/img/marchenko-pastur.nb
+++ b/img/marchenko-pastur.nb
@@ -0,0 +1,422 @@
+(* Content-type: application/vnd.wolfram.mathematica *)
+
+(*** Wolfram Notebook File ***)
+(* http://www.wolfram.com/nb *)
+
+(* CreatedBy='Mathematica 12.1' *)
+
+(*CacheID: 234*)
+(* Internal cache information:
+NotebookFileLineBreakTest
+NotebookFileLineBreakTest
+NotebookDataPosition[       158,          7]
+NotebookDataLength[     18991,        414]
+NotebookOptionsPosition[     18069,        390]
+NotebookOutlinePosition[     18498,        407]
+CellTagsIndexPosition[     18455,        404]
+WindowFrame->Normal*)
+
+(* Beginning of Notebook Content *)
+Notebook[{
+Cell[BoxData[{
+ RowBox[{
+  RowBox[{
+   RowBox[{"n", "=", "10000"}], ";"}], " ", 
+  RowBox[{"(*", " ", "rows", " ", "*)"}]}], "\[IndentingNewLine]", 
+ RowBox[{
+  RowBox[{
+   RowBox[{"p", "=", "1000"}], ";"}], " ", 
+  RowBox[{"(*", " ", "columns", " ", "*)"}]}]}], "Input",
+ CellChangeTimes->{{3.814164590690421*^9, 3.814164608377831*^9}, {
+  3.8141648314838877`*^9, 3.8141648430424347`*^9}, {3.814165688469067*^9, 
+  3.814165694713544*^9}},
+ CellLabel->"In[1]:=",ExpressionUUID->"e07ea3b1-6853-427d-a42f-55af95fd957a"],
+
+Cell[BoxData[{
+ RowBox[{
+  RowBox[{
+   RowBox[{"eig", "=", 
+    RowBox[{"RandomVariate", "[", 
+     RowBox[{"MatrixPropertyDistribution", "[", 
+      RowBox[{
+       RowBox[{
+        RowBox[{"Eigenvalues", "[", "x", "]"}], "/", "n"}], ",", 
+       RowBox[{"Distributed", "[", 
+        RowBox[{"x", ",", 
+         RowBox[{"WishartMatrixDistribution", "[", 
+          RowBox[{"n", ",", 
+           RowBox[{"IdentityMatrix", "[", "p", "]"}]}], "]"}]}], "]"}]}], 
+      "]"}], "]"}]}], ";"}], " ", 
+  RowBox[{"(*", " ", 
+   RowBox[{"eigenvalues", " ", "distribution"}], " ", 
+   "*)"}]}], "\[IndentingNewLine]", 
+ RowBox[{
+  RowBox[{"eighist", "=", 
+   RowBox[{"Histogram", "[", 
+    RowBox[{"eig", ",", 
+     RowBox[{"{", "0.05", "}"}], ",", "\"\<PDF\>\""}], "]"}]}], 
+  ";"}], "\[IndentingNewLine]", 
+ RowBox[{
+  RowBox[{
+   RowBox[{"marcpast", "=", 
+    RowBox[{"Plot", "[", 
+     RowBox[{
+      RowBox[{"PDF", "[", 
+       RowBox[{
+        RowBox[{"MarchenkoPasturDistribution", "[", 
+         RowBox[{"p", "/", "n"}], "]"}], ",", "x"}], "]"}], ",", 
+      RowBox[{"{", 
+       RowBox[{"x", ",", "0", ",", "1.8"}], "}"}], ",", 
+      RowBox[{"PlotStyle", "\[Rule]", "Thick"}], ",", 
+      RowBox[{"PlotRange", "\[Rule]", "All"}], ",", 
+      RowBox[{"PlotLegends", "\[Rule]", "None"}], ",", 
+      RowBox[{"Exclusions", "\[Rule]", "None"}]}], "]"}]}], ";"}], " ", 
+  RowBox[{"(*", " ", 
+   RowBox[{"MP", " ", "distribution"}], " ", "*)"}]}]}], "Input",
+ CellChangeTimes->{{3.81416457947685*^9, 3.814164825544231*^9}, {
+   3.814164859931432*^9, 3.8141648687684603`*^9}, {3.814164899060389*^9, 
+   3.8141649473920717`*^9}, {3.814165059908477*^9, 3.814165064986815*^9}, {
+   3.8141652656624937`*^9, 3.814165289294182*^9}, 3.814165482471281*^9, {
+   3.814165576485319*^9, 3.814165589899877*^9}, {3.814165670536396*^9, 
+   3.814165713057708*^9}},
+ CellLabel->"In[3]:=",ExpressionUUID->"888d2e13-08b5-4891-9154-b2c963d80b02"],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"mpplot", "=", 
+  RowBox[{"Labeled", "[", 
+   RowBox[{
+    RowBox[{"Show", "[", 
+     RowBox[{
+      RowBox[{"{", 
+       RowBox[{"eighist", ",", "marcpast"}], "}"}], ",", 
+      RowBox[{"ImageSize", "\[Rule]", "Large"}], ",", 
+      RowBox[{
+      "PlotLabel", "\[Rule]", 
+       "\"\<Distribution of the Eigenvalues of a Wishart Data Matrix and \
+Marchenko-Pastur Limiting Distribution\>\""}]}], "]"}], ",", 
+    "\"\<n = \!\(\*SuperscriptBox[\(10\), \(4\)]\), p = \!\(\*SuperscriptBox[\
+\(10\), \(3\)]\)\>\""}], "]"}]}]], "Input",
+ CellChangeTimes->{{3.814164924131678*^9, 3.814164957440431*^9}, {
+   3.814165071108502*^9, 3.814165168626462*^9}, {3.81416519957753*^9, 
+   3.8141652259696913`*^9}, 3.814165261963818*^9, {3.8141657313926287`*^9, 
+   3.81416573420116*^9}, {3.814167350854034*^9, 3.8141673998497458`*^9}},
+ CellLabel->"In[6]:=",ExpressionUUID->"8a44eccc-ab0a-4669-9976-f4eef79642cb"],
+
+Cell[BoxData[
+ TemplateBox[{
+   GraphicsBox[{{{
+       Directive[
+        EdgeForm[
+         Directive[
+          Thickness[Small], 
+          Opacity[0.504]]], 
+        RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`]], {{}, {
+         Directive[
+          EdgeForm[
+           Directive[
+            Thickness[Small], 
+            Opacity[0.504]]], 
+          RGBColor[0.987148, 0.8073604000000001, 0.49470040000000004`]], {{{
+            
+            RectangleBox[{0.45, 0}, {0.5, 0.2600000000000001}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.5, 0}, {0.55, 0.7999999999999994}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.55, 0}, {0.6000000000000001, 0.9799999999999992}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.6000000000000001, 0}, {0.65, 1.0600000000000014`},
+              "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.65, 0}, {0.7, 1.1000000000000014`}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.7, 0}, {0.75, 1.119999999999999}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.75, 0}, {0.8, 1.119999999999999}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.8, 0}, {0.8500000000000001, 1.119999999999999}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.8500000000000001, 0}, {0.9, 1.0800000000000014`}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.9, 0}, {0.95, 1.0600000000000014`}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{0.95, 0}, {1., 0.9599999999999992}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1., 0}, {1.05, 0.9999999999999991}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.05, 0}, {1.1, 0.9199999999999993}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.1, 0}, {1.1500000000000001`, 0.8599999999999993}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.1500000000000001`, 0}, {1.2, 0.860000000000003}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.2, 0}, {1.25, 0.8199999999999993}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.25, 0}, {1.3, 0.7599999999999993}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.3, 0}, {1.35, 0.6999999999999994}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.35, 0}, {1.4000000000000001`, 0.6399999999999995},
+              "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.4000000000000001`, 0}, {1.45, 0.6200000000000022},
+              "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.45, 0}, {1.5, 0.5999999999999995}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.5, 0}, {1.55, 0.45999999999999963`}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.55, 0}, {1.6, 0.3999999999999997}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.6, 0}, {1.6500000000000001`, 0.3599999999999997}, 
+             "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.6500000000000001`, 0}, {1.7, 
+             0.24000000000000082`}, "RoundingRadius" -> 0]}}, {{
+            
+            RectangleBox[{1.7, 0}, {1.75, 0.09999999999999992}, 
+             "RoundingRadius" -> 
+             0]}}}}, {}, {}}}, {{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, \
+{}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}, {}}}, {{{{}, {}, 
+        TagBox[{
+          Directive[
+           Opacity[1.], 
+           RGBColor[0.368417, 0.506779, 0.709798], 
+           AbsoluteThickness[1.6], 
+           Thickness[Large]], 
+          LineBox[CompressedData["
+1:eJxF2HlUzN//B/BKktImrSIiEQkRCs+UREVJolQURaGUiiQtlkIKKVmitKCN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+           "]]}, 
+         Annotation[#, "Charting`Private`Tag$2636#1"]& ]}}, {}}}, {
+    ImageSize -> Large, PlotLabel -> 
+     FormBox["\"Distribution of the Eigenvalues of a Wishart Data Matrix and \
+Marchenko-Pastur Limiting Distribution\"", TraditionalForm], AspectRatio -> 
+     NCache[GoldenRatio^(-1), 0.6180339887498948], Axes -> {True, True}, 
+     AxesLabel -> {None, None}, AxesOrigin -> {0.424, 0}, 
+     FrameLabel -> {{None, None}, {None, None}}, 
+     FrameTicks -> {{Automatic, Automatic}, {Automatic, Automatic}}, 
+     GridLines -> {None, None}, GridLinesStyle -> Directive[
+       GrayLevel[0.5, 0.4]], PlotRange -> {{0.45, 1.75}, {All, All}}, 
+     PlotRangePadding -> {{
+        Scaled[0.02], 
+        Scaled[0.02]}, {
+        Scaled[0.02], 
+        Scaled[0.05]}}, Ticks -> {Automatic, Automatic}}], 
+   "\"n = \\!\\(\\*SuperscriptBox[\\(10\\), \\(4\\)]\\), p = \
+\\!\\(\\*SuperscriptBox[\\(10\\), \\(3\\)]\\)\""},
+  "Labeled",
+  DisplayFunction->(GridBox[{{
+      TagBox[
+       ItemBox[
+        PaneBox[
+         TagBox[#, "SkipImageSizeLevel"], Alignment -> {Center, Baseline}, 
+         BaselinePosition -> Baseline], DefaultBaseStyle -> "Labeled"], 
+       "SkipImageSizeLevel"]}, {
+      ItemBox[#2, DefaultBaseStyle -> "LabeledLabel"]}}, 
+    GridBoxAlignment -> {"Columns" -> {{Center}}, "Rows" -> {{Center}}}, 
+    AutoDelete -> False, 
+    GridBoxItemSize -> {"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, 
+    BaselinePosition -> {1, 1}]& ),
+  InterpretationFunction->(RowBox[{"Labeled", "[", 
+     RowBox[{#, ",", #2}], "]"}]& )]], "Output",
+ CellChangeTimes->{{3.8141649350400352`*^9, 3.814164957792368*^9}, {
+   3.814165074474577*^9, 3.814165106326329*^9}, {3.8141651475548*^9, 
+   3.814165169222849*^9}, {3.8141652623143*^9, 3.814165302492792*^9}, 
+   3.814165533438848*^9, {3.814165585850946*^9, 3.8141655989169416`*^9}, 
+   3.814165681040098*^9, 3.814165734914089*^9, 3.8141674097658978`*^9},
+ CellLabel->"Out[6]=",ExpressionUUID->"cfd9fe62-75fc-4a7d-a43e-e70326d3e274"]
+}, Open  ]],
+
+Cell[CellGroupData[{
+
+Cell[BoxData[
+ RowBox[{"Export", "[", 
+  RowBox[{
+   RowBox[{"FileNameJoin", "[", 
+    RowBox[{"{", 
+     RowBox[{
+      RowBox[{"NotebookDirectory", "[", "]"}], ",", 
+      "\"\<marchenko-pastur.pdf\>\""}], "}"}], "]"}], ",", "mpplot", ",", 
+   "\"\<PDF\>\""}], "]"}]], "Input",
+ CellChangeTimes->{{3.8141657274690123`*^9, 3.814165866323689*^9}},
+ CellLabel->"In[7]:=",ExpressionUUID->"353f095a-0bff-4b63-bbe1-d484dee8f355"],
+
+Cell[BoxData["\<\"/home/riccardo/documents/thesis_beamer/img/marchenko-pastur.\
+pdf\"\>"], "Output",
+ CellChangeTimes->{
+  3.814165786375692*^9, {3.8141658293920183`*^9, 3.814165839439434*^9}, 
+   3.8141658714284563`*^9, 3.814167414286223*^9},
+ CellLabel->"Out[7]=",ExpressionUUID->"b0e30f48-ebd4-4574-9684-78b2d9ef9265"]
+}, Open  ]]
+},
+WindowSize->{1097.25, 573.75},
+WindowMargins->{{0, Automatic}, {-0.75, Automatic}},
+Magnification:>0.8 Inherited,
+FrontEndVersion->"12.1 for Linux x86 (64-bit) (June 19, 2020)",
+StyleDefinitions->"Default.nb",
+ExpressionUUID->"330d46d3-10f2-431d-beb3-d05b4008f49d"
+]
+(* End of Notebook Content *)
+
+(* Internal cache information *)
+(*CellTagsOutline
+CellTagsIndex->{}
+*)
+(*CellTagsIndex
+CellTagsIndex->{}
+*)
+(*NotebookFileOutline
+Notebook[{
+Cell[558, 20, 517, 12, 40, "Input",ExpressionUUID->"e07ea3b1-6853-427d-a42f-55af95fd957a"],
+Cell[1078, 34, 1919, 47, 75, "Input",ExpressionUUID->"888d2e13-08b5-4891-9154-b2c963d80b02"],
+Cell[CellGroupData[{
+Cell[3022, 85, 922, 19, 58, "Input",ExpressionUUID->"8a44eccc-ab0a-4669-9976-f4eef79642cb"],
+Cell[3947, 106, 13320, 259, 348, "Output",ExpressionUUID->"cfd9fe62-75fc-4a7d-a43e-e70326d3e274"]
+}, Open  ]],
+Cell[CellGroupData[{
+Cell[17304, 370, 425, 10, 25, "Input",ExpressionUUID->"353f095a-0bff-4b63-bbe1-d484dee8f355"],
+Cell[17732, 382, 321, 5, 26, "Output",ExpressionUUID->"b0e30f48-ebd4-4574-9684-78b2d9ef9265"]
+}, Open  ]]
+}
+]
+*)
+