diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle index e98ec9078..440359e7a 100644 Binary files a/doc/LectureNotes/_build/.doctrees/environment.pickle and b/doc/LectureNotes/_build/.doctrees/environment.pickle differ diff --git a/doc/LectureNotes/_build/.doctrees/week36.doctree b/doc/LectureNotes/_build/.doctrees/week36.doctree index 3e1faeb48..c117f6ca1 100644 Binary files a/doc/LectureNotes/_build/.doctrees/week36.doctree and b/doc/LectureNotes/_build/.doctrees/week36.doctree differ diff --git a/doc/LectureNotes/_build/html/_sources/week36.ipynb b/doc/LectureNotes/_build/html/_sources/week36.ipynb index 991b0f568..bdaee795a 100644 --- a/doc/LectureNotes/_build/html/_sources/week36.ipynb +++ b/doc/LectureNotes/_build/html/_sources/week36.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "b49fd9eb", + "id": "7411173e", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "138fe268", + "id": "f50db913", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "c20f461b", + "id": "4a90e380", "metadata": { "editable": true }, @@ -53,7 +53,7 @@ }, { "cell_type": "markdown", - "id": "55f0d080", + "id": "351d36fd", "metadata": { "editable": true }, @@ -65,7 +65,7 @@ }, { "cell_type": "markdown", - "id": "69ce62a6", + "id": "0f9f4d2f", "metadata": { "editable": true }, @@ -77,7 +77,7 @@ }, { "cell_type": "markdown", - "id": "a9aaa130", + "id": "9ec36bdd", "metadata": { "editable": true }, @@ -89,7 +89,7 @@ }, { "cell_type": "markdown", - "id": "2b1c0902", + "id": "614493c6", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "093e3a8c", + "id": "dc4be423", "metadata": { "editable": true }, @@ -111,7 +111,7 @@ }, { "cell_type": "markdown", - "id": "c0102cd2", + "id": "b262332c", "metadata": { "editable": true }, @@ -121,7 +121,7 @@ }, { "cell_type": "markdown", - "id": "084138a1", + "id": "ab1829a4", "metadata": { "editable": true }, @@ -138,7 +138,7 @@ }, { "cell_type": "markdown", - "id": "cb8ed4f2", + "id": "7c82dd9e", "metadata": { "editable": true }, @@ -148,7 +148,7 @@ }, { "cell_type": "markdown", - "id": "d7c1c21b", + "id": "5bab8721", "metadata": { "editable": true }, @@ -160,7 +160,7 @@ }, { "cell_type": "markdown", - "id": "791e9c6d", + "id": "72273a4f", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "924b8081", + "id": "d4af03ea", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "949f6658", + "id": "c0d35853", "metadata": { "editable": true }, @@ -196,7 +196,7 @@ }, { "cell_type": "markdown", - "id": "46a4a83c", + "id": "db5ea827", "metadata": { "editable": true }, @@ -208,7 +208,7 @@ }, { "cell_type": "markdown", - "id": "ce019aa6", + "id": "a6b341ce", "metadata": { "editable": true }, @@ -230,7 +230,7 @@ }, { "cell_type": "markdown", - "id": "c662d90f", + "id": "db723709", "metadata": { "editable": true }, @@ -242,7 +242,7 @@ }, { "cell_type": "markdown", - "id": "7df45c97", + "id": "1e600268", "metadata": { "editable": true }, @@ -257,7 +257,7 @@ }, { "cell_type": "markdown", - "id": "f0741872", + "id": "b9452938", "metadata": { "editable": true }, @@ -269,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "05530ccb", + "id": "53fea1e6", "metadata": { "editable": true }, @@ -281,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "3048f911", + "id": "ed10baad", "metadata": { "editable": true }, @@ -291,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9e3c2de1", + "id": "a4d0db96", "metadata": { "editable": true }, @@ -303,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "491923c1", + "id": "fae3cadc", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "77d1dbe5", + "id": "e973c775", "metadata": { "editable": true }, @@ -325,7 +325,7 @@ }, { "cell_type": "markdown", - "id": "fb7e2323", + "id": "f88a33f4", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "c9cca57a", + "id": "5bc7214d", "metadata": { "editable": true }, @@ -347,7 +347,7 @@ }, { "cell_type": "markdown", - "id": "04e4b486", + "id": "561d4c91", "metadata": { "editable": true }, @@ -364,7 +364,7 @@ }, { "cell_type": "markdown", - "id": "2b3c22fb", + "id": "0eda2d1e", "metadata": { "editable": true }, @@ -372,25 +372,27 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is" ] }, { "cell_type": "code", "execution_count": 1, - "id": "12637a17", + "id": "c95dbc32", "metadata": { "collapsed": false, "editable": true }, "outputs": [], "source": [ - "Ainv = np.linlag.pinv(A)" + "import numpy as np\n", + "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n", + "Xinv = np.linlag.pinv(X)" ] }, { "cell_type": "markdown", - "id": "904adaf8", + "id": "77079d25", "metadata": { "editable": true }, @@ -401,7 +403,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "6b5465dc", + "id": "cd0ecc44", "metadata": { "collapsed": false, "editable": true @@ -441,7 +443,7 @@ }, { "cell_type": "markdown", - "id": "a80dcee4", + "id": "41d896dc", "metadata": { "editable": true }, @@ -461,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "0c4052d3", + "id": "f594852c", "metadata": { "editable": true }, @@ -473,7 +475,7 @@ }, { "cell_type": "markdown", - "id": "185d433f", + "id": "b78264f2", "metadata": { "editable": true }, @@ -484,7 +486,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "418f8797", + "id": "d05eff95", "metadata": { "collapsed": false, "editable": true @@ -518,7 +520,7 @@ }, { "cell_type": "markdown", - "id": "bf5051ea", + "id": "2fae32c6", "metadata": { "editable": true }, @@ -528,7 +530,7 @@ }, { "cell_type": "markdown", - "id": "202d1f90", + "id": "8631011a", "metadata": { "editable": true }, @@ -541,7 +543,7 @@ }, { "cell_type": "markdown", - "id": "df0a8718", + "id": "8e7701de", "metadata": { "editable": true }, @@ -553,7 +555,7 @@ }, { "cell_type": "markdown", - "id": "4ebd1d20", + "id": "aea7230b", "metadata": { "editable": true }, @@ -563,7 +565,7 @@ }, { "cell_type": "markdown", - "id": "40df3859", + "id": "c5029213", "metadata": { "editable": true }, @@ -576,7 +578,7 @@ }, { "cell_type": "markdown", - "id": "980001f5", + "id": "535ad70e", "metadata": { "editable": true }, @@ -586,7 +588,7 @@ }, { "cell_type": "markdown", - "id": "699b1198", + "id": "159ae526", "metadata": { "editable": true }, @@ -598,7 +600,7 @@ }, { "cell_type": "markdown", - "id": "ff01ef41", + "id": "2f25dfa4", "metadata": { "editable": true }, @@ -613,7 +615,7 @@ }, { "cell_type": "markdown", - "id": "d7b9188f", + "id": "9d187806", "metadata": { "editable": true }, @@ -626,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "778790dc", + "id": "6c692db5", "metadata": { "editable": true }, @@ -640,7 +642,7 @@ }, { "cell_type": "markdown", - "id": "38dd7428", + "id": "3404e0b4", "metadata": { "editable": true }, @@ -652,7 +654,7 @@ }, { "cell_type": "markdown", - "id": "6ed3deb0", + "id": "4bc49666", "metadata": { "editable": true }, @@ -662,7 +664,7 @@ }, { "cell_type": "markdown", - "id": "e0860754", + "id": "3f08dfa1", "metadata": { "editable": true }, @@ -675,7 +677,7 @@ }, { "cell_type": "markdown", - "id": "66751140", + "id": "30bcc0cd", "metadata": { "editable": true }, @@ -687,7 +689,7 @@ }, { "cell_type": "markdown", - "id": "e1fa2bdb", + "id": "d311a7d8", "metadata": { "editable": true }, @@ -699,7 +701,7 @@ }, { "cell_type": "markdown", - "id": "32131982", + "id": "32c46b73", "metadata": { "editable": true }, @@ -711,7 +713,7 @@ }, { "cell_type": "markdown", - "id": "c1979796", + "id": "3cceda87", "metadata": { "editable": true }, @@ -723,7 +725,7 @@ }, { "cell_type": "markdown", - "id": "d7eb56d7", + "id": "10a16675", "metadata": { "editable": true }, @@ -737,7 +739,7 @@ }, { "cell_type": "markdown", - "id": "7c42f326", + "id": "6dba20bf", "metadata": { "editable": true }, @@ -749,7 +751,7 @@ }, { "cell_type": "markdown", - "id": "6496e5c0", + "id": "9325f924", "metadata": { "editable": true }, @@ -759,7 +761,7 @@ }, { "cell_type": "markdown", - "id": "837f7c04", + "id": "745d61e9", "metadata": { "editable": true }, @@ -771,7 +773,7 @@ }, { "cell_type": "markdown", - "id": "55e9395f", + "id": "3866af64", "metadata": { "editable": true }, @@ -781,7 +783,7 @@ }, { "cell_type": "markdown", - "id": "a943f34a", + "id": "710de288", "metadata": { "editable": true }, @@ -793,7 +795,7 @@ }, { "cell_type": "markdown", - "id": "63a227a9", + "id": "5893fba0", "metadata": { "editable": true }, @@ -805,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "8be2bd7b", + "id": "5ef8c59f", "metadata": { "editable": true }, @@ -815,7 +817,7 @@ }, { "cell_type": "markdown", - "id": "35509652", + "id": "79b257ae", "metadata": { "editable": true }, @@ -826,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "2fbe21d0", + "id": "46270996", "metadata": { "editable": true }, @@ -838,7 +840,7 @@ }, { "cell_type": "markdown", - "id": "71e986f8", + "id": "785aebe9", "metadata": { "editable": true }, @@ -854,7 +856,7 @@ }, { "cell_type": "markdown", - "id": "3a5e5f40", + "id": "443f3a3f", "metadata": { "editable": true }, @@ -867,7 +869,7 @@ }, { "cell_type": "markdown", - "id": "752b6ba2", + "id": "cfb8d4c0", "metadata": { "editable": true }, @@ -879,7 +881,7 @@ }, { "cell_type": "markdown", - "id": "4e724327", + "id": "2be1ca1f", "metadata": { "editable": true }, @@ -889,7 +891,7 @@ }, { "cell_type": "markdown", - "id": "b15c2c90", + "id": "9ed0aec2", "metadata": { "editable": true }, @@ -901,7 +903,7 @@ }, { "cell_type": "markdown", - "id": "82e70fa5", + "id": "943b1cd5", "metadata": { "editable": true }, @@ -911,7 +913,7 @@ }, { "cell_type": "markdown", - "id": "3394e2e7", + "id": "4a8a6bff", "metadata": { "editable": true }, @@ -923,7 +925,7 @@ }, { "cell_type": "markdown", - "id": "722ffa86", + "id": "b0371965", "metadata": { "editable": true }, @@ -935,7 +937,7 @@ }, { "cell_type": "markdown", - "id": "720ed773", + "id": "38e7b497", "metadata": { "editable": true }, @@ -951,7 +953,7 @@ }, { "cell_type": "markdown", - "id": "3b57864f", + "id": "682f90db", "metadata": { "editable": true }, @@ -963,7 +965,7 @@ }, { "cell_type": "markdown", - "id": "8f36ff27", + "id": "b5fb7771", "metadata": { "editable": true }, @@ -975,7 +977,7 @@ }, { "cell_type": "markdown", - "id": "70adc9ce", + "id": "0c228525", "metadata": { "editable": true }, @@ -985,7 +987,7 @@ }, { "cell_type": "markdown", - "id": "eb4a995f", + "id": "08a400af", "metadata": { "editable": true }, @@ -997,7 +999,7 @@ }, { "cell_type": "markdown", - "id": "e3cd4b92", + "id": "ec0808b1", "metadata": { "editable": true }, @@ -1007,7 +1009,7 @@ }, { "cell_type": "markdown", - "id": "bbf9b1de", + "id": "92d3d92f", "metadata": { "editable": true }, @@ -1019,7 +1021,7 @@ }, { "cell_type": "markdown", - "id": "e30eff48", + "id": "07486e29", "metadata": { "editable": true }, @@ -1036,7 +1038,7 @@ }, { "cell_type": "markdown", - "id": "9a8f8e2e", + "id": "9868d4f4", "metadata": { "editable": true }, @@ -1048,7 +1050,7 @@ }, { "cell_type": "markdown", - "id": "3cb7928c", + "id": "1793a225", "metadata": { "editable": true }, @@ -1060,7 +1062,7 @@ }, { "cell_type": "markdown", - "id": "b1c6c813", + "id": "c7d3fa52", "metadata": { "editable": true }, @@ -1070,7 +1072,7 @@ }, { "cell_type": "markdown", - "id": "da336775", + "id": "47894977", "metadata": { "editable": true }, @@ -1082,7 +1084,7 @@ }, { "cell_type": "markdown", - "id": "2e0de141", + "id": "d1a11dc8", "metadata": { "editable": true }, @@ -1092,7 +1094,7 @@ }, { "cell_type": "markdown", - "id": "40985ec4", + "id": "a6cf82de", "metadata": { "editable": true }, @@ -1104,7 +1106,7 @@ }, { "cell_type": "markdown", - "id": "31091d7b", + "id": "bbf7ccc7", "metadata": { "editable": true }, @@ -1114,7 +1116,7 @@ }, { "cell_type": "markdown", - "id": "e81d965e", + "id": "bc6dbb83", "metadata": { "editable": true }, @@ -1126,7 +1128,7 @@ }, { "cell_type": "markdown", - "id": "9984b3a4", + "id": "ef5a8292", "metadata": { "editable": true }, @@ -1136,7 +1138,7 @@ }, { "cell_type": "markdown", - "id": "21ec3768", + "id": "d77bf73a", "metadata": { "editable": true }, @@ -1151,7 +1153,7 @@ }, { "cell_type": "markdown", - "id": "6be0ec63", + "id": "265ce93f", "metadata": { "editable": true }, @@ -1163,7 +1165,7 @@ }, { "cell_type": "markdown", - "id": "44c7278c", + "id": "c19e720c", "metadata": { "editable": true }, @@ -1173,7 +1175,7 @@ }, { "cell_type": "markdown", - "id": "918a8cc3", + "id": "8cad2962", "metadata": { "editable": true }, @@ -1185,7 +1187,7 @@ }, { "cell_type": "markdown", - "id": "3949af51", + "id": "bf6316a2", "metadata": { "editable": true }, @@ -1197,7 +1199,7 @@ }, { "cell_type": "markdown", - "id": "aa856d18", + "id": "78413ab6", "metadata": { "editable": true }, @@ -1209,7 +1211,7 @@ }, { "cell_type": "markdown", - "id": "bd30d1d4", + "id": "a77ca41b", "metadata": { "editable": true }, @@ -1219,7 +1221,7 @@ }, { "cell_type": "markdown", - "id": "142de535", + "id": "67b3f67e", "metadata": { "editable": true }, @@ -1231,7 +1233,7 @@ }, { "cell_type": "markdown", - "id": "1efee0c3", + "id": "d3571c8b", "metadata": { "editable": true }, @@ -1243,7 +1245,7 @@ }, { "cell_type": "markdown", - "id": "a6db019a", + "id": "943f822e", "metadata": { "editable": true }, @@ -1255,7 +1257,7 @@ }, { "cell_type": "markdown", - "id": "0f38deda", + "id": "2ec0ba01", "metadata": { "editable": true }, @@ -1265,7 +1267,7 @@ }, { "cell_type": "markdown", - "id": "7f9738a5", + "id": "85bf4c9b", "metadata": { "editable": true }, @@ -1277,7 +1279,7 @@ }, { "cell_type": "markdown", - "id": "4e8460f8", + "id": "c81d05f4", "metadata": { "editable": true }, @@ -1287,7 +1289,7 @@ }, { "cell_type": "markdown", - "id": "ae49ddce", + "id": "ce36390d", "metadata": { "editable": true }, @@ -1301,7 +1303,7 @@ }, { "cell_type": "markdown", - "id": "1efb96d3", + "id": "a47dd697", "metadata": { "editable": true }, @@ -1311,7 +1313,7 @@ }, { "cell_type": "markdown", - "id": "1a030291", + "id": "15fe5a0c", "metadata": { "editable": true }, @@ -1323,7 +1325,7 @@ }, { "cell_type": "markdown", - "id": "af486213", + "id": "62d7f0e9", "metadata": { "editable": true }, @@ -1335,7 +1337,7 @@ }, { "cell_type": "markdown", - "id": "87e5470d", + "id": "f2d4576f", "metadata": { "editable": true }, @@ -1345,7 +1347,7 @@ }, { "cell_type": "markdown", - "id": "182f9f51", + "id": "fb2bc0ff", "metadata": { "editable": true }, @@ -1357,7 +1359,7 @@ }, { "cell_type": "markdown", - "id": "edc0ee3d", + "id": "807ac009", "metadata": { "editable": true }, @@ -1367,7 +1369,7 @@ }, { "cell_type": "markdown", - "id": "673c64a0", + "id": "fe72889b", "metadata": { "editable": true }, @@ -1379,7 +1381,7 @@ }, { "cell_type": "markdown", - "id": "c0554476", + "id": "48844e10", "metadata": { "editable": true }, @@ -1391,7 +1393,7 @@ }, { "cell_type": "markdown", - "id": "cc9c761b", + "id": "72a4bc4c", "metadata": { "editable": true }, @@ -1401,7 +1403,7 @@ }, { "cell_type": "markdown", - "id": "b5c84ed1", + "id": "cfe1db6f", "metadata": { "editable": true }, @@ -1413,7 +1415,7 @@ }, { "cell_type": "markdown", - "id": "a8d2ba86", + "id": "05b6e6f6", "metadata": { "editable": true }, @@ -1423,7 +1425,7 @@ }, { "cell_type": "markdown", - "id": "9e407326", + "id": "0e58f37a", "metadata": { "editable": true }, @@ -1435,7 +1437,7 @@ }, { "cell_type": "markdown", - "id": "36ce4758", + "id": "78593de5", "metadata": { "editable": true }, @@ -1447,7 +1449,7 @@ }, { "cell_type": "markdown", - "id": "8665e33b", + "id": "26b08a56", "metadata": { "editable": true }, @@ -1457,7 +1459,7 @@ }, { "cell_type": "markdown", - "id": "16dbcd82", + "id": "3df661a4", "metadata": { "editable": true }, @@ -1469,7 +1471,7 @@ }, { "cell_type": "markdown", - "id": "642f7768", + "id": "68460e68", "metadata": { "editable": true }, @@ -1482,7 +1484,7 @@ }, { "cell_type": "markdown", - "id": "1784c79c", + "id": "6c660112", "metadata": { "editable": true }, @@ -1494,7 +1496,7 @@ }, { "cell_type": "markdown", - "id": "72b7600f", + "id": "6ff54cf1", "metadata": { "editable": true }, @@ -1506,7 +1508,7 @@ }, { "cell_type": "markdown", - "id": "84ca3171", + "id": "f3f88f94", "metadata": { "editable": true }, @@ -1518,7 +1520,7 @@ }, { "cell_type": "markdown", - "id": "d31aa45e", + "id": "6918fc7c", "metadata": { "editable": true }, @@ -1528,7 +1530,7 @@ }, { "cell_type": "markdown", - "id": "1298ca6d", + "id": "1b2321a3", "metadata": { "editable": true }, @@ -1540,7 +1542,7 @@ }, { "cell_type": "markdown", - "id": "fa2455d6", + "id": "d5e21457", "metadata": { "editable": true }, @@ -1550,7 +1552,7 @@ }, { "cell_type": "markdown", - "id": "8cd50073", + "id": "e03c3617", "metadata": { "editable": true }, @@ -1562,7 +1564,7 @@ }, { "cell_type": "markdown", - "id": "1f7e12c3", + "id": "6c4d9a4a", "metadata": { "editable": true }, @@ -1572,7 +1574,7 @@ }, { "cell_type": "markdown", - "id": "a79821fd", + "id": "584bef2e", "metadata": { "editable": true }, @@ -1584,7 +1586,7 @@ }, { "cell_type": "markdown", - "id": "2ac13531", + "id": "87273242", "metadata": { "editable": true }, @@ -1594,7 +1596,7 @@ }, { "cell_type": "markdown", - "id": "eed25f68", + "id": "f7e8277f", "metadata": { "editable": true }, @@ -1607,7 +1609,7 @@ }, { "cell_type": "markdown", - "id": "f74d71ec", + "id": "e7167c28", "metadata": { "editable": true }, @@ -1619,7 +1621,7 @@ }, { "cell_type": "markdown", - "id": "f0797067", + "id": "6a3fa7b2", "metadata": { "editable": true }, @@ -1631,7 +1633,7 @@ }, { "cell_type": "markdown", - "id": "3cc5d5e0", + "id": "92c33067", "metadata": { "editable": true }, @@ -1641,7 +1643,7 @@ }, { "cell_type": "markdown", - "id": "04250e5c", + "id": "1b7be5cc", "metadata": { "editable": true }, @@ -1653,7 +1655,7 @@ }, { "cell_type": "markdown", - "id": "2b24d2fa", + "id": "9e2db51b", "metadata": { "editable": true }, @@ -1670,7 +1672,7 @@ }, { "cell_type": "markdown", - "id": "aa950760", + "id": "7ea5c023", "metadata": { "editable": true }, @@ -1682,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "475389ed", + "id": "efff1845", "metadata": { "editable": true }, @@ -1694,7 +1696,7 @@ }, { "cell_type": "markdown", - "id": "086558d9", + "id": "80496d8c", "metadata": { "editable": true }, @@ -1704,7 +1706,7 @@ }, { "cell_type": "markdown", - "id": "1a3b2ba5", + "id": "bb33463a", "metadata": { "editable": true }, @@ -1716,7 +1718,7 @@ }, { "cell_type": "markdown", - "id": "c928bc46", + "id": "f16b1457", "metadata": { "editable": true }, @@ -1726,7 +1728,7 @@ }, { "cell_type": "markdown", - "id": "809a58f7", + "id": "4bee5a43", "metadata": { "editable": true }, @@ -1738,7 +1740,7 @@ }, { "cell_type": "markdown", - "id": "3b937bf4", + "id": "17176990", "metadata": { "editable": true }, @@ -1748,7 +1750,7 @@ }, { "cell_type": "markdown", - "id": "6e5b2527", + "id": "ec2c8f34", "metadata": { "editable": true }, @@ -1760,7 +1762,7 @@ }, { "cell_type": "markdown", - "id": "c5f88a45", + "id": "5934144d", "metadata": { "editable": true }, @@ -1770,7 +1772,7 @@ }, { "cell_type": "markdown", - "id": "c57753b4", + "id": "fe097c51", "metadata": { "editable": true }, @@ -1785,7 +1787,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "392f218d", + "id": "92cd1361", "metadata": { "collapsed": false, "editable": true @@ -1847,7 +1849,7 @@ }, { "cell_type": "markdown", - "id": "dd134ab1", + "id": "9e4426f4", "metadata": { "editable": true }, @@ -1857,7 +1859,7 @@ }, { "cell_type": "markdown", - "id": "8d9a61db", + "id": "5f2b6935", "metadata": { "editable": true }, @@ -1868,7 +1870,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "afba7c4a", + "id": "409bbeac", "metadata": { "collapsed": false, "editable": true @@ -1935,7 +1937,7 @@ }, { "cell_type": "markdown", - "id": "4d63db0b", + "id": "67a24e77", "metadata": { "editable": true }, @@ -1946,7 +1948,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "81115144", + "id": "8da16a74", "metadata": { "collapsed": false, "editable": true @@ -2035,7 +2037,7 @@ }, { "cell_type": "markdown", - "id": "e076a968", + "id": "9d2563fa", "metadata": { "editable": true }, @@ -2045,7 +2047,7 @@ }, { "cell_type": "markdown", - "id": "db822c65", + "id": "1232037e", "metadata": { "editable": true }, @@ -2074,7 +2076,7 @@ }, { "cell_type": "markdown", - "id": "fc4a06b6", + "id": "e50205be", "metadata": { "editable": true }, @@ -2090,7 +2092,7 @@ }, { "cell_type": "markdown", - "id": "43299fb3", + "id": "70eccda9", "metadata": { "editable": true }, @@ -2109,7 +2111,7 @@ }, { "cell_type": "markdown", - "id": "6cba00c7", + "id": "e6bdcbf2", "metadata": { "editable": true }, @@ -2123,7 +2125,7 @@ }, { "cell_type": "markdown", - "id": "8c85f61d", + "id": "d74471b8", "metadata": { "editable": true }, @@ -2135,7 +2137,7 @@ }, { "cell_type": "markdown", - "id": "a1901c32", + "id": "f7b178d9", "metadata": { "editable": true }, @@ -2146,7 +2148,7 @@ }, { "cell_type": "markdown", - "id": "c59e8272", + "id": "8ea16c62", "metadata": { "editable": true }, @@ -2158,7 +2160,7 @@ }, { "cell_type": "markdown", - "id": "8cba4bae", + "id": "4a8d7bfd", "metadata": { "editable": true }, @@ -2170,7 +2172,7 @@ }, { "cell_type": "markdown", - "id": "5e18ebb5", + "id": "5367d943", "metadata": { "editable": true }, @@ -2186,7 +2188,7 @@ }, { "cell_type": "markdown", - "id": "10d86c58", + "id": "c7eefd85", "metadata": { "editable": true }, @@ -2197,7 +2199,7 @@ }, { "cell_type": "markdown", - "id": "efba1ba9", + "id": "27ce343a", "metadata": { "editable": true }, @@ -2220,7 +2222,7 @@ }, { "cell_type": "markdown", - "id": "a3eccdab", + "id": "c67ea20b", "metadata": { "editable": true }, @@ -2231,7 +2233,7 @@ }, { "cell_type": "markdown", - "id": "70b901b4", + "id": "b4da473a", "metadata": { "editable": true }, @@ -2243,7 +2245,7 @@ }, { "cell_type": "markdown", - "id": "752b72d2", + "id": "af812d38", "metadata": { "editable": true }, @@ -2255,7 +2257,7 @@ }, { "cell_type": "markdown", - "id": "108aba79", + "id": "9d3aaff5", "metadata": { "editable": true }, @@ -2269,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "65801f3b", + "id": "eebea8b7", "metadata": { "editable": true }, @@ -2300,7 +2302,7 @@ }, { "cell_type": "markdown", - "id": "faf8da78", + "id": "714f281a", "metadata": { "editable": true }, @@ -2322,7 +2324,7 @@ }, { "cell_type": "markdown", - "id": "1c71921e", + "id": "66015ca6", "metadata": { "editable": true }, @@ -2334,7 +2336,7 @@ }, { "cell_type": "markdown", - "id": "cfb57087", + "id": "267f01c0", "metadata": { "editable": true }, @@ -2347,7 +2349,7 @@ }, { "cell_type": "markdown", - "id": "be4ad159", + "id": "998de4b6", "metadata": { "editable": true }, @@ -2359,7 +2361,7 @@ }, { "cell_type": "markdown", - "id": "f5e11f36", + "id": "06a8d452", "metadata": { "editable": true }, @@ -2371,7 +2373,7 @@ }, { "cell_type": "markdown", - "id": "dbf2700f", + "id": "6cd3ed9b", "metadata": { "editable": true }, @@ -2383,7 +2385,7 @@ }, { "cell_type": "markdown", - "id": "0f6d760d", + "id": "70f9aa69", "metadata": { "editable": true }, @@ -2395,7 +2397,7 @@ }, { "cell_type": "markdown", - "id": "ce614956", + "id": "3bb18388", "metadata": { "editable": true }, @@ -2418,7 +2420,7 @@ }, { "cell_type": "markdown", - "id": "086419d8", + "id": "84b546af", "metadata": { "editable": true }, @@ -2430,7 +2432,7 @@ }, { "cell_type": "markdown", - "id": "a228dcf4", + "id": "0e90935e", "metadata": { "editable": true }, @@ -2443,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "f04f6b76", + "id": "b40b33f5", "metadata": { "editable": true }, @@ -2455,7 +2457,7 @@ }, { "cell_type": "markdown", - "id": "642936a8", + "id": "23e0c7ac", "metadata": { "editable": true }, @@ -2467,7 +2469,7 @@ }, { "cell_type": "markdown", - "id": "4f137023", + "id": "dcd4ed9c", "metadata": { "editable": true }, @@ -2479,7 +2481,7 @@ }, { "cell_type": "markdown", - "id": "48fdd9f7", + "id": "5c37adb5", "metadata": { "editable": true }, @@ -2490,7 +2492,7 @@ }, { "cell_type": "markdown", - "id": "ae36f857", + "id": "27e4eea9", "metadata": { "editable": true }, @@ -2502,7 +2504,7 @@ }, { "cell_type": "markdown", - "id": "d312d461", + "id": "5dd95110", "metadata": { "editable": true }, @@ -2513,7 +2515,7 @@ }, { "cell_type": "markdown", - "id": "59ea8b57", + "id": "bc7cfda7", "metadata": { "editable": true }, @@ -2525,7 +2527,7 @@ }, { "cell_type": "markdown", - "id": "b9be32b5", + "id": "2bf6dbf4", "metadata": { "editable": true }, @@ -2535,7 +2537,7 @@ }, { "cell_type": "markdown", - "id": "b378b2c9", + "id": "e8ff9386", "metadata": { "editable": true }, @@ -2566,7 +2568,7 @@ }, { "cell_type": "markdown", - "id": "6ddc5457", + "id": "cd665e76", "metadata": { "editable": true }, @@ -2578,7 +2580,7 @@ }, { "cell_type": "markdown", - "id": "bb0f5795", + "id": "144c5c88", "metadata": { "editable": true }, @@ -2590,7 +2592,7 @@ }, { "cell_type": "markdown", - "id": "c969ba54", + "id": "76c7c5c6", "metadata": { "editable": true }, @@ -2600,7 +2602,7 @@ }, { "cell_type": "markdown", - "id": "630a5421", + "id": "f78405b0", "metadata": { "editable": true }, @@ -2612,7 +2614,7 @@ }, { "cell_type": "markdown", - "id": "49969ea1", + "id": "dd7b69b5", "metadata": { "editable": true }, @@ -2622,7 +2624,7 @@ }, { "cell_type": "markdown", - "id": "93b3d19b", + "id": "a1430a20", "metadata": { "editable": true }, @@ -2634,7 +2636,7 @@ }, { "cell_type": "markdown", - "id": "f61d06bf", + "id": "dfcd0c7b", "metadata": { "editable": true }, @@ -2644,7 +2646,7 @@ }, { "cell_type": "markdown", - "id": "8c25aa29", + "id": "043d37a6", "metadata": { "editable": true }, @@ -2656,7 +2658,7 @@ }, { "cell_type": "markdown", - "id": "add3ac78", + "id": "28fe8edc", "metadata": { "editable": true }, @@ -2666,7 +2668,7 @@ }, { "cell_type": "markdown", - "id": "2cb62527", + "id": "b600d21c", "metadata": { "editable": true }, @@ -2686,7 +2688,7 @@ }, { "cell_type": "markdown", - "id": "3704ebb3", + "id": "d513cf2c", "metadata": { "editable": true }, @@ -2698,7 +2700,7 @@ }, { "cell_type": "markdown", - "id": "76f1d92c", + "id": "d18008b5", "metadata": { "editable": true }, @@ -2708,7 +2710,7 @@ }, { "cell_type": "markdown", - "id": "31ef87a3", + "id": "22f55ec4", "metadata": { "editable": true }, @@ -2720,7 +2722,7 @@ }, { "cell_type": "markdown", - "id": "9f87e53d", + "id": "29e1bff6", "metadata": { "editable": true }, @@ -2732,7 +2734,7 @@ }, { "cell_type": "markdown", - "id": "b465eba1", + "id": "333869a5", "metadata": { "editable": true }, @@ -2744,7 +2746,7 @@ }, { "cell_type": "markdown", - "id": "b522f15f", + "id": "bf8de993", "metadata": { "editable": true }, @@ -2756,7 +2758,7 @@ }, { "cell_type": "markdown", - "id": "a8bf634f", + "id": "1c5349d1", "metadata": { "editable": true }, @@ -2768,7 +2770,7 @@ }, { "cell_type": "markdown", - "id": "51e8bfb6", + "id": "dbede038", "metadata": { "editable": true }, @@ -2780,7 +2782,7 @@ }, { "cell_type": "markdown", - "id": "bfb3c814", + "id": "c1334a9c", "metadata": { "editable": true }, @@ -2792,7 +2794,7 @@ }, { "cell_type": "markdown", - "id": "dfff0c7c", + "id": "ac8dcf04", "metadata": { "editable": true }, @@ -2804,7 +2806,7 @@ }, { "cell_type": "markdown", - "id": "cad0634f", + "id": "1992e354", "metadata": { "editable": true }, @@ -2814,7 +2816,7 @@ }, { "cell_type": "markdown", - "id": "1de5e07a", + "id": "24e5179f", "metadata": { "editable": true }, @@ -2826,7 +2828,7 @@ }, { "cell_type": "markdown", - "id": "43f1714e", + "id": "42979e5a", "metadata": { "editable": true }, @@ -2836,7 +2838,7 @@ }, { "cell_type": "markdown", - "id": "166406ab", + "id": "86a8cd07", "metadata": { "editable": true }, @@ -2855,7 +2857,7 @@ }, { "cell_type": "markdown", - "id": "d547f8dd", + "id": "b9a4f9a9", "metadata": { "editable": true }, @@ -2876,7 +2878,7 @@ }, { "cell_type": "markdown", - "id": "f848f875", + "id": "888c64ec", "metadata": { "editable": true }, @@ -2888,7 +2890,7 @@ }, { "cell_type": "markdown", - "id": "8c506872", + "id": "5086b046", "metadata": { "editable": true }, @@ -2899,7 +2901,7 @@ }, { "cell_type": "markdown", - "id": "317558b2", + "id": "609ea2b0", "metadata": { "editable": true }, @@ -2912,7 +2914,7 @@ }, { "cell_type": "markdown", - "id": "8f8f8c5c", + "id": "687bfdf3", "metadata": { "editable": true }, @@ -2924,7 +2926,7 @@ }, { "cell_type": "markdown", - "id": "acb7607d", + "id": "c2af1d2d", "metadata": { "editable": true }, @@ -2934,7 +2936,7 @@ }, { "cell_type": "markdown", - "id": "c5866ceb", + "id": "b8ec8d93", "metadata": { "editable": true }, @@ -2946,7 +2948,7 @@ }, { "cell_type": "markdown", - "id": "e3648f6b", + "id": "48a6fb5b", "metadata": { "editable": true }, @@ -2956,7 +2958,7 @@ }, { "cell_type": "markdown", - "id": "0b2cef16", + "id": "ba37d4fd", "metadata": { "editable": true }, @@ -2968,7 +2970,7 @@ }, { "cell_type": "markdown", - "id": "94233096", + "id": "824f1789", "metadata": { "editable": true }, @@ -2978,7 +2980,7 @@ }, { "cell_type": "markdown", - "id": "91284f6d", + "id": "07f89fd5", "metadata": { "editable": true }, @@ -2995,7 +2997,7 @@ }, { "cell_type": "markdown", - "id": "3debb528", + "id": "8fcd73d9", "metadata": { "editable": true }, @@ -3014,7 +3016,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "a4cf2a42", + "id": "4402e113", "metadata": { "collapsed": false, "editable": true @@ -3088,7 +3090,7 @@ }, { "cell_type": "markdown", - "id": "26270da6", + "id": "b700e45e", "metadata": { "editable": true }, @@ -3098,7 +3100,7 @@ }, { "cell_type": "markdown", - "id": "d8f0e20a", + "id": "69659d9d", "metadata": { "editable": true }, @@ -3112,7 +3114,7 @@ }, { "cell_type": "markdown", - "id": "6c93bc4b", + "id": "80b423eb", "metadata": { "editable": true }, @@ -3124,7 +3126,7 @@ }, { "cell_type": "markdown", - "id": "8c8b6cc8", + "id": "de547949", "metadata": { "editable": true }, @@ -3134,7 +3136,7 @@ }, { "cell_type": "markdown", - "id": "f76e3a40", + "id": "99d9476a", "metadata": { "editable": true }, @@ -3146,7 +3148,7 @@ }, { "cell_type": "markdown", - "id": "af323d99", + "id": "aa5613ea", "metadata": { "editable": true }, @@ -3156,7 +3158,7 @@ }, { "cell_type": "markdown", - "id": "49a0988c", + "id": "7dea4e11", "metadata": { "editable": true }, @@ -3168,7 +3170,7 @@ }, { "cell_type": "markdown", - "id": "38568afe", + "id": "99ab1d57", "metadata": { "editable": true }, @@ -3178,7 +3180,7 @@ }, { "cell_type": "markdown", - "id": "c3d0eeb9", + "id": "170ee68b", "metadata": { "editable": true }, @@ -3190,7 +3192,7 @@ }, { "cell_type": "markdown", - "id": "1d0ba152", + "id": "3388d6a4", "metadata": { "editable": true }, @@ -3200,7 +3202,7 @@ }, { "cell_type": "markdown", - "id": "f936c1ce", + "id": "a5bdaa16", "metadata": { "editable": true }, @@ -3216,7 +3218,7 @@ }, { "cell_type": "markdown", - "id": "dd10cea6", + "id": "cbf8e7e6", "metadata": { "editable": true }, @@ -3228,7 +3230,7 @@ }, { "cell_type": "markdown", - "id": "2534eb1d", + "id": "52c9e48f", "metadata": { "editable": true }, @@ -3238,7 +3240,7 @@ }, { "cell_type": "markdown", - "id": "104af119", + "id": "1ae37a89", "metadata": { "editable": true }, @@ -3250,7 +3252,7 @@ }, { "cell_type": "markdown", - "id": "5e8e89cf", + "id": "dd8fee27", "metadata": { "editable": true }, @@ -3263,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "467910cd", + "id": "de603b44", "metadata": { "editable": true }, @@ -3275,7 +3277,7 @@ }, { "cell_type": "markdown", - "id": "16f15265", + "id": "3f08e4a0", "metadata": { "editable": true }, @@ -3285,7 +3287,7 @@ }, { "cell_type": "markdown", - "id": "1cd22464", + "id": "b65ee2f0", "metadata": { "editable": true }, @@ -3297,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "90676d1e", 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. 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Linear Regression","14. Building a Feed Forward Neural Network","15. Solving Differential Equations with Deep Learning","16. Convolutional Neural Networks","17. Recurrent neural networks: Overarching view","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","8. Support Vector Machines, overarching aims","9. Decision trees, overarching aims","10. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","11. Basic ideas of the Principal Component Analysis (PCA)","13. Neural networks","7. Optimization, the central part of any Machine Learning algortithm","12. Clustering and Unsupervised Learning","Exercises week 34","Exercises week 35","Exercises week 36","Applied Data Analysis and Machine Learning","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. Elements of Probability Theory and Statistical Data Analysis","Teachers and Grading","Textbooks","Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso Regression","Week 36: Statistical interpretation of Linear Regression and Resampling 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/html/week36.html b/doc/LectureNotes/_build/html/week36.html index b7a101b2a..0c17d27f3 100644 --- a/doc/LectureNotes/_build/html/week36.html +++ b/doc/LectureNotes/_build/html/week36.html @@ -1005,20 +1005,30 @@ values and the column vectors of \(\b

Code for SVD and Inversion of Matrices¶

How do we use the SVD to invert a matrix \(\boldsymbol{X}^\boldsymbol{X}\) which is singular or near singular? -The simple answer is to use the linear algebra function for pseudoinvers, that is

+The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix \(\boldsymbol{X}\), that is

-
Ainv = np.linlag.pinv(A)
+
import numpy as np
+X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+Xinv = np.linlag.pinv(X)
 

 

 
---------------------------------------------------------------------------
-NameError                                 Traceback (most recent call last)
-Input In [1], in <cell line: 1>()
-----> 1 Ainv = np.linlag.pinv(A)
+AttributeError                            Traceback (most recent call last)
+Input In [1], in <cell line: 3>()
+      1 import numpy as np
+      2 X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])
+----> 3 Xinv = np.linlag.pinv(X)
 
-NameError: name 'np' is not defined
+File ~/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/__init__.py:313, in __getattr__(attr)
+    310     from .testing import Tester
+    311     return Tester
+--> 313 raise AttributeError("module {!r} has no attribute "
+    314                      "{!r}".format(__name__, attr))
+
+AttributeError: module 'numpy' has no attribute 'linlag'
diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb index 20ea50cf2..c68516565 100644 --- a/doc/LectureNotes/_build/jupyter_execute/week36.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/week36.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "b49fd9eb", + "id": "7411173e", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "138fe268", + "id": "f50db913", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "c20f461b", + "id": "4a90e380", "metadata": { "editable": true }, @@ -53,7 +53,7 @@ }, { "cell_type": "markdown", - "id": "55f0d080", + "id": "351d36fd", "metadata": { "editable": true }, @@ -65,7 +65,7 @@ }, { "cell_type": "markdown", - "id": "69ce62a6", + "id": "0f9f4d2f", "metadata": { "editable": true }, @@ -77,7 +77,7 @@ }, { "cell_type": "markdown", - "id": "a9aaa130", + "id": "9ec36bdd", "metadata": { "editable": true }, @@ -89,7 +89,7 @@ }, { "cell_type": "markdown", - "id": "2b1c0902", + "id": "614493c6", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "093e3a8c", + "id": "dc4be423", "metadata": { "editable": true }, @@ -111,7 +111,7 @@ }, { "cell_type": "markdown", - "id": "c0102cd2", + "id": "b262332c", "metadata": { "editable": true }, @@ -121,7 +121,7 @@ }, { "cell_type": "markdown", - "id": "084138a1", + "id": "ab1829a4", "metadata": { "editable": true }, @@ -138,7 +138,7 @@ }, { "cell_type": "markdown", - "id": "cb8ed4f2", + "id": "7c82dd9e", "metadata": { "editable": true }, @@ -148,7 +148,7 @@ }, { "cell_type": "markdown", - "id": "d7c1c21b", + "id": "5bab8721", "metadata": { "editable": true }, @@ -160,7 +160,7 @@ }, { "cell_type": "markdown", - "id": "791e9c6d", + "id": "72273a4f", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "924b8081", + "id": "d4af03ea", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "949f6658", + "id": "c0d35853", "metadata": { "editable": true }, @@ -196,7 +196,7 @@ }, { "cell_type": "markdown", - "id": "46a4a83c", + "id": "db5ea827", "metadata": { "editable": true }, @@ -208,7 +208,7 @@ }, { "cell_type": "markdown", - "id": "ce019aa6", + "id": "a6b341ce", "metadata": { "editable": true }, @@ -230,7 +230,7 @@ }, { "cell_type": "markdown", - "id": "c662d90f", + "id": "db723709", "metadata": { "editable": true }, @@ -242,7 +242,7 @@ }, { "cell_type": "markdown", - "id": "7df45c97", + "id": "1e600268", "metadata": { "editable": true }, @@ -257,7 +257,7 @@ }, { "cell_type": "markdown", - "id": "f0741872", + "id": "b9452938", "metadata": { "editable": true }, @@ -269,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "05530ccb", + "id": "53fea1e6", "metadata": { "editable": true }, @@ -281,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "3048f911", + "id": "ed10baad", "metadata": { "editable": true }, @@ -291,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9e3c2de1", + "id": "a4d0db96", "metadata": { "editable": true }, @@ -303,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "491923c1", + "id": "fae3cadc", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "77d1dbe5", + "id": "e973c775", "metadata": { "editable": true }, @@ -325,7 +325,7 @@ }, { "cell_type": "markdown", - "id": "fb7e2323", + "id": "f88a33f4", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "c9cca57a", + "id": "5bc7214d", "metadata": { "editable": true }, @@ -347,7 +347,7 @@ }, { "cell_type": "markdown", - "id": "04e4b486", + "id": "561d4c91", "metadata": { "editable": true }, @@ -364,7 +364,7 @@ }, { "cell_type": "markdown", - "id": "2b3c22fb", + "id": "0eda2d1e", "metadata": { "editable": true }, @@ -372,37 +372,40 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is" ] }, { "cell_type": "code", "execution_count": 1, - "id": "12637a17", + "id": "c95dbc32", "metadata": { "collapsed": false, "editable": true }, "outputs": [ { - "ename": "NameError", - "evalue": "name 'np' is not defined", + "ename": "AttributeError", + "evalue": "module 'numpy' has no attribute 'linlag'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", - "Input \u001b[0;32mIn [1]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m Ainv \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mlinlag\u001b[38;5;241m.\u001b[39mpinv(A)\n", - "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined" + "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", + "Input \u001b[0;32mIn [1]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[38;5;28;01mimport\u001b[39;00m \u001b[38;5;21;01mnumpy\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m \u001b[38;5;21;01mnp\u001b[39;00m\n\u001b[1;32m 2\u001b[0m X \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marray( [ [\u001b[38;5;241m1\u001b[39m,\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m3\u001b[39m],[\u001b[38;5;241m2\u001b[39m,\u001b[38;5;241m4\u001b[39m,\u001b[38;5;241m5\u001b[39m],[\u001b[38;5;241m3\u001b[39m,\u001b[38;5;241m5\u001b[39m,\u001b[38;5;241m6\u001b[39m]])\n\u001b[0;32m----> 3\u001b[0m Xinv \u001b[38;5;241m=\u001b[39m \u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mlinlag\u001b[49m\u001b[38;5;241m.\u001b[39mpinv(X)\n", + "File \u001b[0;32m~/miniforge3/envs/myenv/lib/python3.9/site-packages/numpy/__init__.py:313\u001b[0m, in \u001b[0;36m__getattr__\u001b[0;34m(attr)\u001b[0m\n\u001b[1;32m 310\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mtesting\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m Tester\n\u001b[1;32m 311\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m Tester\n\u001b[0;32m--> 313\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mAttributeError\u001b[39;00m(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mmodule \u001b[39m\u001b[38;5;132;01m{!r}\u001b[39;00m\u001b[38;5;124m has no attribute \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 314\u001b[0m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;132;01m{!r}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;241m.\u001b[39mformat(\u001b[38;5;18m__name__\u001b[39m, attr))\n", + "\u001b[0;31mAttributeError\u001b[0m: module 'numpy' has no attribute 'linlag'" ] } ], "source": [ - "Ainv = np.linlag.pinv(A)" + "import numpy as np\n", + "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n", + "Xinv = np.linlag.pinv(X)" ] }, { "cell_type": "markdown", - "id": "904adaf8", + "id": "77079d25", "metadata": { "editable": true }, @@ -413,7 +416,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "6b5465dc", + "id": "cd0ecc44", "metadata": { "collapsed": false, "editable": true @@ -453,7 +456,7 @@ }, { "cell_type": "markdown", - "id": "a80dcee4", + "id": "41d896dc", "metadata": { "editable": true }, @@ -473,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "0c4052d3", + "id": "f594852c", "metadata": { "editable": true }, @@ -485,7 +488,7 @@ }, { "cell_type": "markdown", - "id": "185d433f", + "id": "b78264f2", "metadata": { "editable": true }, @@ -496,7 +499,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "418f8797", + "id": "d05eff95", "metadata": { "collapsed": false, "editable": true @@ -530,7 +533,7 @@ }, { "cell_type": "markdown", - "id": "bf5051ea", + "id": "2fae32c6", "metadata": { "editable": true }, @@ -540,7 +543,7 @@ }, { "cell_type": "markdown", - "id": "202d1f90", + "id": "8631011a", "metadata": { "editable": true }, @@ -553,7 +556,7 @@ }, { "cell_type": "markdown", - "id": "df0a8718", + "id": "8e7701de", "metadata": { "editable": true }, @@ -565,7 +568,7 @@ }, { "cell_type": "markdown", - "id": "4ebd1d20", + "id": "aea7230b", "metadata": { "editable": true }, @@ -575,7 +578,7 @@ }, { "cell_type": "markdown", - "id": "40df3859", + "id": "c5029213", "metadata": { "editable": true }, @@ -588,7 +591,7 @@ }, { "cell_type": "markdown", - "id": "980001f5", + "id": "535ad70e", "metadata": { "editable": true }, @@ -598,7 +601,7 @@ }, { "cell_type": "markdown", - "id": "699b1198", + "id": "159ae526", "metadata": { "editable": true }, @@ -610,7 +613,7 @@ }, { "cell_type": "markdown", - "id": "ff01ef41", + "id": "2f25dfa4", "metadata": { "editable": true }, @@ -625,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "d7b9188f", + "id": "9d187806", "metadata": { "editable": true }, @@ -638,7 +641,7 @@ }, { "cell_type": "markdown", - "id": "778790dc", + "id": "6c692db5", "metadata": { "editable": true }, @@ -652,7 +655,7 @@ }, { "cell_type": "markdown", - "id": "38dd7428", + "id": "3404e0b4", "metadata": { "editable": true }, @@ -664,7 +667,7 @@ }, { "cell_type": "markdown", - "id": "6ed3deb0", + "id": "4bc49666", "metadata": { "editable": true }, @@ -674,7 +677,7 @@ }, { "cell_type": "markdown", - "id": "e0860754", + "id": "3f08dfa1", "metadata": { "editable": true }, @@ -687,7 +690,7 @@ }, { "cell_type": "markdown", - "id": "66751140", + "id": "30bcc0cd", "metadata": { "editable": true }, @@ -699,7 +702,7 @@ }, { "cell_type": "markdown", - "id": "e1fa2bdb", + "id": "d311a7d8", "metadata": { "editable": true }, @@ -711,7 +714,7 @@ }, { "cell_type": "markdown", - "id": "32131982", + "id": "32c46b73", "metadata": { "editable": true }, @@ -723,7 +726,7 @@ }, { "cell_type": "markdown", - "id": "c1979796", + "id": "3cceda87", "metadata": { "editable": true }, @@ -735,7 +738,7 @@ }, { "cell_type": "markdown", - "id": "d7eb56d7", + "id": "10a16675", "metadata": { "editable": true }, @@ -749,7 +752,7 @@ }, { "cell_type": "markdown", - "id": "7c42f326", + "id": "6dba20bf", "metadata": { "editable": true }, @@ -761,7 +764,7 @@ }, { "cell_type": "markdown", - "id": "6496e5c0", + "id": "9325f924", "metadata": { "editable": true }, @@ -771,7 +774,7 @@ }, { "cell_type": "markdown", - "id": "837f7c04", + "id": "745d61e9", "metadata": { "editable": true }, @@ -783,7 +786,7 @@ }, { "cell_type": "markdown", - "id": "55e9395f", + "id": "3866af64", "metadata": { "editable": true }, @@ -793,7 +796,7 @@ }, { "cell_type": "markdown", - "id": "a943f34a", + "id": "710de288", "metadata": { "editable": true }, @@ -805,7 +808,7 @@ }, { "cell_type": "markdown", - "id": "63a227a9", + "id": "5893fba0", "metadata": { "editable": true }, @@ -817,7 +820,7 @@ }, { "cell_type": "markdown", - "id": "8be2bd7b", + "id": "5ef8c59f", "metadata": { "editable": true }, @@ -827,7 +830,7 @@ }, { "cell_type": "markdown", - "id": "35509652", + "id": "79b257ae", "metadata": { "editable": true }, @@ -838,7 +841,7 @@ }, { "cell_type": "markdown", - "id": "2fbe21d0", + "id": "46270996", "metadata": { "editable": true }, @@ -850,7 +853,7 @@ }, { "cell_type": "markdown", - "id": "71e986f8", + "id": "785aebe9", "metadata": { "editable": true }, @@ -866,7 +869,7 @@ }, { "cell_type": "markdown", - "id": "3a5e5f40", + "id": "443f3a3f", "metadata": { "editable": true }, @@ -879,7 +882,7 @@ }, { "cell_type": "markdown", - "id": "752b6ba2", + "id": "cfb8d4c0", "metadata": { "editable": true }, @@ -891,7 +894,7 @@ }, { "cell_type": "markdown", - "id": "4e724327", + "id": "2be1ca1f", "metadata": { "editable": true }, @@ -901,7 +904,7 @@ }, { "cell_type": "markdown", - "id": "b15c2c90", + "id": "9ed0aec2", "metadata": { "editable": true }, @@ -913,7 +916,7 @@ }, { "cell_type": "markdown", - "id": "82e70fa5", + "id": "943b1cd5", "metadata": { "editable": true }, @@ -923,7 +926,7 @@ }, { "cell_type": "markdown", - "id": "3394e2e7", + "id": "4a8a6bff", "metadata": { "editable": true }, @@ -935,7 +938,7 @@ }, { "cell_type": "markdown", - "id": "722ffa86", + "id": "b0371965", "metadata": { "editable": true }, @@ -947,7 +950,7 @@ }, { "cell_type": "markdown", - "id": "720ed773", + "id": "38e7b497", "metadata": { "editable": true }, @@ -963,7 +966,7 @@ }, { "cell_type": "markdown", - "id": "3b57864f", + "id": "682f90db", "metadata": { "editable": true }, @@ -975,7 +978,7 @@ }, { "cell_type": "markdown", - "id": "8f36ff27", + "id": "b5fb7771", "metadata": { "editable": true }, @@ -987,7 +990,7 @@ }, { "cell_type": "markdown", - "id": "70adc9ce", + "id": "0c228525", "metadata": { "editable": true }, @@ -997,7 +1000,7 @@ }, { "cell_type": "markdown", - "id": "eb4a995f", + "id": "08a400af", "metadata": { "editable": true }, @@ -1009,7 +1012,7 @@ }, { "cell_type": "markdown", - "id": "e3cd4b92", + "id": "ec0808b1", "metadata": { "editable": true }, @@ -1019,7 +1022,7 @@ }, { "cell_type": "markdown", - "id": "bbf9b1de", + "id": "92d3d92f", "metadata": { "editable": true }, @@ -1031,7 +1034,7 @@ }, { "cell_type": "markdown", - "id": "e30eff48", + "id": "07486e29", "metadata": { "editable": true }, @@ -1048,7 +1051,7 @@ }, { "cell_type": "markdown", - "id": "9a8f8e2e", + "id": "9868d4f4", "metadata": { "editable": true }, @@ -1060,7 +1063,7 @@ }, { "cell_type": "markdown", - "id": "3cb7928c", + "id": "1793a225", "metadata": { "editable": true }, @@ -1072,7 +1075,7 @@ }, { "cell_type": "markdown", - "id": "b1c6c813", + "id": "c7d3fa52", "metadata": { "editable": true }, @@ -1082,7 +1085,7 @@ }, { "cell_type": "markdown", - "id": "da336775", + "id": "47894977", "metadata": { "editable": true }, @@ -1094,7 +1097,7 @@ }, { "cell_type": "markdown", - "id": "2e0de141", + "id": "d1a11dc8", "metadata": { "editable": true }, @@ -1104,7 +1107,7 @@ }, { "cell_type": "markdown", - "id": "40985ec4", + "id": "a6cf82de", "metadata": { "editable": true }, @@ -1116,7 +1119,7 @@ }, { "cell_type": "markdown", - "id": "31091d7b", + "id": "bbf7ccc7", "metadata": { "editable": true }, @@ -1126,7 +1129,7 @@ }, { "cell_type": "markdown", - "id": "e81d965e", + "id": "bc6dbb83", "metadata": { "editable": true }, @@ -1138,7 +1141,7 @@ }, { "cell_type": "markdown", - "id": "9984b3a4", + "id": "ef5a8292", "metadata": { "editable": true }, @@ -1148,7 +1151,7 @@ }, { "cell_type": "markdown", - "id": "21ec3768", + "id": "d77bf73a", "metadata": { "editable": true }, @@ -1163,7 +1166,7 @@ }, { "cell_type": "markdown", - "id": "6be0ec63", + "id": "265ce93f", "metadata": { "editable": true }, @@ -1175,7 +1178,7 @@ }, { "cell_type": "markdown", - "id": "44c7278c", + "id": "c19e720c", "metadata": { "editable": true }, @@ -1185,7 +1188,7 @@ }, { "cell_type": "markdown", - "id": "918a8cc3", + "id": "8cad2962", "metadata": { "editable": true }, @@ -1197,7 +1200,7 @@ }, { "cell_type": "markdown", - "id": "3949af51", + "id": "bf6316a2", "metadata": { "editable": true }, @@ -1209,7 +1212,7 @@ }, { "cell_type": "markdown", - "id": "aa856d18", + "id": "78413ab6", "metadata": { "editable": true }, @@ -1221,7 +1224,7 @@ }, { "cell_type": "markdown", - "id": "bd30d1d4", + "id": "a77ca41b", "metadata": { "editable": true }, @@ -1231,7 +1234,7 @@ }, { "cell_type": "markdown", - "id": "142de535", + "id": "67b3f67e", "metadata": { "editable": true }, @@ -1243,7 +1246,7 @@ }, { "cell_type": "markdown", - "id": "1efee0c3", + "id": "d3571c8b", "metadata": { "editable": true }, @@ -1255,7 +1258,7 @@ }, { "cell_type": "markdown", - "id": "a6db019a", + "id": "943f822e", "metadata": { "editable": true }, @@ -1267,7 +1270,7 @@ }, { "cell_type": "markdown", - "id": "0f38deda", + "id": "2ec0ba01", "metadata": { "editable": true }, @@ -1277,7 +1280,7 @@ }, { "cell_type": "markdown", - "id": "7f9738a5", + "id": "85bf4c9b", "metadata": { "editable": true }, @@ -1289,7 +1292,7 @@ }, { "cell_type": "markdown", - "id": "4e8460f8", + "id": "c81d05f4", "metadata": { "editable": true }, @@ -1299,7 +1302,7 @@ }, { "cell_type": "markdown", - "id": "ae49ddce", + "id": "ce36390d", "metadata": { "editable": true }, @@ -1313,7 +1316,7 @@ }, { "cell_type": "markdown", - "id": "1efb96d3", + "id": "a47dd697", "metadata": { "editable": true }, @@ -1323,7 +1326,7 @@ }, { "cell_type": "markdown", - "id": "1a030291", + "id": "15fe5a0c", "metadata": { "editable": true }, @@ -1335,7 +1338,7 @@ }, { "cell_type": "markdown", - "id": "af486213", + "id": "62d7f0e9", "metadata": { "editable": true }, @@ -1347,7 +1350,7 @@ }, { "cell_type": "markdown", - "id": "87e5470d", + "id": "f2d4576f", "metadata": { "editable": true }, @@ -1357,7 +1360,7 @@ }, { "cell_type": "markdown", - "id": "182f9f51", + "id": "fb2bc0ff", "metadata": { "editable": true }, @@ -1369,7 +1372,7 @@ }, { "cell_type": "markdown", - "id": "edc0ee3d", + "id": "807ac009", "metadata": { "editable": true }, @@ -1379,7 +1382,7 @@ }, { "cell_type": "markdown", - "id": "673c64a0", + "id": "fe72889b", "metadata": { "editable": true }, @@ -1391,7 +1394,7 @@ }, { "cell_type": "markdown", - "id": "c0554476", + "id": "48844e10", "metadata": { "editable": true }, @@ -1403,7 +1406,7 @@ }, { "cell_type": "markdown", - "id": "cc9c761b", + "id": "72a4bc4c", "metadata": { "editable": true }, @@ -1413,7 +1416,7 @@ }, { "cell_type": "markdown", - "id": "b5c84ed1", + "id": "cfe1db6f", "metadata": { "editable": true }, @@ -1425,7 +1428,7 @@ }, { "cell_type": "markdown", - "id": "a8d2ba86", + "id": "05b6e6f6", "metadata": { "editable": true }, @@ -1435,7 +1438,7 @@ }, { "cell_type": "markdown", - "id": "9e407326", + "id": "0e58f37a", "metadata": { "editable": true }, @@ -1447,7 +1450,7 @@ }, { "cell_type": "markdown", - "id": "36ce4758", + "id": "78593de5", "metadata": { "editable": true }, @@ -1459,7 +1462,7 @@ }, { "cell_type": "markdown", - "id": "8665e33b", + "id": "26b08a56", "metadata": { "editable": true }, @@ -1469,7 +1472,7 @@ }, { "cell_type": "markdown", - "id": "16dbcd82", + "id": "3df661a4", "metadata": { "editable": true }, @@ -1481,7 +1484,7 @@ }, { "cell_type": "markdown", - "id": "642f7768", + "id": "68460e68", "metadata": { "editable": true }, @@ -1494,7 +1497,7 @@ }, { "cell_type": "markdown", - "id": "1784c79c", + "id": "6c660112", "metadata": { "editable": true }, @@ -1506,7 +1509,7 @@ }, { "cell_type": "markdown", - "id": "72b7600f", + "id": "6ff54cf1", "metadata": { "editable": true }, @@ -1518,7 +1521,7 @@ }, { "cell_type": "markdown", - "id": "84ca3171", + "id": "f3f88f94", "metadata": { "editable": true }, @@ -1530,7 +1533,7 @@ }, { "cell_type": "markdown", - "id": "d31aa45e", + "id": "6918fc7c", "metadata": { "editable": true }, @@ -1540,7 +1543,7 @@ }, { "cell_type": "markdown", - "id": "1298ca6d", + "id": "1b2321a3", "metadata": { "editable": true }, @@ -1552,7 +1555,7 @@ }, { "cell_type": "markdown", - "id": "fa2455d6", + "id": "d5e21457", "metadata": { "editable": true }, @@ -1562,7 +1565,7 @@ }, { "cell_type": "markdown", - "id": "8cd50073", + "id": "e03c3617", "metadata": { "editable": true }, @@ -1574,7 +1577,7 @@ }, { "cell_type": "markdown", - "id": "1f7e12c3", + "id": "6c4d9a4a", "metadata": { "editable": true }, @@ -1584,7 +1587,7 @@ }, { "cell_type": "markdown", - "id": "a79821fd", + "id": "584bef2e", "metadata": { "editable": true }, @@ -1596,7 +1599,7 @@ }, { "cell_type": "markdown", - "id": "2ac13531", + "id": "87273242", "metadata": { "editable": true }, @@ -1606,7 +1609,7 @@ }, { "cell_type": "markdown", - "id": "eed25f68", + "id": "f7e8277f", "metadata": { "editable": true }, @@ -1619,7 +1622,7 @@ }, { "cell_type": "markdown", - "id": "f74d71ec", + "id": "e7167c28", "metadata": { "editable": true }, @@ -1631,7 +1634,7 @@ }, { "cell_type": "markdown", - "id": "f0797067", + "id": "6a3fa7b2", "metadata": { "editable": true }, @@ -1643,7 +1646,7 @@ }, { "cell_type": "markdown", - "id": "3cc5d5e0", + "id": "92c33067", "metadata": { "editable": true }, @@ -1653,7 +1656,7 @@ }, { "cell_type": "markdown", - "id": "04250e5c", + "id": "1b7be5cc", "metadata": { "editable": true }, @@ -1665,7 +1668,7 @@ }, { "cell_type": "markdown", - "id": "2b24d2fa", + "id": "9e2db51b", "metadata": { "editable": true }, @@ -1682,7 +1685,7 @@ }, { "cell_type": "markdown", - "id": "aa950760", + "id": "7ea5c023", "metadata": { "editable": true }, @@ -1694,7 +1697,7 @@ }, { "cell_type": "markdown", - "id": "475389ed", + "id": "efff1845", "metadata": { "editable": true }, @@ -1706,7 +1709,7 @@ }, { "cell_type": "markdown", - "id": "086558d9", + "id": "80496d8c", "metadata": { "editable": true }, @@ -1716,7 +1719,7 @@ }, { "cell_type": "markdown", - "id": "1a3b2ba5", + "id": "bb33463a", "metadata": { "editable": true }, @@ -1728,7 +1731,7 @@ }, { "cell_type": "markdown", - "id": "c928bc46", + "id": "f16b1457", "metadata": { "editable": true }, @@ -1738,7 +1741,7 @@ }, { "cell_type": "markdown", - "id": "809a58f7", + "id": "4bee5a43", "metadata": { "editable": true }, @@ -1750,7 +1753,7 @@ }, { "cell_type": "markdown", - "id": "3b937bf4", + "id": "17176990", "metadata": { "editable": true }, @@ -1760,7 +1763,7 @@ }, { "cell_type": "markdown", - "id": "6e5b2527", + "id": "ec2c8f34", "metadata": { "editable": true }, @@ -1772,7 +1775,7 @@ }, { "cell_type": "markdown", - "id": "c5f88a45", + "id": "5934144d", "metadata": { "editable": true }, @@ -1782,7 +1785,7 @@ }, { "cell_type": "markdown", - "id": "c57753b4", + "id": "fe097c51", "metadata": { "editable": true }, @@ -1797,7 +1800,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "392f218d", + "id": "92cd1361", "metadata": { "collapsed": false, "editable": true @@ -1859,7 +1862,7 @@ }, { "cell_type": "markdown", - "id": "dd134ab1", + "id": "9e4426f4", "metadata": { "editable": true }, @@ -1869,7 +1872,7 @@ }, { "cell_type": "markdown", - "id": "8d9a61db", + "id": "5f2b6935", "metadata": { "editable": true }, @@ -1880,7 +1883,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "afba7c4a", + "id": "409bbeac", "metadata": { "collapsed": false, "editable": true @@ -1947,7 +1950,7 @@ }, { "cell_type": "markdown", - "id": "4d63db0b", + "id": "67a24e77", "metadata": { "editable": true }, @@ -1958,7 +1961,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "81115144", + "id": "8da16a74", "metadata": { "collapsed": false, "editable": true @@ -2047,7 +2050,7 @@ }, { "cell_type": "markdown", - "id": "e076a968", + "id": "9d2563fa", "metadata": { "editable": true }, @@ -2057,7 +2060,7 @@ }, { "cell_type": "markdown", - "id": "db822c65", + "id": "1232037e", "metadata": { "editable": true }, @@ -2086,7 +2089,7 @@ }, { "cell_type": "markdown", - "id": "fc4a06b6", + "id": "e50205be", "metadata": { "editable": true }, @@ -2102,7 +2105,7 @@ }, { "cell_type": "markdown", - "id": "43299fb3", + "id": "70eccda9", "metadata": { "editable": true }, @@ -2121,7 +2124,7 @@ }, { "cell_type": "markdown", - "id": "6cba00c7", + "id": "e6bdcbf2", "metadata": { "editable": true }, @@ -2135,7 +2138,7 @@ }, { "cell_type": "markdown", - "id": "8c85f61d", + "id": "d74471b8", "metadata": { "editable": true }, @@ -2147,7 +2150,7 @@ }, { "cell_type": "markdown", - "id": "a1901c32", + "id": "f7b178d9", "metadata": { "editable": true }, @@ -2158,7 +2161,7 @@ }, { "cell_type": "markdown", - "id": "c59e8272", + "id": "8ea16c62", "metadata": { "editable": true }, @@ -2170,7 +2173,7 @@ }, { "cell_type": "markdown", - "id": "8cba4bae", + "id": "4a8d7bfd", "metadata": { "editable": true }, @@ -2182,7 +2185,7 @@ }, { "cell_type": "markdown", - "id": "5e18ebb5", + "id": "5367d943", "metadata": { "editable": true }, @@ -2198,7 +2201,7 @@ }, { "cell_type": "markdown", - "id": "10d86c58", + "id": "c7eefd85", "metadata": { "editable": true }, @@ -2209,7 +2212,7 @@ }, { "cell_type": "markdown", - "id": "efba1ba9", + "id": "27ce343a", "metadata": { "editable": true }, @@ -2232,7 +2235,7 @@ }, { "cell_type": "markdown", - "id": "a3eccdab", + "id": "c67ea20b", "metadata": { "editable": true }, @@ -2243,7 +2246,7 @@ }, { "cell_type": "markdown", - "id": "70b901b4", + "id": "b4da473a", "metadata": { "editable": true }, @@ -2255,7 +2258,7 @@ }, { "cell_type": "markdown", - "id": "752b72d2", + "id": "af812d38", "metadata": { "editable": true }, @@ -2267,7 +2270,7 @@ }, { "cell_type": "markdown", - "id": "108aba79", + "id": "9d3aaff5", "metadata": { "editable": true }, @@ -2281,7 +2284,7 @@ }, { "cell_type": "markdown", - "id": "65801f3b", + "id": "eebea8b7", "metadata": { "editable": true }, @@ -2312,7 +2315,7 @@ }, { "cell_type": "markdown", - "id": "faf8da78", + "id": "714f281a", "metadata": { "editable": true }, @@ -2334,7 +2337,7 @@ }, { "cell_type": "markdown", - "id": "1c71921e", + "id": "66015ca6", "metadata": { "editable": true }, @@ -2346,7 +2349,7 @@ }, { "cell_type": "markdown", - "id": "cfb57087", + "id": "267f01c0", "metadata": { "editable": true }, @@ -2359,7 +2362,7 @@ }, { "cell_type": "markdown", - "id": "be4ad159", + "id": "998de4b6", "metadata": { "editable": true }, @@ -2371,7 +2374,7 @@ }, { "cell_type": "markdown", - "id": "f5e11f36", + "id": "06a8d452", "metadata": { "editable": true }, @@ -2383,7 +2386,7 @@ }, { "cell_type": "markdown", - "id": "dbf2700f", + "id": "6cd3ed9b", "metadata": { "editable": true }, @@ -2395,7 +2398,7 @@ }, { "cell_type": "markdown", - "id": "0f6d760d", + "id": "70f9aa69", "metadata": { "editable": true }, @@ -2407,7 +2410,7 @@ }, { "cell_type": "markdown", - "id": "ce614956", + "id": "3bb18388", "metadata": { "editable": true }, @@ -2430,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "086419d8", + "id": "84b546af", "metadata": { "editable": true }, @@ -2442,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "a228dcf4", + "id": "0e90935e", "metadata": { "editable": true }, @@ -2455,7 +2458,7 @@ }, { "cell_type": "markdown", - "id": "f04f6b76", + "id": "b40b33f5", "metadata": { "editable": true }, @@ -2467,7 +2470,7 @@ }, { "cell_type": "markdown", - "id": "642936a8", + "id": "23e0c7ac", "metadata": { "editable": true }, @@ -2479,7 +2482,7 @@ }, { "cell_type": "markdown", - "id": "4f137023", + "id": "dcd4ed9c", "metadata": { "editable": true }, @@ -2491,7 +2494,7 @@ }, { "cell_type": "markdown", - "id": "48fdd9f7", + "id": "5c37adb5", "metadata": { "editable": true }, @@ -2502,7 +2505,7 @@ }, { "cell_type": "markdown", - "id": "ae36f857", + "id": "27e4eea9", "metadata": { "editable": true }, @@ -2514,7 +2517,7 @@ }, { "cell_type": "markdown", - "id": "d312d461", + "id": "5dd95110", "metadata": { "editable": true }, @@ -2525,7 +2528,7 @@ }, { "cell_type": "markdown", - "id": "59ea8b57", + "id": "bc7cfda7", "metadata": { "editable": true }, @@ -2537,7 +2540,7 @@ }, { "cell_type": "markdown", - "id": "b9be32b5", + "id": "2bf6dbf4", "metadata": { "editable": true }, @@ -2547,7 +2550,7 @@ }, { "cell_type": "markdown", - "id": "b378b2c9", + "id": "e8ff9386", "metadata": { "editable": true }, @@ -2578,7 +2581,7 @@ }, { "cell_type": "markdown", - "id": "6ddc5457", + "id": "cd665e76", "metadata": { "editable": true }, @@ -2590,7 +2593,7 @@ }, { "cell_type": "markdown", - "id": "bb0f5795", + "id": "144c5c88", "metadata": { "editable": true }, @@ -2602,7 +2605,7 @@ }, { "cell_type": "markdown", - "id": "c969ba54", + "id": "76c7c5c6", "metadata": { "editable": true }, @@ -2612,7 +2615,7 @@ }, { "cell_type": "markdown", - "id": "630a5421", + "id": "f78405b0", "metadata": { "editable": true }, @@ -2624,7 +2627,7 @@ }, { "cell_type": "markdown", - "id": "49969ea1", + "id": "dd7b69b5", "metadata": { "editable": true }, @@ -2634,7 +2637,7 @@ }, { "cell_type": "markdown", - "id": "93b3d19b", + "id": "a1430a20", "metadata": { "editable": true }, @@ -2646,7 +2649,7 @@ }, { "cell_type": "markdown", - "id": "f61d06bf", + "id": "dfcd0c7b", "metadata": { "editable": true }, @@ -2656,7 +2659,7 @@ }, { "cell_type": "markdown", - "id": "8c25aa29", + "id": "043d37a6", "metadata": { "editable": true }, @@ -2668,7 +2671,7 @@ }, { "cell_type": "markdown", - "id": "add3ac78", + "id": "28fe8edc", "metadata": { "editable": true }, @@ -2678,7 +2681,7 @@ }, { "cell_type": "markdown", - "id": "2cb62527", + "id": "b600d21c", "metadata": { "editable": true }, @@ -2698,7 +2701,7 @@ }, { "cell_type": "markdown", - "id": "3704ebb3", + "id": "d513cf2c", "metadata": { "editable": true }, @@ -2710,7 +2713,7 @@ }, { "cell_type": "markdown", - "id": "76f1d92c", + "id": "d18008b5", "metadata": { "editable": true }, @@ -2720,7 +2723,7 @@ }, { "cell_type": "markdown", - "id": "31ef87a3", + "id": "22f55ec4", "metadata": { "editable": true }, @@ -2732,7 +2735,7 @@ }, { "cell_type": "markdown", - "id": "9f87e53d", + "id": "29e1bff6", "metadata": { "editable": true }, @@ -2744,7 +2747,7 @@ }, { "cell_type": "markdown", - "id": "b465eba1", + "id": "333869a5", "metadata": { "editable": true }, @@ -2756,7 +2759,7 @@ }, { "cell_type": "markdown", - "id": "b522f15f", + "id": "bf8de993", "metadata": { "editable": true }, @@ -2768,7 +2771,7 @@ }, { "cell_type": "markdown", - "id": "a8bf634f", + "id": "1c5349d1", "metadata": { "editable": true }, @@ -2780,7 +2783,7 @@ }, { "cell_type": "markdown", - "id": "51e8bfb6", + "id": "dbede038", "metadata": { "editable": true }, @@ -2792,7 +2795,7 @@ }, { "cell_type": "markdown", - "id": "bfb3c814", + "id": "c1334a9c", "metadata": { "editable": true }, @@ -2804,7 +2807,7 @@ }, { "cell_type": "markdown", - "id": "dfff0c7c", + "id": "ac8dcf04", "metadata": { "editable": true }, @@ -2816,7 +2819,7 @@ }, { "cell_type": "markdown", - "id": "cad0634f", + "id": "1992e354", "metadata": { "editable": true }, @@ -2826,7 +2829,7 @@ }, { "cell_type": "markdown", - "id": "1de5e07a", + "id": "24e5179f", "metadata": { "editable": true }, @@ -2838,7 +2841,7 @@ }, { "cell_type": "markdown", - "id": "43f1714e", + "id": "42979e5a", "metadata": { "editable": true }, @@ -2848,7 +2851,7 @@ }, { "cell_type": "markdown", - "id": "166406ab", + "id": "86a8cd07", "metadata": { "editable": true }, @@ -2867,7 +2870,7 @@ }, { "cell_type": "markdown", - "id": "d547f8dd", + "id": "b9a4f9a9", "metadata": { "editable": true }, @@ -2888,7 +2891,7 @@ }, { "cell_type": "markdown", - "id": "f848f875", + "id": "888c64ec", "metadata": { "editable": true }, @@ -2900,7 +2903,7 @@ }, { "cell_type": "markdown", - "id": "8c506872", + "id": "5086b046", "metadata": { "editable": true }, @@ -2911,7 +2914,7 @@ }, { "cell_type": "markdown", - "id": "317558b2", + "id": "609ea2b0", "metadata": { "editable": true }, @@ -2924,7 +2927,7 @@ }, { "cell_type": "markdown", - "id": "8f8f8c5c", + "id": "687bfdf3", "metadata": { "editable": true }, @@ -2936,7 +2939,7 @@ }, { "cell_type": "markdown", - "id": "acb7607d", + "id": "c2af1d2d", "metadata": { "editable": true }, @@ -2946,7 +2949,7 @@ }, { "cell_type": "markdown", - "id": "c5866ceb", + "id": "b8ec8d93", "metadata": { "editable": true }, @@ -2958,7 +2961,7 @@ }, { "cell_type": "markdown", - "id": "e3648f6b", + "id": "48a6fb5b", "metadata": { "editable": true }, @@ -2968,7 +2971,7 @@ }, { "cell_type": "markdown", - "id": "0b2cef16", + "id": "ba37d4fd", "metadata": { "editable": true }, @@ -2980,7 +2983,7 @@ }, { "cell_type": "markdown", - "id": "94233096", + "id": "824f1789", "metadata": { "editable": true }, @@ -2990,7 +2993,7 @@ }, { "cell_type": "markdown", - "id": "91284f6d", + "id": "07f89fd5", "metadata": { "editable": true }, @@ -3007,7 +3010,7 @@ }, { "cell_type": "markdown", - "id": "3debb528", + "id": "8fcd73d9", "metadata": { "editable": true }, @@ -3026,7 +3029,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "a4cf2a42", + "id": "4402e113", "metadata": { "collapsed": false, "editable": true @@ -3100,7 +3103,7 @@ }, { "cell_type": "markdown", - "id": "26270da6", + "id": "b700e45e", "metadata": { "editable": true }, @@ -3110,7 +3113,7 @@ }, { "cell_type": "markdown", - "id": "d8f0e20a", + "id": "69659d9d", "metadata": { "editable": true }, @@ -3124,7 +3127,7 @@ }, { "cell_type": "markdown", - "id": "6c93bc4b", + "id": "80b423eb", "metadata": { "editable": true }, @@ -3136,7 +3139,7 @@ }, { "cell_type": "markdown", - "id": "8c8b6cc8", + "id": "de547949", "metadata": { "editable": true }, @@ -3146,7 +3149,7 @@ }, { "cell_type": "markdown", - "id": "f76e3a40", + "id": "99d9476a", "metadata": { "editable": true }, @@ -3158,7 +3161,7 @@ }, { "cell_type": "markdown", - "id": "af323d99", + "id": "aa5613ea", "metadata": { "editable": true }, @@ -3168,7 +3171,7 @@ }, { "cell_type": "markdown", - "id": "49a0988c", + "id": "7dea4e11", "metadata": { "editable": true }, @@ -3180,7 +3183,7 @@ }, { "cell_type": "markdown", - "id": "38568afe", + "id": "99ab1d57", "metadata": { "editable": true }, @@ -3190,7 +3193,7 @@ }, { "cell_type": "markdown", - "id": "c3d0eeb9", + "id": "170ee68b", "metadata": { "editable": true }, @@ -3202,7 +3205,7 @@ }, { "cell_type": "markdown", - "id": "1d0ba152", + "id": "3388d6a4", "metadata": { "editable": true }, @@ -3212,7 +3215,7 @@ }, { "cell_type": "markdown", - "id": "f936c1ce", + "id": "a5bdaa16", "metadata": { "editable": true }, @@ -3228,7 +3231,7 @@ }, { "cell_type": "markdown", - "id": "dd10cea6", + "id": "cbf8e7e6", "metadata": { "editable": true }, @@ -3240,7 +3243,7 @@ }, { "cell_type": "markdown", - "id": "2534eb1d", + "id": "52c9e48f", "metadata": { "editable": true }, @@ -3250,7 +3253,7 @@ }, { "cell_type": "markdown", - "id": "104af119", + "id": "1ae37a89", "metadata": { "editable": true }, @@ -3262,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "5e8e89cf", + "id": "dd8fee27", "metadata": { "editable": true }, @@ -3275,7 +3278,7 @@ }, { "cell_type": "markdown", - "id": "467910cd", + "id": "de603b44", "metadata": { "editable": true }, @@ -3287,7 +3290,7 @@ }, { "cell_type": "markdown", - "id": "16f15265", + "id": "3f08e4a0", "metadata": { "editable": true }, @@ -3297,7 +3300,7 @@ }, { "cell_type": "markdown", - "id": "1cd22464", + "id": "b65ee2f0", "metadata": { "editable": true }, @@ -3309,7 +3312,7 @@ }, { "cell_type": "markdown", - "id": "90676d1e", + "id": "6526b426", "metadata": { "editable": true }, @@ -3319,7 +3322,7 @@ }, { "cell_type": "markdown", - "id": "2082d057", + "id": "29190fe5", "metadata": { "editable": true }, @@ -3331,7 +3334,7 @@ }, { "cell_type": "markdown", - "id": "544b15d3", + "id": "71b1585c", "metadata": { "editable": true }, @@ -3343,7 +3346,7 @@ }, { "cell_type": "markdown", - "id": "592804c7", + "id": "e18baaa9", "metadata": { "editable": true }, @@ -3353,7 +3356,7 @@ }, { "cell_type": "markdown", - "id": "c11e6e06", + "id": "1c58b8db", "metadata": { "editable": true }, @@ -3365,7 +3368,7 @@ }, { "cell_type": "markdown", - "id": "fdc6e9b5", + "id": "238ee8c5", "metadata": { "editable": true }, @@ -3377,7 +3380,7 @@ }, { "cell_type": "markdown", - "id": "a0e8571d", + "id": "46389b82", "metadata": { "editable": true }, @@ -3389,7 +3392,7 @@ }, { "cell_type": "markdown", - "id": "84eca415", + "id": "0ecbb2d2", "metadata": { "editable": true }, @@ -3399,7 +3402,7 @@ }, { "cell_type": "markdown", - "id": "0924d8d4", + "id": "27368dfb", "metadata": { "editable": true }, @@ -3411,7 +3414,7 @@ }, { "cell_type": "markdown", - "id": "7270fef7", + "id": "27082f48", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/_build/jupyter_execute/week36.py b/doc/LectureNotes/_build/jupyter_execute/week36.py index 912ea6dbe..0ecedfda3 100644 --- a/doc/LectureNotes/_build/jupyter_execute/week36.py +++ b/doc/LectureNotes/_build/jupyter_execute/week36.py @@ -142,12 +142,14 @@ # ## Code for SVD and Inversion of Matrices # # How do we use the SVD to invert a matrix $\boldsymbol{X}^\boldsymbol{X}$ which is singular or near singular? -# The simple answer is to use the linear algebra function for pseudoinvers, that is +# The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\boldsymbol{X}$, that is # In[1]: -Ainv = np.linlag.pinv(A) +import numpy as np +X = np.array( [ [1,2,3],[2,4,5],[3,5,6]]) +Xinv = np.linlag.pinv(X) # Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD. diff --git a/doc/LectureNotes/week36.ipynb b/doc/LectureNotes/week36.ipynb index 991b0f568..bdaee795a 100644 --- a/doc/LectureNotes/week36.ipynb +++ b/doc/LectureNotes/week36.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "b49fd9eb", + "id": "7411173e", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "138fe268", + "id": "f50db913", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "c20f461b", + "id": "4a90e380", "metadata": { "editable": true }, @@ -53,7 +53,7 @@ }, { "cell_type": "markdown", - "id": "55f0d080", + "id": "351d36fd", "metadata": { "editable": true }, @@ -65,7 +65,7 @@ }, { "cell_type": "markdown", - "id": "69ce62a6", + "id": "0f9f4d2f", "metadata": { "editable": true }, @@ -77,7 +77,7 @@ }, { "cell_type": "markdown", - "id": "a9aaa130", + "id": "9ec36bdd", "metadata": { "editable": true }, @@ -89,7 +89,7 @@ }, { "cell_type": "markdown", - "id": "2b1c0902", + "id": "614493c6", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "093e3a8c", + "id": "dc4be423", "metadata": { "editable": true }, @@ -111,7 +111,7 @@ }, { "cell_type": "markdown", - "id": "c0102cd2", + "id": "b262332c", "metadata": { "editable": true }, @@ -121,7 +121,7 @@ }, { "cell_type": "markdown", - "id": "084138a1", + "id": "ab1829a4", "metadata": { "editable": true }, @@ -138,7 +138,7 @@ }, { "cell_type": "markdown", - "id": "cb8ed4f2", + "id": "7c82dd9e", "metadata": { "editable": true }, @@ -148,7 +148,7 @@ }, { "cell_type": "markdown", - "id": "d7c1c21b", + "id": "5bab8721", "metadata": { "editable": true }, @@ -160,7 +160,7 @@ }, { "cell_type": "markdown", - "id": "791e9c6d", + "id": "72273a4f", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "924b8081", + "id": "d4af03ea", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "949f6658", + "id": "c0d35853", "metadata": { "editable": true }, @@ -196,7 +196,7 @@ }, { "cell_type": "markdown", - "id": "46a4a83c", + "id": "db5ea827", "metadata": { "editable": true }, @@ -208,7 +208,7 @@ }, { "cell_type": "markdown", - "id": "ce019aa6", + "id": "a6b341ce", "metadata": { "editable": true }, @@ -230,7 +230,7 @@ }, { "cell_type": "markdown", - "id": "c662d90f", + "id": "db723709", "metadata": { "editable": true }, @@ -242,7 +242,7 @@ }, { "cell_type": "markdown", - "id": "7df45c97", + "id": "1e600268", "metadata": { "editable": true }, @@ -257,7 +257,7 @@ }, { "cell_type": "markdown", - "id": "f0741872", + "id": "b9452938", "metadata": { "editable": true }, @@ -269,7 +269,7 @@ }, { "cell_type": "markdown", - "id": "05530ccb", + "id": "53fea1e6", "metadata": { "editable": true }, @@ -281,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "3048f911", + "id": "ed10baad", "metadata": { "editable": true }, @@ -291,7 +291,7 @@ }, { "cell_type": "markdown", - "id": "9e3c2de1", + "id": "a4d0db96", "metadata": { "editable": true }, @@ -303,7 +303,7 @@ }, { "cell_type": "markdown", - "id": "491923c1", + "id": "fae3cadc", "metadata": { "editable": true }, @@ -313,7 +313,7 @@ }, { "cell_type": "markdown", - "id": "77d1dbe5", + "id": "e973c775", "metadata": { "editable": true }, @@ -325,7 +325,7 @@ }, { "cell_type": "markdown", - "id": "fb7e2323", + "id": "f88a33f4", "metadata": { "editable": true }, @@ -335,7 +335,7 @@ }, { "cell_type": "markdown", - "id": "c9cca57a", + "id": "5bc7214d", "metadata": { "editable": true }, @@ -347,7 +347,7 @@ }, { "cell_type": "markdown", - "id": "04e4b486", + "id": "561d4c91", "metadata": { "editable": true }, @@ -364,7 +364,7 @@ }, { "cell_type": "markdown", - "id": "2b3c22fb", + "id": "0eda2d1e", "metadata": { "editable": true }, @@ -372,25 +372,27 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the computation of the pseudoinverse of a given matrix $\\boldsymbol{X}$, that is" ] }, { "cell_type": "code", "execution_count": 1, - "id": "12637a17", + "id": "c95dbc32", "metadata": { "collapsed": false, "editable": true }, "outputs": [], "source": [ - "Ainv = np.linlag.pinv(A)" + "import numpy as np\n", + "X = np.array( [ [1,2,3],[2,4,5],[3,5,6]])\n", + "Xinv = np.linlag.pinv(X)" ] }, { "cell_type": "markdown", - "id": "904adaf8", + "id": "77079d25", "metadata": { "editable": true }, @@ -401,7 +403,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "6b5465dc", + "id": "cd0ecc44", "metadata": { "collapsed": false, "editable": true @@ -441,7 +443,7 @@ }, { "cell_type": "markdown", - "id": "a80dcee4", + "id": "41d896dc", "metadata": { "editable": true }, @@ -461,7 +463,7 @@ }, { "cell_type": "markdown", - "id": "0c4052d3", + "id": "f594852c", "metadata": { "editable": true }, @@ -473,7 +475,7 @@ }, { "cell_type": "markdown", - "id": "185d433f", + "id": "b78264f2", "metadata": { "editable": true }, @@ -484,7 +486,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "418f8797", + "id": "d05eff95", "metadata": { "collapsed": false, "editable": true @@ -518,7 +520,7 @@ }, { "cell_type": "markdown", - "id": "bf5051ea", + "id": "2fae32c6", "metadata": { "editable": true }, @@ -528,7 +530,7 @@ }, { "cell_type": "markdown", - "id": "202d1f90", + "id": "8631011a", "metadata": { "editable": true }, @@ -541,7 +543,7 @@ }, { "cell_type": "markdown", - "id": "df0a8718", + "id": "8e7701de", "metadata": { "editable": true }, @@ -553,7 +555,7 @@ }, { "cell_type": "markdown", - "id": "4ebd1d20", + "id": "aea7230b", "metadata": { "editable": true }, @@ -563,7 +565,7 @@ }, { "cell_type": "markdown", - "id": "40df3859", + "id": "c5029213", "metadata": { "editable": true }, @@ -576,7 +578,7 @@ }, { "cell_type": "markdown", - "id": "980001f5", + "id": "535ad70e", "metadata": { "editable": true }, @@ -586,7 +588,7 @@ }, { "cell_type": "markdown", - "id": "699b1198", + "id": "159ae526", "metadata": { "editable": true }, @@ -598,7 +600,7 @@ }, { "cell_type": "markdown", - "id": "ff01ef41", + "id": "2f25dfa4", "metadata": { "editable": true }, @@ -613,7 +615,7 @@ }, { "cell_type": "markdown", - "id": "d7b9188f", + "id": "9d187806", "metadata": { "editable": true }, @@ -626,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "778790dc", + "id": "6c692db5", "metadata": { "editable": true }, @@ -640,7 +642,7 @@ }, { "cell_type": "markdown", - "id": "38dd7428", + "id": "3404e0b4", "metadata": { "editable": true }, @@ -652,7 +654,7 @@ }, { "cell_type": "markdown", - "id": "6ed3deb0", + "id": "4bc49666", "metadata": { "editable": true }, @@ -662,7 +664,7 @@ }, { "cell_type": "markdown", - "id": "e0860754", + "id": "3f08dfa1", "metadata": { "editable": true }, @@ -675,7 +677,7 @@ }, { "cell_type": "markdown", - "id": "66751140", + "id": "30bcc0cd", "metadata": { "editable": true }, @@ -687,7 +689,7 @@ }, { "cell_type": "markdown", - "id": "e1fa2bdb", + "id": "d311a7d8", "metadata": { "editable": true }, @@ -699,7 +701,7 @@ }, { "cell_type": "markdown", - "id": "32131982", + "id": "32c46b73", "metadata": { "editable": true }, @@ -711,7 +713,7 @@ }, { "cell_type": "markdown", - "id": "c1979796", + "id": "3cceda87", "metadata": { "editable": true }, @@ -723,7 +725,7 @@ }, { "cell_type": "markdown", - "id": "d7eb56d7", + "id": "10a16675", "metadata": { "editable": true }, @@ -737,7 +739,7 @@ }, { "cell_type": "markdown", - "id": "7c42f326", + "id": "6dba20bf", "metadata": { "editable": true }, @@ -749,7 +751,7 @@ }, { "cell_type": "markdown", - "id": "6496e5c0", + "id": "9325f924", "metadata": { "editable": true }, @@ -759,7 +761,7 @@ }, { "cell_type": "markdown", - "id": "837f7c04", + "id": "745d61e9", "metadata": { "editable": true }, @@ -771,7 +773,7 @@ }, { "cell_type": "markdown", - "id": "55e9395f", + "id": "3866af64", "metadata": { "editable": true }, @@ -781,7 +783,7 @@ }, { "cell_type": "markdown", - "id": "a943f34a", + "id": "710de288", "metadata": { "editable": true }, @@ -793,7 +795,7 @@ }, { "cell_type": "markdown", - "id": "63a227a9", + "id": "5893fba0", "metadata": { "editable": true }, @@ -805,7 +807,7 @@ }, { "cell_type": "markdown", - "id": "8be2bd7b", + "id": "5ef8c59f", "metadata": { "editable": true }, @@ -815,7 +817,7 @@ }, { "cell_type": "markdown", - "id": "35509652", + "id": "79b257ae", "metadata": { "editable": true }, @@ -826,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "2fbe21d0", + "id": "46270996", "metadata": { "editable": true }, @@ -838,7 +840,7 @@ }, { "cell_type": "markdown", - "id": "71e986f8", + "id": "785aebe9", "metadata": { "editable": true }, @@ -854,7 +856,7 @@ }, { "cell_type": "markdown", - "id": "3a5e5f40", + "id": "443f3a3f", "metadata": { "editable": true }, @@ -867,7 +869,7 @@ }, { "cell_type": "markdown", - "id": "752b6ba2", + "id": "cfb8d4c0", "metadata": { "editable": true }, @@ -879,7 +881,7 @@ }, { "cell_type": "markdown", - "id": "4e724327", + "id": "2be1ca1f", "metadata": { "editable": true }, @@ -889,7 +891,7 @@ }, { "cell_type": "markdown", - "id": "b15c2c90", + "id": "9ed0aec2", "metadata": { "editable": true }, @@ -901,7 +903,7 @@ }, { "cell_type": "markdown", - "id": "82e70fa5", + "id": "943b1cd5", "metadata": { "editable": true }, @@ -911,7 +913,7 @@ }, { "cell_type": "markdown", - "id": "3394e2e7", + "id": "4a8a6bff", "metadata": { "editable": true }, @@ -923,7 +925,7 @@ }, { "cell_type": "markdown", - "id": "722ffa86", + "id": "b0371965", "metadata": { "editable": true }, @@ -935,7 +937,7 @@ }, { "cell_type": "markdown", - "id": "720ed773", + "id": "38e7b497", "metadata": { "editable": true }, @@ -951,7 +953,7 @@ }, { "cell_type": "markdown", - "id": "3b57864f", + "id": "682f90db", "metadata": { "editable": true }, @@ -963,7 +965,7 @@ }, { "cell_type": "markdown", - "id": "8f36ff27", + "id": "b5fb7771", "metadata": { "editable": true }, @@ -975,7 +977,7 @@ }, { "cell_type": "markdown", - "id": "70adc9ce", + "id": "0c228525", "metadata": { "editable": true }, @@ -985,7 +987,7 @@ }, { "cell_type": "markdown", - "id": "eb4a995f", + "id": "08a400af", "metadata": { "editable": true }, @@ -997,7 +999,7 @@ }, { "cell_type": "markdown", - "id": "e3cd4b92", + "id": "ec0808b1", "metadata": { "editable": true }, @@ -1007,7 +1009,7 @@ }, { "cell_type": "markdown", - "id": "bbf9b1de", + "id": "92d3d92f", "metadata": { "editable": true }, @@ -1019,7 +1021,7 @@ }, { "cell_type": "markdown", - "id": "e30eff48", + "id": "07486e29", "metadata": { "editable": true }, @@ -1036,7 +1038,7 @@ }, { "cell_type": "markdown", - "id": "9a8f8e2e", + "id": "9868d4f4", "metadata": { "editable": true }, @@ -1048,7 +1050,7 @@ }, { "cell_type": "markdown", - "id": "3cb7928c", + "id": "1793a225", "metadata": { "editable": true }, @@ -1060,7 +1062,7 @@ }, { "cell_type": "markdown", - "id": "b1c6c813", + "id": "c7d3fa52", "metadata": { "editable": true }, @@ -1070,7 +1072,7 @@ }, { "cell_type": "markdown", - "id": "da336775", + "id": "47894977", "metadata": { "editable": true }, @@ -1082,7 +1084,7 @@ }, { "cell_type": "markdown", - "id": "2e0de141", + "id": "d1a11dc8", "metadata": { "editable": true }, @@ -1092,7 +1094,7 @@ }, { "cell_type": "markdown", - "id": "40985ec4", + "id": "a6cf82de", "metadata": { "editable": true }, @@ -1104,7 +1106,7 @@ }, { "cell_type": "markdown", - "id": "31091d7b", + "id": "bbf7ccc7", "metadata": { "editable": true }, @@ -1114,7 +1116,7 @@ }, { "cell_type": "markdown", - "id": "e81d965e", + "id": "bc6dbb83", "metadata": { "editable": true }, @@ -1126,7 +1128,7 @@ }, { "cell_type": "markdown", - "id": "9984b3a4", + "id": "ef5a8292", "metadata": { "editable": true }, @@ -1136,7 +1138,7 @@ }, { "cell_type": "markdown", - "id": "21ec3768", + "id": "d77bf73a", "metadata": { "editable": true }, @@ -1151,7 +1153,7 @@ }, { "cell_type": "markdown", - "id": "6be0ec63", + "id": "265ce93f", "metadata": { "editable": true }, @@ -1163,7 +1165,7 @@ }, { "cell_type": "markdown", - "id": "44c7278c", + "id": "c19e720c", "metadata": { "editable": true }, @@ -1173,7 +1175,7 @@ }, { "cell_type": "markdown", - "id": "918a8cc3", + "id": "8cad2962", "metadata": { "editable": true }, @@ -1185,7 +1187,7 @@ }, { "cell_type": "markdown", - "id": "3949af51", + "id": "bf6316a2", "metadata": { "editable": true }, @@ -1197,7 +1199,7 @@ }, { "cell_type": "markdown", - "id": "aa856d18", + "id": "78413ab6", "metadata": { "editable": true }, @@ -1209,7 +1211,7 @@ }, { "cell_type": "markdown", - "id": "bd30d1d4", + "id": "a77ca41b", "metadata": { "editable": true }, @@ -1219,7 +1221,7 @@ }, { "cell_type": "markdown", - "id": "142de535", + "id": "67b3f67e", "metadata": { "editable": true }, @@ -1231,7 +1233,7 @@ }, { "cell_type": "markdown", - "id": "1efee0c3", + "id": "d3571c8b", "metadata": { "editable": true }, @@ -1243,7 +1245,7 @@ }, { "cell_type": "markdown", - "id": "a6db019a", + "id": "943f822e", "metadata": { "editable": true }, @@ -1255,7 +1257,7 @@ }, { "cell_type": "markdown", - "id": "0f38deda", + "id": "2ec0ba01", "metadata": { "editable": true }, @@ -1265,7 +1267,7 @@ }, { "cell_type": "markdown", - "id": "7f9738a5", + "id": "85bf4c9b", "metadata": { "editable": true }, @@ -1277,7 +1279,7 @@ }, { "cell_type": "markdown", - "id": "4e8460f8", + "id": "c81d05f4", "metadata": { "editable": true }, @@ -1287,7 +1289,7 @@ }, { "cell_type": "markdown", - "id": "ae49ddce", + "id": "ce36390d", "metadata": { "editable": true }, @@ -1301,7 +1303,7 @@ }, { "cell_type": "markdown", - "id": "1efb96d3", + "id": "a47dd697", "metadata": { "editable": true }, @@ -1311,7 +1313,7 @@ }, { "cell_type": "markdown", - "id": "1a030291", + "id": "15fe5a0c", "metadata": { "editable": true }, @@ -1323,7 +1325,7 @@ }, { "cell_type": "markdown", - "id": "af486213", + "id": "62d7f0e9", "metadata": { "editable": true }, @@ -1335,7 +1337,7 @@ }, { "cell_type": "markdown", - "id": "87e5470d", + "id": "f2d4576f", "metadata": { "editable": true }, @@ -1345,7 +1347,7 @@ }, { "cell_type": "markdown", - "id": "182f9f51", + "id": "fb2bc0ff", "metadata": { "editable": true }, @@ -1357,7 +1359,7 @@ }, { "cell_type": "markdown", - "id": "edc0ee3d", + "id": "807ac009", "metadata": { "editable": true }, @@ -1367,7 +1369,7 @@ }, { "cell_type": "markdown", - "id": "673c64a0", + "id": "fe72889b", "metadata": { "editable": true }, @@ -1379,7 +1381,7 @@ }, { "cell_type": "markdown", - "id": "c0554476", + "id": "48844e10", "metadata": { "editable": true }, @@ -1391,7 +1393,7 @@ }, { "cell_type": "markdown", - "id": "cc9c761b", + "id": "72a4bc4c", "metadata": { "editable": true }, @@ -1401,7 +1403,7 @@ }, { "cell_type": "markdown", - "id": "b5c84ed1", + "id": "cfe1db6f", "metadata": { "editable": true }, @@ -1413,7 +1415,7 @@ }, { "cell_type": "markdown", - "id": "a8d2ba86", + "id": "05b6e6f6", "metadata": { "editable": true }, @@ -1423,7 +1425,7 @@ }, { "cell_type": "markdown", - "id": "9e407326", + "id": "0e58f37a", "metadata": { "editable": true }, @@ -1435,7 +1437,7 @@ }, { "cell_type": "markdown", - "id": "36ce4758", + "id": "78593de5", "metadata": { "editable": true }, @@ -1447,7 +1449,7 @@ }, { "cell_type": "markdown", - "id": "8665e33b", + "id": "26b08a56", "metadata": { "editable": true }, @@ -1457,7 +1459,7 @@ }, { "cell_type": "markdown", - "id": "16dbcd82", + "id": "3df661a4", "metadata": { "editable": true }, @@ -1469,7 +1471,7 @@ }, { "cell_type": "markdown", - "id": "642f7768", + "id": "68460e68", "metadata": { "editable": true }, @@ -1482,7 +1484,7 @@ }, { "cell_type": "markdown", - "id": "1784c79c", + "id": "6c660112", "metadata": { "editable": true }, @@ -1494,7 +1496,7 @@ }, { "cell_type": "markdown", - "id": "72b7600f", + "id": "6ff54cf1", "metadata": { "editable": true }, @@ -1506,7 +1508,7 @@ }, { "cell_type": "markdown", - "id": "84ca3171", + "id": "f3f88f94", "metadata": { "editable": true }, @@ -1518,7 +1520,7 @@ }, { "cell_type": "markdown", - "id": "d31aa45e", + "id": "6918fc7c", "metadata": { "editable": true }, @@ -1528,7 +1530,7 @@ }, { "cell_type": "markdown", - "id": "1298ca6d", + "id": "1b2321a3", "metadata": { "editable": true }, @@ -1540,7 +1542,7 @@ }, { "cell_type": "markdown", - "id": "fa2455d6", + "id": "d5e21457", "metadata": { "editable": true }, @@ -1550,7 +1552,7 @@ }, { "cell_type": "markdown", - "id": "8cd50073", + "id": "e03c3617", "metadata": { "editable": true }, @@ -1562,7 +1564,7 @@ }, { "cell_type": "markdown", - "id": "1f7e12c3", + "id": "6c4d9a4a", "metadata": { "editable": true }, @@ -1572,7 +1574,7 @@ }, { "cell_type": "markdown", - "id": "a79821fd", + "id": "584bef2e", "metadata": { "editable": true }, @@ -1584,7 +1586,7 @@ }, { "cell_type": "markdown", - "id": "2ac13531", + "id": "87273242", "metadata": { "editable": true }, @@ -1594,7 +1596,7 @@ }, { "cell_type": "markdown", - "id": "eed25f68", + "id": "f7e8277f", "metadata": { "editable": true }, @@ -1607,7 +1609,7 @@ }, { "cell_type": "markdown", - "id": "f74d71ec", + "id": "e7167c28", "metadata": { "editable": true }, @@ -1619,7 +1621,7 @@ }, { "cell_type": "markdown", - "id": "f0797067", + "id": "6a3fa7b2", "metadata": { "editable": true }, @@ -1631,7 +1633,7 @@ }, { "cell_type": "markdown", - "id": "3cc5d5e0", + "id": "92c33067", "metadata": { "editable": true }, @@ -1641,7 +1643,7 @@ }, { "cell_type": "markdown", - "id": "04250e5c", + "id": "1b7be5cc", "metadata": { "editable": true }, @@ -1653,7 +1655,7 @@ }, { "cell_type": "markdown", - "id": "2b24d2fa", + "id": "9e2db51b", "metadata": { "editable": true }, @@ -1670,7 +1672,7 @@ }, { "cell_type": "markdown", - "id": "aa950760", + "id": "7ea5c023", "metadata": { "editable": true }, @@ -1682,7 +1684,7 @@ }, { "cell_type": "markdown", - "id": "475389ed", + "id": "efff1845", "metadata": { "editable": true }, @@ -1694,7 +1696,7 @@ }, { "cell_type": "markdown", - "id": "086558d9", + "id": "80496d8c", "metadata": { "editable": true }, @@ -1704,7 +1706,7 @@ }, { "cell_type": "markdown", - "id": "1a3b2ba5", + "id": "bb33463a", "metadata": { "editable": true }, @@ -1716,7 +1718,7 @@ }, { "cell_type": "markdown", - "id": "c928bc46", + "id": "f16b1457", "metadata": { "editable": true }, @@ -1726,7 +1728,7 @@ }, { "cell_type": "markdown", - "id": "809a58f7", + "id": "4bee5a43", "metadata": { "editable": true }, @@ -1738,7 +1740,7 @@ }, { "cell_type": "markdown", - "id": "3b937bf4", + "id": "17176990", "metadata": { "editable": true }, @@ -1748,7 +1750,7 @@ }, { "cell_type": "markdown", - "id": "6e5b2527", + "id": "ec2c8f34", "metadata": { "editable": true }, @@ -1760,7 +1762,7 @@ }, { "cell_type": "markdown", - "id": "c5f88a45", + "id": "5934144d", "metadata": { "editable": true }, @@ -1770,7 +1772,7 @@ }, { "cell_type": "markdown", - "id": "c57753b4", + "id": "fe097c51", "metadata": { "editable": true }, @@ -1785,7 +1787,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "392f218d", + "id": "92cd1361", "metadata": { "collapsed": false, "editable": true @@ -1847,7 +1849,7 @@ }, { "cell_type": "markdown", - "id": "dd134ab1", + "id": "9e4426f4", "metadata": { "editable": true }, @@ -1857,7 +1859,7 @@ }, { "cell_type": "markdown", - "id": "8d9a61db", + "id": "5f2b6935", "metadata": { "editable": true }, @@ -1868,7 +1870,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "afba7c4a", + "id": "409bbeac", "metadata": { "collapsed": false, "editable": true @@ -1935,7 +1937,7 @@ }, { "cell_type": "markdown", - "id": "4d63db0b", + "id": "67a24e77", "metadata": { "editable": true }, @@ -1946,7 +1948,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "81115144", + "id": "8da16a74", "metadata": { "collapsed": false, "editable": true @@ -2035,7 +2037,7 @@ }, { "cell_type": "markdown", - "id": "e076a968", + "id": "9d2563fa", "metadata": { "editable": true }, @@ -2045,7 +2047,7 @@ }, { "cell_type": "markdown", - "id": "db822c65", + "id": "1232037e", "metadata": { "editable": true }, @@ -2074,7 +2076,7 @@ }, { "cell_type": "markdown", - "id": "fc4a06b6", + "id": "e50205be", "metadata": { "editable": true }, @@ -2090,7 +2092,7 @@ }, { "cell_type": "markdown", - "id": "43299fb3", + "id": "70eccda9", "metadata": { "editable": true }, @@ -2109,7 +2111,7 @@ }, { "cell_type": "markdown", - "id": "6cba00c7", + "id": "e6bdcbf2", "metadata": { "editable": true }, @@ -2123,7 +2125,7 @@ }, { "cell_type": "markdown", - "id": "8c85f61d", + "id": "d74471b8", "metadata": { "editable": true }, @@ -2135,7 +2137,7 @@ }, { "cell_type": "markdown", - "id": "a1901c32", + "id": "f7b178d9", "metadata": { "editable": true }, @@ -2146,7 +2148,7 @@ }, { "cell_type": "markdown", - "id": "c59e8272", + "id": "8ea16c62", "metadata": { "editable": true }, @@ -2158,7 +2160,7 @@ }, { "cell_type": "markdown", - "id": "8cba4bae", + "id": "4a8d7bfd", "metadata": { "editable": true }, @@ -2170,7 +2172,7 @@ }, { "cell_type": "markdown", - "id": "5e18ebb5", + "id": "5367d943", "metadata": { "editable": true }, @@ -2186,7 +2188,7 @@ }, { "cell_type": "markdown", - "id": "10d86c58", + "id": "c7eefd85", "metadata": { "editable": true }, @@ -2197,7 +2199,7 @@ }, { "cell_type": "markdown", - "id": "efba1ba9", + "id": "27ce343a", "metadata": { "editable": true }, @@ -2220,7 +2222,7 @@ }, { "cell_type": "markdown", - "id": "a3eccdab", + "id": "c67ea20b", "metadata": { "editable": true }, @@ -2231,7 +2233,7 @@ }, { "cell_type": "markdown", - "id": "70b901b4", + "id": "b4da473a", "metadata": { "editable": true }, @@ -2243,7 +2245,7 @@ }, { "cell_type": "markdown", - "id": "752b72d2", + "id": "af812d38", "metadata": { "editable": true }, @@ -2255,7 +2257,7 @@ }, { "cell_type": "markdown", - "id": "108aba79", + "id": "9d3aaff5", "metadata": { "editable": true }, @@ -2269,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "65801f3b", + "id": "eebea8b7", "metadata": { "editable": true }, @@ -2300,7 +2302,7 @@ }, { "cell_type": "markdown", - "id": "faf8da78", + "id": "714f281a", "metadata": { "editable": true }, @@ -2322,7 +2324,7 @@ }, { "cell_type": "markdown", - "id": "1c71921e", + "id": "66015ca6", "metadata": { "editable": true }, @@ -2334,7 +2336,7 @@ }, { "cell_type": "markdown", - "id": "cfb57087", + "id": "267f01c0", "metadata": { "editable": true }, @@ -2347,7 +2349,7 @@ }, { "cell_type": "markdown", - "id": "be4ad159", + "id": "998de4b6", "metadata": { "editable": true }, @@ -2359,7 +2361,7 @@ }, { "cell_type": "markdown", - "id": "f5e11f36", + "id": "06a8d452", "metadata": { "editable": true }, @@ -2371,7 +2373,7 @@ }, { "cell_type": "markdown", - "id": "dbf2700f", + "id": "6cd3ed9b", "metadata": { "editable": true }, @@ -2383,7 +2385,7 @@ }, { "cell_type": "markdown", - "id": "0f6d760d", + "id": "70f9aa69", "metadata": { "editable": true }, @@ -2395,7 +2397,7 @@ }, { "cell_type": "markdown", - "id": "ce614956", + "id": "3bb18388", "metadata": { "editable": true }, @@ -2418,7 +2420,7 @@ }, { "cell_type": "markdown", - "id": "086419d8", + "id": "84b546af", "metadata": { "editable": true }, @@ -2430,7 +2432,7 @@ }, { "cell_type": "markdown", - "id": "a228dcf4", + "id": "0e90935e", "metadata": { "editable": true }, @@ -2443,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "f04f6b76", + "id": "b40b33f5", "metadata": { "editable": true }, @@ -2455,7 +2457,7 @@ }, { "cell_type": "markdown", - "id": "642936a8", + "id": "23e0c7ac", "metadata": { "editable": true }, @@ -2467,7 +2469,7 @@ }, { "cell_type": "markdown", - "id": "4f137023", + "id": "dcd4ed9c", "metadata": { "editable": true }, @@ -2479,7 +2481,7 @@ }, { "cell_type": "markdown", - "id": "48fdd9f7", + "id": "5c37adb5", "metadata": { "editable": true }, @@ -2490,7 +2492,7 @@ }, { "cell_type": "markdown", - "id": "ae36f857", + "id": "27e4eea9", "metadata": { "editable": true }, @@ -2502,7 +2504,7 @@ }, { "cell_type": "markdown", - "id": "d312d461", + "id": "5dd95110", "metadata": { "editable": true }, @@ -2513,7 +2515,7 @@ }, { "cell_type": "markdown", - "id": "59ea8b57", + "id": "bc7cfda7", "metadata": { "editable": true }, @@ -2525,7 +2527,7 @@ }, { "cell_type": "markdown", - "id": "b9be32b5", + "id": "2bf6dbf4", "metadata": { "editable": true }, @@ -2535,7 +2537,7 @@ }, { "cell_type": "markdown", - "id": "b378b2c9", + "id": "e8ff9386", "metadata": { "editable": true }, @@ -2566,7 +2568,7 @@ }, { "cell_type": "markdown", - "id": "6ddc5457", + "id": "cd665e76", "metadata": { "editable": true }, @@ -2578,7 +2580,7 @@ }, { "cell_type": "markdown", - "id": "bb0f5795", + "id": "144c5c88", "metadata": { "editable": true }, @@ -2590,7 +2592,7 @@ }, { "cell_type": "markdown", - "id": "c969ba54", + "id": "76c7c5c6", "metadata": { "editable": true }, @@ -2600,7 +2602,7 @@ }, { "cell_type": "markdown", - "id": "630a5421", + "id": "f78405b0", "metadata": { "editable": true }, @@ -2612,7 +2614,7 @@ }, { "cell_type": "markdown", - "id": "49969ea1", + "id": "dd7b69b5", "metadata": { "editable": true }, @@ -2622,7 +2624,7 @@ }, { "cell_type": "markdown", - "id": "93b3d19b", + "id": "a1430a20", "metadata": { "editable": true }, @@ -2634,7 +2636,7 @@ }, { "cell_type": "markdown", - "id": "f61d06bf", + "id": "dfcd0c7b", "metadata": { "editable": true }, @@ -2644,7 +2646,7 @@ }, { "cell_type": "markdown", - "id": "8c25aa29", + "id": "043d37a6", "metadata": { "editable": true }, @@ -2656,7 +2658,7 @@ }, { "cell_type": "markdown", - "id": "add3ac78", + "id": "28fe8edc", "metadata": { "editable": true }, @@ -2666,7 +2668,7 @@ }, { "cell_type": "markdown", - "id": "2cb62527", + "id": "b600d21c", "metadata": { "editable": true }, @@ -2686,7 +2688,7 @@ }, { "cell_type": "markdown", - "id": "3704ebb3", + "id": "d513cf2c", "metadata": { "editable": true }, @@ -2698,7 +2700,7 @@ }, { "cell_type": "markdown", - "id": "76f1d92c", + "id": "d18008b5", "metadata": { "editable": true }, @@ -2708,7 +2710,7 @@ }, { "cell_type": "markdown", - "id": "31ef87a3", + "id": "22f55ec4", "metadata": { "editable": true }, @@ -2720,7 +2722,7 @@ }, { "cell_type": "markdown", - "id": "9f87e53d", + "id": "29e1bff6", "metadata": { "editable": true }, @@ -2732,7 +2734,7 @@ }, { "cell_type": "markdown", - "id": "b465eba1", + "id": "333869a5", "metadata": { "editable": true }, @@ -2744,7 +2746,7 @@ }, { "cell_type": "markdown", - "id": "b522f15f", + "id": "bf8de993", "metadata": { "editable": true }, @@ -2756,7 +2758,7 @@ }, { "cell_type": "markdown", - "id": "a8bf634f", + "id": "1c5349d1", "metadata": { "editable": true }, @@ -2768,7 +2770,7 @@ }, { "cell_type": "markdown", - "id": "51e8bfb6", + "id": "dbede038", "metadata": { "editable": true }, @@ -2780,7 +2782,7 @@ }, { "cell_type": "markdown", - "id": "bfb3c814", + "id": "c1334a9c", "metadata": { "editable": true }, @@ -2792,7 +2794,7 @@ }, { "cell_type": "markdown", - "id": "dfff0c7c", + "id": "ac8dcf04", "metadata": { "editable": true }, @@ -2804,7 +2806,7 @@ }, { "cell_type": "markdown", - "id": "cad0634f", + "id": "1992e354", "metadata": { "editable": true }, @@ -2814,7 +2816,7 @@ }, { "cell_type": "markdown", - "id": "1de5e07a", + "id": "24e5179f", "metadata": { "editable": true }, @@ -2826,7 +2828,7 @@ }, { "cell_type": "markdown", - "id": "43f1714e", + "id": "42979e5a", "metadata": { "editable": true }, @@ -2836,7 +2838,7 @@ }, { "cell_type": "markdown", - "id": "166406ab", + "id": "86a8cd07", "metadata": { "editable": true }, @@ -2855,7 +2857,7 @@ }, { "cell_type": "markdown", - "id": "d547f8dd", + "id": "b9a4f9a9", "metadata": { "editable": true }, @@ -2876,7 +2878,7 @@ }, { "cell_type": "markdown", - "id": "f848f875", + "id": "888c64ec", "metadata": { "editable": true }, @@ -2888,7 +2890,7 @@ }, { "cell_type": "markdown", - "id": "8c506872", + "id": "5086b046", "metadata": { "editable": true }, @@ -2899,7 +2901,7 @@ }, { "cell_type": "markdown", - "id": "317558b2", + "id": "609ea2b0", "metadata": { "editable": true }, @@ -2912,7 +2914,7 @@ }, { "cell_type": "markdown", - "id": "8f8f8c5c", + "id": "687bfdf3", "metadata": { "editable": true }, @@ -2924,7 +2926,7 @@ }, { "cell_type": "markdown", - "id": "acb7607d", + "id": "c2af1d2d", "metadata": { "editable": true }, @@ -2934,7 +2936,7 @@ }, { "cell_type": "markdown", - "id": "c5866ceb", + "id": "b8ec8d93", "metadata": { "editable": true }, @@ -2946,7 +2948,7 @@ }, { "cell_type": "markdown", - "id": "e3648f6b", + "id": "48a6fb5b", "metadata": { "editable": true }, @@ -2956,7 +2958,7 @@ }, { "cell_type": "markdown", - "id": "0b2cef16", + "id": "ba37d4fd", "metadata": { "editable": true }, @@ -2968,7 +2970,7 @@ }, { "cell_type": "markdown", - "id": "94233096", + "id": "824f1789", "metadata": { "editable": true }, @@ -2978,7 +2980,7 @@ }, { "cell_type": "markdown", - "id": "91284f6d", + "id": "07f89fd5", "metadata": { "editable": true }, @@ -2995,7 +2997,7 @@ }, { "cell_type": "markdown", - "id": "3debb528", + "id": "8fcd73d9", "metadata": { "editable": true }, @@ -3014,7 +3016,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "a4cf2a42", + "id": "4402e113", "metadata": { "collapsed": false, "editable": true @@ -3088,7 +3090,7 @@ }, { "cell_type": "markdown", - "id": "26270da6", + "id": "b700e45e", "metadata": { "editable": true }, @@ -3098,7 +3100,7 @@ }, { "cell_type": "markdown", - "id": "d8f0e20a", + "id": "69659d9d", "metadata": { "editable": true }, @@ -3112,7 +3114,7 @@ }, { "cell_type": "markdown", - "id": "6c93bc4b", + "id": "80b423eb", "metadata": { "editable": true }, @@ -3124,7 +3126,7 @@ }, { "cell_type": "markdown", - "id": "8c8b6cc8", + "id": "de547949", "metadata": { "editable": true }, @@ -3134,7 +3136,7 @@ }, { "cell_type": "markdown", - "id": "f76e3a40", + "id": "99d9476a", "metadata": { "editable": true }, @@ -3146,7 +3148,7 @@ }, { "cell_type": "markdown", - "id": "af323d99", + "id": "aa5613ea", "metadata": { "editable": true }, @@ -3156,7 +3158,7 @@ }, { "cell_type": "markdown", - "id": "49a0988c", + "id": "7dea4e11", "metadata": { "editable": true }, @@ -3168,7 +3170,7 @@ }, { "cell_type": "markdown", - "id": "38568afe", + "id": "99ab1d57", "metadata": { "editable": true }, @@ -3178,7 +3180,7 @@ }, { "cell_type": "markdown", - "id": "c3d0eeb9", + "id": "170ee68b", "metadata": { "editable": true }, @@ -3190,7 +3192,7 @@ }, { "cell_type": "markdown", - "id": "1d0ba152", + "id": "3388d6a4", "metadata": { "editable": true }, @@ -3200,7 +3202,7 @@ }, { "cell_type": "markdown", - "id": "f936c1ce", + "id": "a5bdaa16", "metadata": { "editable": true }, @@ -3216,7 +3218,7 @@ }, { "cell_type": "markdown", - "id": "dd10cea6", + "id": "cbf8e7e6", "metadata": { "editable": true }, @@ -3228,7 +3230,7 @@ }, { "cell_type": "markdown", - "id": "2534eb1d", + "id": "52c9e48f", "metadata": { "editable": true }, @@ -3238,7 +3240,7 @@ }, { "cell_type": "markdown", - "id": "104af119", + "id": "1ae37a89", "metadata": { "editable": true }, @@ -3250,7 +3252,7 @@ }, { "cell_type": "markdown", - "id": "5e8e89cf", + "id": "dd8fee27", "metadata": { "editable": true }, @@ -3263,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "467910cd", + "id": "de603b44", "metadata": { "editable": true }, @@ -3275,7 +3277,7 @@ }, { "cell_type": "markdown", - "id": "16f15265", + "id": "3f08e4a0", "metadata": { "editable": true }, @@ -3285,7 +3287,7 @@ }, { "cell_type": "markdown", - "id": "1cd22464", + "id": "b65ee2f0", "metadata": { "editable": true }, @@ -3297,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "90676d1e", + "id": "6526b426", "metadata": { "editable": true }, @@ -3307,7 +3309,7 @@ }, { "cell_type": "markdown", - "id": "2082d057", + "id": "29190fe5", "metadata": { "editable": true }, @@ -3319,7 +3321,7 @@ }, { "cell_type": "markdown", - "id": "544b15d3", + "id": "71b1585c", "metadata": { "editable": true }, @@ -3331,7 +3333,7 @@ }, { "cell_type": "markdown", - "id": "592804c7", + "id": "e18baaa9", "metadata": { "editable": true }, @@ -3341,7 +3343,7 @@ }, { "cell_type": "markdown", - "id": "c11e6e06", + "id": "1c58b8db", "metadata": { "editable": true }, @@ -3353,7 +3355,7 @@ }, { "cell_type": "markdown", - "id": "fdc6e9b5", + "id": "238ee8c5", "metadata": { "editable": true }, @@ -3365,7 +3367,7 @@ }, { "cell_type": "markdown", - "id": "a0e8571d", + "id": "46389b82", "metadata": { "editable": true }, @@ -3377,7 +3379,7 @@ }, { "cell_type": "markdown", - "id": "84eca415", + "id": "0ecbb2d2", "metadata": { "editable": true }, @@ -3387,7 +3389,7 @@ }, { "cell_type": "markdown", - "id": "0924d8d4", + "id": "27368dfb", "metadata": { "editable": true }, @@ -3399,7 +3401,7 @@ }, { "cell_type": "markdown", - "id": "7270fef7", + "id": "27082f48", "metadata": { "editable": true },