diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle index a4e09dc13..f1a7a1fa9 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/week35.doctree b/doc/LectureNotes/_build/.doctrees/week35.doctree index 1c99d3ba1..4c8675ee6 100644 Binary files a/doc/LectureNotes/_build/.doctrees/week35.doctree and b/doc/LectureNotes/_build/.doctrees/week35.doctree differ diff --git a/doc/LectureNotes/_build/html/_sources/week35.ipynb b/doc/LectureNotes/_build/html/_sources/week35.ipynb index a6f7e09a8..5139bd065 100644 --- a/doc/LectureNotes/_build/html/_sources/week35.ipynb +++ b/doc/LectureNotes/_build/html/_sources/week35.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "c02138a8", + "id": "0b8a0ca4", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "919baf6d", + "id": "04b82164", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "4d5e7179", + "id": "b4569878", "metadata": { "editable": true }, @@ -49,7 +49,7 @@ }, { "cell_type": "markdown", - "id": "3b3f68ae", + "id": "8a651090", "metadata": { "editable": true }, @@ -58,14 +58,14 @@ "\n", "1. See lecture notes for week 35 at \n", "\n", - "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n", + "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n", "\n", - "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4" + "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)." ] }, { "cell_type": "markdown", - "id": "3dfade93", + "id": "5ed41979", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "e7c3b1f5", + "id": "bac89af1", "metadata": { "editable": true }, @@ -115,7 +115,7 @@ }, { "cell_type": "markdown", - "id": "4b7ace0a", + "id": "0c42ff8a", "metadata": { "editable": true }, @@ -127,7 +127,7 @@ }, { "cell_type": "markdown", - "id": "6fa41e5e", + "id": "7a45848d", "metadata": { "editable": true }, @@ -137,7 +137,7 @@ }, { "cell_type": "markdown", - "id": "86724624", + "id": "07ba01c4", "metadata": { "editable": true }, @@ -149,7 +149,7 @@ }, { "cell_type": "markdown", - "id": "f4e16a64", + "id": "335ae896", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "00f70fc7", + "id": "2e4fcb2a", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "859d4683", + "id": "12dc5709", "metadata": { "editable": true }, @@ -195,7 +195,7 @@ }, { "cell_type": "markdown", - "id": "ab473c08", + "id": "c3480a1c", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "a3ccaf3a", + "id": "90295c79", "metadata": { "editable": true }, @@ -219,7 +219,7 @@ }, { "cell_type": "markdown", - "id": "238df8ba", + "id": "219f5e30", "metadata": { "editable": true }, @@ -229,7 +229,7 @@ }, { "cell_type": "markdown", - "id": "e114bb77", + "id": "1abd132b", "metadata": { "editable": true }, @@ -241,7 +241,7 @@ }, { "cell_type": "markdown", - "id": "67833b96", + "id": "ec78644f", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "68d1debe", + "id": "04c74624", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "bd4914a7", + "id": "af25e1c6", "metadata": { "editable": true }, @@ -276,7 +276,7 @@ }, { "cell_type": "markdown", - "id": "9d6509bf", + "id": "2125067f", "metadata": { "editable": true }, @@ -288,7 +288,7 @@ }, { "cell_type": "markdown", - "id": "fa97bb74", + "id": "fce7cbce", "metadata": { "editable": true }, @@ -300,7 +300,7 @@ }, { "cell_type": "markdown", - "id": "fdc65871", + "id": "11ee37c4", "metadata": { "editable": true }, @@ -311,7 +311,7 @@ }, { "cell_type": "markdown", - "id": "6fe43d59", + "id": "d1dd61e2", "metadata": { "editable": true }, @@ -323,7 +323,7 @@ }, { "cell_type": "markdown", - "id": "7f0f5fc2", + "id": "f6ac3f1d", "metadata": { "editable": true }, @@ -342,7 +342,7 @@ }, { "cell_type": "markdown", - "id": "8c7ce425", + "id": "e62b9a8e", "metadata": { "editable": true }, @@ -355,7 +355,7 @@ }, { "cell_type": "markdown", - "id": "23cd06a6", + "id": "25564459", "metadata": { "editable": true }, @@ -365,7 +365,7 @@ }, { "cell_type": "markdown", - "id": "d40244fa", + "id": "4826fa0c", "metadata": { "editable": true }, @@ -377,7 +377,7 @@ }, { "cell_type": "markdown", - "id": "003814bc", + "id": "ad91cfea", "metadata": { "editable": true }, @@ -387,7 +387,7 @@ }, { "cell_type": "markdown", - "id": "765cdab4", + "id": "334c4b5c", "metadata": { "editable": true }, @@ -399,7 +399,7 @@ }, { "cell_type": "markdown", - "id": "c10530e6", + "id": "d37069d0", "metadata": { "editable": true }, @@ -409,7 +409,7 @@ }, { "cell_type": "markdown", - "id": "b1ce0925", + "id": "baf3d564", "metadata": { "editable": true }, @@ -421,7 +421,7 @@ }, { "cell_type": "markdown", - "id": "18e850a5", + "id": "50787505", "metadata": { "editable": true }, @@ -432,7 +432,7 @@ }, { "cell_type": "markdown", - "id": "9fe90791", + "id": "7f360392", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "320e1aef", + "id": "028cf8d2", "metadata": { "editable": true }, @@ -454,7 +454,7 @@ }, { "cell_type": "markdown", - "id": "b6ea67f8", + "id": "3b53ffda", "metadata": { "editable": true }, @@ -466,7 +466,7 @@ }, { "cell_type": "markdown", - "id": "f374c3b2", + "id": "65ff8127", "metadata": { "editable": true }, @@ -476,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "4a879ff3", + "id": "e1d9050c", "metadata": { "editable": true }, @@ -488,28 +488,27 @@ }, { "cell_type": "markdown", - "id": "71ca39b0", + "id": "8e9fce23", "metadata": { "editable": true }, "source": [ "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", - "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", - "in our case $p=5$ meaning that we end up with inverting a small\n", + "{\\mathbb{R}}^{p\\times p}$. In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n", "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", "matrices to invert. The methods discussed here and for many other\n", "supervised learning algorithms like classification with logistic\n", "regression or support vector machines, exhibit dimensionalities which\n", "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", - "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$. This is discussed on Thursday this week.\n", "\n", "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?" ] }, { "cell_type": "markdown", - "id": "0228f656", + "id": "a749fab8", "metadata": { "editable": true }, @@ -536,7 +535,7 @@ }, { "cell_type": "markdown", - "id": "1a6493a2", + "id": "6881afbf", "metadata": { "editable": true }, @@ -548,7 +547,7 @@ }, { "cell_type": "markdown", - "id": "48e5ce45", + "id": "9c2fb92f", "metadata": { "editable": true }, @@ -560,7 +559,7 @@ }, { "cell_type": "markdown", - "id": "2025325b", + "id": "d6d835d5", "metadata": { "editable": true }, @@ -576,7 +575,7 @@ }, { "cell_type": "markdown", - "id": "c4c51c01", + "id": "fed1c91b", "metadata": { "editable": true }, @@ -594,7 +593,7 @@ }, { "cell_type": "markdown", - "id": "5e0ba978", + "id": "7d1dfcf8", "metadata": { "editable": true }, @@ -606,7 +605,7 @@ }, { "cell_type": "markdown", - "id": "4fd652f8", + "id": "bc898e72", "metadata": { "editable": true }, @@ -618,7 +617,7 @@ }, { "cell_type": "markdown", - "id": "694f5db3", + "id": "0f280d98", "metadata": { "editable": true }, @@ -629,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "52a90b73", + "id": "e5f930a8", "metadata": { "editable": true }, @@ -641,7 +640,7 @@ }, { "cell_type": "markdown", - "id": "afab9d50", + "id": "0e0cb390", "metadata": { "editable": true }, @@ -651,7 +650,7 @@ }, { "cell_type": "markdown", - "id": "3dd5ebc2", + "id": "c9a5f03a", "metadata": { "editable": true }, @@ -663,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "d8571fd8", + "id": "932bc862", "metadata": { "editable": true }, @@ -677,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "50f37b40", + "id": "0ca4b0d9", "metadata": { "editable": true }, @@ -689,7 +688,7 @@ }, { "cell_type": "markdown", - "id": "b4e94db7", + "id": "63e2dad1", "metadata": { "editable": true }, @@ -701,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "328ce3c0", + "id": "64ecbf35", "metadata": { "editable": true }, @@ -713,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "3c549113", + "id": "92c17ad7", "metadata": { "editable": true }, @@ -723,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "62a61819", + "id": "2a69e7f9", "metadata": { "editable": true }, @@ -735,7 +734,7 @@ }, { "cell_type": "markdown", - "id": "0d8917d2", + "id": "9b5936bb", "metadata": { "editable": true }, @@ -745,7 +744,7 @@ }, { "cell_type": "markdown", - "id": "d27089b7", + "id": "5bfa3f58", "metadata": { "editable": true }, @@ -757,7 +756,7 @@ }, { "cell_type": "markdown", - "id": "efbc5228", + "id": "38584971", "metadata": { "editable": true }, @@ -771,7 +770,7 @@ }, { "cell_type": "markdown", - "id": "1539dfd3", + "id": "5eaa19f8", "metadata": { "editable": true }, @@ -783,7 +782,7 @@ }, { "cell_type": "markdown", - "id": "56942822", + "id": "3b257952", "metadata": { "editable": true }, @@ -795,7 +794,7 @@ }, { "cell_type": "markdown", - "id": "444a2229", + "id": "cd62fd4a", "metadata": { "editable": true }, @@ -807,7 +806,7 @@ }, { "cell_type": "markdown", - "id": "c19b8e9a", + "id": "63201b06", "metadata": { "editable": true }, @@ -817,7 +816,7 @@ }, { "cell_type": "markdown", - "id": "c3da7629", + "id": "44bcd980", "metadata": { "editable": true }, @@ -829,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "bc0bb616", + "id": "c69cb929", "metadata": { "editable": true }, @@ -839,7 +838,7 @@ }, { "cell_type": "markdown", - "id": "81f469c7", + "id": "d0209cbb", "metadata": { "editable": true }, @@ -851,7 +850,7 @@ }, { "cell_type": "markdown", - "id": "e7dbc066", + "id": "85c2a9b1", "metadata": { "editable": true }, @@ -861,7 +860,7 @@ }, { "cell_type": "markdown", - "id": "e15843da", + "id": "05ab900a", "metadata": { "editable": true }, @@ -873,7 +872,7 @@ }, { "cell_type": "markdown", - "id": "31a95917", + "id": "f8c87433", "metadata": { "editable": true }, @@ -885,7 +884,7 @@ }, { "cell_type": "markdown", - "id": "a3c00979", + "id": "d8d58be6", "metadata": { "editable": true }, @@ -897,7 +896,7 @@ }, { "cell_type": "markdown", - "id": "03321be3", + "id": "cd5493d2", "metadata": { "editable": true }, @@ -912,7 +911,7 @@ }, { "cell_type": "markdown", - "id": "59854eb8", + "id": "69c57766", "metadata": { "editable": true }, @@ -924,7 +923,7 @@ }, { "cell_type": "markdown", - "id": "1f475d05", + "id": "1592a99e", "metadata": { "editable": true }, @@ -934,7 +933,7 @@ }, { "cell_type": "markdown", - "id": "256476a9", + "id": "18378073", "metadata": { "editable": true }, @@ -946,7 +945,7 @@ }, { "cell_type": "markdown", - "id": "f23c2136", + "id": "0833d084", "metadata": { "editable": true }, @@ -956,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "afeafdcb", + "id": "07f35361", "metadata": { "editable": true }, @@ -968,7 +967,7 @@ }, { "cell_type": "markdown", - "id": "4ed3eead", + "id": "3a96219f", "metadata": { "editable": true }, @@ -978,7 +977,7 @@ }, { "cell_type": "markdown", - "id": "9674c619", + "id": "2f0e8ebf", "metadata": { "editable": true }, @@ -990,7 +989,7 @@ }, { "cell_type": "markdown", - "id": "82ae15ae", + "id": "323e987a", "metadata": { "editable": true }, @@ -1002,7 +1001,7 @@ }, { "cell_type": "markdown", - "id": "688a7168", + "id": "4cfaf887", "metadata": { "editable": true }, @@ -1014,7 +1013,7 @@ }, { "cell_type": "markdown", - "id": "01f03acf", + "id": "6288d6fe", "metadata": { "editable": true }, @@ -1024,7 +1023,7 @@ }, { "cell_type": "markdown", - "id": "a30bd0f2", + "id": "878ffa04", "metadata": { "editable": true }, @@ -1036,7 +1035,7 @@ }, { "cell_type": "markdown", - "id": "5593e084", + "id": "efe2818a", "metadata": { "editable": true }, @@ -1044,12 +1043,12 @@ "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n", "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n", "\n", - "The mean squared error is a scalar and if we use the results from the last example, we define a new vector" + "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector" ] }, { "cell_type": "markdown", - "id": "0ecc87b7", + "id": "016b7005", "metadata": { "editable": true }, @@ -1061,7 +1060,7 @@ }, { "cell_type": "markdown", - "id": "d87cf1be", + "id": "47ccce3f", "metadata": { "editable": true }, @@ -1071,7 +1070,7 @@ }, { "cell_type": "markdown", - "id": "c8ac2b85", + "id": "e25deaf7", "metadata": { "editable": true }, @@ -1083,7 +1082,7 @@ }, { "cell_type": "markdown", - "id": "2a1896de", + "id": "ce050c2b", "metadata": { "editable": true }, @@ -1093,7 +1092,7 @@ }, { "cell_type": "markdown", - "id": "f07be920", + "id": "b65cd933", "metadata": { "editable": true }, @@ -1105,7 +1104,7 @@ }, { "cell_type": "markdown", - "id": "8351f7a8", + "id": "6bf5c873", "metadata": { "editable": true }, @@ -1115,7 +1114,7 @@ }, { "cell_type": "markdown", - "id": "b321536f", + "id": "2ad8304d", "metadata": { "editable": true }, @@ -1127,7 +1126,7 @@ }, { "cell_type": "markdown", - "id": "7dea2f6e", + "id": "86e780ca", "metadata": { "editable": true }, @@ -1137,7 +1136,7 @@ }, { "cell_type": "markdown", - "id": "fc8335b3", + "id": "8b76d3f7", "metadata": { "editable": true }, @@ -1149,7 +1148,7 @@ }, { "cell_type": "markdown", - "id": "690c48cc", + "id": "3a1a52ac", "metadata": { "editable": true }, @@ -1159,7 +1158,7 @@ }, { "cell_type": "markdown", - "id": "dbf4e99e", + "id": "da435327", "metadata": { "editable": true }, @@ -1171,7 +1170,7 @@ }, { "cell_type": "markdown", - "id": "20066c64", + "id": "f44db4df", "metadata": { "editable": true }, @@ -1183,7 +1182,7 @@ }, { "cell_type": "markdown", - "id": "512705d5", + "id": "819c0e66", "metadata": { "editable": true }, @@ -1195,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "898589c9", + "id": "510a011e", "metadata": { "editable": true }, @@ -1207,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "04fdb8da", + "id": "fcd571ce", "metadata": { "editable": true }, @@ -1219,7 +1218,7 @@ }, { "cell_type": "markdown", - "id": "fb0d6555", + "id": "b2515b95", "metadata": { "editable": true }, @@ -1235,7 +1234,7 @@ }, { "cell_type": "markdown", - "id": "61ca107d", + "id": "1ccfd4c9", "metadata": { "editable": true }, @@ -1247,7 +1246,7 @@ }, { "cell_type": "markdown", - "id": "71be4e48", + "id": "6f868746", "metadata": { "editable": true }, @@ -1257,7 +1256,7 @@ }, { "cell_type": "markdown", - "id": "9fc474c3", + "id": "541d352f", "metadata": { "editable": true }, @@ -1269,7 +1268,7 @@ }, { "cell_type": "markdown", - "id": "23a27406", + "id": "92874dd6", "metadata": { "editable": true }, @@ -1287,7 +1286,7 @@ }, { "cell_type": "markdown", - "id": "9aa37656", + "id": "19159246", "metadata": { "editable": true }, @@ -1299,7 +1298,7 @@ }, { "cell_type": "markdown", - "id": "1458e79a", + "id": "ed39ed6b", "metadata": { "editable": true }, @@ -1311,7 +1310,7 @@ }, { "cell_type": "markdown", - "id": "c3fcfcdf", + "id": "8adde242", "metadata": { "editable": true }, @@ -1321,7 +1320,7 @@ }, { "cell_type": "markdown", - "id": "9f169fad", + "id": "f1e3f1c2", "metadata": { "editable": true }, @@ -1333,7 +1332,7 @@ }, { "cell_type": "markdown", - "id": "81e7723a", + "id": "39960290", "metadata": { "editable": true }, @@ -1343,7 +1342,7 @@ }, { "cell_type": "markdown", - "id": "b2551cd1", + "id": "db52e3e6", "metadata": { "editable": true }, @@ -1355,7 +1354,7 @@ }, { "cell_type": "markdown", - "id": "601003c8", + "id": "e3d6ac8b", "metadata": { "editable": true }, @@ -1365,7 +1364,7 @@ }, { "cell_type": "markdown", - "id": "1f2fdfa6", + "id": "e4b30664", "metadata": { "editable": true }, @@ -1379,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "ea1518ad", + "id": "4cbc0c2c", "metadata": { "editable": true }, @@ -1391,18 +1390,18 @@ }, { "cell_type": "markdown", - "id": "758b843b", + "id": "5eebcce6", "metadata": { "editable": true }, "source": [ - "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", - "$p\\times n$ matrix $\\boldsymbol{X}$." + "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n", + "$n\\times p$ matrix $\\boldsymbol{X}$." ] }, { "cell_type": "markdown", - "id": "24715db8", + "id": "47074a74", "metadata": { "editable": true }, @@ -1415,7 +1414,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "f8b57463", + "id": "ebc41f6f", "metadata": { "collapsed": false, "editable": true @@ -1423,6 +1422,18 @@ "outputs": [], "source": [ "# matrix inversion to find beta\n", + "# First we set up the data\n", + "import numpy as np\n", + "x = np.random.rand(100)\n", + "y = 2.0+5*x*x+0.1*np.random.randn(100)\n", + "# and then the design matrix X including the intercept\n", + "# The design matrix now as function of a fourth-order polynomial\n", + "X = np.zeros((len(x),5))\n", + "X[:,0] = 1.0\n", + "X[:,1] = x\n", + "X[:,2] = x**2\n", + "X[:,3] = x**3\n", + "X[:,4] = x**4\n", "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n", "# and then make the prediction\n", "ytilde = X @ beta" @@ -1430,7 +1441,7 @@ }, { "cell_type": "markdown", - "id": "70c97572", + "id": "8ea9846d", "metadata": { "editable": true }, @@ -1441,7 +1452,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "1aa95889", + "id": "e996841a", "metadata": { "collapsed": false, "editable": true @@ -1454,7 +1465,7 @@ }, { "cell_type": "markdown", - "id": "e250b2e9", + "id": "39f01129", "metadata": { "editable": true }, @@ -1468,7 +1479,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "a86bd577", + "id": "12e52de5", "metadata": { "collapsed": false, "editable": true @@ -1481,7 +1492,7 @@ }, { "cell_type": "markdown", - "id": "d6ccd7b8", + "id": "8a0cc628", "metadata": { "editable": true }, @@ -1492,7 +1503,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "fb6f7c19", + "id": "8c1508b7", "metadata": { "collapsed": false, "editable": true @@ -1504,7 +1515,7 @@ }, { "cell_type": "markdown", - "id": "eca18427", + "id": "a3f2f3b1", "metadata": { "editable": true }, @@ -1515,7 +1526,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "0fd2b39c", + "id": "334780f0", "metadata": { "collapsed": false, "editable": true @@ -1531,7 +1542,7 @@ }, { "cell_type": "markdown", - "id": "db3fe7d9", + "id": "1f7cfede", "metadata": { "editable": true }, @@ -1542,7 +1553,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "9f59c14e", + "id": "1579c506", "metadata": { "collapsed": false, "editable": true @@ -1556,7 +1567,7 @@ }, { "cell_type": "markdown", - "id": "2fdd5915", + "id": "88c6c130", "metadata": { "editable": true }, @@ -1577,7 +1588,7 @@ }, { "cell_type": "markdown", - "id": "9863e063", + "id": "af0baf7f", "metadata": { "editable": true }, @@ -1588,7 +1599,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "85fc6a78", + "id": "89aa39b5", "metadata": { "collapsed": false, "editable": true @@ -1641,7 +1652,7 @@ }, { "cell_type": "markdown", - "id": "6671297c", + "id": "1d86c024", "metadata": { "editable": true }, @@ -1652,7 +1663,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "810f32f7", + "id": "2ba16c25", "metadata": { "collapsed": false, "editable": true @@ -1677,7 +1688,7 @@ }, { "cell_type": "markdown", - "id": "1ca5e45d", + "id": "8e363f00", "metadata": { "editable": true }, @@ -1689,7 +1700,7 @@ }, { "cell_type": "markdown", - "id": "b6530670", + "id": "d5ef1be0", "metadata": { "editable": true }, @@ -1718,7 +1729,7 @@ }, { "cell_type": "markdown", - "id": "4be4158c", + "id": "7165f38b", "metadata": { "editable": true }, @@ -1743,7 +1754,7 @@ }, { "cell_type": "markdown", - "id": "aa6f9ff6", + "id": "baea18ad", "metadata": { "editable": true }, @@ -1763,7 +1774,7 @@ }, { "cell_type": "markdown", - "id": "856dee18", + "id": "830be0f9", "metadata": { "editable": true }, @@ -1790,7 +1801,7 @@ }, { "cell_type": "markdown", - "id": "3497598c", + "id": "72473f2c", "metadata": { "editable": true }, @@ -1803,7 +1814,7 @@ }, { "cell_type": "markdown", - "id": "fe24fd3f", + "id": "ad92637f", "metadata": { "editable": true }, @@ -1815,7 +1826,7 @@ }, { "cell_type": "markdown", - "id": "55bccc6e", + "id": "928978ac", "metadata": { "editable": true }, @@ -1826,7 +1837,7 @@ }, { "cell_type": "markdown", - "id": "e01c4d09", + "id": "31b09189", "metadata": { "editable": true }, @@ -1842,7 +1853,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "6098d5d9", + "id": "c09e81a9", "metadata": { "collapsed": false, "editable": true @@ -1876,7 +1887,7 @@ }, { "cell_type": "markdown", - "id": "653340bf", + "id": "8ffeb0f1", "metadata": { "editable": true }, @@ -1886,7 +1897,7 @@ }, { "cell_type": "markdown", - "id": "673c8917", + "id": "2b1f32da", "metadata": { "editable": true }, @@ -1901,7 +1912,7 @@ }, { "cell_type": "markdown", - "id": "4f69865b", + "id": "3c2f9703", "metadata": { "editable": true }, @@ -1913,7 +1924,7 @@ }, { "cell_type": "markdown", - "id": "f39433a0", + "id": "15382f87", "metadata": { "editable": true }, @@ -1923,7 +1934,7 @@ }, { "cell_type": "markdown", - "id": "f2f5cb76", + "id": "a04fe6a8", "metadata": { "editable": true }, @@ -1939,7 +1950,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "12dbcdcb", + "id": "de5a8426", "metadata": { "collapsed": false, "editable": true @@ -1956,7 +1967,7 @@ }, { "cell_type": "markdown", - "id": "38cb3698", + "id": "815c116c", "metadata": { "editable": true }, @@ -1969,7 +1980,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "a0cd549b", + "id": "c9049720", "metadata": { "collapsed": false, "editable": true @@ -2016,7 +2027,7 @@ }, { "cell_type": "markdown", - "id": "b4d2ab87", + "id": "c6fb36f0", "metadata": { "editable": true }, @@ -2027,7 +2038,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "9f5d7996", + "id": "d02afd53", "metadata": { "collapsed": false, "editable": true @@ -2130,7 +2141,7 @@ }, { "cell_type": "markdown", - "id": "5ea41ab9", + "id": "4d3d0315", "metadata": { "editable": true }, @@ -2158,7 +2169,7 @@ }, { "cell_type": "markdown", - "id": "ce0afdb4", + "id": "1bf26037", "metadata": { "editable": true }, @@ -2193,7 +2204,7 @@ }, { "cell_type": "markdown", - "id": "ad92f42b", + "id": "1a9303d9", "metadata": { "editable": true }, @@ -2207,7 +2218,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "fae66265", + "id": "add41c28", "metadata": { "collapsed": false, "editable": true @@ -2232,7 +2243,7 @@ }, { "cell_type": "markdown", - "id": "1bf672d8", + "id": "cbad62c2", "metadata": { "editable": true }, @@ -2248,7 +2259,7 @@ }, { "cell_type": "markdown", - "id": "30834557", + "id": "77b3f4a0", "metadata": { "editable": true }, @@ -2260,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "5c05a427", + "id": "2054bdc7", "metadata": { "editable": true }, @@ -2275,7 +2286,7 @@ }, { "cell_type": "markdown", - "id": "902391ce", + "id": "d7f64d04", "metadata": { "editable": true }, @@ -2287,7 +2298,7 @@ }, { "cell_type": "markdown", - "id": "a68a6228", + "id": "6ebe45a1", "metadata": { "editable": true }, @@ -2297,7 +2308,7 @@ }, { "cell_type": "markdown", - "id": "212cf295", + "id": "8cc21b6a", "metadata": { "editable": true }, @@ -2309,7 +2320,7 @@ }, { "cell_type": "markdown", - "id": "e7db36e1", + "id": "71b87ff9", "metadata": { "editable": true }, @@ -2319,7 +2330,7 @@ }, { "cell_type": "markdown", - "id": "5d7f590c", + "id": "b15ee5de", "metadata": { "editable": true }, @@ -2331,7 +2342,7 @@ }, { "cell_type": "markdown", - "id": "bbf7c89a", + "id": "56e808e5", "metadata": { "editable": true }, @@ -2344,7 +2355,7 @@ }, { "cell_type": "markdown", - "id": "05692612", + "id": "59b1fcf5", "metadata": { "editable": true }, @@ -2356,7 +2367,7 @@ }, { "cell_type": "markdown", - "id": "3dc53330", + "id": "fe9af8e4", "metadata": { "editable": true }, @@ -2366,7 +2377,7 @@ }, { "cell_type": "markdown", - "id": "562d655d", + "id": "39d0535d", "metadata": { "editable": true }, @@ -2378,7 +2389,7 @@ }, { "cell_type": "markdown", - "id": "528c869d", + "id": "b05e083e", "metadata": { "editable": true }, @@ -2388,7 +2399,7 @@ }, { "cell_type": "markdown", - "id": "82a20d9e", + "id": "255d5f87", "metadata": { "editable": true }, @@ -2400,7 +2411,7 @@ }, { "cell_type": "markdown", - "id": "5e37316e", + "id": "863f9c03", "metadata": { "editable": true }, @@ -2410,7 +2421,7 @@ }, { "cell_type": "markdown", - "id": "90c44e36", + "id": "fe1719c8", "metadata": { "editable": true }, @@ -2422,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "be72a7d3", + "id": "11bf9275", "metadata": { "editable": true }, @@ -2432,7 +2443,7 @@ }, { "cell_type": "markdown", - "id": "0a435bf5", + "id": "58a73e37", "metadata": { "editable": true }, @@ -2444,7 +2455,7 @@ }, { "cell_type": "markdown", - "id": "a67864cb", + "id": "2ee6e0ab", "metadata": { "editable": true }, @@ -2454,7 +2465,7 @@ }, { "cell_type": "markdown", - "id": "9d8602b7", + "id": "4aa55bc2", "metadata": { "editable": true }, @@ -2466,7 +2477,7 @@ }, { "cell_type": "markdown", - "id": "32caef0d", + "id": "e7dd951c", "metadata": { "editable": true }, @@ -2476,7 +2487,7 @@ }, { "cell_type": "markdown", - "id": "f24465a9", + "id": "b0b56f12", "metadata": { "editable": true }, @@ -2488,7 +2499,7 @@ }, { "cell_type": "markdown", - "id": "f64212da", + "id": "2fa4cfc7", "metadata": { "editable": true }, @@ -2500,7 +2511,7 @@ }, { "cell_type": "markdown", - "id": "8d0bf5ae", + "id": "cc6fe7e2", "metadata": { "editable": true }, @@ -2512,7 +2523,7 @@ }, { "cell_type": "markdown", - "id": "e8d36ff8", + "id": "d121682c", "metadata": { "editable": true }, @@ -2525,7 +2536,7 @@ }, { "cell_type": "markdown", - "id": "aea58aac", + "id": "94aef78e", "metadata": { "editable": true }, @@ -2537,7 +2548,7 @@ }, { "cell_type": "markdown", - "id": "bc4b6f91", + "id": "8d595a8d", "metadata": { "editable": true }, @@ -2547,7 +2558,7 @@ }, { "cell_type": "markdown", - "id": "349b03e2", + "id": "67d81f01", "metadata": { "editable": true }, @@ -2561,7 +2572,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "5cfc62db", + "id": "39da9a38", "metadata": { "collapsed": false, "editable": true @@ -2658,7 +2669,7 @@ }, { "cell_type": "markdown", - "id": "64c5fa5b", + "id": "257ce2f2", "metadata": { "editable": true }, @@ -2679,7 +2690,7 @@ }, { "cell_type": "markdown", - "id": "67496f35", + "id": "3769097f", "metadata": { "editable": true }, @@ -2691,7 +2702,7 @@ }, { "cell_type": "markdown", - "id": "ceae31e0", + "id": "82fc22ad", "metadata": { "editable": true }, @@ -2701,7 +2712,7 @@ }, { "cell_type": "markdown", - "id": "743c68c8", + "id": "dea71d52", "metadata": { "editable": true }, @@ -2713,7 +2724,7 @@ }, { "cell_type": "markdown", - "id": "4af68378", + "id": "113f6e87", "metadata": { "editable": true }, @@ -2723,7 +2734,7 @@ }, { "cell_type": "markdown", - "id": "8228af90", + "id": "1e3b78e5", "metadata": { "editable": true }, @@ -2735,7 +2746,7 @@ }, { "cell_type": "markdown", - "id": "0a4285bc", + "id": "d936547b", "metadata": { "editable": true }, @@ -2751,7 +2762,7 @@ }, { "cell_type": "markdown", - "id": "406af60a", + "id": "7c57b1e6", "metadata": { "editable": true }, @@ -2795,7 +2806,7 @@ }, { "cell_type": "markdown", - "id": "0f05d1e1", + "id": "41d26913", "metadata": { "editable": true }, @@ -2807,7 +2818,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "70f8a406", + "id": "bf2d2a51", "metadata": { "collapsed": false, "editable": true @@ -2823,7 +2834,7 @@ }, { "cell_type": "markdown", - "id": "10a6162c", + "id": "d7d26ad9", "metadata": { "editable": true }, @@ -2834,7 +2845,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "be236243", + "id": "7b725195", "metadata": { "collapsed": false, "editable": true @@ -2852,7 +2863,7 @@ }, { "cell_type": "markdown", - "id": "88a7376f", + "id": "4b578dbb", "metadata": { "editable": true }, @@ -2863,7 +2874,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "ec268779", + "id": "670a51a0", "metadata": { "collapsed": false, "editable": true @@ -2877,7 +2888,7 @@ }, { "cell_type": "markdown", - "id": "b45f3733", + "id": "ad921def", "metadata": { "editable": true }, @@ -2888,7 +2899,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "8b9423c6", + "id": "2d0a0044", "metadata": { "collapsed": false, "editable": true @@ -2901,7 +2912,7 @@ }, { "cell_type": "markdown", - "id": "1061cb1f", + "id": "8ec52805", "metadata": { "editable": true }, @@ -2912,7 +2923,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "00d79fad", + "id": "a7648df4", "metadata": { "collapsed": false, "editable": true @@ -2929,7 +2940,7 @@ }, { "cell_type": "markdown", - "id": "18871b23", + "id": "8e911148", "metadata": { "editable": true }, @@ -2940,7 +2951,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "abfd55f9", + "id": "4420c7f9", "metadata": { "collapsed": false, "editable": true @@ -2956,7 +2967,7 @@ }, { "cell_type": "markdown", - "id": "231f35c5", + "id": "b7580acd", "metadata": { "editable": true }, @@ -2967,7 +2978,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "5bb68d00", + "id": "2a0930b1", "metadata": { "collapsed": false, "editable": true @@ -2991,7 +3002,7 @@ }, { "cell_type": "markdown", - "id": "9c2f9f6b", + "id": "49423ebd", "metadata": { "editable": true }, @@ -3002,7 +3013,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "df3cc855", + "id": "5841e8ee", "metadata": { "collapsed": false, "editable": true @@ -3015,7 +3026,7 @@ }, { "cell_type": "markdown", - "id": "da67457b", + "id": "de1c7a1d", "metadata": { "editable": true }, @@ -3026,7 +3037,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "f9784866", + "id": "8d3ce90a", "metadata": { "collapsed": false, "editable": true @@ -3046,7 +3057,7 @@ }, { "cell_type": "markdown", - "id": "3429e67f", + "id": "b243c849", "metadata": { "editable": true }, @@ -3057,7 +3068,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "9ec7ba47", + "id": "83abda14", "metadata": { "collapsed": false, "editable": true @@ -3100,7 +3111,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "0c0be681", + "id": "c1bda3a1", "metadata": { "collapsed": false, "editable": true @@ -3115,7 +3126,7 @@ }, { "cell_type": "markdown", - "id": "ed12f12d", + "id": "494e99e1", "metadata": { "editable": true }, @@ -3125,7 +3136,7 @@ }, { "cell_type": "markdown", - "id": "50c91ade", + "id": "d496f2b6", "metadata": { "editable": true }, @@ -3139,7 +3150,7 @@ }, { "cell_type": "markdown", - "id": "682a501b", + "id": "617544ab", "metadata": { "editable": true }, @@ -3151,7 +3162,7 @@ }, { "cell_type": "markdown", - "id": "057bbc5d", + "id": "deb3a5c0", "metadata": { "editable": true }, @@ -3163,7 +3174,7 @@ }, { "cell_type": "markdown", - "id": "cddb899f", + "id": "1e29d04b", "metadata": { "editable": true }, @@ -3175,7 +3186,7 @@ }, { "cell_type": "markdown", - "id": "ab325ebc", + "id": "e7f06cd3", "metadata": { "editable": true }, @@ -3185,7 +3196,7 @@ }, { "cell_type": "markdown", - "id": "095ebc5b", + "id": "61fbb179", "metadata": { "editable": true }, @@ -3197,7 +3208,7 @@ }, { "cell_type": "markdown", - "id": "08e642f6", + "id": "715d0808", "metadata": { "editable": true }, @@ -3207,7 +3218,7 @@ }, { "cell_type": "markdown", - "id": "da5c1625", + "id": "e9af9b81", "metadata": { "editable": true }, @@ -3219,7 +3230,7 @@ }, { "cell_type": "markdown", - "id": "2bf04890", + "id": "ce0a6494", "metadata": { "editable": true }, @@ -3230,7 +3241,7 @@ }, { "cell_type": "markdown", - "id": "5ba172cc", + "id": "a9680e79", "metadata": { "editable": true }, @@ -3242,7 +3253,7 @@ }, { "cell_type": "markdown", - "id": "7d1f0b59", + "id": "77b2c7f1", "metadata": { "editable": true }, @@ -3254,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "467e185d", + "id": "00bc0981", "metadata": { "editable": true }, @@ -3264,7 +3275,7 @@ }, { "cell_type": "markdown", - "id": "74822363", + "id": "a5870049", "metadata": { "editable": true }, @@ -3276,7 +3287,7 @@ }, { "cell_type": "markdown", - "id": "5414a1d2", + "id": "8f53c1ca", "metadata": { "editable": true }, @@ -3288,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "08ce1402", + "id": "ac5e957f", "metadata": { "editable": true }, @@ -3298,7 +3309,7 @@ }, { "cell_type": "markdown", - "id": "7bbb949f", + "id": "b949e361", "metadata": { "editable": true }, @@ -3310,7 +3321,7 @@ }, { "cell_type": "markdown", - "id": "4007519d", + "id": "bf9ebc2b", "metadata": { "editable": true }, @@ -3320,7 +3331,7 @@ }, { "cell_type": "markdown", - "id": "45061e33", + "id": "8f15c266", "metadata": { "editable": true }, @@ -3332,7 +3343,7 @@ }, { "cell_type": "markdown", - "id": "f5afee48", + "id": "4f791b65", "metadata": { "editable": true }, @@ -3342,7 +3353,7 @@ }, { "cell_type": "markdown", - "id": "6a3d6440", + "id": "e8017b1c", "metadata": { "editable": true }, @@ -3382,7 +3393,7 @@ }, { "cell_type": "markdown", - "id": "ef8c39b7", + "id": "2996bc1f", "metadata": { "editable": true }, @@ -3399,7 +3410,7 @@ }, { "cell_type": "markdown", - "id": "6edd77ca", + "id": "2b6b0946", "metadata": { "editable": true }, @@ -3422,7 +3433,7 @@ }, { "cell_type": "markdown", - "id": "cba40ade", + "id": "93e7e61f", "metadata": { "editable": true }, @@ -3439,7 +3450,7 @@ }, { "cell_type": "markdown", - "id": "20f096a2", + "id": "fe419623", "metadata": { "editable": true }, @@ -3458,7 +3469,7 @@ }, { "cell_type": "markdown", - "id": "49f6f27c", + "id": "e2140a9e", "metadata": { "editable": true }, @@ -3469,7 +3480,7 @@ }, { "cell_type": "markdown", - "id": "577f3338", + "id": "ffcb729e", "metadata": { "editable": true }, @@ -3481,7 +3492,7 @@ }, { "cell_type": "markdown", - "id": "f456df81", + "id": "5140ddae", "metadata": { "editable": true }, @@ -3499,7 +3510,7 @@ }, { "cell_type": "markdown", - "id": "ed7db14f", + "id": "7bb98440", "metadata": { "editable": true }, @@ -3515,7 +3526,7 @@ }, { "cell_type": "markdown", - "id": "a876de90", + "id": "5611679a", "metadata": { "editable": true }, @@ -3527,7 +3538,7 @@ }, { "cell_type": "markdown", - "id": "3b2eee8d", + "id": "f32c0eb4", "metadata": { "editable": true }, @@ -3537,7 +3548,7 @@ }, { "cell_type": "markdown", - "id": "910d3f09", + "id": "415f02af", "metadata": { "editable": true }, @@ -3552,7 +3563,7 @@ }, { "cell_type": "markdown", - "id": "94bd4c00", + "id": "c28567b8", "metadata": { "editable": true }, @@ -3564,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "56d91a09", + "id": "936af9ce", "metadata": { "editable": true }, @@ -3574,7 +3585,7 @@ }, { "cell_type": "markdown", - "id": "8e59a435", + "id": "308f2aa4", "metadata": { "editable": true }, @@ -3586,7 +3597,7 @@ }, { "cell_type": "markdown", - "id": "9c2e6b9f", + "id": "d74a5600", "metadata": { "editable": true }, @@ -3596,7 +3607,7 @@ }, { "cell_type": "markdown", - "id": "e7188a39", + "id": "3d5f8482", "metadata": { "editable": true }, @@ -3608,7 +3619,7 @@ }, { "cell_type": "markdown", - "id": "daac715a", + "id": "b4eaa6e3", "metadata": { "editable": true }, @@ -3620,7 +3631,7 @@ }, { "cell_type": "markdown", - "id": "66322f83", + "id": "0b821498", "metadata": { "editable": true }, @@ -3635,7 +3646,7 @@ }, { "cell_type": "markdown", - "id": "a03be784", + "id": "d38da336", "metadata": { "editable": true }, @@ -3646,7 +3657,7 @@ }, { "cell_type": "markdown", - "id": "ff3964a4", + "id": "98b0ae03", "metadata": { "editable": true }, @@ -3666,7 +3677,7 @@ }, { "cell_type": "markdown", - "id": "04d411cc", + "id": "d4ba8f69", "metadata": { "editable": true }, @@ -3678,7 +3689,7 @@ }, { "cell_type": "markdown", - "id": "bf2ae91a", + "id": "52bb4456", "metadata": { "editable": true }, @@ -3688,7 +3699,7 @@ }, { "cell_type": "markdown", - "id": "843d113e", + "id": "6ec33a01", "metadata": { "editable": true }, @@ -3700,7 +3711,7 @@ }, { "cell_type": "markdown", - "id": "f59af5f8", + "id": "b39cac82", "metadata": { "editable": true }, @@ -3729,7 +3740,7 @@ }, { "cell_type": "markdown", - "id": "22f1ed07", + "id": "d2595a2e", "metadata": { "editable": true }, @@ -3756,7 +3767,7 @@ }, { "cell_type": "markdown", - "id": "69efb08d", + "id": "427fb028", "metadata": { "editable": true }, @@ -3767,7 +3778,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "640d3bdf", + "id": "0bbd4699", "metadata": { "collapsed": false, "editable": true @@ -3807,7 +3818,7 @@ }, { "cell_type": "markdown", - "id": "98d61790", + "id": "2430babb", "metadata": { "editable": true }, @@ -3824,7 +3835,7 @@ }, { "cell_type": "markdown", - "id": "95b1ffa6", + "id": "4761f74a", "metadata": { "editable": true }, @@ -3847,7 +3858,7 @@ }, { "cell_type": "markdown", - "id": "4605f137", + "id": "ffe52fdc", "metadata": { "editable": true }, @@ -3861,7 +3872,7 @@ }, { "cell_type": "markdown", - "id": "3125bc92", + "id": "c810c7eb", "metadata": { "editable": true }, @@ -3880,7 +3891,7 @@ }, { "cell_type": "markdown", - "id": "32cf3a9e", + "id": "fa17e262", "metadata": { "editable": true }, @@ -3890,7 +3901,7 @@ }, { "cell_type": "markdown", - "id": "573acb2d", + "id": "e7411ce2", "metadata": { "editable": true }, @@ -3902,7 +3913,7 @@ }, { "cell_type": "markdown", - "id": "22320566", + "id": "40d9144e", "metadata": { "editable": true }, @@ -3916,7 +3927,7 @@ }, { "cell_type": "markdown", - "id": "af0ee993", + "id": "28cd0268", "metadata": { "editable": true }, @@ -3928,7 +3939,7 @@ }, { "cell_type": "markdown", - "id": "0213ed87", + "id": "e72c1658", "metadata": { "editable": true }, @@ -3938,7 +3949,7 @@ }, { "cell_type": "markdown", - "id": "06266ee6", + "id": "9e8284ec", "metadata": { "editable": true }, @@ -3950,7 +3961,7 @@ }, { "cell_type": "markdown", - "id": "23e27fdf", + "id": "39c798c9", "metadata": { "editable": true }, @@ -3967,7 +3978,7 @@ }, { "cell_type": "markdown", - "id": "2539e582", + "id": "93cf6fd3", "metadata": { "editable": true }, @@ -3977,7 +3988,7 @@ }, { "cell_type": "markdown", - "id": "bef2cf95", + "id": "c99559d1", "metadata": { "editable": true }, @@ -3993,7 +4004,7 @@ }, { "cell_type": "markdown", - "id": "fe385753", + "id": "199187d7", "metadata": { "editable": true }, @@ -4003,7 +4014,7 @@ }, { "cell_type": "markdown", - "id": "cc270435", + "id": "164f8c17", "metadata": { "editable": true }, @@ -4019,7 +4030,7 @@ }, { "cell_type": "markdown", - "id": "31932192", + "id": "5e9fd974", "metadata": { "editable": true }, @@ -4029,7 +4040,7 @@ }, { "cell_type": "markdown", - "id": "a972a944", + "id": "8e07a82d", "metadata": { "editable": true }, @@ -4045,7 +4056,7 @@ }, { "cell_type": "markdown", - "id": "d6ef62ee", + "id": "b0bf5db6", "metadata": { "editable": true }, @@ -4055,7 +4066,7 @@ }, { "cell_type": "markdown", - "id": "61e10478", + "id": "aecfc56d", "metadata": { "editable": true }, @@ -4072,7 +4083,7 @@ }, { "cell_type": "markdown", - "id": "9e8580da", + "id": "73b4475e", "metadata": { "editable": true }, @@ -4084,7 +4095,7 @@ }, { "cell_type": "markdown", - "id": "0ac00991", + "id": "54057ff8", "metadata": { "editable": true }, @@ -4096,7 +4107,7 @@ }, { "cell_type": "markdown", - "id": "05f8073b", + "id": "ff974462", "metadata": { "editable": true }, @@ -4108,7 +4119,7 @@ }, { "cell_type": "markdown", - "id": "622a8443", + "id": "e5daa192", "metadata": { "editable": true }, @@ -4118,7 +4129,7 @@ }, { "cell_type": "markdown", - "id": "580e143c", + "id": "dc9f0f7b", "metadata": { "editable": true }, @@ -4130,7 +4141,7 @@ }, { "cell_type": "markdown", - "id": "30b75cea", + "id": "f9b10560", "metadata": { "editable": true }, @@ -4142,7 +4153,7 @@ }, { "cell_type": "markdown", - "id": "cdc103a7", + "id": "66a948a0", "metadata": { "editable": true }, @@ -4154,7 +4165,7 @@ }, { "cell_type": "markdown", - "id": "ea01abf6", + "id": "55d9dbd1", "metadata": { "editable": true }, @@ -4164,7 +4175,7 @@ }, { "cell_type": "markdown", - "id": "e4b60d5d", + "id": "c2b7da0d", "metadata": { "editable": true }, @@ -4176,7 +4187,7 @@ }, { "cell_type": "markdown", - "id": "deb0bbd3", + "id": "ff1c658b", "metadata": { "editable": true }, @@ -4186,7 +4197,7 @@ }, { "cell_type": "markdown", - "id": "f1f15de8", + "id": "3d7050b0", "metadata": { "editable": true }, @@ -4198,7 +4209,7 @@ }, { "cell_type": "markdown", - "id": "7c54c407", + "id": "dced3eca", "metadata": { "editable": true }, @@ -4212,7 +4223,7 @@ }, { "cell_type": "markdown", - "id": "31e31b03", + "id": "44ec48f0", "metadata": { "editable": true }, @@ -4224,7 +4235,7 @@ }, { "cell_type": "markdown", - "id": "7d1cef4f", + "id": "ef5a3989", "metadata": { "editable": true }, @@ -4236,7 +4247,7 @@ }, { "cell_type": "markdown", - "id": "aa8cedef", + "id": "6d7071a0", "metadata": { "editable": true }, @@ -4246,7 +4257,7 @@ }, { "cell_type": "markdown", - "id": "022623a1", + "id": "a37c0651", "metadata": { "editable": true }, @@ -4258,7 +4269,7 @@ }, { "cell_type": "markdown", - "id": "c9885020", + "id": "9cd481cc", "metadata": { "editable": true }, @@ -4269,7 +4280,7 @@ }, { "cell_type": "markdown", - "id": "f378f76e", + "id": "e8cd56b7", "metadata": { "editable": true }, @@ -4281,7 +4292,7 @@ }, { "cell_type": "markdown", - "id": "4add7e8a", + "id": "3a629b1b", "metadata": { "editable": true }, @@ -4291,7 +4302,7 @@ }, { "cell_type": "markdown", - "id": "51ea1a29", + "id": "2fbf21c7", "metadata": { "editable": true }, @@ -4303,7 +4314,7 @@ }, { "cell_type": "markdown", - "id": "54a250b6", + "id": "5951205a", "metadata": { "editable": true }, @@ -4313,7 +4324,7 @@ }, { "cell_type": "markdown", - "id": "ebaf68f5", + "id": "53caa2e2", "metadata": { "editable": true }, @@ -4325,7 +4336,7 @@ }, { "cell_type": "markdown", - "id": "5ff687e8", + "id": "64b903b4", "metadata": { "editable": true }, @@ -4336,7 +4347,7 @@ }, { "cell_type": "markdown", - "id": "ea61a360", + "id": "df6a66b3", "metadata": { "editable": true }, @@ -4348,7 +4359,7 @@ }, { "cell_type": "markdown", - "id": "bc168f34", + "id": "ffe2f537", "metadata": { "editable": true }, @@ -4366,7 +4377,7 @@ }, { "cell_type": "markdown", - "id": "39d89895", + "id": "3c291471", "metadata": { "editable": true }, @@ -4382,7 +4393,7 @@ }, { "cell_type": "markdown", - "id": "c229143d", + "id": "0a3f9a94", "metadata": { "editable": true }, @@ -4394,7 +4405,7 @@ }, { "cell_type": "markdown", - "id": "0a569456", + "id": "9e121ab9", "metadata": { "editable": true }, @@ -4406,7 +4417,7 @@ }, { "cell_type": "markdown", - "id": "0c0b9212", + "id": "8af98d79", "metadata": { "editable": true }, @@ -4418,7 +4429,7 @@ }, { "cell_type": "markdown", - "id": "b05a1b83", + "id": "f8df9cb1", "metadata": { "editable": true }, @@ -4431,7 +4442,7 @@ }, { "cell_type": "markdown", - "id": "d64818ce", + "id": "19bfd403", "metadata": { "editable": true }, @@ -4447,7 +4458,7 @@ }, { "cell_type": "markdown", - "id": "32df0f89", + "id": "c1ec9cb3", "metadata": { "editable": true }, @@ -4461,7 +4472,7 @@ }, { "cell_type": "markdown", - "id": "a9d9108b", + "id": "4346796e", "metadata": { "editable": true }, @@ -4471,7 +4482,7 @@ }, { "cell_type": "markdown", - "id": "c6d76715", + "id": "cda3d8a7", "metadata": { "editable": true }, @@ -4483,7 +4494,7 @@ }, { "cell_type": "markdown", - "id": "5cc35821", + "id": "7cf1e6d8", "metadata": { "editable": true }, @@ -4493,7 +4504,7 @@ }, { "cell_type": "markdown", - "id": "27c8eac0", + "id": "c783275d", "metadata": { "editable": true }, @@ -4505,7 +4516,7 @@ }, { "cell_type": "markdown", - "id": "49361e47", + "id": "dac43f32", "metadata": { "editable": true }, @@ -4515,7 +4526,7 @@ }, { "cell_type": "markdown", - "id": "5eb15920", + "id": "012a62f2", "metadata": { "editable": true }, @@ -4529,7 +4540,7 @@ }, { "cell_type": "markdown", - "id": "2160cb07", + "id": "f02bf402", "metadata": { "editable": true }, @@ -4546,7 +4557,7 @@ }, { "cell_type": "markdown", - "id": "a5deaa8e", + "id": "63805f7d", "metadata": { "editable": true }, @@ -4562,7 +4573,7 @@ }, { "cell_type": "markdown", - "id": "f6868141", + "id": "e84b7d85", "metadata": { "editable": true }, @@ -4574,7 +4585,7 @@ }, { "cell_type": "markdown", - "id": "9c17b156", + "id": "3ef0ad9c", "metadata": { "editable": true }, @@ -4587,7 +4598,7 @@ }, { "cell_type": "markdown", - "id": "1b9e8b7e", + "id": "5aa2f0f4", "metadata": { "editable": true }, @@ -4601,7 +4612,7 @@ }, { "cell_type": "markdown", - "id": "c00ac20a", + "id": "5cedd672", "metadata": { "editable": true }, @@ -4611,7 +4622,7 @@ }, { "cell_type": "markdown", - "id": "91729a44", + "id": "cdb4b214", "metadata": { "editable": true }, @@ -4624,7 +4635,7 @@ }, { "cell_type": "markdown", - "id": "62403241", + "id": "830c23c5", "metadata": { "editable": true }, @@ -4643,7 +4654,7 @@ }, { "cell_type": "markdown", - "id": "6d0de657", + "id": "7971e263", "metadata": { "editable": true }, @@ -4655,7 +4666,7 @@ }, { "cell_type": "markdown", - "id": "108adf6b", + "id": "aab14206", "metadata": { "editable": true }, @@ -4667,7 +4678,7 @@ }, { "cell_type": "markdown", - "id": "1b2ba4f7", + "id": "27b6b6fc", "metadata": { "editable": true }, @@ -4677,7 +4688,7 @@ }, { "cell_type": "markdown", - "id": "1711d54c", + "id": "3e702413", "metadata": { "editable": true }, @@ -4689,7 +4700,7 @@ }, { "cell_type": "markdown", - "id": "f88b0c88", + "id": "fd91d200", "metadata": { "editable": true }, @@ -4702,7 +4713,7 @@ }, { "cell_type": "markdown", - "id": "d1835d40", + "id": "35c6dc25", "metadata": { "editable": true }, @@ -4721,7 +4732,7 @@ }, { "cell_type": "markdown", - "id": "7d0f1648", + "id": "e733d668", "metadata": { "editable": true }, @@ -4731,7 +4742,7 @@ }, { "cell_type": "markdown", - "id": "d16ea5cf", + "id": "4feaa7d5", "metadata": { "editable": true }, @@ -4750,7 +4761,7 @@ }, { "cell_type": "markdown", - "id": "634130de", + "id": "2fd87051", "metadata": { "editable": true }, @@ -4768,7 +4779,7 @@ }, { "cell_type": "markdown", - "id": "cecf6c09", + "id": "8a8c7cb0", "metadata": { "editable": true }, @@ -4782,7 +4793,7 @@ }, { "cell_type": "markdown", - "id": "1703b000", + "id": "b5426cb7", "metadata": { "editable": true }, @@ -4797,7 +4808,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "5da5890f", + "id": "45d8c427", "metadata": { "collapsed": false, "editable": true @@ -4818,7 +4829,7 @@ }, { "cell_type": "markdown", - "id": "6b10b22a", + "id": "5cb5bc08", "metadata": { "editable": true }, @@ -4835,7 +4846,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "da699527", + "id": "b43fd84d", "metadata": { "collapsed": false, "editable": true @@ -4867,7 +4878,7 @@ }, { "cell_type": "markdown", - "id": "a40fb51e", + "id": "c303e01f", "metadata": { "editable": true }, @@ -4881,7 +4892,7 @@ }, { "cell_type": "markdown", - "id": "413db31e", + "id": "d56895dd", "metadata": { "editable": true }, @@ -4894,7 +4905,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "53927e5d", + "id": "cfda2184", "metadata": { "collapsed": false, "editable": true @@ -4919,7 +4930,7 @@ }, { "cell_type": "markdown", - "id": "74fc3107", + "id": "3cec050e", "metadata": { "editable": true }, @@ -4929,7 +4940,7 @@ }, { "cell_type": "markdown", - "id": "030881ab", + "id": "8afc4799", "metadata": { "editable": true }, @@ -4940,7 +4951,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "c107aa55", + "id": "1372e459", "metadata": { "collapsed": false, "editable": true @@ -4994,7 +5005,7 @@ }, { "cell_type": "markdown", - "id": "3f47378e", + "id": "aed87895", "metadata": { "editable": true }, @@ -5011,7 +5022,7 @@ }, { "cell_type": "markdown", - "id": "dba29ab3", + "id": "7e1e5b8d", "metadata": { "editable": true }, @@ -5023,7 +5034,7 @@ }, { "cell_type": "markdown", - "id": "0eb702bd", + "id": "b0d462fa", "metadata": { "editable": true }, @@ -5035,7 +5046,7 @@ }, { "cell_type": "markdown", - "id": "1d355153", + "id": "b8b65150", "metadata": { "editable": true }, @@ -5045,7 +5056,7 @@ }, { "cell_type": "markdown", - "id": "a2316749", + "id": "7e8f5c1f", "metadata": { "editable": true }, @@ -5062,7 +5073,7 @@ }, { "cell_type": "markdown", - "id": "8f6a654b", + "id": "1cc1396a", "metadata": { "editable": true }, @@ -5072,7 +5083,7 @@ }, { "cell_type": "markdown", - "id": "72291085", + "id": "5df6ba80", "metadata": { "editable": true }, @@ -5087,7 +5098,7 @@ }, { "cell_type": "markdown", - "id": "2b28a5fe", + "id": "59e9b6e2", "metadata": { "editable": true }, @@ -5097,7 +5108,7 @@ }, { "cell_type": "markdown", - "id": "49efe427", + "id": "ca2d3f9a", "metadata": { "editable": true }, @@ -5111,7 +5122,7 @@ }, { "cell_type": "markdown", - "id": "7edcb842", + "id": "be78dcbb", "metadata": { "editable": true }, @@ -5123,7 +5134,7 @@ }, { "cell_type": "markdown", - "id": "50ec44da", + "id": "0a45fb7f", "metadata": { "editable": true }, @@ -5135,7 +5146,7 @@ }, { "cell_type": "markdown", - "id": "f4010e4a", + "id": "8c6fdb24", "metadata": { "editable": true }, @@ -5147,7 +5158,7 @@ }, { "cell_type": "markdown", - "id": "c0e95eb2", + "id": "a419206b", "metadata": { "editable": true }, @@ -5157,7 +5168,7 @@ }, { "cell_type": "markdown", - "id": "46abefce", + "id": "765c4cbc", "metadata": { "editable": true }, @@ -5169,7 +5180,7 @@ }, { "cell_type": "markdown", - "id": "30544899", + "id": "760d01a8", "metadata": { "editable": true }, @@ -5179,7 +5190,7 @@ }, { "cell_type": "markdown", - "id": "feaeedbe", + "id": "a1f8eeea", "metadata": { "editable": true }, @@ -5196,7 +5207,7 @@ }, { "cell_type": "markdown", - "id": "3a2c0f0e", + "id": "b6fb003a", "metadata": { "editable": true }, @@ -5206,7 +5217,7 @@ }, { "cell_type": "markdown", - "id": "dc33d7ee", + "id": "75e17819", "metadata": { "editable": true }, @@ -5218,7 +5229,7 @@ }, { "cell_type": "markdown", - "id": "0ffb1c46", + "id": "bd29d1d6", "metadata": { "editable": true }, @@ -5228,7 +5239,7 @@ }, { "cell_type": "markdown", - "id": "6cf92025", + "id": "150b7923", "metadata": { "editable": true }, @@ -5240,7 +5251,7 @@ }, { "cell_type": "markdown", - "id": "c21c0c0a", + "id": "b4ffe151", "metadata": { "editable": true }, @@ -5254,7 +5265,7 @@ }, { "cell_type": "markdown", - "id": "48dbf778", + "id": "d3ba3779", "metadata": { "editable": true }, @@ -5266,7 +5277,7 @@ }, { "cell_type": "markdown", - "id": "3f8f57cc", + "id": "512596a9", "metadata": { "editable": true }, @@ -5288,7 +5299,7 @@ }, { "cell_type": "markdown", - "id": "199ea962", + "id": "7559f882", "metadata": { "editable": true }, @@ -5300,7 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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","19. Exercises week 34","Exercises week 35","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","20. 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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","19. Exercises week 34","Exercises week 35","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","20. Week 34: Introduction to the course, Logistics and Practicalities","Week 35: From Ordinary Linear Regression to Ridge and Lasso 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html index ac1b034c7..810325b71 100644 --- a/doc/LectureNotes/_build/html/week35.html +++ b/doc/LectureNotes/_build/html/week35.html @@ -1137,8 +1137,8 @@ doconce format html week35.do.txt --no_mako -->

Reading recommendations:¶

  1. See lecture notes for week 35 at https://compphysics.github.io/MachineLearning/doc/web/course.html

    2. -

  2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics

    3. -

  3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4

    4. +

  4. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)

    5. +

  5. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression).

@@ -1278,14 +1278,13 @@ will treat \(y_i\) as our exac \]

We note also that since our design matrix is defined as \(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\), the product \(\boldsymbol{X}^T\boldsymbol{X} \in -{\mathbb{R}}^{p\times p}\). In the above case we have that \(p \ll n\), -in our case \(p=5\) meaning that we end up with inverting a small +{\mathbb{R}}^{p\times p}\). In most cases we have that \(p \ll n\). In our example case below we have \(p=5\) meaning. We end up with inverting a small \(5\times 5\) matrix. This is a rather common situation, in many cases we end up with low-dimensional matrices to invert. The methods discussed here and for many other supervised learning algorithms like classification with logistic regression or support vector machines, exhibit dimensionalities which allow for the usage of direct linear algebra methods such as LU decomposition or Singular Value Decomposition (SVD) for finding the inverse of the matrix -\(\boldsymbol{X}^T\boldsymbol{X}\).

+\(\boldsymbol{X}^T\boldsymbol{X}\). This is discussed on Thursday this week.

Small question: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix \(\boldsymbol{X}^T\boldsymbol{X}\)? What kind of problems can we expect?

@@ -1453,7 +1452,7 @@ C(\boldsymbol{\beta})=\frac{1}{n}\left\{\left(\boldsymbol{y}-\boldsymbol{X}\bold \]

We note that the design matrix \(\boldsymbol{X}\) does not depend on the unknown parameters defined by the vector \(\boldsymbol{\beta}\). We are now interested in minimizing the cost function with respect to the unknown parameters \(\boldsymbol{\beta}\).

-

The mean squared error is a scalar and if we use the results from the last example, we define a new vector

+

The mean squared error is a scalar and if we use the results from example three above, we can define a new vector

\[ \boldsymbol{w}=\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, @@ -1553,8 +1552,8 @@ We assume our data can represented by a fourth-order polynomial. For the -

we have five predictors/features. The first is the intercept \(\beta_0\). The other terms are \(\beta_i\) with \(i=1,2,3,4\). Furthermore we have \(n\) entries for each predictor. It means that our design matrix is a -\(p\times n\) matrix \(\boldsymbol{X}\).

+

we have five predictors/features. The first is the intercept \(\beta_0\). The other terms are \(\beta_i\) with \(i=1,2,3,4\). Furthermore we have \(n\) entries for each predictor. It means that our design matrix is an +\(n\times p\) matrix \(\boldsymbol{X}\).

Own code for Ordinary Least Squares¶

@@ -1562,25 +1561,24 @@ We assume our data can represented by a fourth-order polynomial. For the 
# matrix inversion to find beta
+# First we set up the data
+import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
-
-
---------------------------------------------------------------------------
-NameError                                 Traceback (most recent call last)
-Input In [1], in <cell line: 2>()
-      1 # matrix inversion to find beta
-----> 2 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
-      3 # and then make the prediction
-      4 ytilde = X @ beta
-
-NameError: name 'np' is not defined
-
-
-

Alternatively, you can use the least squares functionality in Numpy as

@@ -1590,6 +1588,17 @@ We assume our data can represented by a fourth-order polynomial. For the
+
+
---------------------------------------------------------------------------
+NameError                                 Traceback (most recent call last)
+Input In [2], in <cell line: 1>()
+----> 1 fit = np.linalg.lstsq(X, Energies, rcond =None)[0]
+      2 ytildenp = np.dot(fit,X.T)
+
+NameError: name 'Energies' is not defined
+
+
+
diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb index 12c5d0cea..52784c8b7 100644 --- a/doc/LectureNotes/_build/jupyter_execute/week35.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/week35.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "c02138a8", + "id": "0b8a0ca4", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "919baf6d", + "id": "04b82164", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "4d5e7179", + "id": "b4569878", "metadata": { "editable": true }, @@ -49,7 +49,7 @@ }, { "cell_type": "markdown", - "id": "3b3f68ae", + "id": "8a651090", "metadata": { "editable": true }, @@ -58,14 +58,14 @@ "\n", "1. See lecture notes for week 35 at \n", "\n", - "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n", + "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n", "\n", - "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4" + "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)." ] }, { "cell_type": "markdown", - "id": "3dfade93", + "id": "5ed41979", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "e7c3b1f5", + "id": "bac89af1", "metadata": { "editable": true }, @@ -115,7 +115,7 @@ }, { "cell_type": "markdown", - "id": "4b7ace0a", + "id": "0c42ff8a", "metadata": { "editable": true }, @@ -127,7 +127,7 @@ }, { "cell_type": "markdown", - "id": "6fa41e5e", + "id": "7a45848d", "metadata": { "editable": true }, @@ -137,7 +137,7 @@ }, { "cell_type": "markdown", - "id": "86724624", + "id": "07ba01c4", "metadata": { "editable": true }, @@ -149,7 +149,7 @@ }, { "cell_type": "markdown", - "id": "f4e16a64", + "id": "335ae896", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "00f70fc7", + "id": "2e4fcb2a", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "859d4683", + "id": "12dc5709", "metadata": { "editable": true }, @@ -195,7 +195,7 @@ }, { "cell_type": "markdown", - "id": "ab473c08", + "id": "c3480a1c", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "a3ccaf3a", + "id": "90295c79", "metadata": { "editable": true }, @@ -219,7 +219,7 @@ }, { "cell_type": "markdown", - "id": "238df8ba", + "id": "219f5e30", "metadata": { "editable": true }, @@ -229,7 +229,7 @@ }, { "cell_type": "markdown", - "id": "e114bb77", + "id": "1abd132b", "metadata": { "editable": true }, @@ -241,7 +241,7 @@ }, { "cell_type": "markdown", - "id": "67833b96", + "id": "ec78644f", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "68d1debe", + "id": "04c74624", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "bd4914a7", + "id": "af25e1c6", "metadata": { "editable": true }, @@ -276,7 +276,7 @@ }, { "cell_type": "markdown", - "id": "9d6509bf", + "id": "2125067f", "metadata": { "editable": true }, @@ -288,7 +288,7 @@ }, { "cell_type": "markdown", - "id": "fa97bb74", + "id": "fce7cbce", "metadata": { "editable": true }, @@ -300,7 +300,7 @@ }, { "cell_type": "markdown", - "id": "fdc65871", + "id": "11ee37c4", "metadata": { "editable": true }, @@ -311,7 +311,7 @@ }, { "cell_type": "markdown", - "id": "6fe43d59", + "id": "d1dd61e2", "metadata": { "editable": true }, @@ -323,7 +323,7 @@ }, { "cell_type": "markdown", - "id": "7f0f5fc2", + "id": "f6ac3f1d", "metadata": { "editable": true }, @@ -342,7 +342,7 @@ }, { "cell_type": "markdown", - "id": "8c7ce425", + "id": "e62b9a8e", "metadata": { "editable": true }, @@ -355,7 +355,7 @@ }, { "cell_type": "markdown", - "id": "23cd06a6", + "id": "25564459", "metadata": { "editable": true }, @@ -365,7 +365,7 @@ }, { "cell_type": "markdown", - "id": "d40244fa", + "id": "4826fa0c", "metadata": { "editable": true }, @@ -377,7 +377,7 @@ }, { "cell_type": "markdown", - "id": "003814bc", + "id": "ad91cfea", "metadata": { "editable": true }, @@ -387,7 +387,7 @@ }, { "cell_type": "markdown", - "id": "765cdab4", + "id": "334c4b5c", "metadata": { "editable": true }, @@ -399,7 +399,7 @@ }, { "cell_type": "markdown", - "id": "c10530e6", + "id": "d37069d0", "metadata": { "editable": true }, @@ -409,7 +409,7 @@ }, { "cell_type": "markdown", - "id": "b1ce0925", + "id": "baf3d564", "metadata": { "editable": true }, @@ -421,7 +421,7 @@ }, { "cell_type": "markdown", - "id": "18e850a5", + "id": "50787505", "metadata": { "editable": true }, @@ -432,7 +432,7 @@ }, { "cell_type": "markdown", - "id": "9fe90791", + "id": "7f360392", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "320e1aef", + "id": "028cf8d2", "metadata": { "editable": true }, @@ -454,7 +454,7 @@ }, { "cell_type": "markdown", - "id": "b6ea67f8", + "id": "3b53ffda", "metadata": { "editable": true }, @@ -466,7 +466,7 @@ }, { "cell_type": "markdown", - "id": "f374c3b2", + "id": "65ff8127", "metadata": { "editable": true }, @@ -476,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "4a879ff3", + "id": "e1d9050c", "metadata": { "editable": true }, @@ -488,28 +488,27 @@ }, { "cell_type": "markdown", - "id": "71ca39b0", + "id": "8e9fce23", "metadata": { "editable": true }, "source": [ "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", - "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", - "in our case $p=5$ meaning that we end up with inverting a small\n", + "{\\mathbb{R}}^{p\\times p}$. In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n", "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", "matrices to invert. The methods discussed here and for many other\n", "supervised learning algorithms like classification with logistic\n", "regression or support vector machines, exhibit dimensionalities which\n", "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", - "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$. This is discussed on Thursday this week.\n", "\n", "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?" ] }, { "cell_type": "markdown", - "id": "0228f656", + "id": "a749fab8", "metadata": { "editable": true }, @@ -536,7 +535,7 @@ }, { "cell_type": "markdown", - "id": "1a6493a2", + "id": "6881afbf", "metadata": { "editable": true }, @@ -548,7 +547,7 @@ }, { "cell_type": "markdown", - "id": "48e5ce45", + "id": "9c2fb92f", "metadata": { "editable": true }, @@ -560,7 +559,7 @@ }, { "cell_type": "markdown", - "id": "2025325b", + "id": "d6d835d5", "metadata": { "editable": true }, @@ -576,7 +575,7 @@ }, { "cell_type": "markdown", - "id": "c4c51c01", + "id": "fed1c91b", "metadata": { "editable": true }, @@ -594,7 +593,7 @@ }, { "cell_type": "markdown", - "id": "5e0ba978", + "id": "7d1dfcf8", "metadata": { "editable": true }, @@ -606,7 +605,7 @@ }, { "cell_type": "markdown", - "id": "4fd652f8", + "id": "bc898e72", "metadata": { "editable": true }, @@ -618,7 +617,7 @@ }, { "cell_type": "markdown", - "id": "694f5db3", + "id": "0f280d98", "metadata": { "editable": true }, @@ -629,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "52a90b73", + "id": "e5f930a8", "metadata": { "editable": true }, @@ -641,7 +640,7 @@ }, { "cell_type": "markdown", - "id": "afab9d50", + "id": "0e0cb390", "metadata": { "editable": true }, @@ -651,7 +650,7 @@ }, { "cell_type": "markdown", - "id": "3dd5ebc2", + "id": "c9a5f03a", "metadata": { "editable": true }, @@ -663,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "d8571fd8", + "id": "932bc862", "metadata": { "editable": true }, @@ -677,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "50f37b40", + "id": "0ca4b0d9", "metadata": { "editable": true }, @@ -689,7 +688,7 @@ }, { "cell_type": "markdown", - "id": "b4e94db7", + "id": "63e2dad1", "metadata": { "editable": true }, @@ -701,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "328ce3c0", + "id": "64ecbf35", "metadata": { "editable": true }, @@ -713,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "3c549113", + "id": "92c17ad7", "metadata": { "editable": true }, @@ -723,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "62a61819", + "id": "2a69e7f9", "metadata": { "editable": true }, @@ -735,7 +734,7 @@ }, { "cell_type": "markdown", - "id": "0d8917d2", + "id": "9b5936bb", "metadata": { "editable": true }, @@ -745,7 +744,7 @@ }, { "cell_type": "markdown", - "id": "d27089b7", + "id": "5bfa3f58", "metadata": { "editable": true }, @@ -757,7 +756,7 @@ }, { "cell_type": "markdown", - "id": "efbc5228", + "id": "38584971", "metadata": { "editable": true }, @@ -771,7 +770,7 @@ }, { "cell_type": "markdown", - "id": "1539dfd3", + "id": "5eaa19f8", "metadata": { "editable": true }, @@ -783,7 +782,7 @@ }, { "cell_type": "markdown", - "id": "56942822", + "id": "3b257952", "metadata": { "editable": true }, @@ -795,7 +794,7 @@ }, { "cell_type": "markdown", - "id": "444a2229", + "id": "cd62fd4a", "metadata": { "editable": true }, @@ -807,7 +806,7 @@ }, { "cell_type": "markdown", - "id": "c19b8e9a", + "id": "63201b06", "metadata": { "editable": true }, @@ -817,7 +816,7 @@ }, { "cell_type": "markdown", - "id": "c3da7629", + "id": "44bcd980", "metadata": { "editable": true }, @@ -829,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "bc0bb616", + "id": "c69cb929", "metadata": { "editable": true }, @@ -839,7 +838,7 @@ }, { "cell_type": "markdown", - "id": "81f469c7", + "id": "d0209cbb", "metadata": { "editable": true }, @@ -851,7 +850,7 @@ }, { "cell_type": "markdown", - "id": "e7dbc066", + "id": "85c2a9b1", "metadata": { "editable": true }, @@ -861,7 +860,7 @@ }, { "cell_type": "markdown", - "id": "e15843da", + "id": "05ab900a", "metadata": { "editable": true }, @@ -873,7 +872,7 @@ }, { "cell_type": "markdown", - "id": "31a95917", + "id": "f8c87433", "metadata": { "editable": true }, @@ -885,7 +884,7 @@ }, { "cell_type": "markdown", - "id": "a3c00979", + "id": "d8d58be6", "metadata": { "editable": true }, @@ -897,7 +896,7 @@ }, { "cell_type": "markdown", - "id": "03321be3", + "id": "cd5493d2", "metadata": { "editable": true }, @@ -912,7 +911,7 @@ }, { "cell_type": "markdown", - "id": "59854eb8", + "id": "69c57766", "metadata": { "editable": true }, @@ -924,7 +923,7 @@ }, { "cell_type": "markdown", - "id": "1f475d05", + "id": "1592a99e", "metadata": { "editable": true }, @@ -934,7 +933,7 @@ }, { "cell_type": "markdown", - "id": "256476a9", + "id": "18378073", "metadata": { "editable": true }, @@ -946,7 +945,7 @@ }, { "cell_type": "markdown", - "id": "f23c2136", + "id": "0833d084", "metadata": { "editable": true }, @@ -956,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "afeafdcb", + "id": "07f35361", "metadata": { "editable": true }, @@ -968,7 +967,7 @@ }, { "cell_type": "markdown", - "id": "4ed3eead", + "id": "3a96219f", "metadata": { "editable": true }, @@ -978,7 +977,7 @@ }, { "cell_type": "markdown", - "id": "9674c619", + "id": "2f0e8ebf", "metadata": { "editable": true }, @@ -990,7 +989,7 @@ }, { "cell_type": "markdown", - "id": "82ae15ae", + "id": "323e987a", "metadata": { "editable": true }, @@ -1002,7 +1001,7 @@ }, { "cell_type": "markdown", - "id": "688a7168", + "id": "4cfaf887", "metadata": { "editable": true }, @@ -1014,7 +1013,7 @@ }, { "cell_type": "markdown", - "id": "01f03acf", + "id": "6288d6fe", "metadata": { "editable": true }, @@ -1024,7 +1023,7 @@ }, { "cell_type": "markdown", - "id": "a30bd0f2", + "id": "878ffa04", "metadata": { "editable": true }, @@ -1036,7 +1035,7 @@ }, { "cell_type": "markdown", - "id": "5593e084", + "id": "efe2818a", "metadata": { "editable": true }, @@ -1044,12 +1043,12 @@ "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n", "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n", "\n", - "The mean squared error is a scalar and if we use the results from the last example, we define a new vector" + "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector" ] }, { "cell_type": "markdown", - "id": "0ecc87b7", + "id": "016b7005", "metadata": { "editable": true }, @@ -1061,7 +1060,7 @@ }, { "cell_type": "markdown", - "id": "d87cf1be", + "id": "47ccce3f", "metadata": { "editable": true }, @@ -1071,7 +1070,7 @@ }, { "cell_type": "markdown", - "id": "c8ac2b85", + "id": "e25deaf7", "metadata": { "editable": true }, @@ -1083,7 +1082,7 @@ }, { "cell_type": "markdown", - "id": "2a1896de", + "id": "ce050c2b", "metadata": { "editable": true }, @@ -1093,7 +1092,7 @@ }, { "cell_type": "markdown", - "id": "f07be920", + "id": "b65cd933", "metadata": { "editable": true }, @@ -1105,7 +1104,7 @@ }, { "cell_type": "markdown", - "id": "8351f7a8", + "id": "6bf5c873", "metadata": { "editable": true }, @@ -1115,7 +1114,7 @@ }, { "cell_type": "markdown", - "id": "b321536f", + "id": "2ad8304d", "metadata": { "editable": true }, @@ -1127,7 +1126,7 @@ }, { "cell_type": "markdown", - "id": "7dea2f6e", + "id": "86e780ca", "metadata": { "editable": true }, @@ -1137,7 +1136,7 @@ }, { "cell_type": "markdown", - "id": "fc8335b3", + "id": "8b76d3f7", "metadata": { "editable": true }, @@ -1149,7 +1148,7 @@ }, { "cell_type": "markdown", - "id": "690c48cc", + "id": "3a1a52ac", "metadata": { "editable": true }, @@ -1159,7 +1158,7 @@ }, { "cell_type": "markdown", - "id": "dbf4e99e", + "id": "da435327", "metadata": { "editable": true }, @@ -1171,7 +1170,7 @@ }, { "cell_type": "markdown", - "id": "20066c64", + "id": "f44db4df", "metadata": { "editable": true }, @@ -1183,7 +1182,7 @@ }, { "cell_type": "markdown", - "id": "512705d5", + "id": "819c0e66", "metadata": { "editable": true }, @@ -1195,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "898589c9", + "id": "510a011e", "metadata": { "editable": true }, @@ -1207,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "04fdb8da", + "id": "fcd571ce", "metadata": { "editable": true }, @@ -1219,7 +1218,7 @@ }, { "cell_type": "markdown", - "id": "fb0d6555", + "id": "b2515b95", "metadata": { "editable": true }, @@ -1235,7 +1234,7 @@ }, { "cell_type": "markdown", - "id": "61ca107d", + "id": "1ccfd4c9", "metadata": { "editable": true }, @@ -1247,7 +1246,7 @@ }, { "cell_type": "markdown", - "id": "71be4e48", + "id": "6f868746", "metadata": { "editable": true }, @@ -1257,7 +1256,7 @@ }, { "cell_type": "markdown", - "id": "9fc474c3", + "id": "541d352f", "metadata": { "editable": true }, @@ -1269,7 +1268,7 @@ }, { "cell_type": "markdown", - "id": "23a27406", + "id": "92874dd6", "metadata": { "editable": true }, @@ -1287,7 +1286,7 @@ }, { "cell_type": "markdown", - "id": "9aa37656", + "id": "19159246", "metadata": { "editable": true }, @@ -1299,7 +1298,7 @@ }, { "cell_type": "markdown", - "id": "1458e79a", + "id": "ed39ed6b", "metadata": { "editable": true }, @@ -1311,7 +1310,7 @@ }, { "cell_type": "markdown", - "id": "c3fcfcdf", + "id": "8adde242", "metadata": { "editable": true }, @@ -1321,7 +1320,7 @@ }, { "cell_type": "markdown", - "id": "9f169fad", + "id": "f1e3f1c2", "metadata": { "editable": true }, @@ -1333,7 +1332,7 @@ }, { "cell_type": "markdown", - "id": "81e7723a", + "id": "39960290", "metadata": { "editable": true }, @@ -1343,7 +1342,7 @@ }, { "cell_type": "markdown", - "id": "b2551cd1", + "id": "db52e3e6", "metadata": { "editable": true }, @@ -1355,7 +1354,7 @@ }, { "cell_type": "markdown", - "id": "601003c8", + "id": "e3d6ac8b", "metadata": { "editable": true }, @@ -1365,7 +1364,7 @@ }, { "cell_type": "markdown", - "id": "1f2fdfa6", + "id": "e4b30664", "metadata": { "editable": true }, @@ -1379,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "ea1518ad", + "id": "4cbc0c2c", "metadata": { "editable": true }, @@ -1391,18 +1390,18 @@ }, { "cell_type": "markdown", - "id": "758b843b", + "id": "5eebcce6", "metadata": { "editable": true }, "source": [ - "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", - "$p\\times n$ matrix $\\boldsymbol{X}$." + "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n", + "$n\\times p$ matrix $\\boldsymbol{X}$." ] }, { "cell_type": "markdown", - "id": "24715db8", + "id": "47074a74", "metadata": { "editable": true }, @@ -1415,26 +1414,26 @@ { "cell_type": "code", "execution_count": 1, - "id": "f8b57463", + "id": "ebc41f6f", "metadata": { "collapsed": false, "editable": true }, - "outputs": [ - { - "ename": "NameError", - "evalue": "name 'np' is not defined", - "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[1;32m 1\u001b[0m \u001b[38;5;66;03m# matrix inversion to find beta\u001b[39;00m\n\u001b[0;32m----> 2\u001b[0m beta \u001b[38;5;241m=\u001b[39m (\u001b[43mnp\u001b[49m\u001b[38;5;241m.\u001b[39mlinalg\u001b[38;5;241m.\u001b[39minv(X\u001b[38;5;241m.\u001b[39mT \u001b[38;5;241m@\u001b[39m X) \u001b[38;5;241m@\u001b[39m X\u001b[38;5;241m.\u001b[39mT ) \u001b[38;5;241m@\u001b[39m y\n\u001b[1;32m 3\u001b[0m \u001b[38;5;66;03m# and then make the prediction\u001b[39;00m\n\u001b[1;32m 4\u001b[0m ytilde \u001b[38;5;241m=\u001b[39m X \u001b[38;5;241m@\u001b[39m beta\n", - "\u001b[0;31mNameError\u001b[0m: name 'np' is not defined" - ] - } - ], + "outputs": [], "source": [ "# matrix inversion to find beta\n", + "# First we set up the data\n", + "import numpy as np\n", + "x = np.random.rand(100)\n", + "y = 2.0+5*x*x+0.1*np.random.randn(100)\n", + "# and then the design matrix X including the intercept\n", + "# The design matrix now as function of a fourth-order polynomial\n", + "X = np.zeros((len(x),5))\n", + "X[:,0] = 1.0\n", + "X[:,1] = x\n", + "X[:,2] = x**2\n", + "X[:,3] = x**3\n", + "X[:,4] = x**4\n", "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n", "# and then make the prediction\n", "ytilde = X @ beta" @@ -1442,7 +1441,7 @@ }, { "cell_type": "markdown", - "id": "70c97572", + "id": "8ea9846d", "metadata": { "editable": true }, @@ -1453,12 +1452,24 @@ { "cell_type": "code", "execution_count": 2, - "id": "1aa95889", + "id": "e996841a", "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "ename": "NameError", + "evalue": "name 'Energies' is not defined", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m fit \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mlinalg\u001b[38;5;241m.\u001b[39mlstsq(X, \u001b[43mEnergies\u001b[49m, rcond \u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m)[\u001b[38;5;241m0\u001b[39m]\n\u001b[1;32m 2\u001b[0m ytildenp \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mdot(fit,X\u001b[38;5;241m.\u001b[39mT)\n", + "\u001b[0;31mNameError\u001b[0m: name 'Energies' is not defined" + ] + } + ], "source": [ "fit = np.linalg.lstsq(X, Energies, rcond =None)[0]\n", "ytildenp = np.dot(fit,X.T)" @@ -1466,7 +1477,7 @@ }, { "cell_type": "markdown", - "id": "e250b2e9", + "id": "39f01129", "metadata": { "editable": true }, @@ -1480,7 +1491,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "a86bd577", + "id": "12e52de5", "metadata": { "collapsed": false, "editable": true @@ -1493,7 +1504,7 @@ }, { "cell_type": "markdown", - "id": "d6ccd7b8", + "id": "8a0cc628", "metadata": { "editable": true }, @@ -1504,7 +1515,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "fb6f7c19", + "id": "8c1508b7", "metadata": { "collapsed": false, "editable": true @@ -1516,7 +1527,7 @@ }, { "cell_type": "markdown", - "id": "eca18427", + "id": "a3f2f3b1", "metadata": { "editable": true }, @@ -1527,7 +1538,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "0fd2b39c", + "id": "334780f0", "metadata": { "collapsed": false, "editable": true @@ -1543,7 +1554,7 @@ }, { "cell_type": "markdown", - "id": "db3fe7d9", + "id": "1f7cfede", "metadata": { "editable": true }, @@ -1554,7 +1565,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "9f59c14e", + "id": "1579c506", "metadata": { "collapsed": false, "editable": true @@ -1568,7 +1579,7 @@ }, { "cell_type": "markdown", - "id": "2fdd5915", + "id": "88c6c130", "metadata": { "editable": true }, @@ -1589,7 +1600,7 @@ }, { "cell_type": "markdown", - "id": "9863e063", + "id": "af0baf7f", "metadata": { "editable": true }, @@ -1600,7 +1611,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "85fc6a78", + "id": "89aa39b5", "metadata": { "collapsed": false, "editable": true @@ -1653,7 +1664,7 @@ }, { "cell_type": "markdown", - "id": "6671297c", + "id": "1d86c024", "metadata": { "editable": true }, @@ -1664,7 +1675,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "810f32f7", + "id": "2ba16c25", "metadata": { "collapsed": false, "editable": true @@ -1689,7 +1700,7 @@ }, { "cell_type": "markdown", - "id": "1ca5e45d", + "id": "8e363f00", "metadata": { "editable": true }, @@ -1701,7 +1712,7 @@ }, { "cell_type": "markdown", - "id": "b6530670", + "id": "d5ef1be0", "metadata": { "editable": true }, @@ -1730,7 +1741,7 @@ }, { "cell_type": "markdown", - "id": "4be4158c", + "id": "7165f38b", "metadata": { "editable": true }, @@ -1755,7 +1766,7 @@ }, { "cell_type": "markdown", - "id": "aa6f9ff6", + "id": "baea18ad", "metadata": { "editable": true }, @@ -1775,7 +1786,7 @@ }, { "cell_type": "markdown", - "id": "856dee18", + "id": "830be0f9", "metadata": { "editable": true }, @@ -1802,7 +1813,7 @@ }, { "cell_type": "markdown", - "id": "3497598c", + "id": "72473f2c", "metadata": { "editable": true }, @@ -1815,7 +1826,7 @@ }, { "cell_type": "markdown", - "id": "fe24fd3f", + "id": "ad92637f", "metadata": { "editable": true }, @@ -1827,7 +1838,7 @@ }, { "cell_type": "markdown", - "id": "55bccc6e", + "id": "928978ac", "metadata": { "editable": true }, @@ -1838,7 +1849,7 @@ }, { "cell_type": "markdown", - "id": "e01c4d09", + "id": "31b09189", "metadata": { "editable": true }, @@ -1854,7 +1865,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "6098d5d9", + "id": "c09e81a9", "metadata": { "collapsed": false, "editable": true @@ -1888,7 +1899,7 @@ }, { "cell_type": "markdown", - "id": "653340bf", + "id": "8ffeb0f1", "metadata": { "editable": true }, @@ -1898,7 +1909,7 @@ }, { "cell_type": "markdown", - "id": "673c8917", + "id": "2b1f32da", "metadata": { "editable": true }, @@ -1913,7 +1924,7 @@ }, { "cell_type": "markdown", - "id": "4f69865b", + "id": "3c2f9703", "metadata": { "editable": true }, @@ -1925,7 +1936,7 @@ }, { "cell_type": "markdown", - "id": "f39433a0", + "id": "15382f87", "metadata": { "editable": true }, @@ -1935,7 +1946,7 @@ }, { "cell_type": "markdown", - "id": "f2f5cb76", + "id": "a04fe6a8", "metadata": { "editable": true }, @@ -1951,7 +1962,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "12dbcdcb", + "id": "de5a8426", "metadata": { "collapsed": false, "editable": true @@ -1968,7 +1979,7 @@ }, { "cell_type": "markdown", - "id": "38cb3698", + "id": "815c116c", "metadata": { "editable": true }, @@ -1981,7 +1992,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "a0cd549b", + "id": "c9049720", "metadata": { "collapsed": false, "editable": true @@ -2028,7 +2039,7 @@ }, { "cell_type": "markdown", - "id": "b4d2ab87", + "id": "c6fb36f0", "metadata": { "editable": true }, @@ -2039,7 +2050,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "9f5d7996", + "id": "d02afd53", "metadata": { "collapsed": false, "editable": true @@ -2142,7 +2153,7 @@ }, { "cell_type": "markdown", - "id": "5ea41ab9", + "id": "4d3d0315", "metadata": { "editable": true }, @@ -2170,7 +2181,7 @@ }, { "cell_type": "markdown", - "id": "ce0afdb4", + "id": "1bf26037", "metadata": { "editable": true }, @@ -2205,7 +2216,7 @@ }, { "cell_type": "markdown", - "id": "ad92f42b", + "id": "1a9303d9", "metadata": { "editable": true }, @@ -2219,7 +2230,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "fae66265", + "id": "add41c28", "metadata": { "collapsed": false, "editable": true @@ -2244,7 +2255,7 @@ }, { "cell_type": "markdown", - "id": "1bf672d8", + "id": "cbad62c2", "metadata": { "editable": true }, @@ -2260,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "30834557", + "id": "77b3f4a0", "metadata": { "editable": true }, @@ -2272,7 +2283,7 @@ }, { "cell_type": "markdown", - "id": "5c05a427", + "id": "2054bdc7", "metadata": { "editable": true }, @@ -2287,7 +2298,7 @@ }, { "cell_type": "markdown", - "id": "902391ce", + "id": "d7f64d04", "metadata": { "editable": true }, @@ -2299,7 +2310,7 @@ }, { "cell_type": "markdown", - "id": "a68a6228", + "id": "6ebe45a1", "metadata": { "editable": true }, @@ -2309,7 +2320,7 @@ }, { "cell_type": "markdown", - "id": "212cf295", + "id": "8cc21b6a", "metadata": { "editable": true }, @@ -2321,7 +2332,7 @@ }, { "cell_type": "markdown", - "id": "e7db36e1", + "id": "71b87ff9", "metadata": { "editable": true }, @@ -2331,7 +2342,7 @@ }, { "cell_type": "markdown", - "id": "5d7f590c", + "id": "b15ee5de", "metadata": { "editable": true }, @@ -2343,7 +2354,7 @@ }, { "cell_type": "markdown", - "id": "bbf7c89a", + "id": "56e808e5", "metadata": { "editable": true }, @@ -2356,7 +2367,7 @@ }, { "cell_type": "markdown", - "id": "05692612", + "id": "59b1fcf5", "metadata": { "editable": true }, @@ -2368,7 +2379,7 @@ }, { "cell_type": "markdown", - "id": "3dc53330", + "id": "fe9af8e4", "metadata": { "editable": true }, @@ -2378,7 +2389,7 @@ }, { "cell_type": "markdown", - "id": "562d655d", + "id": "39d0535d", "metadata": { "editable": true }, @@ -2390,7 +2401,7 @@ }, { "cell_type": "markdown", - "id": "528c869d", + "id": "b05e083e", "metadata": { "editable": true }, @@ -2400,7 +2411,7 @@ }, { "cell_type": "markdown", - "id": "82a20d9e", + "id": "255d5f87", "metadata": { "editable": true }, @@ -2412,7 +2423,7 @@ }, { "cell_type": "markdown", - "id": "5e37316e", + "id": "863f9c03", "metadata": { "editable": true }, @@ -2422,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "90c44e36", + "id": "fe1719c8", "metadata": { "editable": true }, @@ -2434,7 +2445,7 @@ }, { "cell_type": "markdown", - "id": "be72a7d3", + "id": "11bf9275", "metadata": { "editable": true }, @@ -2444,7 +2455,7 @@ }, { "cell_type": "markdown", - "id": "0a435bf5", + "id": "58a73e37", "metadata": { "editable": true }, @@ -2456,7 +2467,7 @@ }, { "cell_type": "markdown", - "id": "a67864cb", + "id": "2ee6e0ab", "metadata": { "editable": true }, @@ -2466,7 +2477,7 @@ }, { "cell_type": "markdown", - "id": "9d8602b7", + "id": "4aa55bc2", "metadata": { "editable": true }, @@ -2478,7 +2489,7 @@ }, { "cell_type": "markdown", - "id": "32caef0d", + "id": "e7dd951c", "metadata": { "editable": true }, @@ -2488,7 +2499,7 @@ }, { "cell_type": "markdown", - "id": "f24465a9", + "id": "b0b56f12", "metadata": { "editable": true }, @@ -2500,7 +2511,7 @@ }, { "cell_type": "markdown", - "id": "f64212da", + "id": "2fa4cfc7", "metadata": { "editable": true }, @@ -2512,7 +2523,7 @@ }, { "cell_type": "markdown", - "id": "8d0bf5ae", + "id": "cc6fe7e2", "metadata": { "editable": true }, @@ -2524,7 +2535,7 @@ }, { "cell_type": "markdown", - "id": "e8d36ff8", + "id": "d121682c", "metadata": { "editable": true }, @@ -2537,7 +2548,7 @@ }, { "cell_type": "markdown", - "id": "aea58aac", + "id": "94aef78e", "metadata": { "editable": true }, @@ -2549,7 +2560,7 @@ }, { "cell_type": "markdown", - "id": "bc4b6f91", + "id": "8d595a8d", "metadata": { "editable": true }, @@ -2559,7 +2570,7 @@ }, { "cell_type": "markdown", - "id": "349b03e2", + "id": "67d81f01", "metadata": { "editable": true }, @@ -2573,7 +2584,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "5cfc62db", + "id": "39da9a38", "metadata": { "collapsed": false, "editable": true @@ -2670,7 +2681,7 @@ }, { "cell_type": "markdown", - "id": "64c5fa5b", + "id": "257ce2f2", "metadata": { "editable": true }, @@ -2691,7 +2702,7 @@ }, { "cell_type": "markdown", - "id": "67496f35", + "id": "3769097f", "metadata": { "editable": true }, @@ -2703,7 +2714,7 @@ }, { "cell_type": "markdown", - "id": "ceae31e0", + "id": "82fc22ad", "metadata": { "editable": true }, @@ -2713,7 +2724,7 @@ }, { "cell_type": "markdown", - "id": "743c68c8", + "id": "dea71d52", "metadata": { "editable": true }, @@ -2725,7 +2736,7 @@ }, { "cell_type": "markdown", - "id": "4af68378", + "id": "113f6e87", "metadata": { "editable": true }, @@ -2735,7 +2746,7 @@ }, { "cell_type": "markdown", - "id": "8228af90", + "id": "1e3b78e5", "metadata": { "editable": true }, @@ -2747,7 +2758,7 @@ }, { "cell_type": "markdown", - "id": "0a4285bc", + "id": "d936547b", "metadata": { "editable": true }, @@ -2763,7 +2774,7 @@ }, { "cell_type": "markdown", - "id": "406af60a", + "id": "7c57b1e6", "metadata": { "editable": true }, @@ -2807,7 +2818,7 @@ }, { "cell_type": "markdown", - "id": "0f05d1e1", + "id": "41d26913", "metadata": { "editable": true }, @@ -2819,7 +2830,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "70f8a406", + "id": "bf2d2a51", "metadata": { "collapsed": false, "editable": true @@ -2835,7 +2846,7 @@ }, { "cell_type": "markdown", - "id": "10a6162c", + "id": "d7d26ad9", "metadata": { "editable": true }, @@ -2846,7 +2857,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "be236243", + "id": "7b725195", "metadata": { "collapsed": false, "editable": true @@ -2864,7 +2875,7 @@ }, { "cell_type": "markdown", - "id": "88a7376f", + "id": "4b578dbb", "metadata": { "editable": true }, @@ -2875,7 +2886,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "ec268779", + "id": "670a51a0", "metadata": { "collapsed": false, "editable": true @@ -2889,7 +2900,7 @@ }, { "cell_type": "markdown", - "id": "b45f3733", + "id": "ad921def", "metadata": { "editable": true }, @@ -2900,7 +2911,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "8b9423c6", + "id": "2d0a0044", "metadata": { "collapsed": false, "editable": true @@ -2913,7 +2924,7 @@ }, { "cell_type": "markdown", - "id": "1061cb1f", + "id": "8ec52805", "metadata": { "editable": true }, @@ -2924,7 +2935,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "00d79fad", + "id": "a7648df4", "metadata": { "collapsed": false, "editable": true @@ -2941,7 +2952,7 @@ }, { "cell_type": "markdown", - "id": "18871b23", + "id": "8e911148", "metadata": { "editable": true }, @@ -2952,7 +2963,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "abfd55f9", + "id": "4420c7f9", "metadata": { "collapsed": false, "editable": true @@ -2968,7 +2979,7 @@ }, { "cell_type": "markdown", - "id": "231f35c5", + "id": "b7580acd", "metadata": { "editable": true }, @@ -2979,7 +2990,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "5bb68d00", + "id": "2a0930b1", "metadata": { "collapsed": false, "editable": true @@ -3003,7 +3014,7 @@ }, { "cell_type": "markdown", - "id": "9c2f9f6b", + "id": "49423ebd", "metadata": { "editable": true }, @@ -3014,7 +3025,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "df3cc855", + "id": "5841e8ee", "metadata": { "collapsed": false, "editable": true @@ -3027,7 +3038,7 @@ }, { "cell_type": "markdown", - "id": "da67457b", + "id": "de1c7a1d", "metadata": { "editable": true }, @@ -3038,7 +3049,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "f9784866", + "id": "8d3ce90a", "metadata": { "collapsed": false, "editable": true @@ -3058,7 +3069,7 @@ }, { "cell_type": "markdown", - "id": "3429e67f", + "id": "b243c849", "metadata": { "editable": true }, @@ -3069,7 +3080,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "9ec7ba47", + "id": "83abda14", "metadata": { "collapsed": false, "editable": true @@ -3112,7 +3123,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "0c0be681", + "id": "c1bda3a1", "metadata": { "collapsed": false, "editable": true @@ -3127,7 +3138,7 @@ }, { "cell_type": "markdown", - "id": "ed12f12d", + "id": "494e99e1", "metadata": { "editable": true }, @@ -3137,7 +3148,7 @@ }, { "cell_type": "markdown", - "id": "50c91ade", + "id": "d496f2b6", "metadata": { "editable": true }, @@ -3151,7 +3162,7 @@ }, { "cell_type": "markdown", - "id": "682a501b", + "id": "617544ab", "metadata": { "editable": true }, @@ -3163,7 +3174,7 @@ }, { "cell_type": "markdown", - "id": "057bbc5d", + "id": "deb3a5c0", "metadata": { "editable": true }, @@ -3175,7 +3186,7 @@ }, { "cell_type": "markdown", - "id": "cddb899f", + "id": "1e29d04b", "metadata": { "editable": true }, @@ -3187,7 +3198,7 @@ }, { "cell_type": "markdown", - "id": "ab325ebc", + "id": "e7f06cd3", "metadata": { "editable": true }, @@ -3197,7 +3208,7 @@ }, { "cell_type": "markdown", - "id": "095ebc5b", + "id": "61fbb179", "metadata": { "editable": true }, @@ -3209,7 +3220,7 @@ }, { "cell_type": "markdown", - "id": "08e642f6", + "id": "715d0808", "metadata": { "editable": true }, @@ -3219,7 +3230,7 @@ }, { "cell_type": "markdown", - "id": "da5c1625", + "id": "e9af9b81", "metadata": { "editable": true }, @@ -3231,7 +3242,7 @@ }, { "cell_type": "markdown", - "id": "2bf04890", + "id": "ce0a6494", "metadata": { "editable": true }, @@ -3242,7 +3253,7 @@ }, { "cell_type": "markdown", - "id": "5ba172cc", + "id": "a9680e79", "metadata": { "editable": true }, @@ -3254,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "7d1f0b59", + "id": "77b2c7f1", "metadata": { "editable": true }, @@ -3266,7 +3277,7 @@ }, { "cell_type": "markdown", - "id": "467e185d", + "id": "00bc0981", "metadata": { "editable": true }, @@ -3276,7 +3287,7 @@ }, { "cell_type": "markdown", - "id": "74822363", + "id": "a5870049", "metadata": { "editable": true }, @@ -3288,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "5414a1d2", + "id": "8f53c1ca", "metadata": { "editable": true }, @@ -3300,7 +3311,7 @@ }, { "cell_type": "markdown", - "id": "08ce1402", + "id": "ac5e957f", "metadata": { "editable": true }, @@ -3310,7 +3321,7 @@ }, { "cell_type": "markdown", - "id": "7bbb949f", + "id": "b949e361", "metadata": { "editable": true }, @@ -3322,7 +3333,7 @@ }, { "cell_type": "markdown", - "id": "4007519d", + "id": "bf9ebc2b", "metadata": { "editable": true }, @@ -3332,7 +3343,7 @@ }, { "cell_type": "markdown", - "id": "45061e33", + "id": "8f15c266", "metadata": { "editable": true }, @@ -3344,7 +3355,7 @@ }, { "cell_type": "markdown", - "id": "f5afee48", + "id": "4f791b65", "metadata": { "editable": true }, @@ -3354,7 +3365,7 @@ }, { "cell_type": "markdown", - "id": "6a3d6440", + "id": "e8017b1c", "metadata": { "editable": true }, @@ -3394,7 +3405,7 @@ }, { "cell_type": "markdown", - "id": "ef8c39b7", + "id": "2996bc1f", "metadata": { "editable": true }, @@ -3411,7 +3422,7 @@ }, { "cell_type": "markdown", - "id": "6edd77ca", + "id": "2b6b0946", "metadata": { "editable": true }, @@ -3434,7 +3445,7 @@ }, { "cell_type": "markdown", - "id": "cba40ade", + "id": "93e7e61f", "metadata": { "editable": true }, @@ -3451,7 +3462,7 @@ }, { "cell_type": "markdown", - "id": "20f096a2", + "id": "fe419623", "metadata": { "editable": true }, @@ -3470,7 +3481,7 @@ }, { "cell_type": "markdown", - "id": "49f6f27c", + "id": "e2140a9e", "metadata": { "editable": true }, @@ -3481,7 +3492,7 @@ }, { "cell_type": "markdown", - "id": "577f3338", + "id": "ffcb729e", "metadata": { "editable": true }, @@ -3493,7 +3504,7 @@ }, { "cell_type": "markdown", - "id": "f456df81", + "id": "5140ddae", "metadata": { "editable": true }, @@ -3511,7 +3522,7 @@ }, { "cell_type": "markdown", - "id": "ed7db14f", + "id": "7bb98440", "metadata": { "editable": true }, @@ -3527,7 +3538,7 @@ }, { "cell_type": "markdown", - "id": "a876de90", + "id": "5611679a", "metadata": { "editable": true }, @@ -3539,7 +3550,7 @@ }, { "cell_type": "markdown", - "id": "3b2eee8d", + "id": "f32c0eb4", "metadata": { "editable": true }, @@ -3549,7 +3560,7 @@ }, { "cell_type": "markdown", - "id": "910d3f09", + "id": "415f02af", "metadata": { "editable": true }, @@ -3564,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "94bd4c00", + "id": "c28567b8", "metadata": { "editable": true }, @@ -3576,7 +3587,7 @@ }, { "cell_type": "markdown", - "id": "56d91a09", + "id": "936af9ce", "metadata": { "editable": true }, @@ -3586,7 +3597,7 @@ }, { "cell_type": "markdown", - "id": "8e59a435", + "id": "308f2aa4", "metadata": { "editable": true }, @@ -3598,7 +3609,7 @@ }, { "cell_type": "markdown", - "id": "9c2e6b9f", + "id": "d74a5600", "metadata": { "editable": true }, @@ -3608,7 +3619,7 @@ }, { "cell_type": "markdown", - "id": "e7188a39", + "id": "3d5f8482", "metadata": { "editable": true }, @@ -3620,7 +3631,7 @@ }, { "cell_type": "markdown", - "id": "daac715a", + "id": "b4eaa6e3", "metadata": { "editable": true }, @@ -3632,7 +3643,7 @@ }, { "cell_type": "markdown", - "id": "66322f83", + "id": "0b821498", "metadata": { "editable": true }, @@ -3647,7 +3658,7 @@ }, { "cell_type": "markdown", - "id": "a03be784", + "id": "d38da336", "metadata": { "editable": true }, @@ -3658,7 +3669,7 @@ }, { "cell_type": "markdown", - "id": "ff3964a4", + "id": "98b0ae03", "metadata": { "editable": true }, @@ -3678,7 +3689,7 @@ }, { "cell_type": "markdown", - "id": "04d411cc", + "id": "d4ba8f69", "metadata": { "editable": true }, @@ -3690,7 +3701,7 @@ }, { "cell_type": "markdown", - "id": "bf2ae91a", + "id": "52bb4456", "metadata": { "editable": true }, @@ -3700,7 +3711,7 @@ }, { "cell_type": "markdown", - "id": "843d113e", + "id": "6ec33a01", "metadata": { "editable": true }, @@ -3712,7 +3723,7 @@ }, { "cell_type": "markdown", - "id": "f59af5f8", + "id": "b39cac82", "metadata": { "editable": true }, @@ -3741,7 +3752,7 @@ }, { "cell_type": "markdown", - "id": "22f1ed07", + "id": "d2595a2e", "metadata": { "editable": true }, @@ -3768,7 +3779,7 @@ }, { "cell_type": "markdown", - "id": "69efb08d", + "id": "427fb028", "metadata": { "editable": true }, @@ -3779,7 +3790,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "640d3bdf", + "id": "0bbd4699", "metadata": { "collapsed": false, "editable": true @@ -3819,7 +3830,7 @@ }, { "cell_type": "markdown", - "id": "98d61790", + "id": "2430babb", "metadata": { "editable": true }, @@ -3836,7 +3847,7 @@ }, { "cell_type": "markdown", - "id": "95b1ffa6", + "id": "4761f74a", "metadata": { "editable": true }, @@ -3859,7 +3870,7 @@ }, { "cell_type": "markdown", - "id": "4605f137", + "id": "ffe52fdc", "metadata": { "editable": true }, @@ -3873,7 +3884,7 @@ }, { "cell_type": "markdown", - "id": "3125bc92", + "id": "c810c7eb", "metadata": { "editable": true }, @@ -3892,7 +3903,7 @@ }, { "cell_type": "markdown", - "id": "32cf3a9e", + "id": "fa17e262", "metadata": { "editable": true }, @@ -3902,7 +3913,7 @@ }, { "cell_type": "markdown", - "id": "573acb2d", + "id": "e7411ce2", "metadata": { "editable": true }, @@ -3914,7 +3925,7 @@ }, { "cell_type": "markdown", - "id": "22320566", + "id": "40d9144e", "metadata": { "editable": true }, @@ -3928,7 +3939,7 @@ }, { "cell_type": "markdown", - "id": "af0ee993", + "id": "28cd0268", "metadata": { "editable": true }, @@ -3940,7 +3951,7 @@ }, { "cell_type": "markdown", - "id": "0213ed87", + "id": "e72c1658", "metadata": { "editable": true }, @@ -3950,7 +3961,7 @@ }, { "cell_type": "markdown", - "id": "06266ee6", + "id": "9e8284ec", "metadata": { "editable": true }, @@ -3962,7 +3973,7 @@ }, { "cell_type": "markdown", - "id": "23e27fdf", + "id": "39c798c9", "metadata": { "editable": true }, @@ -3979,7 +3990,7 @@ }, { "cell_type": "markdown", - "id": "2539e582", + "id": "93cf6fd3", "metadata": { "editable": true }, @@ -3989,7 +4000,7 @@ }, { "cell_type": "markdown", - "id": "bef2cf95", + "id": "c99559d1", "metadata": { "editable": true }, @@ -4005,7 +4016,7 @@ }, { "cell_type": "markdown", - "id": "fe385753", + "id": "199187d7", "metadata": { "editable": true }, @@ -4015,7 +4026,7 @@ }, { "cell_type": "markdown", - "id": "cc270435", + "id": "164f8c17", "metadata": { "editable": true }, @@ -4031,7 +4042,7 @@ }, { "cell_type": "markdown", - "id": "31932192", + "id": "5e9fd974", "metadata": { "editable": true }, @@ -4041,7 +4052,7 @@ }, { "cell_type": "markdown", - "id": "a972a944", + "id": "8e07a82d", "metadata": { "editable": true }, @@ -4057,7 +4068,7 @@ }, { "cell_type": "markdown", - "id": "d6ef62ee", + "id": "b0bf5db6", "metadata": { "editable": true }, @@ -4067,7 +4078,7 @@ }, { "cell_type": "markdown", - "id": "61e10478", + "id": "aecfc56d", "metadata": { "editable": true }, @@ -4084,7 +4095,7 @@ }, { "cell_type": "markdown", - "id": "9e8580da", + "id": "73b4475e", "metadata": { "editable": true }, @@ -4096,7 +4107,7 @@ }, { "cell_type": "markdown", - "id": "0ac00991", + "id": "54057ff8", "metadata": { "editable": true }, @@ -4108,7 +4119,7 @@ }, { "cell_type": "markdown", - "id": "05f8073b", + "id": "ff974462", "metadata": { "editable": true }, @@ -4120,7 +4131,7 @@ }, { "cell_type": "markdown", - "id": "622a8443", + "id": "e5daa192", "metadata": { "editable": true }, @@ -4130,7 +4141,7 @@ }, { "cell_type": "markdown", - "id": "580e143c", + "id": "dc9f0f7b", "metadata": { "editable": true }, @@ -4142,7 +4153,7 @@ }, { "cell_type": "markdown", - "id": "30b75cea", + "id": "f9b10560", "metadata": { "editable": true }, @@ -4154,7 +4165,7 @@ }, { "cell_type": "markdown", - "id": "cdc103a7", + "id": "66a948a0", "metadata": { "editable": true }, @@ -4166,7 +4177,7 @@ }, { "cell_type": "markdown", - "id": "ea01abf6", + "id": "55d9dbd1", "metadata": { "editable": true }, @@ -4176,7 +4187,7 @@ }, { "cell_type": "markdown", - "id": "e4b60d5d", + "id": "c2b7da0d", "metadata": { "editable": true }, @@ -4188,7 +4199,7 @@ }, { "cell_type": "markdown", - "id": "deb0bbd3", + "id": "ff1c658b", "metadata": { "editable": true }, @@ -4198,7 +4209,7 @@ }, { "cell_type": "markdown", - "id": "f1f15de8", + "id": "3d7050b0", "metadata": { "editable": true }, @@ -4210,7 +4221,7 @@ }, { "cell_type": "markdown", - "id": "7c54c407", + "id": "dced3eca", "metadata": { "editable": true }, @@ -4224,7 +4235,7 @@ }, { "cell_type": "markdown", - "id": "31e31b03", + "id": "44ec48f0", "metadata": { "editable": true }, @@ -4236,7 +4247,7 @@ }, { "cell_type": "markdown", - "id": "7d1cef4f", + "id": "ef5a3989", "metadata": { "editable": true }, @@ -4248,7 +4259,7 @@ }, { "cell_type": "markdown", - "id": "aa8cedef", + "id": "6d7071a0", "metadata": { "editable": true }, @@ -4258,7 +4269,7 @@ }, { "cell_type": "markdown", - "id": "022623a1", + "id": "a37c0651", "metadata": { "editable": true }, @@ -4270,7 +4281,7 @@ }, { "cell_type": "markdown", - "id": "c9885020", + "id": "9cd481cc", "metadata": { "editable": true }, @@ -4281,7 +4292,7 @@ }, { "cell_type": "markdown", - "id": "f378f76e", + "id": "e8cd56b7", "metadata": { "editable": true }, @@ -4293,7 +4304,7 @@ }, { "cell_type": "markdown", - "id": "4add7e8a", + "id": "3a629b1b", "metadata": { "editable": true }, @@ -4303,7 +4314,7 @@ }, { "cell_type": "markdown", - "id": "51ea1a29", + "id": "2fbf21c7", "metadata": { "editable": true }, @@ -4315,7 +4326,7 @@ }, { "cell_type": "markdown", - "id": "54a250b6", + "id": "5951205a", "metadata": { "editable": true }, @@ -4325,7 +4336,7 @@ }, { "cell_type": "markdown", - "id": "ebaf68f5", + "id": "53caa2e2", "metadata": { "editable": true }, @@ -4337,7 +4348,7 @@ }, { "cell_type": "markdown", - "id": "5ff687e8", + "id": "64b903b4", "metadata": { "editable": true }, @@ -4348,7 +4359,7 @@ }, { "cell_type": "markdown", - "id": "ea61a360", + "id": "df6a66b3", "metadata": { "editable": true }, @@ -4360,7 +4371,7 @@ }, { "cell_type": "markdown", - "id": "bc168f34", + "id": "ffe2f537", "metadata": { "editable": true }, @@ -4378,7 +4389,7 @@ }, { "cell_type": "markdown", - "id": "39d89895", + "id": "3c291471", "metadata": { "editable": true }, @@ -4394,7 +4405,7 @@ }, { "cell_type": "markdown", - "id": "c229143d", + "id": "0a3f9a94", "metadata": { "editable": true }, @@ -4406,7 +4417,7 @@ }, { "cell_type": "markdown", - "id": "0a569456", + "id": "9e121ab9", "metadata": { "editable": true }, @@ -4418,7 +4429,7 @@ }, { "cell_type": "markdown", - "id": "0c0b9212", + "id": "8af98d79", "metadata": { "editable": true }, @@ -4430,7 +4441,7 @@ }, { "cell_type": "markdown", - "id": "b05a1b83", + "id": "f8df9cb1", "metadata": { "editable": true }, @@ -4443,7 +4454,7 @@ }, { "cell_type": "markdown", - "id": "d64818ce", + "id": "19bfd403", "metadata": { "editable": true }, @@ -4459,7 +4470,7 @@ }, { "cell_type": "markdown", - "id": "32df0f89", + "id": "c1ec9cb3", "metadata": { "editable": true }, @@ -4473,7 +4484,7 @@ }, { "cell_type": "markdown", - "id": "a9d9108b", + "id": "4346796e", "metadata": { "editable": true }, @@ -4483,7 +4494,7 @@ }, { "cell_type": "markdown", - "id": "c6d76715", + "id": "cda3d8a7", "metadata": { "editable": true }, @@ -4495,7 +4506,7 @@ }, { "cell_type": "markdown", - "id": "5cc35821", + "id": "7cf1e6d8", "metadata": { "editable": true }, @@ -4505,7 +4516,7 @@ }, { "cell_type": "markdown", - "id": "27c8eac0", + "id": "c783275d", "metadata": { "editable": true }, @@ -4517,7 +4528,7 @@ }, { "cell_type": "markdown", - "id": "49361e47", + "id": "dac43f32", "metadata": { "editable": true }, @@ -4527,7 +4538,7 @@ }, { "cell_type": "markdown", - "id": "5eb15920", + "id": "012a62f2", "metadata": { "editable": true }, @@ -4541,7 +4552,7 @@ }, { "cell_type": "markdown", - "id": "2160cb07", + "id": "f02bf402", "metadata": { "editable": true }, @@ -4558,7 +4569,7 @@ }, { "cell_type": "markdown", - "id": "a5deaa8e", + "id": "63805f7d", "metadata": { "editable": true }, @@ -4574,7 +4585,7 @@ }, { "cell_type": "markdown", - "id": "f6868141", + "id": "e84b7d85", "metadata": { "editable": true }, @@ -4586,7 +4597,7 @@ }, { "cell_type": "markdown", - "id": "9c17b156", + "id": "3ef0ad9c", "metadata": { "editable": true }, @@ -4599,7 +4610,7 @@ }, { "cell_type": "markdown", - "id": "1b9e8b7e", + "id": "5aa2f0f4", "metadata": { "editable": true }, @@ -4613,7 +4624,7 @@ }, { "cell_type": "markdown", - "id": "c00ac20a", + "id": "5cedd672", "metadata": { "editable": true }, @@ -4623,7 +4634,7 @@ }, { "cell_type": "markdown", - "id": "91729a44", + "id": "cdb4b214", "metadata": { "editable": true }, @@ -4636,7 +4647,7 @@ }, { "cell_type": "markdown", - "id": "62403241", + "id": "830c23c5", "metadata": { "editable": true }, @@ -4655,7 +4666,7 @@ }, { "cell_type": "markdown", - "id": "6d0de657", + "id": "7971e263", "metadata": { "editable": true }, @@ -4667,7 +4678,7 @@ }, { "cell_type": "markdown", - "id": "108adf6b", + "id": "aab14206", "metadata": { "editable": true }, @@ -4679,7 +4690,7 @@ }, { "cell_type": "markdown", - "id": "1b2ba4f7", + "id": "27b6b6fc", "metadata": { "editable": true }, @@ -4689,7 +4700,7 @@ }, { "cell_type": "markdown", - "id": "1711d54c", + "id": "3e702413", "metadata": { "editable": true }, @@ -4701,7 +4712,7 @@ }, { "cell_type": "markdown", - "id": "f88b0c88", + "id": "fd91d200", "metadata": { "editable": true }, @@ -4714,7 +4725,7 @@ }, { "cell_type": "markdown", - "id": "d1835d40", + "id": "35c6dc25", "metadata": { "editable": true }, @@ -4733,7 +4744,7 @@ }, { "cell_type": "markdown", - "id": "7d0f1648", + "id": "e733d668", "metadata": { "editable": true }, @@ -4743,7 +4754,7 @@ }, { "cell_type": "markdown", - "id": "d16ea5cf", + "id": "4feaa7d5", "metadata": { "editable": true }, @@ -4762,7 +4773,7 @@ }, { "cell_type": "markdown", - "id": "634130de", + "id": "2fd87051", "metadata": { "editable": true }, @@ -4780,7 +4791,7 @@ }, { "cell_type": "markdown", - "id": "cecf6c09", + "id": "8a8c7cb0", "metadata": { "editable": true }, @@ -4794,7 +4805,7 @@ }, { "cell_type": "markdown", - "id": "1703b000", + "id": "b5426cb7", "metadata": { "editable": true }, @@ -4809,7 +4820,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "5da5890f", + "id": "45d8c427", "metadata": { "collapsed": false, "editable": true @@ -4830,7 +4841,7 @@ }, { "cell_type": "markdown", - "id": "6b10b22a", + "id": "5cb5bc08", "metadata": { "editable": true }, @@ -4847,7 +4858,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "da699527", + "id": "b43fd84d", "metadata": { "collapsed": false, "editable": true @@ -4879,7 +4890,7 @@ }, { "cell_type": "markdown", - "id": "a40fb51e", + "id": "c303e01f", "metadata": { "editable": true }, @@ -4893,7 +4904,7 @@ }, { "cell_type": "markdown", - "id": "413db31e", + "id": "d56895dd", "metadata": { "editable": true }, @@ -4906,7 +4917,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "53927e5d", + "id": "cfda2184", "metadata": { "collapsed": false, "editable": true @@ -4931,7 +4942,7 @@ }, { "cell_type": "markdown", - "id": "74fc3107", + "id": "3cec050e", "metadata": { "editable": true }, @@ -4941,7 +4952,7 @@ }, { "cell_type": "markdown", - "id": "030881ab", + "id": "8afc4799", "metadata": { "editable": true }, @@ -4952,7 +4963,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "c107aa55", + "id": "1372e459", "metadata": { "collapsed": false, "editable": true @@ -5006,7 +5017,7 @@ }, { "cell_type": "markdown", - "id": "3f47378e", + "id": "aed87895", "metadata": { "editable": true }, @@ -5023,7 +5034,7 @@ }, { "cell_type": "markdown", - "id": "dba29ab3", + "id": "7e1e5b8d", "metadata": { "editable": true }, @@ -5035,7 +5046,7 @@ }, { "cell_type": "markdown", - "id": "0eb702bd", + "id": "b0d462fa", "metadata": { "editable": true }, @@ -5047,7 +5058,7 @@ }, { "cell_type": "markdown", - "id": "1d355153", + "id": "b8b65150", "metadata": { "editable": true }, @@ -5057,7 +5068,7 @@ }, { "cell_type": "markdown", - "id": "a2316749", + "id": "7e8f5c1f", "metadata": { "editable": true }, @@ -5074,7 +5085,7 @@ }, { "cell_type": "markdown", - "id": "8f6a654b", + "id": "1cc1396a", "metadata": { "editable": true }, @@ -5084,7 +5095,7 @@ }, { "cell_type": "markdown", - "id": "72291085", + "id": "5df6ba80", "metadata": { "editable": true }, @@ -5099,7 +5110,7 @@ }, { "cell_type": "markdown", - "id": "2b28a5fe", + "id": "59e9b6e2", "metadata": { "editable": true }, @@ -5109,7 +5120,7 @@ }, { "cell_type": "markdown", - "id": "49efe427", + "id": "ca2d3f9a", "metadata": { "editable": true }, @@ -5123,7 +5134,7 @@ }, { "cell_type": "markdown", - "id": "7edcb842", + "id": "be78dcbb", "metadata": { "editable": true }, @@ -5135,7 +5146,7 @@ }, { "cell_type": "markdown", - "id": "50ec44da", + "id": "0a45fb7f", "metadata": { "editable": true }, @@ -5147,7 +5158,7 @@ }, { "cell_type": "markdown", - "id": "f4010e4a", + "id": "8c6fdb24", "metadata": { "editable": true }, @@ -5159,7 +5170,7 @@ }, { "cell_type": "markdown", - "id": "c0e95eb2", + "id": "a419206b", "metadata": { "editable": true }, @@ -5169,7 +5180,7 @@ }, { "cell_type": "markdown", - "id": "46abefce", + "id": "765c4cbc", "metadata": { "editable": true }, @@ -5181,7 +5192,7 @@ }, { "cell_type": "markdown", - "id": "30544899", + "id": "760d01a8", "metadata": { "editable": true }, @@ -5191,7 +5202,7 @@ }, { "cell_type": "markdown", - "id": "feaeedbe", + "id": "a1f8eeea", "metadata": { "editable": true }, @@ -5208,7 +5219,7 @@ }, { "cell_type": "markdown", - "id": "3a2c0f0e", + "id": "b6fb003a", "metadata": { "editable": true }, @@ -5218,7 +5229,7 @@ }, { "cell_type": "markdown", - "id": "dc33d7ee", + "id": "75e17819", "metadata": { "editable": true }, @@ -5230,7 +5241,7 @@ }, { "cell_type": "markdown", - "id": "0ffb1c46", + "id": "bd29d1d6", "metadata": { "editable": true }, @@ -5240,7 +5251,7 @@ }, { "cell_type": "markdown", - "id": "6cf92025", + "id": "150b7923", "metadata": { "editable": true }, @@ -5252,7 +5263,7 @@ }, { "cell_type": "markdown", - "id": "c21c0c0a", + "id": "b4ffe151", "metadata": { "editable": true }, @@ -5266,7 +5277,7 @@ }, { "cell_type": "markdown", - "id": "48dbf778", + "id": "d3ba3779", "metadata": { "editable": true }, @@ -5278,7 +5289,7 @@ }, { "cell_type": "markdown", - "id": "3f8f57cc", + "id": "512596a9", "metadata": { "editable": true }, @@ -5300,7 +5311,7 @@ }, { "cell_type": "markdown", - "id": "199ea962", + "id": "7559f882", "metadata": { "editable": true }, @@ -5312,7 +5323,7 @@ }, { "cell_type": "markdown", - "id": "f5433e15", + "id": "8256f707", "metadata": { "editable": true }, @@ -5327,7 +5338,7 @@ }, { "cell_type": "markdown", - "id": "ce1511d3", + "id": "4302d47f", "metadata": { "editable": true }, @@ -5339,7 +5350,7 @@ }, { "cell_type": "markdown", - "id": "31981527", + "id": "b74f22fe", "metadata": { "editable": true }, @@ -5351,7 +5362,7 @@ }, { "cell_type": "markdown", - "id": "ccb547be", + "id": "343c9a10", "metadata": { "editable": true }, @@ -5361,7 +5372,7 @@ }, { "cell_type": "markdown", - "id": "c9a6f756", + "id": "915796dd", "metadata": { "editable": true }, @@ -5373,7 +5384,7 @@ }, { "cell_type": "markdown", - "id": "272b8d77", + "id": "0ab8f900", "metadata": { "editable": true }, @@ -5383,7 +5394,7 @@ }, { "cell_type": "markdown", - "id": "4596a399", + "id": "99167153", "metadata": { "editable": true }, @@ -5395,7 +5406,7 @@ }, { "cell_type": "markdown", - "id": "e15dedbf", + "id": "5d580225", "metadata": { "editable": true }, @@ -5405,7 +5416,7 @@ }, { "cell_type": "markdown", - "id": "fe9dab27", + "id": "004215af", "metadata": { "editable": true }, @@ -5417,7 +5428,7 @@ }, { "cell_type": "markdown", - "id": "894497fb", + "id": "afa31b73", "metadata": { "editable": true }, @@ -5434,7 +5445,7 @@ }, { "cell_type": "markdown", - "id": "843092e6", + "id": "ec4c4745", "metadata": { "editable": true }, @@ -5447,7 +5458,7 @@ }, { "cell_type": "markdown", - "id": "4af49ad4", + "id": "88eba984", "metadata": { "editable": true }, @@ -5459,7 +5470,7 @@ }, { "cell_type": "markdown", - "id": "ed536a1b", + "id": "b57779ac", "metadata": { "editable": true }, @@ -5469,7 +5480,7 @@ }, { "cell_type": "markdown", - "id": "9e199dbf", + "id": "a59f4fd5", "metadata": { "editable": true }, @@ -5482,7 +5493,7 @@ }, { "cell_type": "markdown", - "id": "29d15cda", + "id": "c79793c8", "metadata": { "editable": true }, @@ -5492,7 +5503,7 @@ }, { "cell_type": "markdown", - "id": "a9b91956", + "id": "4f039e96", "metadata": { "editable": true }, @@ -5504,7 +5515,7 @@ }, { "cell_type": "markdown", - "id": "f21d533a", + "id": "737c8a25", "metadata": { "editable": true }, @@ -5517,7 +5528,7 @@ }, { "cell_type": "markdown", - "id": "4f218703", + "id": "39aa3132", "metadata": { "editable": true }, @@ -5530,7 +5541,7 @@ }, { "cell_type": "markdown", - "id": "6bc40dcd", + "id": "b6cea5b0", "metadata": { "editable": true }, @@ -5542,7 +5553,7 @@ }, { "cell_type": "markdown", - "id": "fb27a2e6", + "id": "8fa7e834", "metadata": { "editable": true }, @@ -5554,7 +5565,7 @@ }, { "cell_type": "markdown", - "id": "d285f7c7", + "id": "c7b38950", "metadata": { "editable": true }, @@ -5564,7 +5575,7 @@ }, { "cell_type": "markdown", - "id": "57eb55da", + "id": "06878301", "metadata": { "editable": true }, @@ -5577,7 +5588,7 @@ }, { "cell_type": "markdown", - "id": "94f2f142", + "id": "fbbf597c", "metadata": { "editable": true }, @@ -5589,7 +5600,7 @@ }, { "cell_type": "markdown", - "id": "602c806b", + "id": "bf77f613", "metadata": { "editable": true }, @@ -5601,7 +5612,7 @@ }, { "cell_type": "markdown", - "id": "c01a7be8", + "id": "9fc0cbdb", "metadata": { "editable": true }, @@ -5613,7 +5624,7 @@ }, { "cell_type": "markdown", - "id": "6bbd89f5", + "id": "1d556907", "metadata": { "editable": true }, @@ -5625,7 +5636,7 @@ }, { "cell_type": "markdown", - "id": "d5507c15", + "id": "96d0870b", "metadata": { "editable": true }, @@ -5639,7 +5650,7 @@ }, { "cell_type": "markdown", - "id": "d5eb5c88", + "id": "26ef92fd", "metadata": { "editable": true }, @@ -5651,7 +5662,7 @@ }, { "cell_type": "markdown", - "id": "d59c7c4d", + "id": "77971ba3", "metadata": { "editable": true }, @@ -5661,7 +5672,7 @@ }, { "cell_type": "markdown", - "id": "81d0dd7f", + "id": "cd804d0f", "metadata": { "editable": true }, @@ -5673,7 +5684,7 @@ }, { "cell_type": "markdown", - "id": "6807c376", + "id": "aee042df", "metadata": { "editable": true }, @@ -5685,7 +5696,7 @@ }, { "cell_type": "markdown", - "id": "89c94248", + "id": "0ed46bde", "metadata": { "editable": true }, @@ -5697,7 +5708,7 @@ }, { "cell_type": "markdown", - "id": "24f4e74f", + "id": "8c093c7e", "metadata": { "editable": true }, @@ -5709,7 +5720,7 @@ }, { "cell_type": "markdown", - "id": "eac49311", + "id": "62af183d", "metadata": { "editable": true }, @@ -5721,7 +5732,7 @@ }, { "cell_type": "markdown", - "id": "8ad4bc0d", + "id": "30a8db83", "metadata": { "editable": true }, @@ -5740,7 +5751,7 @@ }, { "cell_type": "markdown", - "id": "af4dd7e2", + "id": "01faaa97", "metadata": { "editable": true }, @@ -5752,7 +5763,7 @@ }, { "cell_type": "markdown", - "id": "92b983c6", + "id": "672de282", "metadata": { "editable": true }, @@ -5762,7 +5773,7 @@ }, { "cell_type": "markdown", - "id": "f9d22662", + "id": "fe52bfff", "metadata": { "editable": true }, @@ -5774,7 +5785,7 @@ }, { "cell_type": "markdown", - "id": "0097f163", + "id": "72b7266d", "metadata": { "editable": true }, @@ -5784,7 +5795,7 @@ }, { "cell_type": "markdown", - "id": "073824a0", + "id": "6681e68d", "metadata": { "editable": true }, @@ -5796,7 +5807,7 @@ }, { "cell_type": "markdown", - "id": "399bdc1b", + "id": "538bde1b", "metadata": { "editable": true }, @@ -5808,7 +5819,7 @@ }, { "cell_type": "markdown", - "id": "d9d9fa7a", + "id": "c3a6e71e", "metadata": { "editable": true }, @@ -5824,7 +5835,7 @@ }, { "cell_type": "markdown", - "id": "40b2c49b", + "id": "b59b021d", "metadata": { "editable": true }, @@ -5836,7 +5847,7 @@ }, { "cell_type": "markdown", - "id": "6e949d98", + "id": "ffa266f1", "metadata": { "editable": true }, @@ -5848,7 +5859,7 @@ }, { "cell_type": "markdown", - "id": "5a004c53", + "id": "e8e97705", "metadata": { "editable": true }, @@ -5858,7 +5869,7 @@ }, { "cell_type": "markdown", - "id": "64e1d6bb", + "id": "73c4369a", "metadata": { "editable": true }, @@ -5870,7 +5881,7 @@ }, { "cell_type": "markdown", - "id": "5cec9b54", + "id": "7727b4fd", "metadata": { "editable": true }, @@ -5880,7 +5891,7 @@ }, { "cell_type": "markdown", - "id": "71516f90", + "id": "3a3e1873", "metadata": { "editable": true }, @@ -5892,7 +5903,7 @@ }, { "cell_type": "markdown", - "id": "50e288e9", + "id": "d1a8b030", "metadata": { "editable": true }, @@ -5909,7 +5920,7 @@ }, { "cell_type": "markdown", - "id": "6c82431c", + "id": "b6f25a34", "metadata": { "editable": true }, @@ -5921,7 +5932,7 @@ }, { "cell_type": "markdown", - "id": "5ebc1428", + "id": "c22a3a5a", "metadata": { "editable": true }, @@ -5933,7 +5944,7 @@ }, { "cell_type": "markdown", - "id": "8f198345", + "id": "9e9b60a5", "metadata": { "editable": true }, @@ -5943,7 +5954,7 @@ }, { "cell_type": "markdown", - "id": "443b3080", + "id": "e253cca2", "metadata": { "editable": true }, @@ -5955,7 +5966,7 @@ }, { "cell_type": "markdown", - "id": "e11ec4ea", + "id": "ee43ad3e", "metadata": { "editable": true }, @@ -5965,7 +5976,7 @@ }, { "cell_type": "markdown", - "id": "1d640caa", + "id": "d8f7c38d", "metadata": { "editable": true }, @@ -5977,7 +5988,7 @@ }, { "cell_type": "markdown", - "id": "e023f076", + "id": "a06e5ab5", "metadata": { "editable": true }, @@ -5987,7 +5998,7 @@ }, { "cell_type": "markdown", - "id": "a4204b50", + "id": "bfb15d60", "metadata": { "editable": true }, @@ -5999,7 +6010,7 @@ }, { "cell_type": "markdown", - "id": "7c9cc257", + "id": "4d1210ce", "metadata": { "editable": true }, diff --git a/doc/LectureNotes/_build/jupyter_execute/week35.py b/doc/LectureNotes/_build/jupyter_execute/week35.py index c28b80fa7..42372a43b 100644 --- a/doc/LectureNotes/_build/jupyter_execute/week35.py +++ b/doc/LectureNotes/_build/jupyter_execute/week35.py @@ -28,9 +28,9 @@ # # 1. See lecture notes for week 35 at # -# 2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics +# 2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background) # -# 3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 +# 3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression). # ## Why Linear Regression (aka Ordinary Least Squares and family), repeat from last week # @@ -193,14 +193,13 @@ # We note also that since our design matrix is defined as $\boldsymbol{X}\in # {\mathbb{R}}^{n\times p}$, the product $\boldsymbol{X}^T\boldsymbol{X} \in -# {\mathbb{R}}^{p\times p}$. In the above case we have that $p \ll n$, -# in our case $p=5$ meaning that we end up with inverting a small +# {\mathbb{R}}^{p\times p}$. In most cases we have that $p \ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small # $5\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional # matrices to invert. The methods discussed here and for many other # supervised learning algorithms like classification with logistic # regression or support vector machines, exhibit dimensionalities which # allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix -# $\boldsymbol{X}^T\boldsymbol{X}$. +# $\boldsymbol{X}^T\boldsymbol{X}$. This is discussed on Thursday this week. # # **Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\boldsymbol{X}^T\boldsymbol{X}$? What kind of problems can we expect? @@ -390,7 +389,7 @@ # We note that the design matrix $\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\boldsymbol{\beta}$. # We are now interested in minimizing the cost function with respect to the unknown parameters $\boldsymbol{\beta}$. # -# The mean squared error is a scalar and if we use the results from the last example, we define a new vector +# The mean squared error is a scalar and if we use the results from example three above, we can define a new vector # $$ # \boldsymbol{w}=\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}, @@ -502,8 +501,8 @@ # \tilde{y}_i = \beta_0+\beta_1x_i+\beta_2x_i^2+\beta_3x_i^3+\beta_4x_i^4. # $$ -# we have five predictors/features. The first is the intercept $\beta_0$. The other terms are $\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a -# $p\times n$ matrix $\boldsymbol{X}$. +# we have five predictors/features. The first is the intercept $\beta_0$. The other terms are $\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an +# $n\times p$ matrix $\boldsymbol{X}$. # ## Own code for Ordinary Least Squares # @@ -513,6 +512,18 @@ # matrix inversion to find beta +# First we set up the data +import numpy as np +x = np.random.rand(100) +y = 2.0+5*x*x+0.1*np.random.randn(100) +# and then the design matrix X including the intercept +# The design matrix now as function of a fourth-order polynomial +X = np.zeros((len(x),5)) +X[:,0] = 1.0 +X[:,1] = x +X[:,2] = x**2 +X[:,3] = x**3 +X[:,4] = x**4 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y # and then make the prediction ytilde = X @ beta diff --git a/doc/LectureNotes/week35.ipynb b/doc/LectureNotes/week35.ipynb index a6f7e09a8..5139bd065 100644 --- a/doc/LectureNotes/week35.ipynb +++ b/doc/LectureNotes/week35.ipynb @@ -2,7 +2,7 @@ "cells": [ { "cell_type": "markdown", - "id": "c02138a8", + "id": "0b8a0ca4", "metadata": { "editable": true }, @@ -14,7 +14,7 @@ }, { "cell_type": "markdown", - "id": "919baf6d", + "id": "04b82164", "metadata": { "editable": true }, @@ -27,7 +27,7 @@ }, { "cell_type": "markdown", - "id": "4d5e7179", + "id": "b4569878", "metadata": { "editable": true }, @@ -49,7 +49,7 @@ }, { "cell_type": "markdown", - "id": "3b3f68ae", + "id": "8a651090", "metadata": { "editable": true }, @@ -58,14 +58,14 @@ "\n", "1. See lecture notes for week 35 at \n", "\n", - "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics\n", + "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra and sections 3.1-3.10 on elements of statistics (background)\n", "\n", - "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4" + "3. Hastie, Tibshirani and Friedman, The elements of statistical learning, sections 3.1-3.4 (on relevance for the discussion of linear regression)." ] }, { "cell_type": "markdown", - "id": "3dfade93", + "id": "5ed41979", "metadata": { "editable": true }, @@ -99,7 +99,7 @@ }, { "cell_type": "markdown", - "id": "e7c3b1f5", + "id": "bac89af1", "metadata": { "editable": true }, @@ -115,7 +115,7 @@ }, { "cell_type": "markdown", - "id": "4b7ace0a", + "id": "0c42ff8a", "metadata": { "editable": true }, @@ -127,7 +127,7 @@ }, { "cell_type": "markdown", - "id": "6fa41e5e", + "id": "7a45848d", "metadata": { "editable": true }, @@ -137,7 +137,7 @@ }, { "cell_type": "markdown", - "id": "86724624", + "id": "07ba01c4", "metadata": { "editable": true }, @@ -149,7 +149,7 @@ }, { "cell_type": "markdown", - "id": "f4e16a64", + "id": "335ae896", "metadata": { "editable": true }, @@ -170,7 +170,7 @@ }, { "cell_type": "markdown", - "id": "00f70fc7", + "id": "2e4fcb2a", "metadata": { "editable": true }, @@ -182,7 +182,7 @@ }, { "cell_type": "markdown", - "id": "859d4683", + "id": "12dc5709", "metadata": { "editable": true }, @@ -195,7 +195,7 @@ }, { "cell_type": "markdown", - "id": "ab473c08", + "id": "c3480a1c", "metadata": { "editable": true }, @@ -207,7 +207,7 @@ }, { "cell_type": "markdown", - "id": "a3ccaf3a", + "id": "90295c79", "metadata": { "editable": true }, @@ -219,7 +219,7 @@ }, { "cell_type": "markdown", - "id": "238df8ba", + "id": "219f5e30", "metadata": { "editable": true }, @@ -229,7 +229,7 @@ }, { "cell_type": "markdown", - "id": "e114bb77", + "id": "1abd132b", "metadata": { "editable": true }, @@ -241,7 +241,7 @@ }, { "cell_type": "markdown", - "id": "67833b96", + "id": "ec78644f", "metadata": { "editable": true }, @@ -254,7 +254,7 @@ }, { "cell_type": "markdown", - "id": "68d1debe", + "id": "04c74624", "metadata": { "editable": true }, @@ -266,7 +266,7 @@ }, { "cell_type": "markdown", - "id": "bd4914a7", + "id": "af25e1c6", "metadata": { "editable": true }, @@ -276,7 +276,7 @@ }, { "cell_type": "markdown", - "id": "9d6509bf", + "id": "2125067f", "metadata": { "editable": true }, @@ -288,7 +288,7 @@ }, { "cell_type": "markdown", - "id": "fa97bb74", + "id": "fce7cbce", "metadata": { "editable": true }, @@ -300,7 +300,7 @@ }, { "cell_type": "markdown", - "id": "fdc65871", + "id": "11ee37c4", "metadata": { "editable": true }, @@ -311,7 +311,7 @@ }, { "cell_type": "markdown", - "id": "6fe43d59", + "id": "d1dd61e2", "metadata": { "editable": true }, @@ -323,7 +323,7 @@ }, { "cell_type": "markdown", - "id": "7f0f5fc2", + "id": "f6ac3f1d", "metadata": { "editable": true }, @@ -342,7 +342,7 @@ }, { "cell_type": "markdown", - "id": "8c7ce425", + "id": "e62b9a8e", "metadata": { "editable": true }, @@ -355,7 +355,7 @@ }, { "cell_type": "markdown", - "id": "23cd06a6", + "id": "25564459", "metadata": { "editable": true }, @@ -365,7 +365,7 @@ }, { "cell_type": "markdown", - "id": "d40244fa", + "id": "4826fa0c", "metadata": { "editable": true }, @@ -377,7 +377,7 @@ }, { "cell_type": "markdown", - "id": "003814bc", + "id": "ad91cfea", "metadata": { "editable": true }, @@ -387,7 +387,7 @@ }, { "cell_type": "markdown", - "id": "765cdab4", + "id": "334c4b5c", "metadata": { "editable": true }, @@ -399,7 +399,7 @@ }, { "cell_type": "markdown", - "id": "c10530e6", + "id": "d37069d0", "metadata": { "editable": true }, @@ -409,7 +409,7 @@ }, { "cell_type": "markdown", - "id": "b1ce0925", + "id": "baf3d564", "metadata": { "editable": true }, @@ -421,7 +421,7 @@ }, { "cell_type": "markdown", - "id": "18e850a5", + "id": "50787505", "metadata": { "editable": true }, @@ -432,7 +432,7 @@ }, { "cell_type": "markdown", - "id": "9fe90791", + "id": "7f360392", "metadata": { "editable": true }, @@ -444,7 +444,7 @@ }, { "cell_type": "markdown", - "id": "320e1aef", + "id": "028cf8d2", "metadata": { "editable": true }, @@ -454,7 +454,7 @@ }, { "cell_type": "markdown", - "id": "b6ea67f8", + "id": "3b53ffda", "metadata": { "editable": true }, @@ -466,7 +466,7 @@ }, { "cell_type": "markdown", - "id": "f374c3b2", + "id": "65ff8127", "metadata": { "editable": true }, @@ -476,7 +476,7 @@ }, { "cell_type": "markdown", - "id": "4a879ff3", + "id": "e1d9050c", "metadata": { "editable": true }, @@ -488,28 +488,27 @@ }, { "cell_type": "markdown", - "id": "71ca39b0", + "id": "8e9fce23", "metadata": { "editable": true }, "source": [ "We note also that since our design matrix is defined as $\\boldsymbol{X}\\in\n", "{\\mathbb{R}}^{n\\times p}$, the product $\\boldsymbol{X}^T\\boldsymbol{X} \\in\n", - "{\\mathbb{R}}^{p\\times p}$. In the above case we have that $p \\ll n$,\n", - "in our case $p=5$ meaning that we end up with inverting a small\n", + "{\\mathbb{R}}^{p\\times p}$. In most cases we have that $p \\ll n$. In our example case below we have $p=5$ meaning. We end up with inverting a small\n", "$5\\times 5$ matrix. This is a rather common situation, in many cases we end up with low-dimensional\n", "matrices to invert. The methods discussed here and for many other\n", "supervised learning algorithms like classification with logistic\n", "regression or support vector machines, exhibit dimensionalities which\n", "allow for the usage of direct linear algebra methods such as **LU** decomposition or **Singular Value Decomposition** (SVD) for finding the inverse of the matrix\n", - "$\\boldsymbol{X}^T\\boldsymbol{X}$.\n", + "$\\boldsymbol{X}^T\\boldsymbol{X}$. This is discussed on Thursday this week.\n", "\n", "**Small question**: Do you think the example we have at hand here (the nuclear binding energies) can lead to problems in inverting the matrix $\\boldsymbol{X}^T\\boldsymbol{X}$? What kind of problems can we expect?" ] }, { "cell_type": "markdown", - "id": "0228f656", + "id": "a749fab8", "metadata": { "editable": true }, @@ -536,7 +535,7 @@ }, { "cell_type": "markdown", - "id": "1a6493a2", + "id": "6881afbf", "metadata": { "editable": true }, @@ -548,7 +547,7 @@ }, { "cell_type": "markdown", - "id": "48e5ce45", + "id": "9c2fb92f", "metadata": { "editable": true }, @@ -560,7 +559,7 @@ }, { "cell_type": "markdown", - "id": "2025325b", + "id": "d6d835d5", "metadata": { "editable": true }, @@ -576,7 +575,7 @@ }, { "cell_type": "markdown", - "id": "c4c51c01", + "id": "fed1c91b", "metadata": { "editable": true }, @@ -594,7 +593,7 @@ }, { "cell_type": "markdown", - "id": "5e0ba978", + "id": "7d1dfcf8", "metadata": { "editable": true }, @@ -606,7 +605,7 @@ }, { "cell_type": "markdown", - "id": "4fd652f8", + "id": "bc898e72", "metadata": { "editable": true }, @@ -618,7 +617,7 @@ }, { "cell_type": "markdown", - "id": "694f5db3", + "id": "0f280d98", "metadata": { "editable": true }, @@ -629,7 +628,7 @@ }, { "cell_type": "markdown", - "id": "52a90b73", + "id": "e5f930a8", "metadata": { "editable": true }, @@ -641,7 +640,7 @@ }, { "cell_type": "markdown", - "id": "afab9d50", + "id": "0e0cb390", "metadata": { "editable": true }, @@ -651,7 +650,7 @@ }, { "cell_type": "markdown", - "id": "3dd5ebc2", + "id": "c9a5f03a", "metadata": { "editable": true }, @@ -663,7 +662,7 @@ }, { "cell_type": "markdown", - "id": "d8571fd8", + "id": "932bc862", "metadata": { "editable": true }, @@ -677,7 +676,7 @@ }, { "cell_type": "markdown", - "id": "50f37b40", + "id": "0ca4b0d9", "metadata": { "editable": true }, @@ -689,7 +688,7 @@ }, { "cell_type": "markdown", - "id": "b4e94db7", + "id": "63e2dad1", "metadata": { "editable": true }, @@ -701,7 +700,7 @@ }, { "cell_type": "markdown", - "id": "328ce3c0", + "id": "64ecbf35", "metadata": { "editable": true }, @@ -713,7 +712,7 @@ }, { "cell_type": "markdown", - "id": "3c549113", + "id": "92c17ad7", "metadata": { "editable": true }, @@ -723,7 +722,7 @@ }, { "cell_type": "markdown", - "id": "62a61819", + "id": "2a69e7f9", "metadata": { "editable": true }, @@ -735,7 +734,7 @@ }, { "cell_type": "markdown", - "id": "0d8917d2", + "id": "9b5936bb", "metadata": { "editable": true }, @@ -745,7 +744,7 @@ }, { "cell_type": "markdown", - "id": "d27089b7", + "id": "5bfa3f58", "metadata": { "editable": true }, @@ -757,7 +756,7 @@ }, { "cell_type": "markdown", - "id": "efbc5228", + "id": "38584971", "metadata": { "editable": true }, @@ -771,7 +770,7 @@ }, { "cell_type": "markdown", - "id": "1539dfd3", + "id": "5eaa19f8", "metadata": { "editable": true }, @@ -783,7 +782,7 @@ }, { "cell_type": "markdown", - "id": "56942822", + "id": "3b257952", "metadata": { "editable": true }, @@ -795,7 +794,7 @@ }, { "cell_type": "markdown", - "id": "444a2229", + "id": "cd62fd4a", "metadata": { "editable": true }, @@ -807,7 +806,7 @@ }, { "cell_type": "markdown", - "id": "c19b8e9a", + "id": "63201b06", "metadata": { "editable": true }, @@ -817,7 +816,7 @@ }, { "cell_type": "markdown", - "id": "c3da7629", + "id": "44bcd980", "metadata": { "editable": true }, @@ -829,7 +828,7 @@ }, { "cell_type": "markdown", - "id": "bc0bb616", + "id": "c69cb929", "metadata": { "editable": true }, @@ -839,7 +838,7 @@ }, { "cell_type": "markdown", - "id": "81f469c7", + "id": "d0209cbb", "metadata": { "editable": true }, @@ -851,7 +850,7 @@ }, { "cell_type": "markdown", - "id": "e7dbc066", + "id": "85c2a9b1", "metadata": { "editable": true }, @@ -861,7 +860,7 @@ }, { "cell_type": "markdown", - "id": "e15843da", + "id": "05ab900a", "metadata": { "editable": true }, @@ -873,7 +872,7 @@ }, { "cell_type": "markdown", - "id": "31a95917", + "id": "f8c87433", "metadata": { "editable": true }, @@ -885,7 +884,7 @@ }, { "cell_type": "markdown", - "id": "a3c00979", + "id": "d8d58be6", "metadata": { "editable": true }, @@ -897,7 +896,7 @@ }, { "cell_type": "markdown", - "id": "03321be3", + "id": "cd5493d2", "metadata": { "editable": true }, @@ -912,7 +911,7 @@ }, { "cell_type": "markdown", - "id": "59854eb8", + "id": "69c57766", "metadata": { "editable": true }, @@ -924,7 +923,7 @@ }, { "cell_type": "markdown", - "id": "1f475d05", + "id": "1592a99e", "metadata": { "editable": true }, @@ -934,7 +933,7 @@ }, { "cell_type": "markdown", - "id": "256476a9", + "id": "18378073", "metadata": { "editable": true }, @@ -946,7 +945,7 @@ }, { "cell_type": "markdown", - "id": "f23c2136", + "id": "0833d084", "metadata": { "editable": true }, @@ -956,7 +955,7 @@ }, { "cell_type": "markdown", - "id": "afeafdcb", + "id": "07f35361", "metadata": { "editable": true }, @@ -968,7 +967,7 @@ }, { "cell_type": "markdown", - "id": "4ed3eead", + "id": "3a96219f", "metadata": { "editable": true }, @@ -978,7 +977,7 @@ }, { "cell_type": "markdown", - "id": "9674c619", + "id": "2f0e8ebf", "metadata": { "editable": true }, @@ -990,7 +989,7 @@ }, { "cell_type": "markdown", - "id": "82ae15ae", + "id": "323e987a", "metadata": { "editable": true }, @@ -1002,7 +1001,7 @@ }, { "cell_type": "markdown", - "id": "688a7168", + "id": "4cfaf887", "metadata": { "editable": true }, @@ -1014,7 +1013,7 @@ }, { "cell_type": "markdown", - "id": "01f03acf", + "id": "6288d6fe", "metadata": { "editable": true }, @@ -1024,7 +1023,7 @@ }, { "cell_type": "markdown", - "id": "a30bd0f2", + "id": "878ffa04", "metadata": { "editable": true }, @@ -1036,7 +1035,7 @@ }, { "cell_type": "markdown", - "id": "5593e084", + "id": "efe2818a", "metadata": { "editable": true }, @@ -1044,12 +1043,12 @@ "We note that the design matrix $\\boldsymbol{X}$ does not depend on the unknown parameters defined by the vector $\\boldsymbol{\\beta}$.\n", "We are now interested in minimizing the cost function with respect to the unknown parameters $\\boldsymbol{\\beta}$.\n", "\n", - "The mean squared error is a scalar and if we use the results from the last example, we define a new vector" + "The mean squared error is a scalar and if we use the results from example three above, we can define a new vector" ] }, { "cell_type": "markdown", - "id": "0ecc87b7", + "id": "016b7005", "metadata": { "editable": true }, @@ -1061,7 +1060,7 @@ }, { "cell_type": "markdown", - "id": "d87cf1be", + "id": "47ccce3f", "metadata": { "editable": true }, @@ -1071,7 +1070,7 @@ }, { "cell_type": "markdown", - "id": "c8ac2b85", + "id": "e25deaf7", "metadata": { "editable": true }, @@ -1083,7 +1082,7 @@ }, { "cell_type": "markdown", - "id": "2a1896de", + "id": "ce050c2b", "metadata": { "editable": true }, @@ -1093,7 +1092,7 @@ }, { "cell_type": "markdown", - "id": "f07be920", + "id": "b65cd933", "metadata": { "editable": true }, @@ -1105,7 +1104,7 @@ }, { "cell_type": "markdown", - "id": "8351f7a8", + "id": "6bf5c873", "metadata": { "editable": true }, @@ -1115,7 +1114,7 @@ }, { "cell_type": "markdown", - "id": "b321536f", + "id": "2ad8304d", "metadata": { "editable": true }, @@ -1127,7 +1126,7 @@ }, { "cell_type": "markdown", - "id": "7dea2f6e", + "id": "86e780ca", "metadata": { "editable": true }, @@ -1137,7 +1136,7 @@ }, { "cell_type": "markdown", - "id": "fc8335b3", + "id": "8b76d3f7", "metadata": { "editable": true }, @@ -1149,7 +1148,7 @@ }, { "cell_type": "markdown", - "id": "690c48cc", + "id": "3a1a52ac", "metadata": { "editable": true }, @@ -1159,7 +1158,7 @@ }, { "cell_type": "markdown", - "id": "dbf4e99e", + "id": "da435327", "metadata": { "editable": true }, @@ -1171,7 +1170,7 @@ }, { "cell_type": "markdown", - "id": "20066c64", + "id": "f44db4df", "metadata": { "editable": true }, @@ -1183,7 +1182,7 @@ }, { "cell_type": "markdown", - "id": "512705d5", + "id": "819c0e66", "metadata": { "editable": true }, @@ -1195,7 +1194,7 @@ }, { "cell_type": "markdown", - "id": "898589c9", + "id": "510a011e", "metadata": { "editable": true }, @@ -1207,7 +1206,7 @@ }, { "cell_type": "markdown", - "id": "04fdb8da", + "id": "fcd571ce", "metadata": { "editable": true }, @@ -1219,7 +1218,7 @@ }, { "cell_type": "markdown", - "id": "fb0d6555", + "id": "b2515b95", "metadata": { "editable": true }, @@ -1235,7 +1234,7 @@ }, { "cell_type": "markdown", - "id": "61ca107d", + "id": "1ccfd4c9", "metadata": { "editable": true }, @@ -1247,7 +1246,7 @@ }, { "cell_type": "markdown", - "id": "71be4e48", + "id": "6f868746", "metadata": { "editable": true }, @@ -1257,7 +1256,7 @@ }, { "cell_type": "markdown", - "id": "9fc474c3", + "id": "541d352f", "metadata": { "editable": true }, @@ -1269,7 +1268,7 @@ }, { "cell_type": "markdown", - "id": "23a27406", + "id": "92874dd6", "metadata": { "editable": true }, @@ -1287,7 +1286,7 @@ }, { "cell_type": "markdown", - "id": "9aa37656", + "id": "19159246", "metadata": { "editable": true }, @@ -1299,7 +1298,7 @@ }, { "cell_type": "markdown", - "id": "1458e79a", + "id": "ed39ed6b", "metadata": { "editable": true }, @@ -1311,7 +1310,7 @@ }, { "cell_type": "markdown", - "id": "c3fcfcdf", + "id": "8adde242", "metadata": { "editable": true }, @@ -1321,7 +1320,7 @@ }, { "cell_type": "markdown", - "id": "9f169fad", + "id": "f1e3f1c2", "metadata": { "editable": true }, @@ -1333,7 +1332,7 @@ }, { "cell_type": "markdown", - "id": "81e7723a", + "id": "39960290", "metadata": { "editable": true }, @@ -1343,7 +1342,7 @@ }, { "cell_type": "markdown", - "id": "b2551cd1", + "id": "db52e3e6", "metadata": { "editable": true }, @@ -1355,7 +1354,7 @@ }, { "cell_type": "markdown", - "id": "601003c8", + "id": "e3d6ac8b", "metadata": { "editable": true }, @@ -1365,7 +1364,7 @@ }, { "cell_type": "markdown", - "id": "1f2fdfa6", + "id": "e4b30664", "metadata": { "editable": true }, @@ -1379,7 +1378,7 @@ }, { "cell_type": "markdown", - "id": "ea1518ad", + "id": "4cbc0c2c", "metadata": { "editable": true }, @@ -1391,18 +1390,18 @@ }, { "cell_type": "markdown", - "id": "758b843b", + "id": "5eebcce6", "metadata": { "editable": true }, "source": [ - "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is a \n", - "$p\\times n$ matrix $\\boldsymbol{X}$." + "we have five predictors/features. The first is the intercept $\\beta_0$. The other terms are $\\beta_i$ with $i=1,2,3,4$. Furthermore we have $n$ entries for each predictor. It means that our design matrix is an \n", + "$n\\times p$ matrix $\\boldsymbol{X}$." ] }, { "cell_type": "markdown", - "id": "24715db8", + "id": "47074a74", "metadata": { "editable": true }, @@ -1415,7 +1414,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "f8b57463", + "id": "ebc41f6f", "metadata": { "collapsed": false, "editable": true @@ -1423,6 +1422,18 @@ "outputs": [], "source": [ "# matrix inversion to find beta\n", + "# First we set up the data\n", + "import numpy as np\n", + "x = np.random.rand(100)\n", + "y = 2.0+5*x*x+0.1*np.random.randn(100)\n", + "# and then the design matrix X including the intercept\n", + "# The design matrix now as function of a fourth-order polynomial\n", + "X = np.zeros((len(x),5))\n", + "X[:,0] = 1.0\n", + "X[:,1] = x\n", + "X[:,2] = x**2\n", + "X[:,3] = x**3\n", + "X[:,4] = x**4\n", "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n", "# and then make the prediction\n", "ytilde = X @ beta" @@ -1430,7 +1441,7 @@ }, { "cell_type": "markdown", - "id": "70c97572", + "id": "8ea9846d", "metadata": { "editable": true }, @@ -1441,7 +1452,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "1aa95889", + "id": "e996841a", "metadata": { "collapsed": false, "editable": true @@ -1454,7 +1465,7 @@ }, { "cell_type": "markdown", - "id": "e250b2e9", + "id": "39f01129", "metadata": { "editable": true }, @@ -1468,7 +1479,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "a86bd577", + "id": "12e52de5", "metadata": { "collapsed": false, "editable": true @@ -1481,7 +1492,7 @@ }, { "cell_type": "markdown", - "id": "d6ccd7b8", + "id": "8a0cc628", "metadata": { "editable": true }, @@ -1492,7 +1503,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "fb6f7c19", + "id": "8c1508b7", "metadata": { "collapsed": false, "editable": true @@ -1504,7 +1515,7 @@ }, { "cell_type": "markdown", - "id": "eca18427", + "id": "a3f2f3b1", "metadata": { "editable": true }, @@ -1515,7 +1526,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "0fd2b39c", + "id": "334780f0", "metadata": { "collapsed": false, "editable": true @@ -1531,7 +1542,7 @@ }, { "cell_type": "markdown", - "id": "db3fe7d9", + "id": "1f7cfede", "metadata": { "editable": true }, @@ -1542,7 +1553,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "9f59c14e", + "id": "1579c506", "metadata": { "collapsed": false, "editable": true @@ -1556,7 +1567,7 @@ }, { "cell_type": "markdown", - "id": "2fdd5915", + "id": "88c6c130", "metadata": { "editable": true }, @@ -1577,7 +1588,7 @@ }, { "cell_type": "markdown", - "id": "9863e063", + "id": "af0baf7f", "metadata": { "editable": true }, @@ -1588,7 +1599,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "85fc6a78", + "id": "89aa39b5", "metadata": { "collapsed": false, "editable": true @@ -1641,7 +1652,7 @@ }, { "cell_type": "markdown", - "id": "6671297c", + "id": "1d86c024", "metadata": { "editable": true }, @@ -1652,7 +1663,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "810f32f7", + "id": "2ba16c25", "metadata": { "collapsed": false, "editable": true @@ -1677,7 +1688,7 @@ }, { "cell_type": "markdown", - "id": "1ca5e45d", + "id": "8e363f00", "metadata": { "editable": true }, @@ -1689,7 +1700,7 @@ }, { "cell_type": "markdown", - "id": "b6530670", + "id": "d5ef1be0", "metadata": { "editable": true }, @@ -1718,7 +1729,7 @@ }, { "cell_type": "markdown", - "id": "4be4158c", + "id": "7165f38b", "metadata": { "editable": true }, @@ -1743,7 +1754,7 @@ }, { "cell_type": "markdown", - "id": "aa6f9ff6", + "id": "baea18ad", "metadata": { "editable": true }, @@ -1763,7 +1774,7 @@ }, { "cell_type": "markdown", - "id": "856dee18", + "id": "830be0f9", "metadata": { "editable": true }, @@ -1790,7 +1801,7 @@ }, { "cell_type": "markdown", - "id": "3497598c", + "id": "72473f2c", "metadata": { "editable": true }, @@ -1803,7 +1814,7 @@ }, { "cell_type": "markdown", - "id": "fe24fd3f", + "id": "ad92637f", "metadata": { "editable": true }, @@ -1815,7 +1826,7 @@ }, { "cell_type": "markdown", - "id": "55bccc6e", + "id": "928978ac", "metadata": { "editable": true }, @@ -1826,7 +1837,7 @@ }, { "cell_type": "markdown", - "id": "e01c4d09", + "id": "31b09189", "metadata": { "editable": true }, @@ -1842,7 +1853,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "6098d5d9", + "id": "c09e81a9", "metadata": { "collapsed": false, "editable": true @@ -1876,7 +1887,7 @@ }, { "cell_type": "markdown", - "id": "653340bf", + "id": "8ffeb0f1", "metadata": { "editable": true }, @@ -1886,7 +1897,7 @@ }, { "cell_type": "markdown", - "id": "673c8917", + "id": "2b1f32da", "metadata": { "editable": true }, @@ -1901,7 +1912,7 @@ }, { "cell_type": "markdown", - "id": "4f69865b", + "id": "3c2f9703", "metadata": { "editable": true }, @@ -1913,7 +1924,7 @@ }, { "cell_type": "markdown", - "id": "f39433a0", + "id": "15382f87", "metadata": { "editable": true }, @@ -1923,7 +1934,7 @@ }, { "cell_type": "markdown", - "id": "f2f5cb76", + "id": "a04fe6a8", "metadata": { "editable": true }, @@ -1939,7 +1950,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "12dbcdcb", + "id": "de5a8426", "metadata": { "collapsed": false, "editable": true @@ -1956,7 +1967,7 @@ }, { "cell_type": "markdown", - "id": "38cb3698", + "id": "815c116c", "metadata": { "editable": true }, @@ -1969,7 +1980,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "a0cd549b", + "id": "c9049720", "metadata": { "collapsed": false, "editable": true @@ -2016,7 +2027,7 @@ }, { "cell_type": "markdown", - "id": "b4d2ab87", + "id": "c6fb36f0", "metadata": { "editable": true }, @@ -2027,7 +2038,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "9f5d7996", + "id": "d02afd53", "metadata": { "collapsed": false, "editable": true @@ -2130,7 +2141,7 @@ }, { "cell_type": "markdown", - "id": "5ea41ab9", + "id": "4d3d0315", "metadata": { "editable": true }, @@ -2158,7 +2169,7 @@ }, { "cell_type": "markdown", - "id": "ce0afdb4", + "id": "1bf26037", "metadata": { "editable": true }, @@ -2193,7 +2204,7 @@ }, { "cell_type": "markdown", - "id": "ad92f42b", + "id": "1a9303d9", "metadata": { "editable": true }, @@ -2207,7 +2218,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "fae66265", + "id": "add41c28", "metadata": { "collapsed": false, "editable": true @@ -2232,7 +2243,7 @@ }, { "cell_type": "markdown", - "id": "1bf672d8", + "id": "cbad62c2", "metadata": { "editable": true }, @@ -2248,7 +2259,7 @@ }, { "cell_type": "markdown", - "id": "30834557", + "id": "77b3f4a0", "metadata": { "editable": true }, @@ -2260,7 +2271,7 @@ }, { "cell_type": "markdown", - "id": "5c05a427", + "id": "2054bdc7", "metadata": { "editable": true }, @@ -2275,7 +2286,7 @@ }, { "cell_type": "markdown", - "id": "902391ce", + "id": "d7f64d04", "metadata": { "editable": true }, @@ -2287,7 +2298,7 @@ }, { "cell_type": "markdown", - "id": "a68a6228", + "id": "6ebe45a1", "metadata": { "editable": true }, @@ -2297,7 +2308,7 @@ }, { "cell_type": "markdown", - "id": "212cf295", + "id": "8cc21b6a", "metadata": { "editable": true }, @@ -2309,7 +2320,7 @@ }, { "cell_type": "markdown", - "id": "e7db36e1", + "id": "71b87ff9", "metadata": { "editable": true }, @@ -2319,7 +2330,7 @@ }, { "cell_type": "markdown", - "id": "5d7f590c", + "id": "b15ee5de", "metadata": { "editable": true }, @@ -2331,7 +2342,7 @@ }, { "cell_type": "markdown", - "id": "bbf7c89a", + "id": "56e808e5", "metadata": { "editable": true }, @@ -2344,7 +2355,7 @@ }, { "cell_type": "markdown", - "id": "05692612", + "id": "59b1fcf5", "metadata": { "editable": true }, @@ -2356,7 +2367,7 @@ }, { "cell_type": "markdown", - "id": "3dc53330", + "id": "fe9af8e4", "metadata": { "editable": true }, @@ -2366,7 +2377,7 @@ }, { "cell_type": "markdown", - "id": "562d655d", + "id": "39d0535d", "metadata": { "editable": true }, @@ -2378,7 +2389,7 @@ }, { "cell_type": "markdown", - "id": "528c869d", + "id": "b05e083e", "metadata": { "editable": true }, @@ -2388,7 +2399,7 @@ }, { "cell_type": "markdown", - "id": "82a20d9e", + "id": "255d5f87", "metadata": { "editable": true }, @@ -2400,7 +2411,7 @@ }, { "cell_type": "markdown", - "id": "5e37316e", + "id": "863f9c03", "metadata": { "editable": true }, @@ -2410,7 +2421,7 @@ }, { "cell_type": "markdown", - "id": "90c44e36", + "id": "fe1719c8", "metadata": { "editable": true }, @@ -2422,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "be72a7d3", + "id": "11bf9275", "metadata": { "editable": true }, @@ -2432,7 +2443,7 @@ }, { "cell_type": "markdown", - "id": "0a435bf5", + "id": "58a73e37", "metadata": { "editable": true }, @@ -2444,7 +2455,7 @@ }, { "cell_type": "markdown", - "id": "a67864cb", + "id": "2ee6e0ab", "metadata": { "editable": true }, @@ -2454,7 +2465,7 @@ }, { "cell_type": "markdown", - "id": "9d8602b7", + "id": "4aa55bc2", "metadata": { "editable": true }, @@ -2466,7 +2477,7 @@ }, { "cell_type": "markdown", - "id": "32caef0d", + "id": "e7dd951c", "metadata": { "editable": true }, @@ -2476,7 +2487,7 @@ }, { "cell_type": "markdown", - "id": "f24465a9", + "id": "b0b56f12", "metadata": { "editable": true }, @@ -2488,7 +2499,7 @@ }, { "cell_type": "markdown", - "id": "f64212da", + "id": "2fa4cfc7", "metadata": { "editable": true }, @@ -2500,7 +2511,7 @@ }, { "cell_type": "markdown", - "id": "8d0bf5ae", + "id": "cc6fe7e2", "metadata": { "editable": true }, @@ -2512,7 +2523,7 @@ }, { "cell_type": "markdown", - "id": "e8d36ff8", + "id": "d121682c", "metadata": { "editable": true }, @@ -2525,7 +2536,7 @@ }, { "cell_type": "markdown", - "id": "aea58aac", + "id": "94aef78e", "metadata": { "editable": true }, @@ -2537,7 +2548,7 @@ }, { "cell_type": "markdown", - "id": "bc4b6f91", + "id": "8d595a8d", "metadata": { "editable": true }, @@ -2547,7 +2558,7 @@ }, { "cell_type": "markdown", - "id": "349b03e2", + "id": "67d81f01", "metadata": { "editable": true }, @@ -2561,7 +2572,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "5cfc62db", + "id": "39da9a38", "metadata": { "collapsed": false, "editable": true @@ -2658,7 +2669,7 @@ }, { "cell_type": "markdown", - "id": "64c5fa5b", + "id": "257ce2f2", "metadata": { "editable": true }, @@ -2679,7 +2690,7 @@ }, { "cell_type": "markdown", - "id": "67496f35", + "id": "3769097f", "metadata": { "editable": true }, @@ -2691,7 +2702,7 @@ }, { "cell_type": "markdown", - "id": "ceae31e0", + "id": "82fc22ad", "metadata": { "editable": true }, @@ -2701,7 +2712,7 @@ }, { "cell_type": "markdown", - "id": "743c68c8", + "id": "dea71d52", "metadata": { "editable": true }, @@ -2713,7 +2724,7 @@ }, { "cell_type": "markdown", - "id": "4af68378", + "id": "113f6e87", "metadata": { "editable": true }, @@ -2723,7 +2734,7 @@ }, { "cell_type": "markdown", - "id": "8228af90", + "id": "1e3b78e5", "metadata": { "editable": true }, @@ -2735,7 +2746,7 @@ }, { "cell_type": "markdown", - "id": "0a4285bc", + "id": "d936547b", "metadata": { "editable": true }, @@ -2751,7 +2762,7 @@ }, { "cell_type": "markdown", - "id": "406af60a", + "id": "7c57b1e6", "metadata": { "editable": true }, @@ -2795,7 +2806,7 @@ }, { "cell_type": "markdown", - "id": "0f05d1e1", + "id": "41d26913", "metadata": { "editable": true }, @@ -2807,7 +2818,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "70f8a406", + "id": "bf2d2a51", "metadata": { "collapsed": false, "editable": true @@ -2823,7 +2834,7 @@ }, { "cell_type": "markdown", - "id": "10a6162c", + "id": "d7d26ad9", "metadata": { "editable": true }, @@ -2834,7 +2845,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "be236243", + "id": "7b725195", "metadata": { "collapsed": false, "editable": true @@ -2852,7 +2863,7 @@ }, { "cell_type": "markdown", - "id": "88a7376f", + "id": "4b578dbb", "metadata": { "editable": true }, @@ -2863,7 +2874,7 @@ { "cell_type": "code", "execution_count": 17, - "id": "ec268779", + "id": "670a51a0", "metadata": { "collapsed": false, "editable": true @@ -2877,7 +2888,7 @@ }, { "cell_type": "markdown", - "id": "b45f3733", + "id": "ad921def", "metadata": { "editable": true }, @@ -2888,7 +2899,7 @@ { "cell_type": "code", "execution_count": 18, - "id": "8b9423c6", + "id": "2d0a0044", "metadata": { "collapsed": false, "editable": true @@ -2901,7 +2912,7 @@ }, { "cell_type": "markdown", - "id": "1061cb1f", + "id": "8ec52805", "metadata": { "editable": true }, @@ -2912,7 +2923,7 @@ { "cell_type": "code", "execution_count": 19, - "id": "00d79fad", + "id": "a7648df4", "metadata": { "collapsed": false, "editable": true @@ -2929,7 +2940,7 @@ }, { "cell_type": "markdown", - "id": "18871b23", + "id": "8e911148", "metadata": { "editable": true }, @@ -2940,7 +2951,7 @@ { "cell_type": "code", "execution_count": 20, - "id": "abfd55f9", + "id": "4420c7f9", "metadata": { "collapsed": false, "editable": true @@ -2956,7 +2967,7 @@ }, { "cell_type": "markdown", - "id": "231f35c5", + "id": "b7580acd", "metadata": { "editable": true }, @@ -2967,7 +2978,7 @@ { "cell_type": "code", "execution_count": 21, - "id": "5bb68d00", + "id": "2a0930b1", "metadata": { "collapsed": false, "editable": true @@ -2991,7 +3002,7 @@ }, { "cell_type": "markdown", - "id": "9c2f9f6b", + "id": "49423ebd", "metadata": { "editable": true }, @@ -3002,7 +3013,7 @@ { "cell_type": "code", "execution_count": 22, - "id": "df3cc855", + "id": "5841e8ee", "metadata": { "collapsed": false, "editable": true @@ -3015,7 +3026,7 @@ }, { "cell_type": "markdown", - "id": "da67457b", + "id": "de1c7a1d", "metadata": { "editable": true }, @@ -3026,7 +3037,7 @@ { "cell_type": "code", "execution_count": 23, - "id": "f9784866", + "id": "8d3ce90a", "metadata": { "collapsed": false, "editable": true @@ -3046,7 +3057,7 @@ }, { "cell_type": "markdown", - "id": "3429e67f", + "id": "b243c849", "metadata": { "editable": true }, @@ -3057,7 +3068,7 @@ { "cell_type": "code", "execution_count": 24, - "id": "9ec7ba47", + "id": "83abda14", "metadata": { "collapsed": false, "editable": true @@ -3100,7 +3111,7 @@ { "cell_type": "code", "execution_count": 25, - "id": "0c0be681", + "id": "c1bda3a1", "metadata": { "collapsed": false, "editable": true @@ -3115,7 +3126,7 @@ }, { "cell_type": "markdown", - "id": "ed12f12d", + "id": "494e99e1", "metadata": { "editable": true }, @@ -3125,7 +3136,7 @@ }, { "cell_type": "markdown", - "id": "50c91ade", + "id": "d496f2b6", "metadata": { "editable": true }, @@ -3139,7 +3150,7 @@ }, { "cell_type": "markdown", - "id": "682a501b", + "id": "617544ab", "metadata": { "editable": true }, @@ -3151,7 +3162,7 @@ }, { "cell_type": "markdown", - "id": "057bbc5d", + "id": "deb3a5c0", "metadata": { "editable": true }, @@ -3163,7 +3174,7 @@ }, { "cell_type": "markdown", - "id": "cddb899f", + "id": "1e29d04b", "metadata": { "editable": true }, @@ -3175,7 +3186,7 @@ }, { "cell_type": "markdown", - "id": "ab325ebc", + "id": "e7f06cd3", "metadata": { "editable": true }, @@ -3185,7 +3196,7 @@ }, { "cell_type": "markdown", - "id": "095ebc5b", + "id": "61fbb179", "metadata": { "editable": true }, @@ -3197,7 +3208,7 @@ }, { "cell_type": "markdown", - "id": "08e642f6", + "id": "715d0808", "metadata": { "editable": true }, @@ -3207,7 +3218,7 @@ }, { "cell_type": "markdown", - "id": "da5c1625", + "id": "e9af9b81", "metadata": { "editable": true }, @@ -3219,7 +3230,7 @@ }, { "cell_type": "markdown", - "id": "2bf04890", + "id": "ce0a6494", "metadata": { "editable": true }, @@ -3230,7 +3241,7 @@ }, { "cell_type": "markdown", - "id": "5ba172cc", + "id": "a9680e79", "metadata": { "editable": true }, @@ -3242,7 +3253,7 @@ }, { "cell_type": "markdown", - "id": "7d1f0b59", + "id": "77b2c7f1", "metadata": { "editable": true }, @@ -3254,7 +3265,7 @@ }, { "cell_type": "markdown", - "id": "467e185d", + "id": "00bc0981", "metadata": { "editable": true }, @@ -3264,7 +3275,7 @@ }, { "cell_type": "markdown", - "id": "74822363", + "id": "a5870049", "metadata": { "editable": true }, @@ -3276,7 +3287,7 @@ }, { "cell_type": "markdown", - "id": "5414a1d2", + "id": "8f53c1ca", "metadata": { "editable": true }, @@ -3288,7 +3299,7 @@ }, { "cell_type": "markdown", - "id": "08ce1402", + "id": "ac5e957f", "metadata": { "editable": true }, @@ -3298,7 +3309,7 @@ }, { "cell_type": "markdown", - "id": "7bbb949f", + "id": "b949e361", "metadata": { "editable": true }, @@ -3310,7 +3321,7 @@ }, { "cell_type": "markdown", - "id": "4007519d", + "id": "bf9ebc2b", "metadata": { "editable": true }, @@ -3320,7 +3331,7 @@ }, { "cell_type": "markdown", - "id": "45061e33", + "id": "8f15c266", "metadata": { "editable": true }, @@ -3332,7 +3343,7 @@ }, { "cell_type": "markdown", - "id": "f5afee48", + "id": "4f791b65", "metadata": { "editable": true }, @@ -3342,7 +3353,7 @@ }, { "cell_type": "markdown", - "id": "6a3d6440", + "id": "e8017b1c", "metadata": { "editable": true }, @@ -3382,7 +3393,7 @@ }, { "cell_type": "markdown", - "id": "ef8c39b7", + "id": "2996bc1f", "metadata": { "editable": true }, @@ -3399,7 +3410,7 @@ }, { "cell_type": "markdown", - "id": "6edd77ca", + "id": "2b6b0946", "metadata": { "editable": true }, @@ -3422,7 +3433,7 @@ }, { "cell_type": "markdown", - "id": "cba40ade", + "id": "93e7e61f", "metadata": { "editable": true }, @@ -3439,7 +3450,7 @@ }, { "cell_type": "markdown", - "id": "20f096a2", + "id": "fe419623", "metadata": { "editable": true }, @@ -3458,7 +3469,7 @@ }, { "cell_type": "markdown", - "id": "49f6f27c", + "id": "e2140a9e", "metadata": { "editable": true }, @@ -3469,7 +3480,7 @@ }, { "cell_type": "markdown", - "id": "577f3338", + "id": "ffcb729e", "metadata": { "editable": true }, @@ -3481,7 +3492,7 @@ }, { "cell_type": "markdown", - "id": "f456df81", + "id": "5140ddae", "metadata": { "editable": true }, @@ -3499,7 +3510,7 @@ }, { "cell_type": "markdown", - "id": "ed7db14f", + "id": "7bb98440", "metadata": { "editable": true }, @@ -3515,7 +3526,7 @@ }, { "cell_type": "markdown", - "id": "a876de90", + "id": "5611679a", "metadata": { "editable": true }, @@ -3527,7 +3538,7 @@ }, { "cell_type": "markdown", - "id": "3b2eee8d", + "id": "f32c0eb4", "metadata": { "editable": true }, @@ -3537,7 +3548,7 @@ }, { "cell_type": "markdown", - "id": "910d3f09", + "id": "415f02af", "metadata": { "editable": true }, @@ -3552,7 +3563,7 @@ }, { "cell_type": "markdown", - "id": "94bd4c00", + "id": "c28567b8", "metadata": { "editable": true }, @@ -3564,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "56d91a09", + "id": "936af9ce", "metadata": { "editable": true }, @@ -3574,7 +3585,7 @@ }, { "cell_type": "markdown", - "id": "8e59a435", + "id": "308f2aa4", "metadata": { "editable": true }, @@ -3586,7 +3597,7 @@ }, { "cell_type": "markdown", - "id": "9c2e6b9f", + "id": "d74a5600", "metadata": { "editable": true }, @@ -3596,7 +3607,7 @@ }, { "cell_type": "markdown", - "id": "e7188a39", + "id": "3d5f8482", "metadata": { "editable": true }, @@ -3608,7 +3619,7 @@ }, { "cell_type": "markdown", - "id": "daac715a", + "id": "b4eaa6e3", "metadata": { "editable": true }, @@ -3620,7 +3631,7 @@ }, { "cell_type": "markdown", - "id": "66322f83", + "id": "0b821498", "metadata": { "editable": true }, @@ -3635,7 +3646,7 @@ }, { "cell_type": "markdown", - "id": "a03be784", + "id": "d38da336", "metadata": { "editable": true }, @@ -3646,7 +3657,7 @@ }, { "cell_type": "markdown", - "id": "ff3964a4", + "id": "98b0ae03", "metadata": { "editable": true }, @@ -3666,7 +3677,7 @@ }, { "cell_type": "markdown", - "id": "04d411cc", + "id": "d4ba8f69", "metadata": { "editable": true }, @@ -3678,7 +3689,7 @@ }, { "cell_type": "markdown", - "id": "bf2ae91a", + "id": "52bb4456", "metadata": { "editable": true }, @@ -3688,7 +3699,7 @@ }, { "cell_type": "markdown", - "id": "843d113e", + "id": "6ec33a01", "metadata": { "editable": true }, @@ -3700,7 +3711,7 @@ }, { "cell_type": "markdown", - "id": "f59af5f8", + "id": "b39cac82", "metadata": { "editable": true }, @@ -3729,7 +3740,7 @@ }, { "cell_type": "markdown", - "id": "22f1ed07", + "id": "d2595a2e", "metadata": { "editable": true }, @@ -3756,7 +3767,7 @@ }, { "cell_type": "markdown", - "id": "69efb08d", + "id": "427fb028", "metadata": { "editable": true }, @@ -3767,7 +3778,7 @@ { "cell_type": "code", "execution_count": 26, - "id": "640d3bdf", + "id": "0bbd4699", "metadata": { "collapsed": false, "editable": true @@ -3807,7 +3818,7 @@ }, { "cell_type": "markdown", - "id": "98d61790", + "id": "2430babb", "metadata": { "editable": true }, @@ -3824,7 +3835,7 @@ }, { "cell_type": "markdown", - "id": "95b1ffa6", + "id": "4761f74a", "metadata": { "editable": true }, @@ -3847,7 +3858,7 @@ }, { "cell_type": "markdown", - "id": "4605f137", + "id": "ffe52fdc", "metadata": { "editable": true }, @@ -3861,7 +3872,7 @@ }, { "cell_type": "markdown", - "id": "3125bc92", + "id": "c810c7eb", "metadata": { "editable": true }, @@ -3880,7 +3891,7 @@ }, { "cell_type": "markdown", - "id": "32cf3a9e", + "id": "fa17e262", "metadata": { "editable": true }, @@ -3890,7 +3901,7 @@ }, { "cell_type": "markdown", - "id": "573acb2d", + "id": "e7411ce2", "metadata": { "editable": true }, @@ -3902,7 +3913,7 @@ }, { "cell_type": "markdown", - "id": "22320566", + "id": "40d9144e", "metadata": { "editable": true }, @@ -3916,7 +3927,7 @@ }, { "cell_type": "markdown", - "id": "af0ee993", + "id": "28cd0268", "metadata": { "editable": true }, @@ -3928,7 +3939,7 @@ }, { "cell_type": "markdown", - "id": "0213ed87", + "id": "e72c1658", "metadata": { "editable": true }, @@ -3938,7 +3949,7 @@ }, { "cell_type": "markdown", - "id": "06266ee6", + "id": "9e8284ec", "metadata": { "editable": true }, @@ -3950,7 +3961,7 @@ }, { "cell_type": "markdown", - "id": "23e27fdf", + "id": "39c798c9", "metadata": { "editable": true }, @@ -3967,7 +3978,7 @@ }, { "cell_type": "markdown", - "id": "2539e582", + "id": "93cf6fd3", "metadata": { "editable": true }, @@ -3977,7 +3988,7 @@ }, { "cell_type": "markdown", - "id": "bef2cf95", + "id": "c99559d1", "metadata": { "editable": true }, @@ -3993,7 +4004,7 @@ }, { "cell_type": "markdown", - "id": "fe385753", + "id": "199187d7", "metadata": { "editable": true }, @@ -4003,7 +4014,7 @@ }, { "cell_type": "markdown", - "id": "cc270435", + "id": "164f8c17", "metadata": { "editable": true }, @@ -4019,7 +4030,7 @@ }, { "cell_type": "markdown", - "id": "31932192", + "id": "5e9fd974", "metadata": { "editable": true }, @@ -4029,7 +4040,7 @@ }, { "cell_type": "markdown", - "id": "a972a944", + "id": "8e07a82d", "metadata": { "editable": true }, @@ -4045,7 +4056,7 @@ }, { "cell_type": "markdown", - "id": "d6ef62ee", + "id": "b0bf5db6", "metadata": { "editable": true }, @@ -4055,7 +4066,7 @@ }, { "cell_type": "markdown", - "id": "61e10478", + "id": "aecfc56d", "metadata": { "editable": true }, @@ -4072,7 +4083,7 @@ }, { "cell_type": "markdown", - "id": "9e8580da", + "id": "73b4475e", "metadata": { "editable": true }, @@ -4084,7 +4095,7 @@ }, { "cell_type": "markdown", - "id": "0ac00991", + "id": "54057ff8", "metadata": { "editable": true }, @@ -4096,7 +4107,7 @@ }, { "cell_type": "markdown", - "id": "05f8073b", + "id": "ff974462", "metadata": { "editable": true }, @@ -4108,7 +4119,7 @@ }, { "cell_type": "markdown", - "id": "622a8443", + "id": "e5daa192", "metadata": { "editable": true }, @@ -4118,7 +4129,7 @@ }, { "cell_type": "markdown", - "id": "580e143c", + "id": "dc9f0f7b", "metadata": { "editable": true }, @@ -4130,7 +4141,7 @@ }, { "cell_type": "markdown", - "id": "30b75cea", + "id": "f9b10560", "metadata": { "editable": true }, @@ -4142,7 +4153,7 @@ }, { "cell_type": "markdown", - "id": "cdc103a7", + "id": "66a948a0", "metadata": { "editable": true }, @@ -4154,7 +4165,7 @@ }, { "cell_type": "markdown", - "id": "ea01abf6", + "id": "55d9dbd1", "metadata": { "editable": true }, @@ -4164,7 +4175,7 @@ }, { "cell_type": "markdown", - "id": "e4b60d5d", + "id": "c2b7da0d", "metadata": { "editable": true }, @@ -4176,7 +4187,7 @@ }, { "cell_type": "markdown", - "id": "deb0bbd3", + "id": "ff1c658b", "metadata": { "editable": true }, @@ -4186,7 +4197,7 @@ }, { "cell_type": "markdown", - "id": "f1f15de8", + "id": "3d7050b0", "metadata": { "editable": true }, @@ -4198,7 +4209,7 @@ }, { "cell_type": "markdown", - "id": "7c54c407", + "id": "dced3eca", "metadata": { "editable": true }, @@ -4212,7 +4223,7 @@ }, { "cell_type": "markdown", - "id": "31e31b03", + "id": "44ec48f0", "metadata": { "editable": true }, @@ -4224,7 +4235,7 @@ }, { "cell_type": "markdown", - "id": "7d1cef4f", + "id": "ef5a3989", "metadata": { "editable": true }, @@ -4236,7 +4247,7 @@ }, { "cell_type": "markdown", - "id": "aa8cedef", + "id": "6d7071a0", "metadata": { "editable": true }, @@ -4246,7 +4257,7 @@ }, { "cell_type": "markdown", - "id": "022623a1", + "id": "a37c0651", "metadata": { "editable": true }, @@ -4258,7 +4269,7 @@ }, { "cell_type": "markdown", - "id": "c9885020", + "id": "9cd481cc", "metadata": { "editable": true }, @@ -4269,7 +4280,7 @@ }, { "cell_type": "markdown", - "id": "f378f76e", + "id": "e8cd56b7", "metadata": { "editable": true }, @@ -4281,7 +4292,7 @@ }, { "cell_type": "markdown", - "id": "4add7e8a", + "id": "3a629b1b", "metadata": { "editable": true }, @@ -4291,7 +4302,7 @@ }, { "cell_type": "markdown", - "id": "51ea1a29", + "id": "2fbf21c7", "metadata": { "editable": true }, @@ -4303,7 +4314,7 @@ }, { "cell_type": "markdown", - "id": "54a250b6", + "id": "5951205a", "metadata": { "editable": true }, @@ -4313,7 +4324,7 @@ }, { "cell_type": "markdown", - "id": "ebaf68f5", + "id": "53caa2e2", "metadata": { "editable": true }, @@ -4325,7 +4336,7 @@ }, { "cell_type": "markdown", - "id": "5ff687e8", + "id": "64b903b4", "metadata": { "editable": true }, @@ -4336,7 +4347,7 @@ }, { "cell_type": "markdown", - "id": "ea61a360", + "id": "df6a66b3", "metadata": { "editable": true }, @@ -4348,7 +4359,7 @@ }, { "cell_type": "markdown", - "id": "bc168f34", + "id": "ffe2f537", "metadata": { "editable": true }, @@ -4366,7 +4377,7 @@ }, { "cell_type": "markdown", - "id": "39d89895", + "id": "3c291471", "metadata": { "editable": true }, @@ -4382,7 +4393,7 @@ }, { "cell_type": "markdown", - "id": "c229143d", + "id": "0a3f9a94", "metadata": { "editable": true }, @@ -4394,7 +4405,7 @@ }, { "cell_type": "markdown", - "id": "0a569456", + "id": "9e121ab9", "metadata": { "editable": true }, @@ -4406,7 +4417,7 @@ }, { "cell_type": "markdown", - "id": "0c0b9212", + "id": "8af98d79", "metadata": { "editable": true }, @@ -4418,7 +4429,7 @@ }, { "cell_type": "markdown", - "id": "b05a1b83", + "id": "f8df9cb1", "metadata": { "editable": true }, @@ -4431,7 +4442,7 @@ }, { "cell_type": "markdown", - "id": "d64818ce", + "id": "19bfd403", "metadata": { "editable": true }, @@ -4447,7 +4458,7 @@ }, { "cell_type": "markdown", - "id": "32df0f89", + "id": "c1ec9cb3", "metadata": { "editable": true }, @@ -4461,7 +4472,7 @@ }, { "cell_type": "markdown", - "id": "a9d9108b", + "id": "4346796e", "metadata": { "editable": true }, @@ -4471,7 +4482,7 @@ }, { "cell_type": "markdown", - "id": "c6d76715", + "id": "cda3d8a7", "metadata": { "editable": true }, @@ -4483,7 +4494,7 @@ }, { "cell_type": "markdown", - "id": "5cc35821", + "id": "7cf1e6d8", "metadata": { "editable": true }, @@ -4493,7 +4504,7 @@ }, { "cell_type": "markdown", - "id": "27c8eac0", + "id": "c783275d", "metadata": { "editable": true }, @@ -4505,7 +4516,7 @@ }, { "cell_type": "markdown", - "id": "49361e47", + "id": "dac43f32", "metadata": { "editable": true }, @@ -4515,7 +4526,7 @@ }, { "cell_type": "markdown", - "id": "5eb15920", + "id": "012a62f2", "metadata": { "editable": true }, @@ -4529,7 +4540,7 @@ }, { "cell_type": "markdown", - "id": "2160cb07", + "id": "f02bf402", "metadata": { "editable": true }, @@ -4546,7 +4557,7 @@ }, { "cell_type": "markdown", - "id": "a5deaa8e", + "id": "63805f7d", "metadata": { "editable": true }, @@ -4562,7 +4573,7 @@ }, { "cell_type": "markdown", - "id": "f6868141", + "id": "e84b7d85", "metadata": { "editable": true }, @@ -4574,7 +4585,7 @@ }, { "cell_type": "markdown", - "id": "9c17b156", + "id": "3ef0ad9c", "metadata": { "editable": true }, @@ -4587,7 +4598,7 @@ }, { "cell_type": "markdown", - "id": "1b9e8b7e", + "id": "5aa2f0f4", "metadata": { "editable": true }, @@ -4601,7 +4612,7 @@ }, { "cell_type": "markdown", - "id": "c00ac20a", + "id": "5cedd672", "metadata": { "editable": true }, @@ -4611,7 +4622,7 @@ }, { "cell_type": "markdown", - "id": "91729a44", + "id": "cdb4b214", "metadata": { "editable": true }, @@ -4624,7 +4635,7 @@ }, { "cell_type": "markdown", - "id": "62403241", + "id": "830c23c5", "metadata": { "editable": true }, @@ -4643,7 +4654,7 @@ }, { "cell_type": "markdown", - "id": "6d0de657", + "id": "7971e263", "metadata": { "editable": true }, @@ -4655,7 +4666,7 @@ }, { "cell_type": "markdown", - "id": "108adf6b", + "id": "aab14206", "metadata": { "editable": true }, @@ -4667,7 +4678,7 @@ }, { "cell_type": "markdown", - "id": "1b2ba4f7", + "id": "27b6b6fc", "metadata": { "editable": true }, @@ -4677,7 +4688,7 @@ }, { "cell_type": "markdown", - "id": "1711d54c", + "id": "3e702413", "metadata": { "editable": true }, @@ -4689,7 +4700,7 @@ }, { "cell_type": "markdown", - "id": "f88b0c88", + "id": "fd91d200", "metadata": { "editable": true }, @@ -4702,7 +4713,7 @@ }, { "cell_type": "markdown", - "id": "d1835d40", + "id": "35c6dc25", "metadata": { "editable": true }, @@ -4721,7 +4732,7 @@ }, { "cell_type": "markdown", - "id": "7d0f1648", + "id": "e733d668", "metadata": { "editable": true }, @@ -4731,7 +4742,7 @@ }, { "cell_type": "markdown", - "id": "d16ea5cf", + "id": "4feaa7d5", "metadata": { "editable": true }, @@ -4750,7 +4761,7 @@ }, { "cell_type": "markdown", - "id": "634130de", + "id": "2fd87051", "metadata": { "editable": true }, @@ -4768,7 +4779,7 @@ }, { "cell_type": "markdown", - "id": "cecf6c09", + "id": "8a8c7cb0", "metadata": { "editable": true }, @@ -4782,7 +4793,7 @@ }, { "cell_type": "markdown", - "id": "1703b000", + "id": "b5426cb7", "metadata": { "editable": true }, @@ -4797,7 +4808,7 @@ { "cell_type": "code", "execution_count": 27, - "id": "5da5890f", + "id": "45d8c427", "metadata": { "collapsed": false, "editable": true @@ -4818,7 +4829,7 @@ }, { "cell_type": "markdown", - "id": "6b10b22a", + "id": "5cb5bc08", "metadata": { "editable": true }, @@ -4835,7 +4846,7 @@ { "cell_type": "code", "execution_count": 28, - "id": "da699527", + "id": "b43fd84d", "metadata": { "collapsed": false, "editable": true @@ -4867,7 +4878,7 @@ }, { "cell_type": "markdown", - "id": "a40fb51e", + "id": "c303e01f", "metadata": { "editable": true }, @@ -4881,7 +4892,7 @@ }, { "cell_type": "markdown", - "id": "413db31e", + "id": "d56895dd", "metadata": { "editable": true }, @@ -4894,7 +4905,7 @@ { "cell_type": "code", "execution_count": 29, - "id": "53927e5d", + "id": "cfda2184", "metadata": { "collapsed": false, "editable": true @@ -4919,7 +4930,7 @@ }, { "cell_type": "markdown", - "id": "74fc3107", + "id": "3cec050e", "metadata": { "editable": true }, @@ -4929,7 +4940,7 @@ }, { "cell_type": "markdown", - "id": "030881ab", + "id": "8afc4799", "metadata": { "editable": true }, @@ -4940,7 +4951,7 @@ { "cell_type": "code", "execution_count": 30, - "id": "c107aa55", + "id": "1372e459", "metadata": { "collapsed": false, "editable": true @@ -4994,7 +5005,7 @@ }, { "cell_type": "markdown", - "id": "3f47378e", + "id": "aed87895", "metadata": { "editable": true }, @@ -5011,7 +5022,7 @@ }, { "cell_type": "markdown", - "id": "dba29ab3", + "id": "7e1e5b8d", "metadata": { "editable": true }, @@ -5023,7 +5034,7 @@ }, { "cell_type": "markdown", - "id": "0eb702bd", + "id": "b0d462fa", "metadata": { "editable": true }, @@ -5035,7 +5046,7 @@ }, { "cell_type": "markdown", - "id": "1d355153", + "id": "b8b65150", "metadata": { "editable": true }, @@ -5045,7 +5056,7 @@ }, { "cell_type": "markdown", - "id": "a2316749", + "id": "7e8f5c1f", "metadata": { "editable": true }, @@ -5062,7 +5073,7 @@ }, { "cell_type": "markdown", - "id": "8f6a654b", + "id": "1cc1396a", "metadata": { "editable": true }, @@ -5072,7 +5083,7 @@ }, { "cell_type": "markdown", - "id": "72291085", + "id": "5df6ba80", "metadata": { "editable": true }, @@ -5087,7 +5098,7 @@ }, { "cell_type": "markdown", - "id": "2b28a5fe", + "id": "59e9b6e2", "metadata": { "editable": true }, @@ -5097,7 +5108,7 @@ }, { "cell_type": "markdown", - "id": "49efe427", + "id": "ca2d3f9a", "metadata": { "editable": true }, @@ -5111,7 +5122,7 @@ }, { "cell_type": "markdown", - "id": "7edcb842", + "id": "be78dcbb", "metadata": { "editable": true }, @@ -5123,7 +5134,7 @@ }, { "cell_type": "markdown", - "id": "50ec44da", + "id": "0a45fb7f", "metadata": { "editable": true }, @@ -5135,7 +5146,7 @@ }, { "cell_type": "markdown", - "id": "f4010e4a", + "id": "8c6fdb24", "metadata": { "editable": true }, @@ -5147,7 +5158,7 @@ }, { "cell_type": "markdown", - "id": "c0e95eb2", + "id": "a419206b", "metadata": { "editable": true }, @@ -5157,7 +5168,7 @@ }, { "cell_type": "markdown", - "id": "46abefce", + "id": "765c4cbc", "metadata": { "editable": true }, @@ -5169,7 +5180,7 @@ }, { "cell_type": "markdown", - "id": "30544899", + "id": "760d01a8", "metadata": { "editable": true }, @@ -5179,7 +5190,7 @@ }, { "cell_type": "markdown", - "id": "feaeedbe", + "id": "a1f8eeea", "metadata": { "editable": true }, @@ -5196,7 +5207,7 @@ }, { "cell_type": "markdown", - "id": "3a2c0f0e", + "id": "b6fb003a", "metadata": { "editable": true }, @@ -5206,7 +5217,7 @@ }, { "cell_type": "markdown", - "id": "dc33d7ee", + "id": "75e17819", "metadata": { "editable": true }, @@ -5218,7 +5229,7 @@ }, { "cell_type": "markdown", - "id": "0ffb1c46", + "id": "bd29d1d6", "metadata": { "editable": true }, @@ -5228,7 +5239,7 @@ }, { "cell_type": "markdown", - "id": "6cf92025", + "id": "150b7923", "metadata": { "editable": true }, @@ -5240,7 +5251,7 @@ }, { "cell_type": "markdown", - "id": "c21c0c0a", + "id": "b4ffe151", "metadata": { "editable": true }, @@ -5254,7 +5265,7 @@ }, { "cell_type": "markdown", - "id": "48dbf778", + "id": "d3ba3779", "metadata": { "editable": true }, @@ -5266,7 +5277,7 @@ }, { "cell_type": "markdown", - "id": "3f8f57cc", + "id": "512596a9", "metadata": { "editable": true }, @@ -5288,7 +5299,7 @@ }, { "cell_type": "markdown", - "id": "199ea962", + "id": "7559f882", "metadata": { "editable": true }, @@ -5300,7 +5311,7 @@ }, { "cell_type": "markdown", - "id": "f5433e15", + "id": "8256f707", "metadata": { "editable": true }, @@ -5315,7 +5326,7 @@ }, { "cell_type": "markdown", - "id": "ce1511d3", + "id": "4302d47f", "metadata": { "editable": true }, @@ -5327,7 +5338,7 @@ }, { "cell_type": "markdown", - "id": "31981527", + "id": "b74f22fe", "metadata": { "editable": true }, @@ -5339,7 +5350,7 @@ }, { "cell_type": "markdown", - "id": "ccb547be", + "id": "343c9a10", "metadata": { "editable": true }, @@ -5349,7 +5360,7 @@ }, { "cell_type": "markdown", - "id": "c9a6f756", + "id": "915796dd", "metadata": { "editable": true }, @@ -5361,7 +5372,7 @@ }, { "cell_type": "markdown", - "id": "272b8d77", + "id": "0ab8f900", "metadata": { "editable": true }, @@ -5371,7 +5382,7 @@ }, { "cell_type": "markdown", - "id": "4596a399", + "id": "99167153", "metadata": { "editable": true }, @@ -5383,7 +5394,7 @@ }, { "cell_type": "markdown", - "id": "e15dedbf", + "id": "5d580225", "metadata": { "editable": true }, @@ -5393,7 +5404,7 @@ }, { "cell_type": "markdown", - "id": "fe9dab27", + "id": "004215af", "metadata": { "editable": true }, @@ -5405,7 +5416,7 @@ }, { "cell_type": "markdown", - "id": "894497fb", + "id": "afa31b73", "metadata": { "editable": true }, @@ -5422,7 +5433,7 @@ }, { "cell_type": "markdown", - "id": "843092e6", + "id": "ec4c4745", "metadata": { "editable": true }, @@ -5435,7 +5446,7 @@ }, { "cell_type": "markdown", - "id": "4af49ad4", + "id": "88eba984", "metadata": { "editable": true }, @@ -5447,7 +5458,7 @@ }, { "cell_type": "markdown", - "id": "ed536a1b", + "id": "b57779ac", "metadata": { "editable": true }, @@ -5457,7 +5468,7 @@ }, { "cell_type": "markdown", - "id": "9e199dbf", + "id": "a59f4fd5", "metadata": { "editable": true }, @@ -5470,7 +5481,7 @@ }, { "cell_type": "markdown", - "id": "29d15cda", + "id": "c79793c8", "metadata": { "editable": true }, @@ -5480,7 +5491,7 @@ }, { "cell_type": "markdown", - "id": "a9b91956", + "id": "4f039e96", "metadata": { "editable": true }, @@ -5492,7 +5503,7 @@ }, { "cell_type": "markdown", - "id": "f21d533a", + "id": "737c8a25", "metadata": { "editable": true }, @@ -5505,7 +5516,7 @@ }, { "cell_type": "markdown", - "id": "4f218703", + "id": "39aa3132", "metadata": { "editable": true }, @@ -5518,7 +5529,7 @@ }, { "cell_type": "markdown", - "id": "6bc40dcd", + "id": "b6cea5b0", "metadata": { "editable": true }, @@ -5530,7 +5541,7 @@ }, { "cell_type": "markdown", - "id": "fb27a2e6", + "id": "8fa7e834", "metadata": { "editable": true }, @@ -5542,7 +5553,7 @@ }, { "cell_type": "markdown", - "id": "d285f7c7", + "id": "c7b38950", "metadata": { "editable": true }, @@ -5552,7 +5563,7 @@ }, { "cell_type": "markdown", - "id": "57eb55da", + "id": "06878301", "metadata": { "editable": true }, @@ -5565,7 +5576,7 @@ }, { "cell_type": "markdown", - "id": "94f2f142", + "id": "fbbf597c", "metadata": { "editable": true }, @@ -5577,7 +5588,7 @@ }, { "cell_type": "markdown", - "id": "602c806b", + "id": "bf77f613", "metadata": { "editable": true }, @@ -5589,7 +5600,7 @@ }, { "cell_type": "markdown", - "id": "c01a7be8", + "id": "9fc0cbdb", "metadata": { "editable": true }, @@ -5601,7 +5612,7 @@ }, { "cell_type": "markdown", - "id": "6bbd89f5", + "id": "1d556907", "metadata": { "editable": true }, @@ -5613,7 +5624,7 @@ }, { "cell_type": "markdown", - "id": "d5507c15", + "id": "96d0870b", "metadata": { "editable": true }, @@ -5627,7 +5638,7 @@ }, { "cell_type": "markdown", - "id": "d5eb5c88", + "id": "26ef92fd", "metadata": { "editable": true }, @@ -5639,7 +5650,7 @@ }, { "cell_type": "markdown", - "id": "d59c7c4d", + "id": "77971ba3", "metadata": { "editable": true }, @@ -5649,7 +5660,7 @@ }, { "cell_type": "markdown", - "id": "81d0dd7f", + "id": "cd804d0f", "metadata": { "editable": true }, @@ -5661,7 +5672,7 @@ }, { "cell_type": "markdown", - "id": "6807c376", + "id": "aee042df", "metadata": { "editable": true }, @@ -5673,7 +5684,7 @@ }, { "cell_type": "markdown", - "id": "89c94248", + "id": "0ed46bde", "metadata": { "editable": true }, @@ -5685,7 +5696,7 @@ }, { "cell_type": "markdown", - "id": "24f4e74f", + "id": "8c093c7e", "metadata": { "editable": true }, @@ -5697,7 +5708,7 @@ }, { "cell_type": "markdown", - "id": "eac49311", + "id": "62af183d", "metadata": { "editable": true }, @@ -5709,7 +5720,7 @@ }, { "cell_type": "markdown", - "id": "8ad4bc0d", + "id": "30a8db83", "metadata": { "editable": true }, @@ -5728,7 +5739,7 @@ }, { "cell_type": "markdown", - "id": "af4dd7e2", + "id": "01faaa97", "metadata": { "editable": true }, @@ -5740,7 +5751,7 @@ }, { "cell_type": "markdown", - "id": "92b983c6", + "id": "672de282", "metadata": { "editable": true }, @@ -5750,7 +5761,7 @@ }, { "cell_type": "markdown", - "id": "f9d22662", + "id": "fe52bfff", "metadata": { "editable": true }, @@ -5762,7 +5773,7 @@ }, { "cell_type": "markdown", - "id": "0097f163", + "id": "72b7266d", "metadata": { "editable": true }, @@ -5772,7 +5783,7 @@ }, { "cell_type": "markdown", - "id": "073824a0", + "id": "6681e68d", "metadata": { "editable": true }, @@ -5784,7 +5795,7 @@ }, { "cell_type": "markdown", - "id": "399bdc1b", + "id": "538bde1b", "metadata": { "editable": true }, @@ -5796,7 +5807,7 @@ }, { "cell_type": "markdown", - "id": "d9d9fa7a", + "id": "c3a6e71e", "metadata": { "editable": true }, @@ -5812,7 +5823,7 @@ }, { "cell_type": "markdown", - "id": "40b2c49b", + "id": "b59b021d", "metadata": { "editable": true }, @@ -5824,7 +5835,7 @@ }, { "cell_type": "markdown", - "id": "6e949d98", + "id": "ffa266f1", "metadata": { "editable": true }, @@ -5836,7 +5847,7 @@ }, { "cell_type": "markdown", - "id": "5a004c53", + "id": "e8e97705", "metadata": { "editable": true }, @@ -5846,7 +5857,7 @@ }, { "cell_type": "markdown", - "id": "64e1d6bb", + "id": "73c4369a", "metadata": { "editable": true }, @@ -5858,7 +5869,7 @@ }, { "cell_type": "markdown", - "id": "5cec9b54", + "id": "7727b4fd", "metadata": { "editable": true }, @@ -5868,7 +5879,7 @@ }, { "cell_type": "markdown", - "id": "71516f90", + "id": "3a3e1873", "metadata": { "editable": true }, @@ -5880,7 +5891,7 @@ }, { "cell_type": "markdown", - "id": "50e288e9", + "id": "d1a8b030", "metadata": { "editable": true }, @@ -5897,7 +5908,7 @@ }, { "cell_type": "markdown", - "id": "6c82431c", + "id": "b6f25a34", "metadata": { "editable": true }, @@ -5909,7 +5920,7 @@ }, { "cell_type": "markdown", - "id": "5ebc1428", + "id": "c22a3a5a", "metadata": { "editable": true }, @@ -5921,7 +5932,7 @@ }, { "cell_type": "markdown", - "id": "8f198345", + "id": "9e9b60a5", "metadata": { "editable": true }, @@ -5931,7 +5942,7 @@ }, { "cell_type": "markdown", - "id": "443b3080", + "id": "e253cca2", "metadata": { "editable": true }, @@ -5943,7 +5954,7 @@ }, { "cell_type": "markdown", - "id": "e11ec4ea", + "id": "ee43ad3e", "metadata": { "editable": true }, @@ -5953,7 +5964,7 @@ }, { "cell_type": "markdown", - "id": "1d640caa", + "id": "d8f7c38d", "metadata": { "editable": true }, @@ -5965,7 +5976,7 @@ }, { "cell_type": "markdown", - "id": "e023f076", + "id": "a06e5ab5", "metadata": { "editable": true }, @@ -5975,7 +5986,7 @@ }, { "cell_type": "markdown", - "id": "a4204b50", + "id": "bfb15d60", "metadata": { "editable": true }, @@ -5987,7 +5998,7 @@ }, { "cell_type": "markdown", - "id": "7c9cc257", + "id": "4d1210ce", "metadata": { "editable": true }, diff --git a/doc/pub/week35/html/._week35-bs018.html b/doc/pub/week35/html/._week35-bs018.html index 666e9d7f9..c542a78f6 100644 --- a/doc/pub/week35/html/._week35-bs018.html +++ b/doc/pub/week35/html/._week35-bs018.html @@ -378,7 +378,18 @@ MathJax.Hub.Config({ 
# matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
diff --git a/doc/pub/week35/html/week35-reveal.html b/doc/pub/week35/html/week35-reveal.html
index dc4fff779..1551dccac 100644
--- a/doc/pub/week35/html/week35-reveal.html
+++ b/doc/pub/week35/html/week35-reveal.html
@@ -828,7 +828,18 @@ $$
       

         

   
# matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html
index 226cfa867..399427f2e 100644
--- a/doc/pub/week35/html/week35-solarized.html
+++ b/doc/pub/week35/html/week35-solarized.html
@@ -838,7 +838,18 @@ $$
       

         

   
# matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html
index 2ff8be1fc..88ed72166 100644
--- a/doc/pub/week35/html/week35.html
+++ b/doc/pub/week35/html/week35.html
@@ -915,7 +915,18 @@ $$
       

         

   
# matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta
diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
index 990e29374..a5fabcd36 100644
Binary files a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz and b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz differ
diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb
index f4d59ecff..5139bd065 100644
--- a/doc/pub/week35/ipynb/week35.ipynb
+++ b/doc/pub/week35/ipynb/week35.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "id": "3369db88",
+   "id": "0b8a0ca4",
    "metadata": {
     "editable": true
    },
@@ -14,7 +14,7 @@
   },
   {
    "cell_type": "markdown",
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@@ -27,7 +27,7 @@
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@@ -49,7 +49,7 @@
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@@ -65,7 +65,7 @@
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@@ -99,7 +99,7 @@
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@@ -115,7 +115,7 @@
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@@ -127,7 +127,7 @@
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@@ -137,7 +137,7 @@
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@@ -149,7 +149,7 @@
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@@ -170,7 +170,7 @@
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@@ -182,7 +182,7 @@
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@@ -195,7 +195,7 @@
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@@ -207,7 +207,7 @@
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@@ -219,7 +219,7 @@
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@@ -229,7 +229,7 @@
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@@ -241,7 +241,7 @@
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@@ -254,7 +254,7 @@
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@@ -266,7 +266,7 @@
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@@ -276,7 +276,7 @@
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@@ -288,7 +288,7 @@
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@@ -300,7 +300,7 @@
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@@ -311,7 +311,7 @@
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@@ -323,7 +323,7 @@
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@@ -342,7 +342,7 @@
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@@ -355,7 +355,7 @@
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@@ -365,7 +365,7 @@
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@@ -377,7 +377,7 @@
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@@ -387,7 +387,7 @@
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@@ -399,7 +399,7 @@
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@@ -409,7 +409,7 @@
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@@ -421,7 +421,7 @@
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@@ -432,7 +432,7 @@
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@@ -444,7 +444,7 @@
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@@ -454,7 +454,7 @@
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@@ -466,7 +466,7 @@
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@@ -476,7 +476,7 @@
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@@ -488,7 +488,7 @@
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@@ -508,7 +508,7 @@
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@@ -535,7 +535,7 @@
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@@ -547,7 +547,7 @@
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@@ -559,7 +559,7 @@
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@@ -575,7 +575,7 @@
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@@ -593,7 +593,7 @@
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@@ -605,7 +605,7 @@
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@@ -617,7 +617,7 @@
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@@ -628,7 +628,7 @@
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@@ -640,7 +640,7 @@
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@@ -650,7 +650,7 @@
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@@ -662,7 +662,7 @@
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@@ -676,7 +676,7 @@
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@@ -688,7 +688,7 @@
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@@ -700,7 +700,7 @@
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@@ -712,7 +712,7 @@
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@@ -722,7 +722,7 @@
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@@ -734,7 +734,7 @@
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@@ -744,7 +744,7 @@
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@@ -756,7 +756,7 @@
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@@ -770,7 +770,7 @@
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@@ -782,7 +782,7 @@
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@@ -794,7 +794,7 @@
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@@ -806,7 +806,7 @@
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@@ -816,7 +816,7 @@
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@@ -828,7 +828,7 @@
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@@ -838,7 +838,7 @@
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@@ -850,7 +850,7 @@
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@@ -860,7 +860,7 @@
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@@ -872,7 +872,7 @@
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@@ -884,7 +884,7 @@
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@@ -896,7 +896,7 @@
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@@ -911,7 +911,7 @@
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@@ -923,7 +923,7 @@
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@@ -933,7 +933,7 @@
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@@ -945,7 +945,7 @@
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@@ -955,7 +955,7 @@
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@@ -967,7 +967,7 @@
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@@ -977,7 +977,7 @@
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@@ -989,7 +989,7 @@
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@@ -1001,7 +1001,7 @@
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@@ -1013,7 +1013,7 @@
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@@ -1023,7 +1023,7 @@
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@@ -1035,7 +1035,7 @@
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@@ -1048,7 +1048,7 @@
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@@ -1060,7 +1060,7 @@
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@@ -1070,7 +1070,7 @@
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@@ -1082,7 +1082,7 @@
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@@ -1092,7 +1092,7 @@
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@@ -1104,7 +1104,7 @@
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@@ -1114,7 +1114,7 @@
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@@ -1126,7 +1126,7 @@
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@@ -1136,7 +1136,7 @@
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@@ -1148,7 +1148,7 @@
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@@ -1158,7 +1158,7 @@
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@@ -1170,7 +1170,7 @@
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@@ -1182,7 +1182,7 @@
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@@ -1194,7 +1194,7 @@
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@@ -1206,7 +1206,7 @@
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@@ -1218,7 +1218,7 @@
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@@ -1234,7 +1234,7 @@
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@@ -1246,7 +1246,7 @@
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@@ -1256,7 +1256,7 @@
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@@ -1268,7 +1268,7 @@
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@@ -1286,7 +1286,7 @@
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@@ -1298,7 +1298,7 @@
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@@ -1310,7 +1310,7 @@
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@@ -1320,7 +1320,7 @@
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-   "id": "9599fd86",
+   "id": "f1e3f1c2",
    "metadata": {
     "editable": true
    },
@@ -1332,7 +1332,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "394818b7",
+   "id": "39960290",
    "metadata": {
     "editable": true
    },
@@ -1342,7 +1342,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "166ed846",
+   "id": "db52e3e6",
    "metadata": {
     "editable": true
    },
@@ -1354,7 +1354,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ea09845c",
+   "id": "e3d6ac8b",
    "metadata": {
     "editable": true
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@@ -1364,7 +1364,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "84d4b4c2",
+   "id": "e4b30664",
    "metadata": {
     "editable": true
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@@ -1378,7 +1378,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "144fbe09",
+   "id": "4cbc0c2c",
    "metadata": {
     "editable": true
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@@ -1390,7 +1390,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbfbfc64",
+   "id": "5eebcce6",
    "metadata": {
     "editable": true
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@@ -1401,7 +1401,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "60a70183",
+   "id": "47074a74",
    "metadata": {
     "editable": true
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@@ -1414,7 +1414,7 @@
   {
    "cell_type": "code",
    "execution_count": 1,
-   "id": "f80c5d34",
+   "id": "ebc41f6f",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1422,7 +1422,18 @@
    "outputs": [],
    "source": [
     "# matrix inversion to find beta\n",
+    "# First we set up the data\n",
     "import numpy as np\n",
+    "x = np.random.rand(100)\n",
+    "y = 2.0+5*x*x+0.1*np.random.randn(100)\n",
+    "# and then the design matrix X including the intercept\n",
+    "#  The design matrix now as function of a fourth-order polynomial\n",
+    "X = np.zeros((len(x),5))\n",
+    "X[:,0] = 1.0\n",
+    "X[:,1] = x\n",
+    "X[:,2] = x**2\n",
+    "X[:,3] = x**3\n",
+    "X[:,4] = x**4\n",
     "beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y\n",
     "# and then make the prediction\n",
     "ytilde = X @ beta"
@@ -1430,7 +1441,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ab38e5f",
+   "id": "8ea9846d",
    "metadata": {
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@@ -1441,7 +1452,7 @@
   {
    "cell_type": "code",
    "execution_count": 2,
-   "id": "59e14ceb",
+   "id": "e996841a",
    "metadata": {
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     "editable": true
@@ -1454,7 +1465,7 @@
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   {
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-   "id": "2f02e87a",
+   "id": "39f01129",
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@@ -1468,7 +1479,7 @@
   {
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    "execution_count": 3,
-   "id": "81a96389",
+   "id": "12e52de5",
    "metadata": {
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     "editable": true
@@ -1481,7 +1492,7 @@
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   {
    "cell_type": "markdown",
-   "id": "702395a0",
+   "id": "8a0cc628",
    "metadata": {
     "editable": true
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@@ -1492,7 +1503,7 @@
   {
    "cell_type": "code",
    "execution_count": 4,
-   "id": "74416523",
+   "id": "8c1508b7",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1504,7 +1515,7 @@
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   {
    "cell_type": "markdown",
-   "id": "cdacfa78",
+   "id": "a3f2f3b1",
    "metadata": {
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@@ -1515,7 +1526,7 @@
   {
    "cell_type": "code",
    "execution_count": 5,
-   "id": "8d416246",
+   "id": "334780f0",
    "metadata": {
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     "editable": true
@@ -1531,7 +1542,7 @@
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   {
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-   "id": "24ce23ef",
+   "id": "1f7cfede",
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@@ -1542,7 +1553,7 @@
   {
    "cell_type": "code",
    "execution_count": 6,
-   "id": "571b5d7c",
+   "id": "1579c506",
    "metadata": {
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     "editable": true
@@ -1556,7 +1567,7 @@
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   {
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-   "id": "c74c8161",
+   "id": "88c6c130",
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@@ -1577,7 +1588,7 @@
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-   "id": "878e6935",
+   "id": "af0baf7f",
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@@ -1588,7 +1599,7 @@
   {
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    "execution_count": 7,
-   "id": "3bf38929",
+   "id": "89aa39b5",
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     "editable": true
@@ -1641,7 +1652,7 @@
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   {
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-   "id": "a82ed329",
+   "id": "1d86c024",
    "metadata": {
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@@ -1652,7 +1663,7 @@
   {
    "cell_type": "code",
    "execution_count": 8,
-   "id": "50637468",
+   "id": "2ba16c25",
    "metadata": {
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     "editable": true
@@ -1677,7 +1688,7 @@
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   {
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-   "id": "a2c86c0d",
+   "id": "8e363f00",
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@@ -1689,7 +1700,7 @@
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   {
    "cell_type": "markdown",
-   "id": "c8eb18df",
+   "id": "d5ef1be0",
    "metadata": {
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@@ -1718,7 +1729,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1f0523f4",
+   "id": "7165f38b",
    "metadata": {
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@@ -1743,7 +1754,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "be02231d",
+   "id": "baea18ad",
    "metadata": {
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@@ -1763,7 +1774,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "932edb5f",
+   "id": "830be0f9",
    "metadata": {
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@@ -1790,7 +1801,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b4867600",
+   "id": "72473f2c",
    "metadata": {
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@@ -1803,7 +1814,7 @@
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   {
    "cell_type": "markdown",
-   "id": "f129ff79",
+   "id": "ad92637f",
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@@ -1815,7 +1826,7 @@
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   {
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-   "id": "c0c407d4",
+   "id": "928978ac",
    "metadata": {
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@@ -1826,7 +1837,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "6ae88f00",
+   "id": "31b09189",
    "metadata": {
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@@ -1842,7 +1853,7 @@
   {
    "cell_type": "code",
    "execution_count": 9,
-   "id": "f58a6c6c",
+   "id": "c09e81a9",
    "metadata": {
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     "editable": true
@@ -1876,7 +1887,7 @@
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   {
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-   "id": "6ed7f38e",
+   "id": "8ffeb0f1",
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@@ -1886,7 +1897,7 @@
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   {
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-   "id": "3f3bd728",
+   "id": "2b1f32da",
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@@ -1901,7 +1912,7 @@
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   {
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-   "id": "122c3875",
+   "id": "3c2f9703",
    "metadata": {
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@@ -1913,7 +1924,7 @@
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   {
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-   "id": "8ba3e788",
+   "id": "15382f87",
    "metadata": {
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@@ -1923,7 +1934,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "fbae43ba",
+   "id": "a04fe6a8",
    "metadata": {
     "editable": true
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@@ -1939,7 +1950,7 @@
   {
    "cell_type": "code",
    "execution_count": 10,
-   "id": "dfae68c3",
+   "id": "de5a8426",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -1956,7 +1967,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "ecae83bf",
+   "id": "815c116c",
    "metadata": {
     "editable": true
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@@ -1969,7 +1980,7 @@
   {
    "cell_type": "code",
    "execution_count": 11,
-   "id": "261f89fe",
+   "id": "c9049720",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2016,7 +2027,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "95e68d5a",
+   "id": "c6fb36f0",
    "metadata": {
     "editable": true
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@@ -2027,7 +2038,7 @@
   {
    "cell_type": "code",
    "execution_count": 12,
-   "id": "1a202842",
+   "id": "d02afd53",
    "metadata": {
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     "editable": true
@@ -2130,7 +2141,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "97e9e0f5",
+   "id": "4d3d0315",
    "metadata": {
     "editable": true
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@@ -2158,7 +2169,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "69aaec0f",
+   "id": "1bf26037",
    "metadata": {
     "editable": true
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@@ -2193,7 +2204,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "22ec93f8",
+   "id": "1a9303d9",
    "metadata": {
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@@ -2207,7 +2218,7 @@
   {
    "cell_type": "code",
    "execution_count": 13,
-   "id": "5954369a",
+   "id": "add41c28",
    "metadata": {
     "collapsed": false,
     "editable": true
@@ -2232,7 +2243,7 @@
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   {
    "cell_type": "markdown",
-   "id": "9aeedd1b",
+   "id": "cbad62c2",
    "metadata": {
     "editable": true
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@@ -2248,7 +2259,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "68a1624c",
+   "id": "77b3f4a0",
    "metadata": {
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@@ -2260,7 +2271,7 @@
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   {
    "cell_type": "markdown",
-   "id": "07461fc8",
+   "id": "2054bdc7",
    "metadata": {
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@@ -2275,7 +2286,7 @@
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   {
    "cell_type": "markdown",
-   "id": "441108dd",
+   "id": "d7f64d04",
    "metadata": {
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@@ -2287,7 +2298,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "04a1abe1",
+   "id": "6ebe45a1",
    "metadata": {
     "editable": true
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@@ -2297,7 +2308,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c5c25018",
+   "id": "8cc21b6a",
    "metadata": {
     "editable": true
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@@ -2309,7 +2320,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1bd8cdeb",
+   "id": "71b87ff9",
    "metadata": {
     "editable": true
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@@ -2319,7 +2330,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63a32eb5",
+   "id": "b15ee5de",
    "metadata": {
     "editable": true
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@@ -2331,7 +2342,7 @@
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   {
    "cell_type": "markdown",
-   "id": "d6b0c424",
+   "id": "56e808e5",
    "metadata": {
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@@ -2344,7 +2355,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cc90b562",
+   "id": "59b1fcf5",
    "metadata": {
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@@ -2356,7 +2367,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "569a1632",
+   "id": "fe9af8e4",
    "metadata": {
     "editable": true
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@@ -2366,7 +2377,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f3a3a977",
+   "id": "39d0535d",
    "metadata": {
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@@ -2378,7 +2389,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cd00ebfe",
+   "id": "b05e083e",
    "metadata": {
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@@ -2388,7 +2399,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "9d613560",
+   "id": "255d5f87",
    "metadata": {
     "editable": true
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@@ -2400,7 +2411,7 @@
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   {
    "cell_type": "markdown",
-   "id": "3e252015",
+   "id": "863f9c03",
    "metadata": {
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@@ -2410,7 +2421,7 @@
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   {
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-   "id": "14eb93b6",
+   "id": "fe1719c8",
    "metadata": {
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@@ -2422,7 +2433,7 @@
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   {
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-   "id": "d729b501",
+   "id": "11bf9275",
    "metadata": {
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@@ -2432,7 +2443,7 @@
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   {
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-   "id": "8b1ebd36",
+   "id": "58a73e37",
    "metadata": {
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@@ -2444,7 +2455,7 @@
   },
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-   "id": "3985b967",
+   "id": "2ee6e0ab",
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@@ -2454,7 +2465,7 @@
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-   "id": "5698f884",
+   "id": "4aa55bc2",
    "metadata": {
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@@ -2466,7 +2477,7 @@
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   {
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-   "id": "4761bc12",
+   "id": "e7dd951c",
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@@ -2476,7 +2487,7 @@
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   {
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-   "id": "0d78ee94",
+   "id": "b0b56f12",
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@@ -2488,7 +2499,7 @@
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-   "id": "5a172cbe",
+   "id": "2fa4cfc7",
    "metadata": {
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@@ -2500,7 +2511,7 @@
   },
   {
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-   "id": "584843c8",
+   "id": "cc6fe7e2",
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@@ -2512,7 +2523,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bbeed632",
+   "id": "d121682c",
    "metadata": {
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@@ -2525,7 +2536,7 @@
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-   "id": "c5475feb",
+   "id": "94aef78e",
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@@ -2537,7 +2548,7 @@
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-   "id": "d18453ba",
+   "id": "8d595a8d",
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@@ -2547,7 +2558,7 @@
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-   "id": "78faca46",
+   "id": "67d81f01",
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@@ -2561,7 +2572,7 @@
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    "execution_count": 14,
-   "id": "cac2d85d",
+   "id": "39da9a38",
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     "editable": true
@@ -2658,7 +2669,7 @@
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+   "id": "257ce2f2",
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@@ -2679,7 +2690,7 @@
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-   "id": "6c3af5b5",
+   "id": "3769097f",
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@@ -2691,7 +2702,7 @@
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+   "id": "82fc22ad",
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@@ -2701,7 +2712,7 @@
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+   "id": "dea71d52",
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@@ -2713,7 +2724,7 @@
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+   "id": "113f6e87",
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@@ -2723,7 +2734,7 @@
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+   "id": "1e3b78e5",
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@@ -2735,7 +2746,7 @@
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+   "id": "d936547b",
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@@ -2751,7 +2762,7 @@
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+   "id": "7c57b1e6",
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@@ -2795,7 +2806,7 @@
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+   "id": "41d26913",
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@@ -2807,7 +2818,7 @@
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-   "id": "f3443da8",
+   "id": "bf2d2a51",
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@@ -2823,7 +2834,7 @@
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+   "id": "d7d26ad9",
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@@ -2834,7 +2845,7 @@
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-   "id": "ad33ea1a",
+   "id": "7b725195",
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@@ -2852,7 +2863,7 @@
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@@ -2863,7 +2874,7 @@
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-   "id": "94eca69e",
+   "id": "670a51a0",
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@@ -2877,7 +2888,7 @@
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+   "id": "ad921def",
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@@ -2888,7 +2899,7 @@
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-   "id": "286c1006",
+   "id": "2d0a0044",
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@@ -2901,7 +2912,7 @@
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+   "id": "8ec52805",
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@@ -2912,7 +2923,7 @@
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-   "id": "6bef6cc8",
+   "id": "a7648df4",
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@@ -2929,7 +2940,7 @@
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@@ -2940,7 +2951,7 @@
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@@ -2956,7 +2967,7 @@
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@@ -2967,7 +2978,7 @@
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@@ -2991,7 +3002,7 @@
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@@ -3002,7 +3013,7 @@
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@@ -3015,7 +3026,7 @@
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@@ -3026,7 +3037,7 @@
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@@ -3046,7 +3057,7 @@
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@@ -3057,7 +3068,7 @@
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@@ -3100,7 +3111,7 @@
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@@ -3115,7 +3126,7 @@
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@@ -3125,7 +3136,7 @@
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@@ -3139,7 +3150,7 @@
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@@ -3151,7 +3162,7 @@
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@@ -3163,7 +3174,7 @@
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@@ -3175,7 +3186,7 @@
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@@ -3185,7 +3196,7 @@
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@@ -3197,7 +3208,7 @@
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@@ -3207,7 +3218,7 @@
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@@ -3219,7 +3230,7 @@
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@@ -3230,7 +3241,7 @@
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@@ -3242,7 +3253,7 @@
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@@ -3254,7 +3265,7 @@
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@@ -3264,7 +3275,7 @@
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@@ -3276,7 +3287,7 @@
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@@ -3288,7 +3299,7 @@
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@@ -3298,7 +3309,7 @@
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@@ -3310,7 +3321,7 @@
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@@ -3320,7 +3331,7 @@
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@@ -3332,7 +3343,7 @@
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@@ -3342,7 +3353,7 @@
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@@ -3382,7 +3393,7 @@
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@@ -3399,7 +3410,7 @@
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@@ -3422,7 +3433,7 @@
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@@ -3439,7 +3450,7 @@
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@@ -3458,7 +3469,7 @@
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@@ -3469,7 +3480,7 @@
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@@ -3481,7 +3492,7 @@
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@@ -3499,7 +3510,7 @@
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@@ -3515,7 +3526,7 @@
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@@ -3527,7 +3538,7 @@
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@@ -3537,7 +3548,7 @@
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@@ -3552,7 +3563,7 @@
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@@ -3564,7 +3575,7 @@
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@@ -3574,7 +3585,7 @@
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@@ -3586,7 +3597,7 @@
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@@ -3596,7 +3607,7 @@
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@@ -3608,7 +3619,7 @@
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@@ -3620,7 +3631,7 @@
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@@ -3635,7 +3646,7 @@
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@@ -3646,7 +3657,7 @@
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@@ -3666,7 +3677,7 @@
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@@ -3678,7 +3689,7 @@
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@@ -3688,7 +3699,7 @@
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@@ -3700,7 +3711,7 @@
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@@ -3729,7 +3740,7 @@
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@@ -3756,7 +3767,7 @@
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@@ -3767,7 +3778,7 @@
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@@ -3807,7 +3818,7 @@
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@@ -3824,7 +3835,7 @@
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@@ -3847,7 +3858,7 @@
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@@ -3861,7 +3872,7 @@
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@@ -3880,7 +3891,7 @@
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@@ -3890,7 +3901,7 @@
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@@ -3902,7 +3913,7 @@
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@@ -3916,7 +3927,7 @@
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@@ -3928,7 +3939,7 @@
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@@ -3938,7 +3949,7 @@
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@@ -3950,7 +3961,7 @@
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@@ -3967,7 +3978,7 @@
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@@ -3977,7 +3988,7 @@
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@@ -3993,7 +4004,7 @@
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@@ -4003,7 +4014,7 @@
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@@ -4019,7 +4030,7 @@
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@@ -4029,7 +4040,7 @@
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@@ -4045,7 +4056,7 @@
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@@ -4055,7 +4066,7 @@
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@@ -4072,7 +4083,7 @@
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@@ -4084,7 +4095,7 @@
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@@ -4096,7 +4107,7 @@
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@@ -4108,7 +4119,7 @@
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@@ -4118,7 +4129,7 @@
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@@ -4130,7 +4141,7 @@
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@@ -4142,7 +4153,7 @@
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@@ -4154,7 +4165,7 @@
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@@ -4164,7 +4175,7 @@
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@@ -4176,7 +4187,7 @@
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@@ -4186,7 +4197,7 @@
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@@ -4198,7 +4209,7 @@
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@@ -4212,7 +4223,7 @@
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@@ -4224,7 +4235,7 @@
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@@ -4236,7 +4247,7 @@
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@@ -4246,7 +4257,7 @@
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@@ -4258,7 +4269,7 @@
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@@ -4269,7 +4280,7 @@
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@@ -4281,7 +4292,7 @@
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@@ -4291,7 +4302,7 @@
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@@ -4303,7 +4314,7 @@
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@@ -4313,7 +4324,7 @@
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@@ -4325,7 +4336,7 @@
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@@ -4336,7 +4347,7 @@
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@@ -4348,7 +4359,7 @@
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@@ -4366,7 +4377,7 @@
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@@ -4382,7 +4393,7 @@
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@@ -4394,7 +4405,7 @@
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@@ -4406,7 +4417,7 @@
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@@ -4418,7 +4429,7 @@
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@@ -4431,7 +4442,7 @@
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@@ -4447,7 +4458,7 @@
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@@ -4461,7 +4472,7 @@
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@@ -4471,7 +4482,7 @@
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@@ -4483,7 +4494,7 @@
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@@ -4493,7 +4504,7 @@
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@@ -4505,7 +4516,7 @@
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@@ -4515,7 +4526,7 @@
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@@ -4529,7 +4540,7 @@
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@@ -4546,7 +4557,7 @@
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@@ -4562,7 +4573,7 @@
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@@ -4574,7 +4585,7 @@
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@@ -4587,7 +4598,7 @@
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@@ -4601,7 +4612,7 @@
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@@ -4611,7 +4622,7 @@
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@@ -4624,7 +4635,7 @@
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@@ -4643,7 +4654,7 @@
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@@ -4655,7 +4666,7 @@
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@@ -4667,7 +4678,7 @@
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@@ -4677,7 +4688,7 @@
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@@ -4689,7 +4700,7 @@
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@@ -4702,7 +4713,7 @@
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@@ -4721,7 +4732,7 @@
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@@ -4731,7 +4742,7 @@
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@@ -4750,7 +4761,7 @@
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@@ -4768,7 +4779,7 @@
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@@ -4782,7 +4793,7 @@
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@@ -4797,7 +4808,7 @@
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@@ -4818,7 +4829,7 @@
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@@ -4835,7 +4846,7 @@
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@@ -4867,7 +4878,7 @@
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@@ -4881,7 +4892,7 @@
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@@ -4894,7 +4905,7 @@
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@@ -4919,7 +4930,7 @@
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@@ -4929,7 +4940,7 @@
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@@ -4940,7 +4951,7 @@
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@@ -4994,7 +5005,7 @@
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@@ -5011,7 +5022,7 @@
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@@ -5023,7 +5034,7 @@
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@@ -5035,7 +5046,7 @@
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@@ -5045,7 +5056,7 @@
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@@ -5062,7 +5073,7 @@
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@@ -5072,7 +5083,7 @@
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@@ -5087,7 +5098,7 @@
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@@ -5097,7 +5108,7 @@
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@@ -5111,7 +5122,7 @@
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@@ -5123,7 +5134,7 @@
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@@ -5135,7 +5146,7 @@
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@@ -5147,7 +5158,7 @@
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@@ -5157,7 +5168,7 @@
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@@ -5169,7 +5180,7 @@
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@@ -5179,7 +5190,7 @@
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@@ -5196,7 +5207,7 @@
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@@ -5206,7 +5217,7 @@
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@@ -5218,7 +5229,7 @@
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@@ -5228,7 +5239,7 @@
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@@ -5240,7 +5251,7 @@
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@@ -5254,7 +5265,7 @@
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@@ -5266,7 +5277,7 @@
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@@ -5288,7 +5299,7 @@
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@@ -5300,7 +5311,7 @@
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@@ -5315,7 +5326,7 @@
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@@ -5327,7 +5338,7 @@
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@@ -5339,7 +5350,7 @@
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@@ -5349,7 +5360,7 @@
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@@ -5361,7 +5372,7 @@
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@@ -5371,7 +5382,7 @@
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@@ -5383,7 +5394,7 @@
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@@ -5393,7 +5404,7 @@
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@@ -5405,7 +5416,7 @@
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@@ -5422,7 +5433,7 @@
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@@ -5435,7 +5446,7 @@
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@@ -5447,7 +5458,7 @@
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@@ -5457,7 +5468,7 @@
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@@ -5470,7 +5481,7 @@
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@@ -5480,7 +5491,7 @@
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@@ -5492,7 +5503,7 @@
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@@ -5505,7 +5516,7 @@
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@@ -5518,7 +5529,7 @@
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@@ -5530,7 +5541,7 @@
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@@ -5542,7 +5553,7 @@
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@@ -5552,7 +5563,7 @@
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@@ -5565,7 +5576,7 @@
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@@ -5577,7 +5588,7 @@
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@@ -5589,7 +5600,7 @@
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@@ -5601,7 +5612,7 @@
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@@ -5613,7 +5624,7 @@
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@@ -5627,7 +5638,7 @@
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@@ -5639,7 +5650,7 @@
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@@ -5649,7 +5660,7 @@
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@@ -5661,7 +5672,7 @@
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@@ -5673,7 +5684,7 @@
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@@ -5685,7 +5696,7 @@
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@@ -5697,7 +5708,7 @@
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@@ -5709,7 +5720,7 @@
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@@ -5728,7 +5739,7 @@
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   {
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-   "id": "522244d0",
+   "id": "01faaa97",
    "metadata": {
     "editable": true
    },
@@ -5740,7 +5751,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "dcd412b2",
+   "id": "672de282",
    "metadata": {
     "editable": true
    },
@@ -5750,7 +5761,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "89d43a47",
+   "id": "fe52bfff",
    "metadata": {
     "editable": true
    },
@@ -5762,7 +5773,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "baebd7ed",
+   "id": "72b7266d",
    "metadata": {
     "editable": true
    },
@@ -5772,7 +5783,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "85dec277",
+   "id": "6681e68d",
    "metadata": {
     "editable": true
    },
@@ -5784,7 +5795,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "1741e7d1",
+   "id": "538bde1b",
    "metadata": {
     "editable": true
    },
@@ -5796,7 +5807,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "8ea96754",
+   "id": "c3a6e71e",
    "metadata": {
     "editable": true
    },
@@ -5812,7 +5823,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "382d468a",
+   "id": "b59b021d",
    "metadata": {
     "editable": true
    },
@@ -5824,7 +5835,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "4f075888",
+   "id": "ffa266f1",
    "metadata": {
     "editable": true
    },
@@ -5836,7 +5847,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a0b0a445",
+   "id": "e8e97705",
    "metadata": {
     "editable": true
    },
@@ -5846,7 +5857,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "bfb5ba4a",
+   "id": "73c4369a",
    "metadata": {
     "editable": true
    },
@@ -5858,7 +5869,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f5aeeb57",
+   "id": "7727b4fd",
    "metadata": {
     "editable": true
    },
@@ -5868,7 +5879,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "cba74039",
+   "id": "3a3e1873",
    "metadata": {
     "editable": true
    },
@@ -5880,7 +5891,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "a23b507f",
+   "id": "d1a8b030",
    "metadata": {
     "editable": true
    },
@@ -5897,7 +5908,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "c5a6865b",
+   "id": "b6f25a34",
    "metadata": {
     "editable": true
    },
@@ -5909,7 +5920,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "27b91235",
+   "id": "c22a3a5a",
    "metadata": {
     "editable": true
    },
@@ -5921,7 +5932,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "f6186fa5",
+   "id": "9e9b60a5",
    "metadata": {
     "editable": true
    },
@@ -5931,7 +5942,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "aaaa5cb1",
+   "id": "e253cca2",
    "metadata": {
     "editable": true
    },
@@ -5943,7 +5954,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "63db8f95",
+   "id": "ee43ad3e",
    "metadata": {
     "editable": true
    },
@@ -5953,7 +5964,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "b7334b98",
+   "id": "d8f7c38d",
    "metadata": {
     "editable": true
    },
@@ -5965,7 +5976,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "5018fc51",
+   "id": "a06e5ab5",
    "metadata": {
     "editable": true
    },
@@ -5975,7 +5986,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "58e283bc",
+   "id": "bfb15d60",
    "metadata": {
     "editable": true
    },
@@ -5987,7 +5998,7 @@
   },
   {
    "cell_type": "markdown",
-   "id": "36abd1a0",
+   "id": "4d1210ce",
    "metadata": {
     "editable": true
    },
diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt
index c34af53f4..bf131f4a6 100644
--- a/doc/src/week35/week35.do.txt
+++ b/doc/src/week35/week35.do.txt
@@ -583,7 +583,18 @@ $n\times p$ matrix $\bm{X}$.
 It is rather straightforward to implement the matrix inversion and obtain the parameters $\bm{\beta}$. After having defined the matrix $\bm{X}$ and the outputs $\bm{y}$ we have 
 !bc pycod 
 # matrix inversion to find beta
+# First we set up the data
 import numpy as np
+x = np.random.rand(100)
+y = 2.0+5*x*x+0.1*np.random.randn(100)
+# and then the design matrix X including the intercept
+#  The design matrix now as function of a fourth-order polynomial
+X = np.zeros((len(x),5))
+X[:,0] = 1.0
+X[:,1] = x
+X[:,2] = x**2
+X[:,3] = x**3
+X[:,4] = x**4
 beta = (np.linalg.inv(X.T @ X) @ X.T ) @ y
 # and then make the prediction
 ytilde = X @ beta