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zst@!3A6eU18oLkv#~b_etnKs0p2*rhZ|o;n+vkma-2b(7_c#OvQ545TGKmd{Vj3H; z4i$-I7=>t*rm}{o(rc+T5}jCpLd6a!RNVhLnwR?)@H=^LKJLZ7u)ndt>Bi%OeX-y2 zD)u{4Kz|Efd(pG=3lw!}&QL|^Vq1iJo&FmTdV)l$kFne06oBfv*5PSM0n*D(k z5PRB-nLW)?#Jaub4gC p=Vs5Qmf5p5T_&h!k)~v^=ZDSg`MMYT(i=6GM|!mv\n", - "\n", "\n", - "2. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n", + "2. Video of lecture at \n", "\n", - "3. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n", + "3. Whiteboard notes at \n", "\n", - "4. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week." + "4. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra\n", + "\n", + "5. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.\n", + "\n", + "6. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL\"https://mml-book.github.io/\" (section 5.5 on derivatives) is very useful for exercise 1 this coming week." ] }, { "cell_type": "markdown", - "id": "507709a4", + "id": "f0040522", "metadata": { "editable": true }, @@ -102,7 +104,7 @@ }, { "cell_type": "markdown", - "id": "bb3b5f67", + "id": "d64fcc78", "metadata": { "editable": true }, @@ -118,7 +120,7 @@ }, { "cell_type": "markdown", - "id": "14444be5", + "id": "a1ada6ce", "metadata": { "editable": true }, @@ -130,7 +132,7 @@ }, { "cell_type": "markdown", - "id": "4edc5258", + "id": "5645817b", "metadata": { "editable": true }, @@ -140,7 +142,7 @@ }, { "cell_type": "markdown", - "id": "4d6f9c43", + "id": "30bee0bb", "metadata": { "editable": true }, @@ -152,7 +154,7 @@ }, { "cell_type": "markdown", - "id": "fb52f7f5", + "id": "e6e20e11", "metadata": { "editable": true }, @@ -173,7 +175,7 @@ }, { "cell_type": "markdown", - "id": "4fff6b39", + "id": "7bfd72c1", "metadata": { "editable": true }, @@ -185,7 +187,7 @@ }, { "cell_type": "markdown", - "id": "54a4cabb", + "id": "69275ea0", "metadata": { "editable": true }, @@ -198,7 +200,7 @@ }, { "cell_type": "markdown", - "id": "d383a507", + "id": "976bb1cb", "metadata": { "editable": true }, @@ -210,7 +212,7 @@ }, { "cell_type": "markdown", - "id": "498fe31e", + "id": "0d85580a", "metadata": { "editable": true }, @@ -222,7 +224,7 @@ }, { "cell_type": "markdown", - "id": "7bbbdcd2", + "id": "5a8ac559", "metadata": { "editable": true }, @@ -232,7 +234,7 @@ }, { "cell_type": "markdown", - "id": "405a61ac", + "id": "64f74e34", "metadata": { "editable": true }, @@ -244,7 +246,7 @@ }, { "cell_type": "markdown", - "id": "e7c07f57", + "id": "8d503698", "metadata": { "editable": true }, @@ -257,7 +259,7 @@ }, { "cell_type": "markdown", - "id": "6054782f", + "id": "787c6923", "metadata": { "editable": true }, @@ -269,7 +271,7 @@ }, { "cell_type": "markdown", - "id": "7b55104f", + "id": "54c2b6dc", "metadata": { "editable": true }, @@ -279,7 +281,7 @@ }, { "cell_type": "markdown", - "id": "4635525e", + "id": "a8114ec8", "metadata": { "editable": true }, @@ -291,7 +293,7 @@ }, { "cell_type": "markdown", - "id": "11f6ffc7", + "id": "650e695e", "metadata": { "editable": true }, @@ -303,7 +305,7 @@ }, { "cell_type": "markdown", - "id": "820ddc3f", + "id": "885a3110", "metadata": { "editable": true }, @@ -314,7 +316,7 @@ }, { "cell_type": "markdown", - "id": "9aa4b864", + "id": "9fbaf50f", "metadata": { "editable": true }, @@ -326,7 +328,7 @@ }, { "cell_type": "markdown", - "id": "c542e149", + "id": "2376fa59", "metadata": { "editable": true }, @@ -345,7 +347,7 @@ }, { "cell_type": "markdown", - "id": "7be13194", + "id": "937df3c9", "metadata": { "editable": true }, @@ -358,7 +360,7 @@ }, { "cell_type": "markdown", - "id": "aad2641f", + "id": "e066a9bc", "metadata": { "editable": true }, @@ -368,7 +370,7 @@ }, { "cell_type": "markdown", - "id": "9dfe7e22", + "id": "af821847", "metadata": { "editable": true }, @@ -380,7 +382,7 @@ }, { "cell_type": "markdown", - "id": "852c82a9", + "id": "5e7ddc8b", "metadata": { "editable": true }, @@ -390,7 +392,7 @@ }, { "cell_type": "markdown", - "id": "84163c68", + "id": "47d5910e", "metadata": { "editable": true }, @@ -402,7 +404,7 @@ }, { "cell_type": "markdown", - "id": "5bfb8965", + "id": "6d62a75a", "metadata": { "editable": true }, @@ -412,7 +414,7 @@ }, { "cell_type": "markdown", - "id": "3822fa25", + "id": "58d10371", "metadata": { "editable": true }, @@ -424,7 +426,7 @@ }, { "cell_type": "markdown", - "id": "2c0f9f09", + "id": "1f349d3b", "metadata": { "editable": true }, @@ -435,7 +437,7 @@ }, { "cell_type": "markdown", - "id": "7ce9a08c", + "id": "14a30c64", "metadata": { "editable": true }, @@ -447,7 +449,7 @@ }, { "cell_type": "markdown", - "id": "3efb75ba", + "id": "b2faea7f", "metadata": { "editable": true }, @@ -457,7 +459,7 @@ }, { "cell_type": "markdown", - "id": "8c4ceb9d", + "id": "5810b814", "metadata": { "editable": true }, @@ -469,7 +471,7 @@ }, { "cell_type": "markdown", - "id": "53a5a827", + "id": "fcf3c26e", "metadata": { "editable": true }, @@ -479,7 +481,7 @@ }, { "cell_type": "markdown", - "id": "12d309d7", + "id": "a4453721", "metadata": { "editable": true }, @@ -491,7 +493,7 @@ }, { "cell_type": "markdown", - "id": "16fdbfbe", + "id": "38da802c", "metadata": { "editable": true }, @@ -511,7 +513,7 @@ }, { "cell_type": "markdown", - "id": "b5651e23", + "id": "bdc1020c", "metadata": { "editable": true }, @@ -538,7 +540,7 @@ }, { "cell_type": "markdown", - "id": "e3289213", + "id": "c69738c8", "metadata": { "editable": true }, @@ -550,7 +552,7 @@ }, { "cell_type": "markdown", - "id": "db9c8386", + "id": "c180c4b8", "metadata": { "editable": true }, @@ -562,7 +564,7 @@ }, { "cell_type": "markdown", - "id": "ea7d7d7e", + "id": "e273b465", "metadata": { "editable": true }, @@ -578,7 +580,7 @@ }, { "cell_type": "markdown", - "id": "7dace63c", + "id": "89031202", "metadata": { "editable": true }, @@ -596,7 +598,7 @@ }, { "cell_type": "markdown", - "id": "b00b4a82", + "id": "a5bfe37a", "metadata": { "editable": true }, @@ -608,7 +610,7 @@ }, { "cell_type": "markdown", - "id": "26392bc2", + "id": "9fe43da2", "metadata": { "editable": true }, @@ -620,7 +622,7 @@ }, { "cell_type": "markdown", - "id": "f5214ffc", + "id": "17f42b20", "metadata": { "editable": true }, @@ -631,7 +633,7 @@ }, { "cell_type": "markdown", - "id": "0eec52de", + "id": "6e12c604", "metadata": { "editable": true }, @@ -643,7 +645,7 @@ }, { "cell_type": "markdown", - "id": "077e4fe0", + "id": "c03524f3", "metadata": { "editable": true }, @@ -653,7 +655,7 @@ }, { "cell_type": "markdown", - "id": "e3ff215c", + "id": "66768181", "metadata": { "editable": true }, @@ -665,7 +667,7 @@ }, { "cell_type": "markdown", - "id": "6cb9ae59", + "id": "23864bbb", "metadata": { "editable": true }, @@ -679,7 +681,7 @@ }, { "cell_type": "markdown", - "id": "7a02062e", + "id": "c39fc9ec", "metadata": { "editable": true }, @@ -691,7 +693,7 @@ }, { "cell_type": "markdown", - "id": "45f73c0f", + "id": "9733262a", "metadata": { "editable": true }, @@ -703,7 +705,7 @@ }, { "cell_type": "markdown", - "id": "6c9d24a2", + "id": "dc160822", "metadata": { "editable": true }, @@ -715,7 +717,7 @@ }, { "cell_type": "markdown", - "id": "241fed59", + "id": "1211e28f", "metadata": { "editable": true }, @@ -725,7 +727,7 @@ }, { "cell_type": "markdown", - "id": "1c67de99", + "id": "939dfa18", "metadata": { "editable": true }, @@ -737,7 +739,7 @@ }, { "cell_type": "markdown", - "id": "36af470a", + "id": "f1d30438", "metadata": { "editable": true }, @@ -749,7 +751,7 @@ }, { "cell_type": "markdown", - "id": "38f9d9cf", + "id": "bdb64e7a", "metadata": { "editable": true }, @@ -761,7 +763,7 @@ }, { "cell_type": "markdown", - "id": "b20c89a1", + "id": "4952d52d", "metadata": { "editable": true }, @@ -775,7 +777,7 @@ }, { "cell_type": "markdown", - "id": "e56ba6c5", + "id": "f7ea9937", "metadata": { "editable": true }, @@ -787,7 +789,7 @@ }, { "cell_type": "markdown", - "id": "1481f969", + "id": "31e1ef19", "metadata": { "editable": true }, @@ -799,7 +801,7 @@ }, { "cell_type": "markdown", - "id": "fbd29e7c", + "id": "22628265", "metadata": { "editable": true }, @@ -811,7 +813,7 @@ }, { "cell_type": "markdown", - "id": "889c22f1", + "id": "d2e9c679", "metadata": { "editable": true }, @@ -821,7 +823,7 @@ }, { "cell_type": "markdown", - "id": "5a753c31", + "id": "c03929d2", "metadata": { "editable": true }, @@ -833,7 +835,7 @@ }, { "cell_type": "markdown", - "id": "4fed4676", + "id": "131c4d51", "metadata": { "editable": true }, @@ -843,7 +845,7 @@ }, { "cell_type": "markdown", - "id": "e1894902", + "id": "27cc18f9", "metadata": { "editable": true }, @@ -855,7 +857,7 @@ }, { "cell_type": "markdown", - "id": "ddca93b1", + "id": "f78fde44", "metadata": { "editable": true }, @@ -865,7 +867,7 @@ }, { "cell_type": "markdown", - "id": "5f9d8901", + "id": "b3117122", "metadata": { "editable": true }, @@ -877,7 +879,7 @@ }, { "cell_type": "markdown", - "id": "726c169c", + "id": "6025f5d9", "metadata": { "editable": true }, @@ -889,7 +891,7 @@ }, { "cell_type": "markdown", - "id": "7d7715d5", + "id": "f6d884e9", "metadata": { "editable": true }, @@ -901,7 +903,7 @@ }, { "cell_type": "markdown", - "id": "057fedd9", + "id": "d7241fbc", "metadata": { "editable": true }, @@ -916,7 +918,7 @@ }, { "cell_type": "markdown", - "id": "4c06fecd", + "id": "30ea6892", "metadata": { "editable": true }, @@ -928,7 +930,7 @@ }, { "cell_type": "markdown", - "id": "490e6469", + "id": "ada2be4f", "metadata": { "editable": true }, @@ -938,7 +940,7 @@ }, { "cell_type": "markdown", - "id": "b6259635", + "id": "131da9a2", "metadata": { "editable": true }, @@ -950,7 +952,7 @@ }, { "cell_type": "markdown", - "id": "0fda0191", + "id": "15d00d29", "metadata": { "editable": true }, @@ -960,7 +962,7 @@ }, { "cell_type": "markdown", - "id": "797e356f", + "id": "bd4077dd", "metadata": { "editable": true }, @@ -972,7 +974,7 @@ }, { "cell_type": "markdown", - "id": "46bdaee6", + "id": "a0fd8aac", "metadata": { "editable": true }, @@ -982,7 +984,7 @@ }, { "cell_type": "markdown", - "id": "cb1f5f36", + "id": "ae370d67", "metadata": { "editable": true }, @@ -994,7 +996,7 @@ }, { "cell_type": "markdown", - "id": "fa70c555", + "id": "687aa950", "metadata": { "editable": true }, @@ -1006,7 +1008,7 @@ }, { "cell_type": "markdown", - "id": "2fdb367f", + "id": "4159fec2", "metadata": { "editable": true }, @@ -1018,7 +1020,7 @@ }, { "cell_type": "markdown", - "id": "ced6532f", + "id": "e5f8e212", "metadata": { "editable": true }, @@ -1028,7 +1030,7 @@ }, { "cell_type": "markdown", - "id": "49c27708", + "id": "c26a9c8f", "metadata": { "editable": true }, @@ -1040,7 +1042,7 @@ }, { "cell_type": "markdown", - "id": "27da1de0", + "id": "859c590b", "metadata": { "editable": true }, @@ -1053,7 +1055,7 @@ }, { "cell_type": "markdown", - "id": "9ddef35d", + "id": "fbb8dd7f", "metadata": { "editable": true }, @@ -1065,7 +1067,7 @@ }, { "cell_type": "markdown", - "id": "ae006a7e", + "id": "bf096607", "metadata": { "editable": true }, @@ -1075,7 +1077,7 @@ }, { "cell_type": "markdown", - "id": "904d2486", + "id": "428e2635", "metadata": { "editable": true }, @@ -1087,7 +1089,7 @@ }, { "cell_type": "markdown", - "id": "525eda84", + "id": "81ff9c77", "metadata": { "editable": true }, @@ -1097,7 +1099,7 @@ }, { "cell_type": "markdown", - "id": "7eadfb46", + "id": "a4b9ca1c", "metadata": { "editable": true }, @@ -1109,7 +1111,7 @@ }, { "cell_type": "markdown", - "id": "8ee45bc9", + "id": "b64449f2", "metadata": { "editable": true }, @@ -1119,7 +1121,7 @@ }, { "cell_type": "markdown", - "id": "73b41248", + "id": "6a104348", "metadata": { "editable": true }, @@ -1131,7 +1133,7 @@ }, { "cell_type": "markdown", - "id": "dc4356aa", + "id": "ce07e978", "metadata": { "editable": true }, @@ -1141,7 +1143,7 @@ }, { "cell_type": "markdown", - "id": "3e2b94a9", + "id": "f81215cb", "metadata": { "editable": true }, @@ -1153,7 +1155,7 @@ }, { "cell_type": "markdown", - "id": "b9716f36", + "id": "54e70573", "metadata": { "editable": true }, @@ -1163,7 +1165,7 @@ }, { "cell_type": "markdown", - "id": "01341a5b", + "id": "2f2f537a", "metadata": { "editable": true }, @@ -1175,7 +1177,7 @@ }, { "cell_type": "markdown", - "id": "83a3f463", + "id": "9fb9df02", "metadata": { "editable": true }, @@ -1191,7 +1193,7 @@ }, { "cell_type": "markdown", - "id": "c2c42203", + "id": "2dff6039", "metadata": { "editable": true }, @@ -1203,7 +1205,7 @@ }, { "cell_type": "markdown", - "id": "94f060b9", + "id": "783debc8", "metadata": { "editable": true }, @@ -1213,7 +1215,7 @@ }, { "cell_type": "markdown", - "id": "d0e5c32c", + "id": "42965f7e", "metadata": { "editable": true }, @@ -1225,7 +1227,7 @@ }, { "cell_type": "markdown", - "id": "0b464ed4", + "id": "57e8d8fe", "metadata": { "editable": true }, @@ -1243,7 +1245,7 @@ }, { "cell_type": "markdown", - "id": "5c952119", + "id": "c43b98ff", "metadata": { "editable": true }, @@ -1255,7 +1257,7 @@ }, { "cell_type": "markdown", - "id": "25e4a5f3", + "id": "6ff581de", "metadata": { "editable": true }, @@ -1267,7 +1269,7 @@ }, { "cell_type": "markdown", - "id": "4f774737", + "id": "8f4462e2", "metadata": { "editable": true }, @@ -1277,7 +1279,7 @@ }, { "cell_type": "markdown", - "id": "59c91b82", + "id": "c935b169", "metadata": { "editable": true }, @@ -1289,7 +1291,7 @@ }, { "cell_type": "markdown", - "id": "33ceec0f", + "id": "50b87db5", "metadata": { "editable": true }, @@ -1299,7 +1301,7 @@ }, { "cell_type": "markdown", - "id": "931da8ef", + "id": "2599f182", "metadata": { "editable": true }, @@ -1311,7 +1313,7 @@ }, { "cell_type": "markdown", - "id": "2f2516b6", + "id": "6c07218b", "metadata": { "editable": true }, @@ -1321,7 +1323,7 @@ }, { "cell_type": "markdown", - "id": "bcb56a20", + "id": "d1058185", "metadata": { "editable": true }, @@ -1335,7 +1337,7 @@ }, { "cell_type": "markdown", - "id": "86333a6f", + "id": "951d1097", "metadata": { "editable": true }, @@ -1347,7 +1349,7 @@ }, { "cell_type": "markdown", - "id": "1513e73c", + "id": "73e7100b", "metadata": { "editable": true }, @@ -1358,7 +1360,7 @@ }, { "cell_type": "markdown", - "id": "c8c47d9a", + "id": "b7caf7f6", "metadata": { "editable": true }, @@ -1371,7 +1373,7 @@ { "cell_type": "code", "execution_count": 1, - "id": "8ec18f31", + "id": "01229cfb", "metadata": { "collapsed": false, "editable": true @@ -1398,7 +1400,7 @@ }, { "cell_type": "markdown", - "id": "b7255330", + "id": "f4a3896e", "metadata": { "editable": true }, @@ -1409,7 +1411,7 @@ { "cell_type": "code", "execution_count": 2, - "id": "9d5f3a70", + "id": "6ca9744d", "metadata": { "collapsed": false, "editable": true @@ -1422,7 +1424,7 @@ }, { "cell_type": "markdown", - "id": "8daa4681", + "id": "c434f72f", "metadata": { "editable": true }, @@ -1436,7 +1438,7 @@ { "cell_type": "code", "execution_count": 3, - "id": "de91ca73", + "id": "d4131dbc", "metadata": { "collapsed": false, "editable": true @@ -1449,7 +1451,7 @@ }, { "cell_type": "markdown", - "id": "442ef680", + "id": "e5cff99e", "metadata": { "editable": true }, @@ -1460,7 +1462,7 @@ { "cell_type": "code", "execution_count": 4, - "id": "9b329ab4", + "id": "c1b15227", "metadata": { "collapsed": false, "editable": true @@ -1472,7 +1474,7 @@ }, { "cell_type": "markdown", - "id": "d7ded522", + "id": "a0a432b4", "metadata": { "editable": true }, @@ -1483,7 +1485,7 @@ { "cell_type": "code", "execution_count": 5, - "id": "34c18b8f", + "id": "319c1a33", "metadata": { "collapsed": false, "editable": true @@ -1499,7 +1501,7 @@ }, { "cell_type": "markdown", - "id": "229d94fa", + "id": "98d492e3", "metadata": { "editable": true }, @@ -1510,7 +1512,7 @@ { "cell_type": "code", "execution_count": 6, - "id": "afb92eb9", + "id": "52ad17b3", "metadata": { "collapsed": false, "editable": true @@ -1524,7 +1526,7 @@ }, { "cell_type": "markdown", - "id": "8a02aaaf", + "id": "29abfbb7", "metadata": { "editable": true }, @@ -1545,7 +1547,7 @@ }, { "cell_type": "markdown", - "id": "b64b7d81", + "id": "503f0621", "metadata": { "editable": true }, @@ -1556,7 +1558,7 @@ { "cell_type": "code", "execution_count": 7, - "id": "094eed3e", + "id": "6927de82", "metadata": { "collapsed": false, "editable": true @@ -1609,7 +1611,7 @@ }, { "cell_type": "markdown", - "id": "76119de5", + "id": "0b4f15f1", "metadata": { "editable": true }, @@ -1620,7 +1622,7 @@ { "cell_type": "code", "execution_count": 8, - "id": "dff69c9a", + "id": "ac968b0f", "metadata": { "collapsed": false, "editable": true @@ -1645,7 +1647,7 @@ }, { "cell_type": "markdown", - "id": "02285c48", + "id": "4ea3a52b", "metadata": { "editable": true }, @@ -1657,7 +1659,7 @@ }, { "cell_type": "markdown", - "id": "71fab9ce", + "id": "a027a9a5", "metadata": { "editable": true }, @@ -1686,7 +1688,7 @@ }, { "cell_type": "markdown", - "id": "424914f7", + "id": "b0b3e66c", "metadata": { "editable": true }, @@ -1711,7 +1713,7 @@ }, { "cell_type": "markdown", - "id": "e486c793", + "id": "8af30d8b", "metadata": { "editable": true }, @@ -1731,7 +1733,7 @@ }, { "cell_type": "markdown", - "id": "1c6a9f34", + "id": "cd639711", "metadata": { "editable": true }, @@ -1758,7 +1760,7 @@ }, { "cell_type": "markdown", - "id": "28f4300e", + "id": "efb38ba0", "metadata": { "editable": true }, @@ -1771,7 +1773,7 @@ }, { "cell_type": "markdown", - "id": "deb48c69", + "id": "bcc3ee33", "metadata": { "editable": true }, @@ -1783,7 +1785,7 @@ }, { "cell_type": "markdown", - "id": "127f4812", + "id": "0c4c440b", "metadata": { "editable": true }, @@ -1794,7 +1796,7 @@ }, { "cell_type": "markdown", - "id": "305b4349", + "id": "b006a855", "metadata": { "editable": true }, @@ -1810,7 +1812,7 @@ { "cell_type": "code", "execution_count": 9, - "id": "9dacf721", + "id": "98079995", "metadata": { "collapsed": false, "editable": true @@ -1844,7 +1846,7 @@ }, { "cell_type": "markdown", - "id": "63596c47", + "id": "4d8bc84b", "metadata": { "editable": true }, @@ -1854,7 +1856,7 @@ }, { "cell_type": "markdown", - "id": "e663049c", + "id": "78d54efa", "metadata": { "editable": true }, @@ -1869,7 +1871,7 @@ }, { "cell_type": "markdown", - "id": "e2b36261", + "id": "2c6126fc", "metadata": { "editable": true }, @@ -1881,7 +1883,7 @@ }, { "cell_type": "markdown", - "id": "69fa292f", + "id": "2cc97a3e", "metadata": { "editable": true }, @@ -1891,7 +1893,7 @@ }, { "cell_type": "markdown", - "id": "23b3dece", + "id": "9e0ea279", "metadata": { "editable": true }, @@ -1907,7 +1909,7 @@ { "cell_type": "code", "execution_count": 10, - "id": "e1630ab8", + "id": "d00a25ea", "metadata": { "collapsed": false, "editable": true @@ -1924,7 +1926,7 @@ }, { "cell_type": "markdown", - "id": "c2d3b936", + "id": "864015a1", "metadata": { "editable": true }, @@ -1937,7 +1939,7 @@ { "cell_type": "code", "execution_count": 11, - "id": "355c6a66", + "id": "5ba30f30", "metadata": { "collapsed": false, "editable": true @@ -1984,7 +1986,7 @@ }, { "cell_type": "markdown", - "id": "ab553d39", + "id": "8dc3cb9f", "metadata": { "editable": true }, @@ -1998,7 +2000,7 @@ }, { "cell_type": "markdown", - "id": "f439180e", + "id": "068d5e0a", "metadata": { "editable": true }, @@ -2010,7 +2012,7 @@ }, { "cell_type": "markdown", - "id": "7ba7386d", + "id": "6feb2964", "metadata": { "editable": true }, @@ -2022,7 +2024,7 @@ }, { "cell_type": "markdown", - "id": "7e0a1440", + "id": "df3d9523", "metadata": { "editable": true }, @@ -2034,7 +2036,7 @@ }, { "cell_type": "markdown", - "id": "2e7611f3", + "id": "7e8785d1", "metadata": { "editable": true }, @@ -2044,7 +2046,7 @@ }, { "cell_type": "markdown", - "id": "414e53ca", + "id": "ef3be4f1", "metadata": { "editable": true }, @@ -2056,7 +2058,7 @@ }, { "cell_type": "markdown", - "id": "31ed8899", + "id": "3dcd8cf6", "metadata": { "editable": true }, @@ -2066,7 +2068,7 @@ }, { "cell_type": "markdown", - "id": "a66641fb", + "id": "7625f933", "metadata": { "editable": true }, @@ -2078,7 +2080,7 @@ }, { "cell_type": "markdown", - "id": "28a50f9e", + "id": "ef6f7710", "metadata": { "editable": true }, @@ -2089,7 +2091,7 @@ }, { "cell_type": "markdown", - "id": "b8c5f507", + "id": "b52129d8", "metadata": { "editable": true }, @@ -2101,7 +2103,7 @@ }, { "cell_type": "markdown", - "id": "445744c8", + "id": "c0670f24", "metadata": { "editable": true }, @@ -2113,7 +2115,7 @@ }, { "cell_type": "markdown", - "id": "ef90b7cb", + "id": "a30df157", "metadata": { "editable": true }, @@ -2123,7 +2125,7 @@ }, { "cell_type": "markdown", - "id": "9d4c2ac7", + "id": "75243d64", "metadata": { "editable": true }, @@ -2135,7 +2137,7 @@ }, { "cell_type": "markdown", - "id": "84e62cfc", + "id": "9c359540", "metadata": { "editable": true }, @@ -2147,7 +2149,7 @@ }, { "cell_type": "markdown", - "id": "41c2ba4b", + "id": "fa82ae4e", "metadata": { "editable": true }, @@ -2157,7 +2159,7 @@ }, { "cell_type": "markdown", - "id": "beb557ef", + "id": "65734ef1", "metadata": { "editable": true }, @@ -2169,7 +2171,7 @@ }, { "cell_type": "markdown", - "id": "b3fdffce", + "id": "650117d0", "metadata": { "editable": true }, @@ -2179,7 +2181,7 @@ }, { "cell_type": "markdown", - "id": "8005292a", + "id": "06771b2d", "metadata": { "editable": true }, @@ -2191,7 +2193,7 @@ }, { "cell_type": "markdown", - "id": "92d9f94b", + "id": "690e56c7", "metadata": { "editable": true }, @@ -2201,7 +2203,7 @@ }, { "cell_type": "markdown", - "id": "17940c71", + "id": "db5e7402", "metadata": { "editable": true }, @@ -2241,7 +2243,7 @@ }, { "cell_type": "markdown", - "id": "6ffb31dd", + "id": "a7813ad4", "metadata": { "editable": true }, @@ -2258,7 +2260,7 @@ }, { "cell_type": "markdown", - "id": "cc767251", + "id": "2108743e", "metadata": { "editable": true }, @@ -2281,7 +2283,7 @@ }, { "cell_type": "markdown", - "id": "347029b1", + "id": "3f63b5de", "metadata": { "editable": true }, @@ -2298,7 +2300,7 @@ }, { "cell_type": "markdown", - "id": "a77dda49", + "id": "05003c17", "metadata": { "editable": true }, @@ -2317,7 +2319,7 @@ }, { "cell_type": "markdown", - "id": "25a0d7d6", + "id": "315a72a0", "metadata": { "editable": true }, @@ -2328,7 +2330,7 @@ }, { "cell_type": "markdown", - "id": "70462fa0", + "id": "66ceac64", "metadata": { "editable": true }, @@ -2340,7 +2342,7 @@ }, { "cell_type": "markdown", - "id": "f8421c8e", + "id": "35b697b9", "metadata": { "editable": true }, @@ -2358,7 +2360,7 @@ }, { "cell_type": "markdown", - "id": "d4ef09ac", + "id": "4712a94c", "metadata": { "editable": true }, @@ -2374,7 +2376,7 @@ }, { "cell_type": "markdown", - "id": "fc6f8651", + "id": "c11c7415", "metadata": { "editable": true }, @@ -2386,7 +2388,7 @@ }, { "cell_type": "markdown", - "id": "ece72dfe", + "id": "7eecbc75", "metadata": { "editable": true }, @@ -2396,7 +2398,7 @@ }, { "cell_type": "markdown", - "id": "3f900d15", + "id": "3577f76a", "metadata": { "editable": true }, @@ -2409,7 +2411,7 @@ }, { "cell_type": "markdown", - "id": "3cb45c74", + "id": "e1cad910", "metadata": { "editable": true }, @@ -2421,7 +2423,7 @@ }, { "cell_type": "markdown", - "id": "81b8790f", + "id": "e1ed7f36", "metadata": { "editable": true }, @@ -2431,7 +2433,7 @@ }, { "cell_type": "markdown", - "id": "9d42c98f", + "id": "d9797b9a", "metadata": { "editable": true }, @@ -2444,7 +2446,7 @@ }, { "cell_type": "markdown", - "id": "ab036419", + "id": "8a7d60f5", "metadata": { "editable": true }, @@ -2454,7 +2456,7 @@ }, { "cell_type": "markdown", - "id": "bba11ff3", + "id": "597db6eb", "metadata": { "editable": true }, @@ -2466,7 +2468,7 @@ }, { "cell_type": "markdown", - "id": "6518184d", + "id": "03d92296", "metadata": { "editable": true }, @@ -2479,7 +2481,7 @@ }, { "cell_type": "markdown", - "id": "ba89e5b0", + "id": "135a3e67", "metadata": { "editable": true }, @@ -2492,7 +2494,7 @@ }, { "cell_type": "markdown", - "id": "47325cf5", + "id": "21a76d33", "metadata": { "editable": true }, @@ -2504,7 +2506,7 @@ }, { "cell_type": "markdown", - "id": "8d9036c1", + "id": "b7797c08", "metadata": { "editable": true }, @@ -2516,7 +2518,7 @@ }, { "cell_type": "markdown", - "id": "889f801b", + "id": "bd4d1e02", "metadata": { "editable": true }, @@ -2526,7 +2528,7 @@ }, { "cell_type": "markdown", - "id": "8a803d4a", + "id": "b46ffa29", "metadata": { "editable": true }, @@ -2539,7 +2541,7 @@ }, { "cell_type": "markdown", - "id": "ec92b840", + "id": "82955735", "metadata": { "editable": true }, @@ -2551,7 +2553,7 @@ }, { "cell_type": "markdown", - "id": "6de4ce40", + "id": "a489bfcb", "metadata": { "editable": true }, @@ -2563,7 +2565,7 @@ }, { "cell_type": "markdown", - "id": "22e97024", + "id": "b79dc723", "metadata": { "editable": true }, @@ -2575,7 +2577,7 @@ }, { "cell_type": "markdown", - "id": "9cd0a124", + "id": "b4e4b19c", "metadata": { "editable": true }, @@ -2587,7 +2589,7 @@ }, { "cell_type": "markdown", - "id": "29b31bed", + "id": "34dc0808", "metadata": { "editable": true }, @@ -2601,7 +2603,7 @@ }, { "cell_type": "markdown", - "id": "d1145ae7", + "id": "9e053bfc", "metadata": { "editable": true }, @@ -2613,7 +2615,7 @@ }, { "cell_type": "markdown", - "id": "f1479849", + "id": "b8cb20a9", "metadata": { "editable": true }, @@ -2623,7 +2625,7 @@ }, { "cell_type": "markdown", - "id": "0c51f5eb", + "id": "d218f1b3", "metadata": { "editable": true }, @@ -2635,7 +2637,7 @@ }, { "cell_type": "markdown", - "id": "2cbb0f42", + "id": "dc3e144e", "metadata": { "editable": true }, @@ -2647,7 +2649,7 @@ }, { "cell_type": "markdown", - "id": "406c4098", + "id": "b9dcf0b9", "metadata": { "editable": true }, @@ -2659,7 +2661,7 @@ }, { "cell_type": "markdown", - "id": "70966948", + "id": "a118bee6", "metadata": { "editable": true }, @@ -2671,7 +2673,7 @@ }, { "cell_type": "markdown", - "id": "627f3d38", + "id": "a18637c1", "metadata": { "editable": true }, @@ -2683,7 +2685,7 @@ }, { "cell_type": "markdown", - "id": "129105f7", + "id": "22ecf42d", "metadata": { "editable": true }, @@ -2706,7 +2708,7 @@ { "cell_type": "code", "execution_count": 12, - "id": "bce32748", + "id": "2c933b95", "metadata": { "collapsed": false, "editable": true @@ -2782,7 +2784,7 @@ }, { "cell_type": "markdown", - "id": "62eccb26", + "id": "27fcd2cd", "metadata": { "editable": true }, @@ -2794,7 +2796,7 @@ }, { "cell_type": "markdown", - "id": "675cc7de", + "id": "fb501ab3", "metadata": { "editable": true }, @@ -2809,7 +2811,7 @@ }, { "cell_type": "markdown", - "id": "ff36cd09", + "id": "bbdc8c03", "metadata": { "editable": true }, @@ -2821,7 +2823,7 @@ }, { "cell_type": "markdown", - "id": "7ae7a42a", + "id": "4f4e1b19", "metadata": { "editable": true }, @@ -2831,7 +2833,7 @@ }, { "cell_type": "markdown", - "id": "607325f0", + "id": "4b53a433", "metadata": { "editable": true }, @@ -2843,7 +2845,7 @@ }, { "cell_type": "markdown", - "id": "cfb328b4", + "id": "243e598d", "metadata": { "editable": true }, @@ -2853,7 +2855,7 @@ }, { "cell_type": "markdown", - "id": "bc44d844", + "id": "9e54e87c", "metadata": { "editable": true }, @@ -2865,7 +2867,7 @@ }, { "cell_type": "markdown", - "id": "17ddbfda", + "id": "ca9080f4", "metadata": { "editable": true }, @@ -2877,7 +2879,7 @@ }, { "cell_type": "markdown", - "id": "f5658751", + "id": "c0da749a", "metadata": { "editable": true }, @@ -2892,7 +2894,7 @@ }, { "cell_type": "markdown", - "id": "2b01e674", + "id": "73106e44", "metadata": { "editable": true }, @@ -2903,7 +2905,7 @@ }, { "cell_type": "markdown", - "id": "c18a65c6", + "id": "47a5df25", "metadata": { "editable": true }, @@ -2923,7 +2925,7 @@ }, { "cell_type": "markdown", - "id": "04258de2", + "id": "f75c5f9d", "metadata": { "editable": true }, @@ -2935,7 +2937,7 @@ }, { "cell_type": "markdown", - "id": "e10d0aad", + "id": "ae49bbbe", "metadata": { "editable": true }, @@ -2945,7 +2947,7 @@ }, { "cell_type": "markdown", - "id": "d198578b", + "id": "9927bcbf", "metadata": { "editable": true }, @@ -2957,7 +2959,7 @@ }, { "cell_type": "markdown", - "id": "2aec2fc1", + "id": "557aa14a", "metadata": { "editable": true }, @@ -2986,7 +2988,7 @@ }, { "cell_type": "markdown", - "id": "81fed6f5", + "id": "b77be05b", "metadata": { "editable": true }, @@ -3013,7 +3015,7 @@ }, { "cell_type": "markdown", - "id": "fe7d1d49", + "id": "dbca187b", "metadata": { "editable": true }, @@ -3024,7 +3026,7 @@ { "cell_type": "code", "execution_count": 13, - "id": "0b32041d", + "id": "52ee2c0a", "metadata": { "collapsed": false, "editable": true @@ -3064,7 +3066,7 @@ }, { "cell_type": "markdown", - "id": "db224b5b", + "id": "9585fb61", "metadata": { "editable": true }, @@ -3081,7 +3083,7 @@ }, { "cell_type": "markdown", - "id": "6b988f39", + "id": "1820219c", "metadata": { "editable": true }, @@ -3104,7 +3106,7 @@ }, { "cell_type": "markdown", - "id": "1214012e", + "id": "61e763b3", "metadata": { "editable": true }, @@ -3118,7 +3120,7 @@ }, { "cell_type": "markdown", - "id": "c7a8cd85", + "id": "d215271f", "metadata": { "editable": true }, @@ -3137,7 +3139,7 @@ }, { "cell_type": "markdown", - "id": "0f2a2211", + "id": "9e097ba9", "metadata": { "editable": true }, @@ -3147,7 +3149,7 @@ }, { "cell_type": "markdown", - "id": "e9863a0f", + "id": "c11e5ea0", "metadata": { "editable": true }, @@ -3159,7 +3161,7 @@ }, { "cell_type": "markdown", - "id": "d88c8316", + "id": "d7a4b260", "metadata": { "editable": true }, @@ -3173,7 +3175,7 @@ }, { "cell_type": "markdown", - "id": "643f1b6d", + "id": "dde8684e", "metadata": { "editable": true }, @@ -3185,7 +3187,7 @@ }, { "cell_type": "markdown", - "id": "0769ec85", + "id": "1ee4ec03", "metadata": { "editable": true }, @@ -3195,7 +3197,7 @@ }, { "cell_type": "markdown", - "id": "f0d61e66", + "id": "5db5fd8b", "metadata": { "editable": true }, @@ -3207,7 +3209,7 @@ }, { "cell_type": "markdown", - "id": "0d7c9d9e", + "id": "315fa545", "metadata": { "editable": true }, @@ -3224,7 +3226,7 @@ }, { "cell_type": "markdown", - "id": "335fb5ed", + "id": "669553b8", "metadata": { "editable": true }, @@ -3234,7 +3236,7 @@ }, { "cell_type": "markdown", - "id": "1dbb3e05", + "id": "a5c69189", "metadata": { "editable": true }, @@ -3250,7 +3252,7 @@ }, { "cell_type": "markdown", - "id": "4a535424", + "id": "c98988f9", "metadata": { "editable": true }, @@ -3260,7 +3262,7 @@ }, { "cell_type": "markdown", - "id": "512418d2", + "id": "ff274815", "metadata": { "editable": true }, @@ -3276,7 +3278,7 @@ }, { "cell_type": "markdown", - "id": "baa56321", + "id": "be481f90", "metadata": { "editable": true }, @@ -3286,7 +3288,7 @@ }, { "cell_type": "markdown", - "id": "1fd5881c", + "id": "5389a4fd", "metadata": { "editable": true }, @@ -3302,7 +3304,7 @@ }, { "cell_type": "markdown", - "id": "546a2bf3", + "id": "38c8b825", "metadata": { "editable": true }, @@ -3312,7 +3314,7 @@ }, { "cell_type": "markdown", - "id": "4e356417", + "id": "854c2801", "metadata": { "editable": true }, @@ -3329,7 +3331,7 @@ }, { "cell_type": "markdown", - "id": "1f3c787b", + "id": "77b24821", "metadata": { "editable": true }, @@ -3341,7 +3343,7 @@ }, { "cell_type": "markdown", - "id": "72018911", + "id": "9cb3e178", "metadata": { "editable": true }, @@ -3353,7 +3355,7 @@ }, { "cell_type": "markdown", - "id": "7d741e98", + "id": "a3e31623", "metadata": { "editable": true }, @@ -3365,7 +3367,7 @@ }, { "cell_type": "markdown", - "id": "f2e4a67f", + "id": "28150e6b", "metadata": { "editable": true }, @@ -3375,7 +3377,7 @@ }, { "cell_type": "markdown", - "id": "302c8edf", + "id": "fc1f871e", "metadata": { "editable": true }, @@ -3387,7 +3389,7 @@ }, { "cell_type": "markdown", - "id": "def3ec8a", + "id": "dd7f98b3", "metadata": { "editable": true }, @@ -3399,7 +3401,7 @@ }, { "cell_type": "markdown", - "id": "37150ee7", + "id": "b20a7cc6", "metadata": { "editable": true }, @@ -3411,7 +3413,7 @@ }, { "cell_type": "markdown", - "id": "e937a396", + "id": "82d661e0", "metadata": { "editable": true }, @@ -3421,7 +3423,7 @@ }, { "cell_type": "markdown", - "id": "07e76c7b", + "id": "705bb92a", "metadata": { "editable": true }, @@ -3433,7 +3435,7 @@ }, { "cell_type": "markdown", - "id": "99f80e22", + "id": "5c983fca", "metadata": { "editable": true }, @@ -3443,7 +3445,7 @@ }, { "cell_type": "markdown", - "id": "7a917bf2", + "id": "5b32695b", "metadata": { "editable": true }, @@ -3455,7 +3457,7 @@ }, { "cell_type": "markdown", - "id": "c11ae5fe", + "id": "c4285bb3", "metadata": { "editable": true }, @@ -3472,7 +3474,7 @@ }, { "cell_type": "markdown", - "id": "f300aa1b", + "id": "1fa01f9a", "metadata": { "editable": true }, @@ -3484,7 +3486,7 @@ }, { "cell_type": "markdown", - "id": "8bd1e2a6", + "id": "c9d412c2", "metadata": { "editable": true }, @@ -3496,7 +3498,7 @@ }, { "cell_type": "markdown", - "id": "62502292", + "id": "5307ebb8", "metadata": { "editable": true }, @@ -3506,7 +3508,7 @@ }, { "cell_type": "markdown", - "id": "efca6f6c", + "id": "ed2b38c3", "metadata": { "editable": true }, @@ -3518,7 +3520,7 @@ }, { "cell_type": "markdown", - "id": "e2cbd6f5", + "id": "5dd9e137", "metadata": { "editable": true }, @@ -3529,7 +3531,7 @@ }, { "cell_type": "markdown", - "id": "5cc9aea6", + "id": "38a3902d", "metadata": { "editable": true }, @@ -3541,7 +3543,7 @@ }, { "cell_type": "markdown", - "id": "61ad8f4c", + "id": "53a81f9f", "metadata": { "editable": true }, @@ -3551,7 +3553,7 @@ }, { "cell_type": "markdown", - "id": "b1cc9449", + "id": "c645a7d0", "metadata": { "editable": true }, @@ -3563,7 +3565,7 @@ }, { "cell_type": "markdown", - "id": "a9b28652", + "id": "a3098a03", "metadata": { "editable": true }, @@ -3573,7 +3575,7 @@ }, { "cell_type": "markdown", - "id": "f26b69cc", + "id": "08b90293", "metadata": { "editable": true }, @@ -3585,7 +3587,7 @@ }, { "cell_type": "markdown", - "id": "53acde0e", + "id": "1842a571", "metadata": { "editable": true }, @@ -3596,7 +3598,7 @@ }, { "cell_type": "markdown", - "id": "512b9f17", + "id": "0290fcb9", "metadata": { "editable": true }, @@ -3608,7 +3610,7 @@ }, { "cell_type": "markdown", - "id": "a1c49f5f", + "id": "ecbc00d0", "metadata": { "editable": true }, @@ -3626,7 +3628,7 @@ }, { "cell_type": "markdown", - "id": "db70d623", + "id": "670c8859", "metadata": { "editable": true }, @@ -3642,7 +3644,7 @@ }, { "cell_type": "markdown", - "id": "95b18b06", + "id": "286fd62d", "metadata": { "editable": true }, @@ -3654,7 +3656,7 @@ }, { "cell_type": "markdown", - "id": "db07fa1e", + "id": "a51c433e", "metadata": { "editable": true }, @@ -3666,7 +3668,7 @@ }, { "cell_type": "markdown", - "id": "9856cea3", + "id": "d19e4786", "metadata": { "editable": true }, @@ -3678,7 +3680,7 @@ }, { "cell_type": "markdown", - "id": "7f6cba4c", + "id": "56caedc2", "metadata": { "editable": true }, @@ -3691,7 +3693,7 @@ }, { "cell_type": "markdown", - "id": "581565a5", + "id": "d71c6ba0", "metadata": { "editable": true }, @@ -3707,7 +3709,7 @@ }, { "cell_type": "markdown", - "id": "22b63d08", + "id": "3dfd726a", "metadata": { "editable": true }, @@ -3721,7 +3723,7 @@ }, { "cell_type": "markdown", - "id": "cd6032aa", + "id": "5ed3d153", "metadata": { "editable": true }, @@ -3731,7 +3733,7 @@ }, { "cell_type": "markdown", - "id": "25b7936a", + "id": "1ac50487", "metadata": { "editable": true }, @@ -3743,7 +3745,7 @@ }, { "cell_type": "markdown", - "id": "9c4ad4d2", + "id": "d3cc0098", "metadata": { "editable": true }, @@ -3753,7 +3755,7 @@ }, { "cell_type": "markdown", - "id": "75bcfcea", + "id": "d1adc309", "metadata": { "editable": true }, @@ -3765,7 +3767,7 @@ }, { "cell_type": "markdown", - "id": "95c9cc40", + "id": "606b3621", "metadata": { "editable": true }, @@ -3775,7 +3777,7 @@ }, { "cell_type": "markdown", - "id": "33c19ed1", + "id": "48dfd7a6", "metadata": { "editable": true }, @@ -3789,7 +3791,7 @@ }, { "cell_type": "markdown", - "id": "c28b0a97", + "id": "f2d389fc", "metadata": { "editable": true }, @@ -3806,7 +3808,7 @@ }, { "cell_type": "markdown", - "id": "d4ece056", + "id": "93bee516", "metadata": { "editable": true }, @@ -3822,7 +3824,7 @@ }, { "cell_type": "markdown", - "id": "22360ee3", + "id": "b19050a0", "metadata": { "editable": true }, @@ -3834,7 +3836,7 @@ }, { "cell_type": "markdown", - "id": "77e8998d", + "id": "d6f57474", "metadata": { "editable": true }, @@ -3847,7 +3849,7 @@ }, { "cell_type": "markdown", - "id": "8ef4225d", + "id": "f7dd4e66", "metadata": { "editable": true }, @@ -3861,7 +3863,7 @@ }, { "cell_type": "markdown", - "id": "627eac2b", + "id": "99d565fb", "metadata": { "editable": true }, @@ -3871,7 +3873,7 @@ }, { "cell_type": "markdown", - "id": "f7ac0db1", + "id": "c65756ab", "metadata": { "editable": true }, @@ -3884,7 +3886,7 @@ }, { "cell_type": "markdown", - "id": "e93eae5d", + "id": "16c3efb7", "metadata": { "editable": true }, @@ -3903,7 +3905,7 @@ }, { "cell_type": "markdown", - "id": "58364673", + "id": "6e4194ff", "metadata": { "editable": true }, @@ -3915,7 +3917,7 @@ }, { "cell_type": "markdown", - "id": "30541096", + "id": "5259c2eb", "metadata": { "editable": true }, @@ -3927,7 +3929,7 @@ }, { "cell_type": "markdown", - "id": "101bb217", + "id": "1b927e23", "metadata": { "editable": true }, @@ -3937,7 +3939,7 @@ }, { "cell_type": "markdown", - "id": "ba8e52b1", + "id": "34ab019a", "metadata": { "editable": true }, @@ -3949,7 +3951,7 @@ }, { "cell_type": "markdown", - "id": "67d4335e", + "id": "21075d57", "metadata": { "editable": true }, @@ -3962,7 +3964,7 @@ }, { "cell_type": "markdown", - "id": "06416768", + "id": "8833f450", "metadata": { "editable": true }, @@ -3981,7 +3983,7 @@ }, { "cell_type": "markdown", - "id": "dd2f405a", + "id": "0813cc4c", "metadata": { "editable": true }, @@ -3991,7 +3993,7 @@ }, { "cell_type": "markdown", - "id": "40ded798", + "id": "50d7582d", "metadata": { "editable": true }, @@ -4010,7 +4012,7 @@ }, { "cell_type": "markdown", - "id": "d3044a01", + "id": "8f85e67d", "metadata": { "editable": true }, @@ -4028,7 +4030,7 @@ }, { "cell_type": "markdown", - "id": "d612ad97", + "id": "385c55d6", "metadata": { "editable": true }, @@ -4042,7 +4044,7 @@ }, { "cell_type": "markdown", - "id": "52f27c53", + "id": "ead461f7", "metadata": { "editable": true }, @@ -4057,7 +4059,7 @@ { "cell_type": "code", "execution_count": 14, - "id": "ca5ce77c", + "id": "80c333dc", "metadata": { "collapsed": false, "editable": true @@ -4078,7 +4080,7 @@ }, { "cell_type": "markdown", - "id": "1c02019d", + "id": "69f8b2df", "metadata": { "editable": true }, @@ -4095,7 +4097,7 @@ { "cell_type": "code", "execution_count": 15, - "id": "58381fbe", + "id": "4a07353d", "metadata": { "collapsed": false, "editable": true @@ -4127,7 +4129,7 @@ }, { "cell_type": "markdown", - "id": "a647044e", + "id": "ee3bce85", "metadata": { "editable": true }, @@ -4141,7 +4143,7 @@ }, { "cell_type": "markdown", - "id": "d7bea0a6", + "id": "3d15065a", "metadata": { "editable": true }, @@ -4154,7 +4156,7 @@ { "cell_type": "code", "execution_count": 16, - "id": "81d407da", + "id": "488f60ab", "metadata": { "collapsed": false, "editable": true @@ -4179,7 +4181,7 @@ }, { "cell_type": "markdown", - "id": "7998701a", + "id": "cafa6fa9", "metadata": { "editable": true }, @@ -4191,7 +4193,7 @@ }, { "cell_type": "markdown", - "id": "e4f65917", + "id": "ce70f3a8", "metadata": { "editable": true }, @@ -4203,7 +4205,7 @@ }, { "cell_type": "markdown", - "id": "73ecb476", + "id": "10c922ba", "metadata": { "editable": true }, @@ -4213,7 +4215,7 @@ }, { "cell_type": "markdown", - "id": "f4f10483", + "id": "0d791d47", "metadata": { "editable": true }, @@ -4230,7 +4232,7 @@ }, { "cell_type": "markdown", - "id": "9e173ba2", + "id": "533a270f", "metadata": { "editable": true }, @@ -4240,7 +4242,7 @@ }, { "cell_type": "markdown", - "id": "6b6f3d02", + "id": "1b8bd026", "metadata": { "editable": true }, @@ -4255,7 +4257,7 @@ }, { "cell_type": "markdown", - "id": "69a2c9b4", + "id": "f3f09455", "metadata": { "editable": true }, @@ -4265,7 +4267,7 @@ }, { "cell_type": "markdown", - "id": "41dedd73", + "id": "bce601da", "metadata": { "editable": true }, @@ -4279,7 +4281,7 @@ }, { "cell_type": "markdown", - "id": "a7b2bb8c", + "id": "85ed9781", "metadata": { "editable": true }, @@ -4291,7 +4293,7 @@ }, { "cell_type": "markdown", - "id": "dba35fc4", + "id": "7f6fcb9f", "metadata": { "editable": true }, @@ -4303,7 +4305,7 @@ }, { "cell_type": "markdown", - "id": "0e30a645", + "id": "6ea6f608", "metadata": { "editable": true }, @@ -4315,7 +4317,7 @@ }, { "cell_type": "markdown", - "id": "6fa482d3", + "id": "005927d3", "metadata": { "editable": true }, @@ -4325,7 +4327,7 @@ }, { "cell_type": "markdown", - "id": "ae9f30ae", + "id": "c1f7c3dd", "metadata": { "editable": true }, @@ -4337,7 +4339,7 @@ }, { "cell_type": "markdown", - "id": "67907a60", + "id": "a5c144b5", "metadata": { "editable": true }, @@ -4347,7 +4349,7 @@ }, { "cell_type": "markdown", - "id": "3a21fe71", + "id": "73802678", "metadata": { "editable": true }, @@ -4364,7 +4366,7 @@ }, { "cell_type": "markdown", - "id": "3b91b5d8", + "id": "aa888252", "metadata": { "editable": true }, @@ -4374,7 +4376,7 @@ }, { "cell_type": "markdown", - "id": "86de4409", + "id": "9d3363b0", "metadata": { "editable": true }, @@ -4386,7 +4388,7 @@ }, { "cell_type": "markdown", - "id": "d0c0608f", + "id": "49a1347a", "metadata": { "editable": true }, @@ -4396,7 +4398,7 @@ }, { "cell_type": "markdown", - "id": "98f544a6", + "id": "1ba2b422", "metadata": { "editable": true }, @@ -4408,7 +4410,7 @@ }, { "cell_type": "markdown", - "id": "24587410", + "id": "5ffae339", "metadata": { "editable": true }, @@ -4422,7 +4424,7 @@ }, { "cell_type": "markdown", - "id": "61fb4275", + "id": "cf729fa0", "metadata": { "editable": true }, @@ -4434,7 +4436,7 @@ }, { "cell_type": "markdown", - "id": "6d2404eb", + "id": "57cf0b92", "metadata": { "editable": true }, @@ -4456,7 +4458,7 @@ }, { "cell_type": "markdown", - "id": "22615b7e", + "id": "c61e0046", "metadata": { "editable": true }, @@ -4468,7 +4470,7 @@ }, { "cell_type": "markdown", - "id": "bee48c87", + "id": "66448cf4", "metadata": { "editable": true }, @@ -4483,7 +4485,7 @@ }, { "cell_type": "markdown", - "id": "7f12f53a", + "id": "53d03309", "metadata": { "editable": true }, @@ -4495,7 +4497,7 @@ }, { "cell_type": "markdown", - "id": "a6acb75c", + "id": "530e38d2", "metadata": { "editable": true }, @@ -4507,7 +4509,7 @@ }, { "cell_type": "markdown", - "id": "131e9d9c", + "id": "f073fbdc", "metadata": { "editable": true }, @@ -4517,7 +4519,7 @@ }, { "cell_type": "markdown", - "id": "85812d77", + "id": "cd18b54e", "metadata": { "editable": true }, @@ -4529,7 +4531,7 @@ }, { "cell_type": "markdown", - "id": "35007c74", + "id": "c21ebf43", "metadata": { "editable": true }, @@ -4539,7 +4541,7 @@ }, { "cell_type": "markdown", - "id": "0100935a", + "id": "7b868e3b", "metadata": { "editable": true }, @@ -4551,7 +4553,7 @@ }, { "cell_type": "markdown", - "id": "7b6ff86e", + "id": "e6e45cd5", "metadata": { "editable": true }, @@ -4561,7 +4563,7 @@ }, { "cell_type": "markdown", - "id": "1f452dfb", + "id": "d54cddec", "metadata": { "editable": true }, @@ -4573,7 +4575,7 @@ }, { "cell_type": "markdown", - "id": "05600fbd", + "id": "3f887335", "metadata": { "editable": true }, @@ -4590,7 +4592,7 @@ }, { "cell_type": "markdown", - "id": "a2af3e04", + "id": "43987fb6", "metadata": { "editable": true }, @@ -4603,7 +4605,7 @@ }, { "cell_type": "markdown", - "id": "0342c08c", + "id": "78741f5d", "metadata": { "editable": true }, @@ -4615,7 +4617,7 @@ }, { "cell_type": "markdown", - "id": "bdd5faeb", + "id": "ecf46f39", "metadata": { "editable": true }, @@ -4625,7 +4627,7 @@ }, { "cell_type": "markdown", - "id": "35873dc6", + "id": "f98d1ea2", "metadata": { "editable": true }, @@ -4638,7 +4640,7 @@ }, { "cell_type": "markdown", - "id": "6ff6ab78", + "id": "56a4e22a", "metadata": { "editable": true }, @@ -4648,7 +4650,7 @@ }, { "cell_type": "markdown", - "id": "77286b5f", + "id": "3087fc2c", "metadata": { "editable": true }, @@ -4660,7 +4662,7 @@ }, { "cell_type": "markdown", - "id": "0ade2616", + "id": "8d386a35", "metadata": { "editable": true }, @@ -4673,7 +4675,7 @@ }, { "cell_type": "markdown", - "id": "da22a168", + "id": "c8d58182", "metadata": { "editable": true }, @@ -4686,7 +4688,7 @@ }, { "cell_type": "markdown", - "id": "caa18da2", + "id": "15fe7dd6", "metadata": { "editable": true }, @@ -4698,7 +4700,7 @@ }, { "cell_type": "markdown", - "id": "f31c8084", + "id": "042bbee6", "metadata": { "editable": true }, @@ -4710,7 +4712,7 @@ }, { "cell_type": "markdown", - "id": "53dc71a2", + "id": "911f8e5c", "metadata": { "editable": true }, @@ -4720,7 +4722,7 @@ }, { "cell_type": "markdown", - "id": "753442d5", + "id": "40d4bbda", "metadata": { "editable": true }, @@ -4733,7 +4735,7 @@ }, { "cell_type": "markdown", - "id": "e0b2f58a", + "id": "8a00d027", "metadata": { "editable": true }, @@ -4745,7 +4747,7 @@ }, { "cell_type": "markdown", - "id": "e4af8e7e", + "id": "ff109317", "metadata": { "editable": true }, @@ -4757,7 +4759,7 @@ }, { "cell_type": "markdown", - "id": "6e353a3a", + "id": "4e5859eb", "metadata": { "editable": true }, @@ -4774,7 +4776,7 @@ }, { "cell_type": "markdown", - "id": "66a4056b", + "id": "75a8694b", "metadata": { "editable": true }, @@ -4786,7 +4788,7 @@ }, { "cell_type": "markdown", - "id": "ea5b3d9f", + "id": "80c47724", "metadata": { "editable": true }, @@ -4796,7 +4798,7 @@ }, { "cell_type": "markdown", - "id": "db50585a", + "id": "6b385376", "metadata": { "editable": true }, @@ -4808,7 +4810,7 @@ }, { "cell_type": "markdown", - "id": "afd3cd57", + "id": "32963935", "metadata": { "editable": true }, @@ -4818,7 +4820,7 @@ }, { "cell_type": "markdown", - "id": "6606774b", + "id": "de8c6756", "metadata": { "editable": true }, @@ -4830,7 +4832,7 @@ }, { "cell_type": "markdown", - "id": "bcfd3adb", + "id": "76b587d1", "metadata": { "editable": true }, @@ -4842,7 +4844,7 @@ }, { "cell_type": "markdown", - "id": "46d7227b", + "id": "8a1fc10c", "metadata": { "editable": true }, @@ -4858,7 +4860,7 @@ }, { "cell_type": "markdown", - "id": "a214b881", + "id": "a6377518", "metadata": { "editable": true }, @@ -4870,7 +4872,7 @@ }, { "cell_type": "markdown", - "id": "727c87de", + "id": "367d1837", "metadata": { "editable": true }, @@ -4882,7 +4884,7 @@ }, { "cell_type": "markdown", - "id": "5083858e", + "id": "2fa779fe", "metadata": { "editable": true }, @@ -4892,7 +4894,7 @@ }, { "cell_type": "markdown", - "id": "973ad2fa", + "id": "e0f99ac6", "metadata": { "editable": true }, @@ -4904,7 +4906,7 @@ }, { "cell_type": "markdown", - "id": "4fbd0028", + "id": "d2134909", "metadata": { "editable": true }, @@ -4914,7 +4916,7 @@ }, { "cell_type": "markdown", - "id": "bee26901", + "id": "09b0f22e", "metadata": { "editable": true }, @@ -4926,7 +4928,7 @@ }, { "cell_type": "markdown", - "id": "da771d96", + "id": "1d1987fc", "metadata": { "editable": true }, @@ -4943,7 +4945,7 @@ }, { "cell_type": "markdown", - "id": "8ecb8a62", + "id": "16f94ea4", "metadata": { "editable": true }, @@ -4955,7 +4957,7 @@ }, { "cell_type": "markdown", - "id": "5cf08e3d", + "id": "21c224fd", "metadata": { "editable": true }, @@ -4967,7 +4969,7 @@ }, { "cell_type": "markdown", - "id": "d984ce12", + "id": "80c9611e", "metadata": { "editable": true }, @@ -4977,7 +4979,7 @@ }, { "cell_type": "markdown", - "id": "837a69b7", + "id": "ca8706ca", "metadata": { "editable": true }, @@ -4989,7 +4991,7 @@ }, { "cell_type": "markdown", - "id": "21dde5ad", + "id": "6bcce614", "metadata": { "editable": true }, @@ -4999,7 +5001,7 @@ }, { "cell_type": "markdown", - "id": "4d49c112", + "id": "857b9ab1", "metadata": { "editable": true }, @@ -5011,7 +5013,7 @@ }, { "cell_type": "markdown", - "id": "5b4eabcf", + "id": "bb8c1546", "metadata": { "editable": true }, @@ -5021,7 +5023,7 @@ }, { "cell_type": "markdown", - "id": "28dcb3c8", + "id": "2f7636ba", "metadata": { "editable": true }, @@ -5033,7 +5035,7 @@ }, { "cell_type": "markdown", - "id": "f315a306", + "id": "07e308e9", "metadata": { "editable": true }, @@ -5043,7 +5045,7 @@ }, { "cell_type": "markdown", - "id": "7b4f990c", + "id": "7a20da6b", "metadata": { "editable": true }, @@ -5055,13 +5057,856 @@ }, { "cell_type": "markdown", - "id": "197daa81", + "id": "404a4814", "metadata": { "editable": true }, "source": [ "This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package [CVXOPT](https://cvxopt.org/). We will discuss how to code LASSO regression next week, when we have introduced gradient methods." ] + }, + { + "cell_type": "markdown", + "id": "c47881af", + "metadata": { + "editable": true + }, + "source": [ + "## Material for exercises week 35" + ] + }, + { + "cell_type": "markdown", + "id": "de625607", + "metadata": { + "editable": true + }, + "source": [ + "## Important technicalities: More on Rescaling data\n", + "\n", + "When you are comparing your own code with for example **Scikit-Learn**'s\n", + "library, there are some technicalities to keep in mind. The examples\n", + "here demonstrate some of these aspects with potential pitfalls.\n", + "\n", + "The discussion here focuses on the role of the intercept, how we can\n", + "set up the design matrix, what scaling we should use and other topics\n", + "which tend confuse us.\n", + "\n", + "The intercept can be interpreted as the expected value of our\n", + "target/output variables when all other predictors are set to zero.\n", + "Thus, if we cannot assume that the expected outputs/targets are zero\n", + "when all predictors are zero (the columns in the design matrix), it\n", + "may be a bad idea to implement a model which penalizes the intercept.\n", + "Furthermore, in for example Ridge and Lasso regression, the default solutions\n", + "from the library **Scikit-Learn** (when not shrinking $\\beta_0$) for the unknown parameters\n", + "$\\boldsymbol{\\beta}$, are derived under the assumption that both $\\boldsymbol{y}$ and\n", + "$\\boldsymbol{X}$ are zero centered, that is we subtract the mean values.\n", + "\n", + "If our predictors represent different scales, then it is important to\n", + "standardize the design matrix $\\boldsymbol{X}$ by subtracting the mean of each\n", + "column from the corresponding column and dividing the column with its\n", + "standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library,\n", + "the results may differ. \n", + "\n", + "The\n", + "[Standardscaler](https://scikit-learn.org/stable/modules/generated/sklearn.preprocessing.StandardScaler.html)\n", + "function in **Scikit-Learn** does this for us. For the data sets we\n", + "have been studying in our various examples, the data are in many cases\n", + "already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a\n", + "survey of your data, with a critical assessment of them in case you need to scale the data.\n", + "\n", + "If you need to scale the data, not doing so will give an *unfair*\n", + "penalization of the parameters since their magnitude depends on the\n", + "scale of their corresponding predictor.\n", + "\n", + "The **Scikit-Learn** site has a good discussion of different ways of preprocessing data.\n", + "\n", + "Suppose as an example that you \n", + "you have an input variable given by the heights of different persons.\n", + "Human height might be measured in inches or meters or\n", + "kilometers. If measured in kilometers, a standard linear regression\n", + "model with this predictor would probably give a much bigger\n", + "coefficient term, than if measured in millimeters.\n", + "This can clearly lead to problems in evaluating the cost/loss functions.\n", + "\n", + "Keep in mind that when you transform your data set before training a model, the same transformation needs to be done\n", + "on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "id": "c00a9805", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "\"\"\"\n", + "#Model training, we compute the mean value of y and X\n", + "y_train_mean = np.mean(y_train)\n", + "X_train_mean = np.mean(X_train,axis=0)\n", + "X_train = X_train - X_train_mean\n", + "y_train = y_train - y_train_mean\n", + "\n", + "# The we fit our model with the training data\n", + "trained_model = some_model.fit(X_train,y_train)\n", + "\n", + "\n", + "#Model prediction, we need also to transform our data set used for the prediction.\n", + "X_test = X_test - X_train_mean #Use mean from training data\n", + "y_pred = trained_model(X_test)\n", + "y_pred = y_pred + y_train_mean\n", + "\"\"\"" + ] + }, + { + "cell_type": "markdown", + "id": "b3fb9183", + "metadata": { + "editable": true + }, + "source": [ + "Let us try to understand what this may imply mathematically when we\n", + "subtract the mean values, also known as *zero centering*. For\n", + "simplicity, we will focus on ordinary regression, as done in the above example.\n", + "\n", + "The cost/loss function for regression is" + ] + }, + { + "cell_type": "markdown", + "id": "07a54955", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\beta_0, \\beta_1, ... , \\beta_{p-1}) = \\frac{1}{n}\\sum_{i=0}^{n} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij}\\beta_j\\right)^2,.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1ac9c31d", + "metadata": { + "editable": true + }, + "source": [ + "Recall also that we use the squared value. This expression can lead to an\n", + "increased penalty for higher differences between predicted and\n", + "output/target values.\n", + "\n", + "What we have done is to single out the $\\beta_0$ term in the\n", + "definition of the mean squared error (MSE). The design matrix $X$\n", + "does in this case not contain any intercept column. When we take the\n", + "derivative with respect to $\\beta_0$, we want the derivative to obey" + ] + }, + { + "cell_type": "markdown", + "id": "ee6c6a4c", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial \\beta_j} = 0,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "beb40c81", + "metadata": { + "editable": true + }, + "source": [ + "for all $j$. For $\\beta_0$ we have" + ] + }, + { + "cell_type": "markdown", + "id": "b051946e", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\frac{\\partial C}{\\partial \\beta_0} = -\\frac{2}{n}\\sum_{i=0}^{n-1} \\left(y_i - \\beta_0 - \\sum_{j=1}^{p-1} X_{ij} \\beta_j\\right).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "885a6b08", + "metadata": { + "editable": true + }, + "source": [ + "Multiplying away the constant $2/n$, we obtain" + ] + }, + { + "cell_type": "markdown", + "id": "5774d6be", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\sum_{i=0}^{n-1} \\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} \\sum_{j=1}^{p-1} X_{ij} \\beta_j.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "73ff6979", + "metadata": { + "editable": true + }, + "source": [ + "Let us specialize first to the case where we have only two parameters $\\beta_0$ and $\\beta_1$.\n", + "Our result for $\\beta_0$ simplifies then to" + ] + }, + { + "cell_type": "markdown", + "id": "792128bf", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "n\\beta_0 = \\sum_{i=0}^{n-1}y_i - \\sum_{i=0}^{n-1} X_{i1} \\beta_1.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2be70ed5", + "metadata": { + "editable": true + }, + "source": [ + "We obtain then" + ] + }, + { + "cell_type": "markdown", + "id": "1d0c4ac1", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\beta_1\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2d4a5936", + "metadata": { + "editable": true + }, + "source": [ + "If we define" + ] + }, + { + "cell_type": "markdown", + "id": "d30abc39", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_{\\boldsymbol{x}_1}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{i1},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "7f945f8b", + "metadata": { + "editable": true + }, + "source": [ + "and the mean value of the outputs as" + ] + }, + { + "cell_type": "markdown", + "id": "dff5bf57", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_y=\\frac{1}{n}\\sum_{i=0}^{n-1}y_i,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "f222bcca", + "metadata": { + "editable": true + }, + "source": [ + "we have" + ] + }, + { + "cell_type": "markdown", + "id": "f1658ccd", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0 = \\mu_y - \\beta_1\\mu_{\\boldsymbol{x}_1}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "394ef3c4", + "metadata": { + "editable": true + }, + "source": [ + "In the general case with more parameters than $\\beta_0$ and $\\beta_1$, we have" + ] + }, + { + "cell_type": "markdown", + "id": "5f1dd1af", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\frac{1}{n}\\sum_{i=0}^{n-1}\\sum_{j=1}^{p-1} X_{ij}\\beta_j.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0cafc2d3", + "metadata": { + "editable": true + }, + "source": [ + "We can rewrite the latter equation as" + ] + }, + { + "cell_type": "markdown", + "id": "2c878f98", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\beta_0 = \\frac{1}{n}\\sum_{i=0}^{n-1}y_i - \\sum_{j=1}^{p-1} \\mu_{\\boldsymbol{x}_j}\\beta_j,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1dab41db", + "metadata": { + "editable": true + }, + "source": [ + "where we have defined" + ] + }, + { + "cell_type": "markdown", + "id": "cf2f2908", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\mu_{\\boldsymbol{x}_j}=\\frac{1}{n}\\sum_{i=0}^{n-1} X_{ij},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "20bac2a4", + "metadata": { + "editable": true + }, + "source": [ + "the mean value for all elements of the column vector $\\boldsymbol{x}_j$.\n", + "\n", + "Replacing $y_i$ with $y_i - y_i - \\overline{\\boldsymbol{y}}$ and centering also our design matrix results in a cost function (in vector-matrix disguise)" + ] + }, + { + "cell_type": "markdown", + "id": "0285afd3", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(\\boldsymbol{\\beta}) = (\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta})^T(\\boldsymbol{\\tilde{y}} - \\tilde{X}\\boldsymbol{\\beta}).\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "ffff8823", + "metadata": { + "editable": true + }, + "source": [ + "If we minimize with respect to $\\boldsymbol{\\beta}$ we have then" + ] + }, + { + "cell_type": "markdown", + "id": "eea815a7", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X})^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}},\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "0d935c60", + "metadata": { + "editable": true + }, + "source": [ + "where $\\boldsymbol{\\tilde{y}} = \\boldsymbol{y} - \\overline{\\boldsymbol{y}}$\n", + "and $\\tilde{X}_{ij} = X_{ij} - \\frac{1}{n}\\sum_{k=0}^{n-1}X_{kj}$.\n", + "\n", + "For Ridge regression we need to add $\\lambda \\boldsymbol{\\beta}^T\\boldsymbol{\\beta}$ to the cost function and get then" + ] + }, + { + "cell_type": "markdown", + "id": "bb4eabb8", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\hat{\\boldsymbol{\\beta}} = (\\tilde{X}^T\\tilde{X} + \\lambda I)^{-1}\\tilde{X}^T\\boldsymbol{\\tilde{y}}.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "1da18d81", + "metadata": { + "editable": true + }, + "source": [ + "What does this mean? And why do we insist on all this? Let us look at some examples.\n", + "\n", + "This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (*code example thanks to Øyvind Sigmundson Schøyen*). Here our scaling of the data is done by subtracting the mean values only.\n", + "Note also that we do not split the data into training and test." + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "id": "a11f0699", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "from sklearn.linear_model import LinearRegression\n", + "\n", + "\n", + "np.random.seed(2021)\n", + "\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "\n", + "def fit_beta(X, y):\n", + " return np.linalg.pinv(X.T @ X) @ X.T @ y\n", + "\n", + "\n", + "true_beta = [2, 0.5, 3.7]\n", + "\n", + "x = np.linspace(0, 1, 11)\n", + "y = np.sum(\n", + " np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0\n", + ") + 0.1 * np.random.normal(size=len(x))\n", + "\n", + "degree = 3\n", + "X = np.zeros((len(x), degree))\n", + "\n", + "# Include the intercept in the design matrix\n", + "for p in range(degree):\n", + " X[:, p] = x ** p\n", + "\n", + "beta = fit_beta(X, y)\n", + "\n", + "# Intercept is included in the design matrix\n", + "skl = LinearRegression(fit_intercept=False).fit(X, y)\n", + "\n", + "print(f\"True beta: {true_beta}\")\n", + "print(f\"Fitted beta: {beta}\")\n", + "print(f\"Sklearn fitted beta: {skl.coef_}\")\n", + "ypredictOwn = X @ beta\n", + "ypredictSKL = skl.predict(X)\n", + "print(f\"MSE with intercept column\")\n", + "print(MSE(y,ypredictOwn))\n", + "print(f\"MSE with intercept column from SKL\")\n", + "print(MSE(y,ypredictSKL))\n", + "\n", + "\n", + "plt.figure()\n", + "plt.scatter(x, y, label=\"Data\")\n", + "plt.plot(x, X @ beta, label=\"Fit\")\n", + "plt.plot(x, skl.predict(X), label=\"Sklearn (fit_intercept=False)\")\n", + "\n", + "\n", + "# Do not include the intercept in the design matrix\n", + "X = np.zeros((len(x), degree - 1))\n", + "\n", + "for p in range(degree - 1):\n", + " X[:, p] = x ** (p + 1)\n", + "\n", + "# Intercept is not included in the design matrix\n", + "skl = LinearRegression(fit_intercept=True).fit(X, y)\n", + "\n", + "# Use centered values for X and y when computing coefficients\n", + "y_offset = np.average(y, axis=0)\n", + "X_offset = np.average(X, axis=0)\n", + "\n", + "beta = fit_beta(X - X_offset, y - y_offset)\n", + "intercept = np.mean(y_offset - X_offset @ beta)\n", + "\n", + "print(f\"Manual intercept: {intercept}\")\n", + "print(f\"Fitted beta (without intercept): {beta}\")\n", + "print(f\"Sklearn intercept: {skl.intercept_}\")\n", + "print(f\"Sklearn fitted beta (without intercept): {skl.coef_}\")\n", + "ypredictOwn = X @ beta\n", + "ypredictSKL = skl.predict(X)\n", + "print(f\"MSE with Manual intercept\")\n", + "print(MSE(y,ypredictOwn+intercept))\n", + "print(f\"MSE with Sklearn intercept\")\n", + "print(MSE(y,ypredictSKL))\n", + "\n", + "plt.plot(x, X @ beta + intercept, \"--\", label=\"Fit (manual intercept)\")\n", + "plt.plot(x, skl.predict(X), \"--\", label=\"Sklearn (fit_intercept=True)\")\n", + "plt.grid()\n", + "plt.legend()\n", + "\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "ed61dd49", + "metadata": { + "editable": true + }, + "source": [ + "The intercept is the value of our output/target variable\n", + "when all our features are zero and our function crosses the $y$-axis (for a one-dimensional case). \n", + "\n", + "Printing the MSE, we see first that both methods give the same MSE, as\n", + "they should. However, when we move to for example Ridge regression,\n", + "the way we treat the intercept may give a larger or smaller MSE,\n", + "meaning that the MSE can be penalized by the value of the\n", + "intercept. Not including the intercept in the fit, means that the\n", + "regularization term does not include $\\beta_0$. For different values\n", + "of $\\lambda$, this may lead to different MSE values. \n", + "\n", + "To remind the reader, the regularization term, with the intercept in Ridge regression, is given by" + ] + }, + { + "cell_type": "markdown", + "id": "5de45190", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=0}^{p-1}\\beta_j^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "6f53e8c1", + "metadata": { + "editable": true + }, + "source": [ + "but when we take out the intercept, this equation becomes" + ] + }, + { + "cell_type": "markdown", + "id": "3b0ffc74", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_2^2 = \\lambda \\sum_{j=1}^{p-1}\\beta_j^2.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "10305919", + "metadata": { + "editable": true + }, + "source": [ + "For Lasso regression we have" + ] + }, + { + "cell_type": "markdown", + "id": "b2ce8814", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "\\lambda \\vert\\vert \\boldsymbol{\\beta} \\vert\\vert_1 = \\lambda \\sum_{j=1}^{p-1}\\vert\\beta_j\\vert.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "e08c9fd7", + "metadata": { + "editable": true + }, + "source": [ + "It means that, when scaling the design matrix and the outputs/targets,\n", + "by subtracting the mean values, we have an optimization problem which\n", + "is not penalized by the intercept. The MSE value can then be smaller\n", + "since it focuses only on the remaining quantities. If we however bring\n", + "back the intercept, we will get a MSE which then contains the\n", + "intercept.\n", + "\n", + "Armed with this wisdom, we attempt first to simply set the intercept equal to **False** in our implementation of Ridge regression for our well-known vanilla data set." + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "5b7e1c63", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn import linear_model\n", + "\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "\n", + "\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "np.random.seed(3155)\n", + "\n", + "n = 100\n", + "x = np.random.rand(n)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n", + "\n", + "Maxpolydegree = 20\n", + "X = np.zeros((n,Maxpolydegree))\n", + "#We include explicitely the intercept column\n", + "for degree in range(Maxpolydegree):\n", + " X[:,degree] = x**degree\n", + "# We split the data in test and training data\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "p = Maxpolydegree\n", + "I = np.eye(p,p)\n", + "# Decide which values of lambda to use\n", + "nlambdas = 6\n", + "MSEOwnRidgePredict = np.zeros(nlambdas)\n", + "MSERidgePredict = np.zeros(nlambdas)\n", + "lambdas = np.logspace(-4, 2, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train\n", + " # Note: we include the intercept column and no scaling\n", + " RegRidge = linear_model.Ridge(lmb,fit_intercept=False)\n", + " RegRidge.fit(X_train,y_train)\n", + " # and then make the prediction\n", + " ytildeOwnRidge = X_train @ OwnRidgeBeta\n", + " ypredictOwnRidge = X_test @ OwnRidgeBeta\n", + " ytildeRidge = RegRidge.predict(X_train)\n", + " ypredictRidge = RegRidge.predict(X_test)\n", + " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n", + " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n", + " print(\"Beta values for own Ridge implementation\")\n", + " print(OwnRidgeBeta)\n", + " print(\"Beta values for Scikit-Learn Ridge implementation\")\n", + " print(RegRidge.coef_)\n", + " print(\"MSE values for own Ridge implementation\")\n", + " print(MSEOwnRidgePredict[i])\n", + " print(\"MSE values for Scikit-Learn Ridge implementation\")\n", + " print(MSERidgePredict[i])\n", + "\n", + "# Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')\n", + "plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')\n", + "\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "bbf9639e", + "metadata": { + "editable": true + }, + "source": [ + "The results here agree when we force **Scikit-Learn**'s Ridge function to include the first column in our design matrix.\n", + "We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix.\n", + "What happens if we do not include the intercept in our fit?\n", + "Let us see how we can change this code by zero centering." + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "id": "3a82ee91", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import pandas as pd\n", + "import matplotlib.pyplot as plt\n", + "from sklearn.model_selection import train_test_split\n", + "from sklearn import linear_model\n", + "from sklearn.preprocessing import StandardScaler\n", + "\n", + "def MSE(y_data,y_model):\n", + " n = np.size(y_model)\n", + " return np.sum((y_data-y_model)**2)/n\n", + "# A seed just to ensure that the random numbers are the same for every run.\n", + "# Useful for eventual debugging.\n", + "np.random.seed(315)\n", + "\n", + "n = 100\n", + "x = np.random.rand(n)\n", + "y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)\n", + "\n", + "Maxpolydegree = 20\n", + "X = np.zeros((n,Maxpolydegree-1))\n", + "\n", + "for degree in range(1,Maxpolydegree): #No intercept column\n", + " X[:,degree-1] = x**(degree)\n", + "\n", + "# We split the data in test and training data\n", + "X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)\n", + "\n", + "#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable\n", + "X_train_mean = np.mean(X_train,axis=0)\n", + "#Center by removing mean from each feature\n", + "X_train_scaled = X_train - X_train_mean \n", + "X_test_scaled = X_test - X_train_mean\n", + "#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)\n", + "#Remove the intercept from the training data.\n", + "y_scaler = np.mean(y_train) \n", + "y_train_scaled = y_train - y_scaler \n", + "\n", + "p = Maxpolydegree-1\n", + "I = np.eye(p,p)\n", + "# Decide which values of lambda to use\n", + "nlambdas = 6\n", + "MSEOwnRidgePredict = np.zeros(nlambdas)\n", + "MSERidgePredict = np.zeros(nlambdas)\n", + "\n", + "lambdas = np.logspace(-4, 2, nlambdas)\n", + "for i in range(nlambdas):\n", + " lmb = lambdas[i]\n", + " OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)\n", + " intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data\n", + " #Add intercept to prediction\n", + " ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler \n", + " RegRidge = linear_model.Ridge(lmb)\n", + " RegRidge.fit(X_train,y_train)\n", + " ypredictRidge = RegRidge.predict(X_test)\n", + " MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)\n", + " MSERidgePredict[i] = MSE(y_test,ypredictRidge)\n", + " print(\"Beta values for own Ridge implementation\")\n", + " print(OwnRidgeBeta) #Intercept is given by mean of target variable\n", + " print(\"Beta values for Scikit-Learn Ridge implementation\")\n", + " print(RegRidge.coef_)\n", + " print('Intercept from own implementation:')\n", + " print(intercept_)\n", + " print('Intercept from Scikit-Learn Ridge implementation')\n", + " print(RegRidge.intercept_)\n", + " print(\"MSE values for own Ridge implementation\")\n", + " print(MSEOwnRidgePredict[i])\n", + " print(\"MSE values for Scikit-Learn Ridge implementation\")\n", + " print(MSERidgePredict[i])\n", + "\n", + "\n", + "# Now plot the results\n", + "plt.figure()\n", + "plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')\n", + "plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')\n", + "plt.xlabel('log10(lambda)')\n", + "plt.ylabel('MSE')\n", + "plt.legend()\n", + "plt.show()" + ] + }, + { + "cell_type": "markdown", + "id": "c061da66", + "metadata": { + "editable": true + }, + "source": [ + "We see here, when compared to the code which includes explicitely the\n", + "intercept column, that our MSE value is actually smaller. This is\n", + "because the regularization term does not include the intercept value\n", + "$\\beta_0$ in the fitting. This applies to Lasso regularization as\n", + "well. It means that our optimization is now done only with the\n", + "centered matrix and/or vector that enter the fitting procedure." + ] } ], "metadata": {}, diff --git a/doc/LectureNotes/_build/html/searchindex.js b/doc/LectureNotes/_build/html/searchindex.js index 8a9f7b94f..005f6fa10 100644 --- a/doc/LectureNotes/_build/html/searchindex.js +++ b/doc/LectureNotes/_build/html/searchindex.js @@ -1 +1 @@ -Search.setIndex({"alltitles": {"A Classification Tree": [[9, "a-classification-tree"]], "A Frequentist approach to data analysis": [[0, "a-frequentist-approach-to-data-analysis"], [24, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[24, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[13, "adam-optimizer"]], "Activation functions": [[12, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[24, "adding-error-analysis-and-training-set-up"], [25, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[24, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[25, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[24, "and-what-about-using-neural-networks"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[18, null]], "Autocorrelation function": [[21, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[25, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[19, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [25, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [25, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[24, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex functions": [[13, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[25, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [25, "correlation-matrix"]], "Correlation Matrix with Pandas": [[25, "correlation-matrix-with-pandas"]], "Course Format": [[24, "course-format"]], "Course setting": [[20, null]], "Covariance Matrix Examples": [[25, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[25, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[24, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep learning methods": [[24, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[25, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the Lasso Regression Equations": [[25, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[24, "discriminative-modeling"]], "Domains and probabilities": [[21, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[25, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[21, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[24, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[25, "example-2"]], "Example 3": [[25, "example-3"]], "Example 4": [[25, "example-4"]], "Example Matrix": [[25, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[25, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[25, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[24, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Expectation values": [[21, "expectation-values"]], "Extremely useful tools, strongly recommended": [[24, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[25, "fixing-the-singularity"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[25, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [25, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[19, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[24, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[24, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [24, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[24, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Grading": [[22, "grading"], [22, "id2"], [24, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[19, "important-matrix-and-vector-handling-packages"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[22, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[24, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[24, "installing-r-c-cython-numba-etc"]], "Instructor information": [[22, "instructor-information"]], "Interpretations and optimizing our parameters": [[24, "interpretations-and-optimizing-our-parameters"], [24, "id2"], [24, "id3"], [25, "interpretations-and-optimizing-our-parameters"], [25, "id1"], [25, "id2"]], "Interpreting the Ridge results": [[25, "interpreting-the-ridge-results"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [25, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [18, "introduction"], [19, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[19, "lu-decomposition-the-inverse-of-a-matrix"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"]], "Learning outcomes": [[18, "learning-outcomes"], [24, "learning-outcomes"]], "Lectures and ComputerLab": [[24, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[19, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[25, "linear-regression-problems"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [25, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[23, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[24, "machine-learning"]], "Machine learning": [[18, "machine-learning"]], "Main textbooks": [[24, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[25, "making-your-own-test-train-splitting"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [25, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [25, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[24, "matrices-in-python"]], "Matrix multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[21, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [25, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[25, "meet-the-hessian-matrix"]], "Meet the Pandas": [[24, "meet-the-pandas"]], "Min-Max Scaling": [[25, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[25, "more-interpretations"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More preprocessing": [[25, "more-preprocessing"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Note about SVD Calculations": [[25, "note-about-svd-calculations"]], "Numerical experiments and the covariance, central limit theorem": [[21, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[19, "numpy-and-arrays"], [24, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[24, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[24, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[24, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Organizing our data": [[0, "organizing-our-data"], [24, "organizing-our-data"]], "Other Matrix and Vector Operations": [[19, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[24, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[24, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[24, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[24, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[24, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[24, "own-code-for-ordinary-least-squares"], [25, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[24, "pandas-ai"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plans for week 35": [[25, "plans-for-week-35"]], "Practical tips": [[13, "practical-tips"]], "Practicalities": [[22, "practicalities"], [22, "id1"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our data": [[25, "preprocessing-our-data"]], "Prerequisites": [[24, "prerequisites"]], "Prerequisites and background": [[18, "prerequisites-and-background"]], "Prerequisites: Collect and pre-process data": [[3, "prerequisites-collect-and-pre-process-data"]], "Probability Distribution Functions": [[21, "probability-distribution-functions"]], "Program for stochastic gradient": [[13, "program-for-stochastic-gradient"]], "Properties of PDFs": [[21, "properties-of-pdfs"]], "Pros and cons of trees, pros": [[9, "pros-and-cons-of-trees-pros"]], "Python installers": [[18, "python-installers"], [24, "python-installers"]], "RMS prop": [[13, "rms-prop"]], "Random Numbers": [[21, "random-numbers"]], "Random forests": [[10, "random-forests"]], "Randomized PCA": [[11, "randomized-pca"]], "Reading material": [[24, "reading-material"]], "Reading recommendations:": [[25, "reading-recommendations"]], "Reading suggestions week 34": [[24, "reading-suggestions-week-34"]], "Recurrent neural networks": [[12, "recurrent-neural-networks"]], "Recurrent neural networks: Overarching view": [[4, null]], "Reducing the number of degrees of freedom, overarching view": [[0, "reducing-the-number-of-degrees-of-freedom-overarching-view"], [25, "reducing-the-number-of-degrees-of-freedom-overarching-view"]], "Reformulating the problem": [[2, "reformulating-the-problem"]], "Regression Case": [[10, "regression-case"]], "Regression analysis, overarching aims": [[24, "regression-analysis-overarching-aims"]], "Regression analysis, overarching aims II": [[24, "regression-analysis-overarching-aims-ii"]], "Regularization": [[1, "regularization"]], "Reminder from last week": [[25, "reminder-from-last-week"]], "Reminder on Statistics": [[6, "reminder-on-statistics"]], "Replace or not": [[13, "replace-or-not"]], "Required Technologies": [[18, "required-technologies"]], "Resampling Methods": [[6, null]], "Resampling methods": [[6, "id1"]], "Residual Error": [[25, "residual-error"]], "Resources on differential equations and deep learning": [[2, "resources-on-differential-equations-and-deep-learning"]], "Revisiting our Linear Regression Solvers": [[13, "revisiting-our-linear-regression-solvers"]], "Rewriting the Covariance and/or Correlation Matrix": [[25, "rewriting-the-covariance-and-or-correlation-matrix"]], "Rewriting the fitting procedure as a linear algebra problem": [[24, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem"]], "Rewriting the fitting procedure as a linear algebra problem, more details": [[24, "rewriting-the-fitting-procedure-as-a-linear-algebra-problem-more-details"]], "Ridge and LASSO Regression": [[25, "ridge-and-lasso-regression"]], "Ridge and Lasso Regression": [[5, null], [5, "id1"]], "Same code but now with momentum gradient descent": [[13, "same-code-but-now-with-momentum-gradient-descent"]], "Schedule first week": [[24, "schedule-first-week"]], "Schematic Regression Procedure": [[9, "schematic-regression-procedure"]], "Setting up the Back propagation algorithm": [[12, "setting-up-the-back-propagation-algorithm"]], "Setting up the Matrix to be inverted": [[25, "setting-up-the-matrix-to-be-inverted"]], "Setting up the network using Autograd; The full program": [[2, "setting-up-the-network-using-autograd-the-full-program"]], "Similar (second order function now) problem but now with AdaGrad": [[13, "similar-second-order-function-now-problem-but-now-with-adagrad"]], "Simple Python Code to read in Data and perform Classification": [[9, "simple-python-code-to-read-in-data-and-perform-classification"]], "Simple case": [[25, "simple-case"]], "Simple linear regression model using scikit-learn": [[0, "simple-linear-regression-model-using-scikit-learn"], [24, "simple-linear-regression-model-using-scikit-learn"]], "Software and needed installations": [[24, "software-and-needed-installations"]], "Solving Differential Equations with Deep Learning": [[2, null]], 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"write": [4, 11], "x": 25, "xgboost": 10, "your": [0, 10, 16, 25]}}) \ No newline at end of file +Search.setIndex({"alltitles": {"A Classification Tree": [[9, "a-classification-tree"]], "A Frequentist approach to data analysis": [[0, "a-frequentist-approach-to-data-analysis"], [24, "a-frequentist-approach-to-data-analysis"]], "A better approach": [[8, "a-better-approach"]], "A first summary": [[24, "a-first-summary"]], "A quick Reminder on Lagrangian Multipliers": [[8, "a-quick-reminder-on-lagrangian-multipliers"]], "A simple example": [[4, "a-simple-example"]], "A soft classifier": [[8, "a-soft-classifier"]], "A top-down perspective on Neural networks": [[1, "a-top-down-perspective-on-neural-networks"]], "ADAM optimizer": [[13, "adam-optimizer"]], "Activation functions": [[12, "activation-functions"]], "Adaptive boosting: AdaBoost, Basic Algorithm": [[10, "adaptive-boosting-adaboost-basic-algorithm"]], "Adding error analysis and training set up": [[24, "adding-error-analysis-and-training-set-up"], [25, "adding-error-analysis-and-training-set-up"]], "Adjust hyperparameters": [[1, "adjust-hyperparameters"]], "Algorithms for Setting up Decision Trees": [[9, "algorithms-for-setting-up-decision-trees"]], "An Overview of Ensemble Methods": [[10, "an-overview-of-ensemble-methods"]], "An extrapolation example": [[4, "an-extrapolation-example"]], "An optimization/minimization problem": [[24, "an-optimization-minimization-problem"]], "And finally \\boldsymbol{X}\\boldsymbol{X}^T": [[25, "and-finally-boldsymbol-x-boldsymbol-x-t"]], "And what about using neural networks?": [[24, "and-what-about-using-neural-networks"]], "Another example, the moons again": [[9, "another-example-the-moons-again"]], "Applied Data Analysis and Machine Learning": [[18, null]], "Autocorrelation function": [[21, "autocorrelation-function"]], "Automatic differentiation": [[13, "automatic-differentiation"]], "Back to Ridge and LASSO Regression": [[25, "back-to-ridge-and-lasso-regression"]], "Back to the Cancer Data": [[11, "back-to-the-cancer-data"]], "Bagging": [[10, "bagging"]], "Bagging Examples": [[10, "bagging-examples"]], "Basic Matrix Features": [[19, "basic-matrix-features"]], "Basic ideas of the Principal Component Analysis (PCA)": [[11, null]], "Basic math of the SVD": [[5, "basic-math-of-the-svd"], [25, "basic-math-of-the-svd"]], "Basics": [[7, "basics"]], "Basics of a tree": [[9, "basics-of-a-tree"]], "Batch Normalization": [[1, "batch-normalization"]], "Bayes\u2019 Theorem and Ridge and Lasso Regression": [[5, "bayes-theorem-and-ridge-and-lasso-regression"]], "Boosting, a Bird\u2019s Eye View": [[10, "boosting-a-bird-s-eye-view"]], "Bootstrap": [[6, "bootstrap"]], "Bringing it together, first back propagation equation": [[12, "bringing-it-together-first-back-propagation-equation"]], "Building a Feed Forward Neural Network": [[1, null]], "Building a tree, regression": [[9, "building-a-tree-regression"]], "Building neural networks in Tensorflow and Keras": [[1, "building-neural-networks-in-tensorflow-and-keras"]], "CNNs in more detail, building convolutional neural networks in Tensorflow and Keras": [[3, "cnns-in-more-detail-building-convolutional-neural-networks-in-tensorflow-and-keras"]], "Cancer Data again now with Decision Trees and other Methods": [[9, "cancer-data-again-now-with-decision-trees-and-other-methods"]], "Choose cost function and optimizer": [[1, "choose-cost-function-and-optimizer"]], "Classical PCA Theorem": [[11, "classical-pca-theorem"]], "Clustering and Unsupervised Learning": [[14, null]], "Code for SVD and Inversion of Matrices": [[5, "code-for-svd-and-inversion-of-matrices"]], "Codes and Approaches": [[14, "codes-and-approaches"]], "Codes for the SVD": [[5, "codes-for-the-svd"], [25, "codes-for-the-svd"]], "Coding Setup and Linear Regression": [[15, "coding-setup-and-linear-regression"]], "Collect and pre-process data": [[1, "collect-and-pre-process-data"]], "Communication channels": [[24, "communication-channels"]], "Compare Bagging on Trees with Random Forests": [[10, "compare-bagging-on-trees-with-random-forests"]], "Comparing with a numerical scheme": [[2, "comparing-with-a-numerical-scheme"]], "Computing the Gini index": [[9, "computing-the-gini-index"]], "Conjugate gradient method": [[13, "conjugate-gradient-method"]], "Convex functions": [[13, "convex-functions"]], "Convolution Examples: Polynomial multiplication": [[3, "convolution-examples-polynomial-multiplication"]], "Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)": [[3, "convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms"]], "Convolutional Neural Network": [[12, "convolutional-neural-network"]], "Convolutional Neural Networks": [[3, null]], "Correlation Function and Design/Feature Matrix": [[25, "correlation-function-and-design-feature-matrix"]], "Correlation Matrix": [[11, "correlation-matrix"], [25, "correlation-matrix"]], "Correlation Matrix with Pandas": [[25, "correlation-matrix-with-pandas"]], "Course Format": [[24, "course-format"]], "Course setting": [[20, null]], "Covariance Matrix Examples": [[25, "covariance-matrix-examples"]], "Covariance and Correlation Matrix": [[25, "covariance-and-correlation-matrix"]], "Cross-validation": [[6, "cross-validation"]], "Deadlines for projects (tentative)": [[24, "deadlines-for-projects-tentative"]], "Decision trees, overarching aims": [[9, null]], "Deep learning methods": [[24, "deep-learning-methods"]], "Define model and architecture": [[1, "define-model-and-architecture"]], "Defining the cost function": [[1, "defining-the-cost-function"]], "Deliverables": [[15, "deliverables"], [16, "deliverables"]], "Derivatives and the chain rule": [[12, "derivatives-and-the-chain-rule"]], "Derivatives, example 1": [[25, "derivatives-example-1"]], "Deriving OLS from a probability distribution": [[5, "deriving-ols-from-a-probability-distribution"]], "Deriving and Implementing Ordinary Least Squares": [[16, "deriving-and-implementing-ordinary-least-squares"]], "Deriving and Implementing Ridge Regression": [[17, "deriving-and-implementing-ridge-regression"]], "Deriving the Lasso Regression Equations": [[25, "deriving-the-lasso-regression-equations"]], "Deriving the Ridge Regression Equations": [[25, "deriving-the-ridge-regression-equations"]], "Deriving the back propagation code for a multilayer perceptron model": [[12, "deriving-the-back-propagation-code-for-a-multilayer-perceptron-model"]], "Developing a code for doing neural networks with back propagation": [[1, "developing-a-code-for-doing-neural-networks-with-back-propagation"]], "Diagonalize the sample covariance matrix to obtain the principal components": [[11, "diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components"]], "Different kernels and Mercer\u2019s theorem": [[8, "different-kernels-and-mercer-s-theorem"]], "Disadvantages": [[9, "disadvantages"]], "Discriminative Modeling": [[24, "discriminative-modeling"]], "Domains and probabilities": [[21, "domains-and-probabilities"]], "Dropout": [[1, "dropout"]], "Economy-size SVD": [[25, "economy-size-svd"]], "Elements of Probability Theory and Statistical Data Analysis": [[21, null]], "Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods": [[10, null]], "Entropy and the ID3 algorithm": [[9, "entropy-and-the-id3-algorithm"]], "Essential elements of ML": [[24, "essential-elements-of-ml"]], "Evaluate model performance on test data": [[1, "evaluate-model-performance-on-test-data"]], "Example 2": [[25, "example-2"]], "Example 3": [[25, "example-3"]], "Example 4": [[25, "example-4"]], "Example Matrix": [[25, "example-matrix"]], "Example of discriminative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-discriminative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of generative modeling, taken from Generative Deep Learning by David Foster": [[24, "example-of-generative-modeling-taken-from-generative-deep-learning-by-david-foster"]], "Example of own Standard scaling": [[25, "example-of-own-standard-scaling"]], "Example relevant for the exercises": [[25, "example-relevant-for-the-exercises"]], "Example: Exponential decay": [[2, "example-exponential-decay"]], "Example: Population growth": [[2, "example-population-growth"]], "Example: The diffusion equation": [[2, "example-the-diffusion-equation"]], "Example: binary classification problem": [[1, "example-binary-classification-problem"]], "Examples": [[24, "examples"]], "Examples of likelihood functions used in logistic regression and neural networks": [[7, "examples-of-likelihood-functions-used-in-logistic-regression-and-neural-networks"]], "Exercise 1 - Choice of model and degrees of freedom": [[17, "exercise-1-choice-of-model-and-degrees-of-freedom"]], "Exercise 1 - Finding the derivative of Matrix-Vector expressions": [[16, "exercise-1-finding-the-derivative-of-matrix-vector-expressions"]], "Exercise 1 - Github Setup": [[15, "exercise-1-github-setup"]], "Exercise 1: Setting up various Python environments": [[0, "exercise-1-setting-up-various-python-environments"]], "Exercise 2 - Deriving the expression for OLS": [[16, "exercise-2-deriving-the-expression-for-ols"]], "Exercise 2 - Deriving the expression for Ridge Regression": [[17, "exercise-2-deriving-the-expression-for-ridge-regression"]], "Exercise 2 - Setting up a Github repository": [[15, "exercise-2-setting-up-a-github-repository"]], "Exercise 2: making your own data and exploring scikit-learn": [[0, "exercise-2-making-your-own-data-and-exploring-scikit-learn"]], "Exercise 3 - Creating feature matrix and implementing OLS using the analytical expression": [[16, "exercise-3-creating-feature-matrix-and-implementing-ols-using-the-analytical-expression"]], "Exercise 3 - Fitting an OLS model to data": [[15, "exercise-3-fitting-an-ols-model-to-data"]], "Exercise 3 - Scaling data": [[17, "exercise-3-scaling-data"]], "Exercise 3 - Setting up a Python virtual environment": [[15, "exercise-3-setting-up-a-python-virtual-environment"]], "Exercise 3: Normalizing our data": [[0, "exercise-3-normalizing-our-data"]], "Exercise 4 - Fitting a polynomial": [[16, "exercise-4-fitting-a-polynomial"]], "Exercise 4 - Implementing Ridge Regression": [[17, "exercise-4-implementing-ridge-regression"]], "Exercise 4 - Testing multiple hyperparameters": [[17, "exercise-4-testing-multiple-hyperparameters"]], "Exercise 4 - The train-test split": [[15, "exercise-4-the-train-test-split"]], "Exercise 4: Adding Ridge Regression": [[0, "exercise-4-adding-ridge-regression"]], "Exercise 5 - Comparing your code with sklearn": [[16, "exercise-5-comparing-your-code-with-sklearn"]], "Exercise 5: Analytical exercises": [[0, "exercise-5-analytical-exercises"]], "Exercise: Cross-validation as resampling techniques, adding more complexity": [[6, "exercise-cross-validation-as-resampling-techniques-adding-more-complexity"]], "Exercise: Analysis of real data": [[6, "exercise-analysis-of-real-data"]], "Exercise: Bias-variance trade-off and resampling techniques": [[6, "exercise-bias-variance-trade-off-and-resampling-techniques"]], "Exercise: Lasso Regression on the Franke function with resampling": [[6, "exercise-lasso-regression-on-the-franke-function-with-resampling"]], "Exercise: Ordinary Least Square (OLS) on the Franke function": [[6, "exercise-ordinary-least-square-ols-on-the-franke-function"]], "Exercise: Ridge Regression on the Franke function with resampling": [[6, "exercise-ridge-regression-on-the-franke-function-with-resampling"]], "Exercises": [[0, "exercises"]], "Exercises and Projects": [[6, "exercises-and-projects"]], "Exercises week 34": [[15, null]], "Exercises week 35": [[16, null]], "Expectation values": [[21, "expectation-values"]], "Extremely useful tools, strongly recommended": [[24, "extremely-useful-tools-strongly-recommended"]], "Feed-forward neural networks": [[12, "feed-forward-neural-networks"]], "Feed-forward pass": [[1, "feed-forward-pass"]], "Final back propagating equation": [[12, "final-back-propagating-equation"]], "Fine-tuning neural network hyperparameters": [[1, "fine-tuning-neural-network-hyperparameters"]], "Fitting an Equation of State for Dense Nuclear Matter": [[0, "fitting-an-equation-of-state-for-dense-nuclear-matter"]], "Fixing the singularity": [[25, "fixing-the-singularity"]], "Frequently used scaling functions": [[25, "frequently-used-scaling-functions"]], "From one to many layers, the universal approximation theorem": [[12, "from-one-to-many-layers-the-universal-approximation-theorem"]], "Functionality in Scikit-Learn": [[25, "functionality-in-scikit-learn"]], "Further Dimensionality Remarks": [[3, "further-dimensionality-remarks"]], "Further properties (important for our analyses later)": [[5, "further-properties-important-for-our-analyses-later"], [25, "further-properties-important-for-our-analyses-later"]], "Gaussian Elimination": [[19, "gaussian-elimination"]], "General Features": [[9, "general-features"]], "General linear models and linear algebra": [[24, "general-linear-models-and-linear-algebra"]], "Generalizing the fitting procedure as a linear algebra problem": [[24, "generalizing-the-fitting-procedure-as-a-linear-algebra-problem"], [24, "id1"]], "Generative Adversarial Networks": [[4, "generative-adversarial-networks"]], "Generative Models": [[4, "generative-models"]], "Generative Versus Discriminative Modeling": [[24, "generative-versus-discriminative-modeling"]], "Geometric Interpretation and link with Singular Value Decomposition": [[11, "geometric-interpretation-and-link-with-singular-value-decomposition"]], "Gradient Boosting, Classification Example": [[10, "gradient-boosting-classification-example"]], "Gradient Boosting, Examples of Regression": [[10, "gradient-boosting-examples-of-regression"]], "Gradient Clipping": [[1, "gradient-clipping"]], "Gradient boosting: Basics with Steepest Descent/Functional Gradient Descent": [[10, "gradient-boosting-basics-with-steepest-descent-functional-gradient-descent"]], "Gradient descent": [[2, "gradient-descent"]], "Grading": [[22, "grading"], [22, "id2"], [24, "grading"]], "How to take derivatives of Matrix-Vector expressions": [[16, "how-to-take-derivatives-of-matrix-vector-expressions"]], "Hyperplanes and all that": [[8, "hyperplanes-and-all-that"]], "Important Matrix and vector handling packages": [[19, "important-matrix-and-vector-handling-packages"]], "Important technicalities: More on Rescaling data": [[25, "important-technicalities-more-on-rescaling-data"]], "Improving performance": [[1, "improving-performance"]], "In summary": [[22, "in-summary"]], "Including Stochastic Gradient Descent with Autograd": [[13, "including-stochastic-gradient-descent-with-autograd"]], "Incremental PCA": [[11, "incremental-pca"]], "Installing R, C++, cython or Julia": [[24, "installing-r-c-cython-or-julia"]], "Installing R, C++, cython, Numba etc": [[24, "installing-r-c-cython-numba-etc"]], "Instructor information": [[22, "instructor-information"]], "Interpretations and optimizing our parameters": [[24, "interpretations-and-optimizing-our-parameters"], [24, "id2"], [24, "id3"], [25, "interpretations-and-optimizing-our-parameters"], [25, "id1"], [25, "id2"]], "Interpreting the Ridge results": [[25, "interpreting-the-ridge-results"]], "Introducing JAX": [[13, "introducing-jax"]], "Introducing the Covariance and Correlation functions": [[11, "introducing-the-covariance-and-correlation-functions"], [25, "introducing-the-covariance-and-correlation-functions"]], "Introduction": [[0, "introduction"], [6, "introduction"], [18, "introduction"], [19, "introduction"]], "Iterative Fitting, Classification and AdaBoost": [[10, "iterative-fitting-classification-and-adaboost"]], "Iterative Fitting, Regression and Squared-error Cost Function": [[10, "iterative-fitting-regression-and-squared-error-cost-function"]], "Kernel PCA": [[11, "kernel-pca"]], "Kernels and non-linearity": [[8, "kernels-and-non-linearity"]], "LU Decomposition, the inverse of a matrix": [[19, "lu-decomposition-the-inverse-of-a-matrix"]], "Layers": [[1, "layers"]], "Layers used to build CNNs": [[3, "layers-used-to-build-cnns"]], "Learning goals": [[15, "learning-goals"], [16, "learning-goals"], [17, "learning-goals"]], "Learning outcomes": [[18, "learning-outcomes"], [24, "learning-outcomes"]], "Lectures and ComputerLab": [[24, "lectures-and-computerlab"]], "Limitations of supervised learning with deep networks": [[1, "limitations-of-supervised-learning-with-deep-networks"]], "Linear Algebra, Handling of Arrays and more Python Features": [[19, null]], "Linear Regression": [[0, null]], "Linear Regression Problems": [[25, "linear-regression-problems"]], "Linear Regression, basic elements": [[0, "linear-regression-basic-elements"]], "Linking Bayes\u2019 Theorem with Ridge and Lasso Regression": [[5, "linking-bayes-theorem-with-ridge-and-lasso-regression"]], "Linking the regression analysis with a statistical interpretation": [[5, "linking-the-regression-analysis-with-a-statistical-interpretation"]], "Linking with the SVD": [[5, "linking-with-the-svd"], [25, "linking-with-the-svd"]], "Links to relevant courses at the University of Oslo": [[23, "links-to-relevant-courses-at-the-university-of-oslo"]], "Logistic Regression": [[7, null], [7, "id1"]], "MNIST and GANs": [[4, "mnist-and-gans"]], "Machine Learning": [[24, "machine-learning"]], "Machine learning": [[18, "machine-learning"]], "Main textbooks": [[24, "main-textbooks"]], "Making a tree": [[9, "making-a-tree"]], "Making your own Bootstrap: Changing the Level of the Decision Tree": [[10, "making-your-own-bootstrap-changing-the-level-of-the-decision-tree"]], "Making your own test-train splitting": [[25, "making-your-own-test-train-splitting"]], "Material for exercises week 35": [[25, "material-for-exercises-week-35"]], "Mathematical Interpretation of Ordinary Least Squares": [[5, "mathematical-interpretation-of-ordinary-least-squares"], [25, "mathematical-interpretation-of-ordinary-least-squares"]], "Mathematical optimization of convex functions": [[8, "mathematical-optimization-of-convex-functions"]], "Mathematics of CNNs": [[3, "mathematics-of-cnns"]], "Mathematics of the SVD and implications": [[5, "mathematics-of-the-svd-and-implications"], [25, "mathematics-of-the-svd-and-implications"]], "Matrices in Python": [[24, "matrices-in-python"]], "Matrix multiplication": [[1, "matrix-multiplication"]], "Matrix-vector notation and activation": [[12, "matrix-vector-notation-and-activation"]], "Meet the covariance!": [[21, "meet-the-covariance"]], "Meet the Covariance Matrix": [[5, "meet-the-covariance-matrix"], [25, "meet-the-covariance-matrix"]], "Meet the Hessian Matrix": [[25, "meet-the-hessian-matrix"]], "Meet the Pandas": [[24, "meet-the-pandas"]], "Min-Max Scaling": [[25, "min-max-scaling"]], "Momentum based GD": [[13, "momentum-based-gd"]], "More complicated Example: The Ising model": [[6, "more-complicated-example-the-ising-model"]], "More interpretations": [[25, "more-interpretations"]], "More on Dimensionalities": [[3, "more-on-dimensionalities"]], "More on Rescaling data": [[6, "more-on-rescaling-data"]], "More preprocessing": [[25, "more-preprocessing"]], "Multilayer perceptrons": [[12, "multilayer-perceptrons"]], "Network requirements": [[2, "network-requirements"]], "Neural Networks vs CNNs": [[3, "neural-networks-vs-cnns"]], "Neural networks": [[12, null]], "Note about SVD Calculations": [[25, "note-about-svd-calculations"]], "Numerical experiments and the covariance, central limit theorem": [[21, "numerical-experiments-and-the-covariance-central-limit-theorem"]], "Numpy and arrays": [[19, "numpy-and-arrays"], [24, "numpy-and-arrays"]], "Numpy examples and Important Matrix and vector handling packages": [[24, "numpy-examples-and-important-matrix-and-vector-handling-packages"]], "Optimization, the central part of any Machine Learning algortithm": [[13, null]], "Optimizing our parameters": [[24, "optimizing-our-parameters"]], "Optimizing our parameters, more details": [[24, "optimizing-our-parameters-more-details"]], "Optimizing the cost function": [[1, "optimizing-the-cost-function"]], "Organizing our data": [[0, "organizing-our-data"], [24, "organizing-our-data"]], "Other Matrix and Vector Operations": [[19, "other-matrix-and-vector-operations"]], "Other Types of Recurrent Neural Networks": [[4, "other-types-of-recurrent-neural-networks"]], "Other courses on Data science and Machine Learning at UiO": [[24, "other-courses-on-data-science-and-machine-learning-at-uio"]], "Other courses on Data science and Machine Learning at UiO, contn": [[24, "other-courses-on-data-science-and-machine-learning-at-uio-contn"]], "Other popular texts": [[24, "other-popular-texts"]], "Other techniques": [[11, "other-techniques"]], "Other types of networks": [[12, "other-types-of-networks"]], "Other ways of visualizing the trees": [[9, "other-ways-of-visualizing-the-trees"]], "Our model for the nuclear binding energies": [[24, "our-model-for-the-nuclear-binding-energies"]], "Overview of first week": [[24, "overview-of-first-week"]], "Own code for Ordinary Least Squares": [[24, "own-code-for-ordinary-least-squares"], [25, "own-code-for-ordinary-least-squares"]], "PCA and scikit-learn": [[11, "pca-and-scikit-learn"]], "Pandas AI": [[24, "pandas-ai"]], "Partial Differential Equations": [[2, "partial-differential-equations"]], "Plans for week 35": [[25, "plans-for-week-35"]], "Practical tips": [[13, "practical-tips"]], "Practicalities": [[22, "practicalities"], [22, "id1"]], "Predicting New Points With A Trained Recurrent Neural Network": [[4, "predicting-new-points-with-a-trained-recurrent-neural-network"]], "Preprocessing our 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"wisconsin": 7, "write": [4, 11], "x": 25, "xgboost": 10, "your": [0, 10, 16, 25]}}) \ No newline at end of file diff --git a/doc/LectureNotes/_build/html/week35.html b/doc/LectureNotes/_build/html/week35.html index 353391878..3f4cb1e47 100644 --- a/doc/LectureNotes/_build/html/week35.html +++ b/doc/LectureNotes/_build/html/week35.html @@ -440,6 +440,8 @@ document.write(`
  • Interpreting the Ridge results
  • More interpretations
  • Deriving the Lasso Regression Equations
    • +
  • Material for exercises week 35
    • +
  • Important technicalities: More on Rescaling data
    • @@ -471,10 +473,8 @@ doconce format html week35.do.txt --no_mako -->

    Reading recommendations:#

    1. These lecture notes

      2. -
    - - -
      +

    1. Video of lecture at https://youtu.be/2mvizAQFST8

      2. +

    2. Whiteboard notes at CompPhysics/MachineLearning

    3. Goodfellow, Bengio and Courville, Deep Learning, chapter 2 on linear algebra

    4. Raschka et al on preprocessing of data, relevant for exercise 3 this week, see chapter 4.

    5. For exercise 1 of week 35, the book by A. Aldo Faisal, Cheng Soon Ong, and Marc Peter Deisenroth on the Mathematics of Machine Learning, may be very relevant. In particular chapter 5 at URL”https://mml-book.github.io/” (section 5.5 on derivatives) is very useful for exercise 1 this coming week.

      6. @@ -2354,6 +2354,452 @@ C(\boldsymbol{X},\boldsymbol{\theta})=\frac{1}{n}\left\{(\boldsymbol{y}-\boldsym \]

      This equation does not lead to a nice analytical equation as in either Ridge regression or ordinary least squares. This equation can however be solved by using standard convex optimization algorithms using for example the Python package CVXOPT. We will discuss how to code LASSO regression next week, when we have introduced gradient methods.

      +
      +

      Material for exercises week 35#

      +
      +
      +

      Important technicalities: More on Rescaling data#

      +

      When you are comparing your own code with for example Scikit-Learn’s +library, there are some technicalities to keep in mind. The examples +here demonstrate some of these aspects with potential pitfalls.

      +

      The discussion here focuses on the role of the intercept, how we can +set up the design matrix, what scaling we should use and other topics +which tend confuse us.

      +

      The intercept can be interpreted as the expected value of our +target/output variables when all other predictors are set to zero. +Thus, if we cannot assume that the expected outputs/targets are zero +when all predictors are zero (the columns in the design matrix), it +may be a bad idea to implement a model which penalizes the intercept. +Furthermore, in for example Ridge and Lasso regression, the default solutions +from the library Scikit-Learn (when not shrinking \(\beta_0\)) for the unknown parameters +\(\boldsymbol{\beta}\), are derived under the assumption that both \(\boldsymbol{y}\) and +\(\boldsymbol{X}\) are zero centered, that is we subtract the mean values.

      +

      If our predictors represent different scales, then it is important to +standardize the design matrix \(\boldsymbol{X}\) by subtracting the mean of each +column from the corresponding column and dividing the column with its +standard deviation. Most machine learning libraries do this as a default. This means that if you compare your code with the results from a given library, +the results may differ.

      +

      The +Standardscaler +function in Scikit-Learn does this for us. For the data sets we +have been studying in our various examples, the data are in many cases +already scaled and there is no need to scale them. You as a user of different machine learning algorithms, should always perform a +survey of your data, with a critical assessment of them in case you need to scale the data.

      +

      If you need to scale the data, not doing so will give an unfair +penalization of the parameters since their magnitude depends on the +scale of their corresponding predictor.

      +

      The Scikit-Learn site https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#plot-all-scaling-standard-scaler-section has a good discussion of different ways of preprocessing data.

      +

      Suppose as an example that you +you have an input variable given by the heights of different persons. +Human height might be measured in inches or meters or +kilometers. If measured in kilometers, a standard linear regression +model with this predictor would probably give a much bigger +coefficient term, than if measured in millimeters. +This can clearly lead to problems in evaluating the cost/loss functions.

      +

      Keep in mind that when you transform your data set before training a model, the same transformation needs to be done +on your eventual new data set before making a prediction. If we translate this into a Python code, it would could be implemented as

      +
      +
      +
      """
+#Model training, we compute the mean value of y and X
+y_train_mean = np.mean(y_train)
+X_train_mean = np.mean(X_train,axis=0)
+X_train = X_train - X_train_mean
+y_train = y_train - y_train_mean
+
+# The we fit our model with the training data
+trained_model = some_model.fit(X_train,y_train)
+
+
+#Model prediction, we need also to transform our data set used for the prediction.
+X_test = X_test - X_train_mean #Use mean from training data
+y_pred = trained_model(X_test)
+y_pred = y_pred + y_train_mean
+"""
+
      +
      +
      +
      +

      Let us try to understand what this may imply mathematically when we +subtract the mean values, also known as zero centering. For +simplicity, we will focus on ordinary regression, as done in the above example.

      +

      The cost/loss function for regression is

      +
      +\[ +C(\beta_0, \beta_1, ... , \beta_{p-1}) = \frac{1}{n}\sum_{i=0}^{n} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij}\beta_j\right)^2,. +\]
      +

      Recall also that we use the squared value. This expression can lead to an +increased penalty for higher differences between predicted and +output/target values.

      +

      What we have done is to single out the \(\beta_0\) term in the +definition of the mean squared error (MSE). The design matrix \(X\) +does in this case not contain any intercept column. When we take the +derivative with respect to \(\beta_0\), we want the derivative to obey

      +
      +\[ +\frac{\partial C}{\partial \beta_j} = 0, +\]
      +

      for all \(j\). For \(\beta_0\) we have

      +
      +\[ +\frac{\partial C}{\partial \beta_0} = -\frac{2}{n}\sum_{i=0}^{n-1} \left(y_i - \beta_0 - \sum_{j=1}^{p-1} X_{ij} \beta_j\right). +\]
      +

      Multiplying away the constant \(2/n\), we obtain

      +
      +\[ +\sum_{i=0}^{n-1} \beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} \sum_{j=1}^{p-1} X_{ij} \beta_j. +\]
      +

      Let us specialize first to the case where we have only two parameters \(\beta_0\) and \(\beta_1\). +Our result for \(\beta_0\) simplifies then to

      +
      +\[ +n\beta_0 = \sum_{i=0}^{n-1}y_i - \sum_{i=0}^{n-1} X_{i1} \beta_1. +\]
      +

      We obtain then

      +
      +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \beta_1\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}. +\]
      +

      If we define

      +
      +\[ +\mu_{\boldsymbol{x}_1}=\frac{1}{n}\sum_{i=0}^{n-1} X_{i1}, +\]
      +

      and the mean value of the outputs as

      +
      +\[ +\mu_y=\frac{1}{n}\sum_{i=0}^{n-1}y_i, +\]
      +

      we have

      +
      +\[ +\beta_0 = \mu_y - \beta_1\mu_{\boldsymbol{x}_1}. +\]
      +

      In the general case with more parameters than \(\beta_0\) and \(\beta_1\), we have

      +
      +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=1}^{p-1} X_{ij}\beta_j. +\]
      +

      We can rewrite the latter equation as

      +
      +\[ +\beta_0 = \frac{1}{n}\sum_{i=0}^{n-1}y_i - \sum_{j=1}^{p-1} \mu_{\boldsymbol{x}_j}\beta_j, +\]
      +

      where we have defined

      +
      +\[ +\mu_{\boldsymbol{x}_j}=\frac{1}{n}\sum_{i=0}^{n-1} X_{ij}, +\]
      +

      the mean value for all elements of the column vector \(\boldsymbol{x}_j\).

      +

      Replacing \(y_i\) with \(y_i - y_i - \overline{\boldsymbol{y}}\) and centering also our design matrix results in a cost function (in vector-matrix disguise)

      +
      +\[ +C(\boldsymbol{\beta}) = (\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta})^T(\boldsymbol{\tilde{y}} - \tilde{X}\boldsymbol{\beta}). +\]
      +

      If we minimize with respect to \(\boldsymbol{\beta}\) we have then

      +
      +\[ +\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X})^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}, +\]
      +

      where \(\boldsymbol{\tilde{y}} = \boldsymbol{y} - \overline{\boldsymbol{y}}\) +and \(\tilde{X}_{ij} = X_{ij} - \frac{1}{n}\sum_{k=0}^{n-1}X_{kj}\).

      +

      For Ridge regression we need to add \(\lambda \boldsymbol{\beta}^T\boldsymbol{\beta}\) to the cost function and get then

      +
      +\[ +\hat{\boldsymbol{\beta}} = (\tilde{X}^T\tilde{X} + \lambda I)^{-1}\tilde{X}^T\boldsymbol{\tilde{y}}. +\]
      +

      What does this mean? And why do we insist on all this? Let us look at some examples.

      +

      This code shows a simple first-order fit to a data set using the above transformed data, where we consider the role of the intercept first, by either excluding it or including it (code example thanks to Øyvind Sigmundson Schøyen). Here our scaling of the data is done by subtracting the mean values only. +Note also that we do not split the data into training and test.

      +
      +
      +
      import numpy as np
+import matplotlib.pyplot as plt
+
+from sklearn.linear_model import LinearRegression
+
+
+np.random.seed(2021)
+
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+def fit_beta(X, y):
+    return np.linalg.pinv(X.T @ X) @ X.T @ y
+
+
+true_beta = [2, 0.5, 3.7]
+
+x = np.linspace(0, 1, 11)
+y = np.sum(
+    np.asarray([x ** p * b for p, b in enumerate(true_beta)]), axis=0
+) + 0.1 * np.random.normal(size=len(x))
+
+degree = 3
+X = np.zeros((len(x), degree))
+
+# Include the intercept in the design matrix
+for p in range(degree):
+    X[:, p] = x ** p
+
+beta = fit_beta(X, y)
+
+# Intercept is included in the design matrix
+skl = LinearRegression(fit_intercept=False).fit(X, y)
+
+print(f"True beta: {true_beta}")
+print(f"Fitted beta: {beta}")
+print(f"Sklearn fitted beta: {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with intercept column")
+print(MSE(y,ypredictOwn))
+print(f"MSE with intercept column from SKL")
+print(MSE(y,ypredictSKL))
+
+
+plt.figure()
+plt.scatter(x, y, label="Data")
+plt.plot(x, X @ beta, label="Fit")
+plt.plot(x, skl.predict(X), label="Sklearn (fit_intercept=False)")
+
+
+# Do not include the intercept in the design matrix
+X = np.zeros((len(x), degree - 1))
+
+for p in range(degree - 1):
+    X[:, p] = x ** (p + 1)
+
+# Intercept is not included in the design matrix
+skl = LinearRegression(fit_intercept=True).fit(X, y)
+
+# Use centered values for X and y when computing coefficients
+y_offset = np.average(y, axis=0)
+X_offset = np.average(X, axis=0)
+
+beta = fit_beta(X - X_offset, y - y_offset)
+intercept = np.mean(y_offset - X_offset @ beta)
+
+print(f"Manual intercept: {intercept}")
+print(f"Fitted beta (without intercept): {beta}")
+print(f"Sklearn intercept: {skl.intercept_}")
+print(f"Sklearn fitted beta (without intercept): {skl.coef_}")
+ypredictOwn = X @ beta
+ypredictSKL = skl.predict(X)
+print(f"MSE with Manual intercept")
+print(MSE(y,ypredictOwn+intercept))
+print(f"MSE with Sklearn intercept")
+print(MSE(y,ypredictSKL))
+
+plt.plot(x, X @ beta + intercept, "--", label="Fit (manual intercept)")
+plt.plot(x, skl.predict(X), "--", label="Sklearn (fit_intercept=True)")
+plt.grid()
+plt.legend()
+
+plt.show()
+
      +
      +
      +
      +

      The intercept is the value of our output/target variable +when all our features are zero and our function crosses the \(y\)-axis (for a one-dimensional case).

      +

      Printing the MSE, we see first that both methods give the same MSE, as +they should. However, when we move to for example Ridge regression, +the way we treat the intercept may give a larger or smaller MSE, +meaning that the MSE can be penalized by the value of the +intercept. Not including the intercept in the fit, means that the +regularization term does not include \(\beta_0\). For different values +of \(\lambda\), this may lead to different MSE values.

      +

      To remind the reader, the regularization term, with the intercept in Ridge regression, is given by

      +
      +\[ +\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=0}^{p-1}\beta_j^2, +\]
      +

      but when we take out the intercept, this equation becomes

      +
      +\[ +\lambda \vert\vert \boldsymbol{\beta} \vert\vert_2^2 = \lambda \sum_{j=1}^{p-1}\beta_j^2. +\]
      +

      For Lasso regression we have

      +
      +\[ +\lambda \vert\vert \boldsymbol{\beta} \vert\vert_1 = \lambda \sum_{j=1}^{p-1}\vert\beta_j\vert. +\]
      +

      It means that, when scaling the design matrix and the outputs/targets, +by subtracting the mean values, we have an optimization problem which +is not penalized by the intercept. The MSE value can then be smaller +since it focuses only on the remaining quantities. If we however bring +back the intercept, we will get a MSE which then contains the +intercept.

      +

      Armed with this wisdom, we attempt first to simply set the intercept equal to False in our implementation of Ridge regression for our well-known vanilla data set.

      +
      +
      +
      import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+
+
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(3155)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree))
+#We include explicitely the intercept column
+for degree in range(Maxpolydegree):
+    X[:,degree] = x**degree
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+p = Maxpolydegree
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
+    # Note: we include the intercept column and no scaling
+    RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
+    RegRidge.fit(X_train,y_train)
+    # and then make the prediction
+    ytildeOwnRidge = X_train @ OwnRidgeBeta
+    ypredictOwnRidge = X_test @ OwnRidgeBeta
+    ytildeRidge = RegRidge.predict(X_train)
+    ypredictRidge = RegRidge.predict(X_test)
+    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    print("Beta values for own Ridge implementation")
+    print(OwnRidgeBeta)
+    print("Beta values for Scikit-Learn Ridge implementation")
+    print(RegRidge.coef_)
+    print("MSE values for own Ridge implementation")
+    print(MSEOwnRidgePredict[i])
+    print("MSE values for Scikit-Learn Ridge implementation")
+    print(MSERidgePredict[i])
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'r', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g', label = 'MSE Ridge Test')
+
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
      +
      +
      +
      +

      The results here agree when we force Scikit-Learn’s Ridge function to include the first column in our design matrix. +We see that the results agree very well. Here we have thus explicitely included the intercept column in the design matrix. +What happens if we do not include the intercept in our fit? +Let us see how we can change this code by zero centering.

      +
      +
      +
      import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+from sklearn.model_selection import train_test_split
+from sklearn import linear_model
+from sklearn.preprocessing import StandardScaler
+
+def MSE(y_data,y_model):
+    n = np.size(y_model)
+    return np.sum((y_data-y_model)**2)/n
+# A seed just to ensure that the random numbers are the same for every run.
+# Useful for eventual debugging.
+np.random.seed(315)
+
+n = 100
+x = np.random.rand(n)
+y = np.exp(-x**2) + 1.5 * np.exp(-(x-2)**2)
+
+Maxpolydegree = 20
+X = np.zeros((n,Maxpolydegree-1))
+
+for degree in range(1,Maxpolydegree): #No intercept column
+    X[:,degree-1] = x**(degree)
+
+# We split the data in test and training data
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
+
+#For our own implementation, we will need to deal with the intercept by centering the design matrix and the target variable
+X_train_mean = np.mean(X_train,axis=0)
+#Center by removing mean from each feature
+X_train_scaled = X_train - X_train_mean 
+X_test_scaled = X_test - X_train_mean
+#The model intercept (called y_scaler) is given by the mean of the target variable (IF X is centered)
+#Remove the intercept from the training data.
+y_scaler = np.mean(y_train)           
+y_train_scaled = y_train - y_scaler   
+
+p = Maxpolydegree-1
+I = np.eye(p,p)
+# Decide which values of lambda to use
+nlambdas = 6
+MSEOwnRidgePredict = np.zeros(nlambdas)
+MSERidgePredict = np.zeros(nlambdas)
+
+lambdas = np.logspace(-4, 2, nlambdas)
+for i in range(nlambdas):
+    lmb = lambdas[i]
+    OwnRidgeBeta = np.linalg.pinv(X_train_scaled.T @ X_train_scaled+lmb*I) @ X_train_scaled.T @ (y_train_scaled)
+    intercept_ = y_scaler - X_train_mean@OwnRidgeBeta #The intercept can be shifted so the model can predict on uncentered data
+    #Add intercept to prediction
+    ypredictOwnRidge = X_test_scaled @ OwnRidgeBeta + y_scaler 
+    RegRidge = linear_model.Ridge(lmb)
+    RegRidge.fit(X_train,y_train)
+    ypredictRidge = RegRidge.predict(X_test)
+    MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
+    MSERidgePredict[i] = MSE(y_test,ypredictRidge)
+    print("Beta values for own Ridge implementation")
+    print(OwnRidgeBeta) #Intercept is given by mean of target variable
+    print("Beta values for Scikit-Learn Ridge implementation")
+    print(RegRidge.coef_)
+    print('Intercept from own implementation:')
+    print(intercept_)
+    print('Intercept from Scikit-Learn Ridge implementation')
+    print(RegRidge.intercept_)
+    print("MSE values for own Ridge implementation")
+    print(MSEOwnRidgePredict[i])
+    print("MSE values for Scikit-Learn Ridge implementation")
+    print(MSERidgePredict[i])
+
+
+# Now plot the results
+plt.figure()
+plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
+plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
+plt.xlabel('log10(lambda)')
+plt.ylabel('MSE')
+plt.legend()
+plt.show()
+
      +
      +
      +
      +

      We see here, when compared to the code which includes explicitely the +intercept column, that our MSE value is actually smaller. This is +because the regularization term does not include the intercept value +\(\beta_0\) in the fitting. This applies to Lasso regularization as +well. It means that our optimization is now done only with the +centered matrix and/or vector that enter the fitting procedure.

      +