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162,67,65,68.6,6,3,female,65.5 162,67,65,68.6,6,4,female,65 162,67,65,68.6,6,5,female,64.5 162,67,65,68.6,6,6,female,63.5 163,67,65.5,68.87,5,1,male,70 163,67,65.5,68.87,5,2,male,69 163,67,65.5,68.87,5,3,female,65.5 163,67,65.5,68.87,5,4,female,65.5 163,67,65.5,68.87,5,5,female,63 164,67,65.5,68.87,4,1,male,70 164,67,65.5,68.87,4,2,male,67.7 164,67,65.5,68.87,4,3,female,63 164,67,65.5,68.87,4,4,female,60 165,67,65,68.6,3,1,male,65 165,67,65,68.6,3,2,female,62 165,67,65,68.6,3,3,female,62 166,67.5,65,68.85,11,1,male,71 166,67.5,65,68.85,11,2,male,69 166,67.5,65,68.85,11,3,female,64 166,67.5,65,68.85,11,4,female,64 166,67.5,65,68.85,11,5,female,63 166,67.5,65,68.85,11,6,female,63 166,67.5,65,68.85,11,7,female,63 166,67.5,65,68.85,11,8,female,63 166,67.5,65,68.85,11,9,female,63 166,67.5,65,68.85,11,10,female,62.5 166,67.5,65,68.85,11,11,female,62 167,67,64,68.06,4,1,male,71.5 167,67,64,68.06,4,2,male,70 167,67,64,68.06,4,3,male,67 167,67,64,68.06,4,4,male,67 168,67,63.5,67.79,8,1,male,71 168,67,63.5,67.79,8,2,male,70.2 168,67,63.5,67.79,8,3,male,69.2 168,67,63.5,67.79,8,4,male,68.5 168,67,63.5,67.79,8,5,male,68 168,67,63.5,67.79,8,6,male,67 168,67,63.5,67.79,8,7,male,65.5 168,67,63.5,67.79,8,8,female,63.5 169,67,63,67.52,3,1,male,69 169,67,63,67.52,3,2,male,68 169,67,63,67.52,3,3,female,63 170,67.5,62,67.23,5,1,male,70 170,67.5,62,67.23,5,2,male,69.5 170,67.5,62,67.23,5,3,male,69 170,67.5,62,67.23,5,4,male,68.5 170,67.5,62,67.23,5,5,female,66 171,67,61,66.44,1,1,male,67 172,66,67,69.18,8,1,male,70.5 172,66,67,69.18,8,2,male,70.5 172,66,67,69.18,8,3,male,67 172,66,67,69.18,8,4,male,66 172,66,67,69.18,8,5,male,66 172,66,67,69.18,8,6,female,62 172,66,67,69.18,8,7,female,62 172,66,67,69.18,8,8,female,61.5 173,66,67,69.18,9,1,male,72 173,66,67,69.18,9,2,male,65 173,66,67,69.18,9,3,male,65 173,66,67,69.18,9,4,female,67 173,66,67,69.18,9,5,female,64 173,66,67,69.18,9,6,female,64 173,66,67,69.18,9,7,female,62 173,66,67,69.18,9,8,female,60 173,66,67,69.18,9,9,female,60 174,66,66,68.64,5,1,male,66 174,66,66,68.64,5,2,male,65 174,66,66,68.64,5,3,female,67 174,66,66,68.64,5,4,female,66.5 174,66,66,68.64,5,5,female,65.5 175,66,66,68.64,6,1,male,72 175,66,66,68.64,6,2,male,68 175,66,66,68.64,6,3,female,66 175,66,66,68.64,6,4,female,65 175,66,66,68.64,6,5,female,62 175,66,66,68.64,6,6,female,61 176,66.5,65,68.35,8,1,male,68.7 176,66.5,65,68.35,8,2,male,68.5 176,66.5,65,68.35,8,3,male,66.5 176,66.5,65,68.35,8,4,male,64.5 176,66.5,65,68.35,8,5,female,62.5 176,66.5,65,68.35,8,6,female,60.5 176,66.5,65,68.35,8,7,female,60.5 176,66.5,65,68.35,8,8,female,57.5 177,66,65.5,68.37,5,1,male,72 177,66,65.5,68.37,5,2,male,71 177,66,65.5,68.37,5,3,male,67 177,66,65.5,68.37,5,4,female,66 177,66,65.5,68.37,5,5,female,65 178,66,63,67.02,1,1,male,70 179,66,63.5,67.29,2,1,female,64.5 179,66,63.5,67.29,2,2,female,62 180,66.5,63,67.27,6,1,male,67.2 180,66.5,63,67.27,6,2,male,67 180,66.5,63,67.27,6,3,male,65 180,66.5,63,67.27,6,4,female,65 180,66.5,63,67.27,6,5,female,65 180,66.5,63,67.27,6,6,female,63 181,66.5,62.5,67,7,1,male,70 181,66.5,62.5,67,7,2,male,68 181,66.5,62.5,67,7,3,female,63.5 181,66.5,62.5,67,7,4,female,62.5 181,66.5,62.5,67,7,5,female,62.5 181,66.5,62.5,67,7,6,female,62.5 181,66.5,62.5,67,7,7,female,62.5 182,66,61.5,66.21,1,1,male,70 183,66,60,65.4,4,1,male,68 183,66,60,65.4,4,2,male,67 183,66,60,65.4,4,3,male,65 183,66,60,65.4,4,4,female,60 184,66,60,65.4,1,1,male,65 185,66,59,64.86,15,1,male,68 185,66,59,64.86,15,2,male,67 185,66,59,64.86,15,3,male,66.5 185,66,59,64.86,15,4,male,66 185,66,59,64.86,15,5,male,65.7 185,66,59,64.86,15,6,male,65.5 185,66,59,64.86,15,7,male,65 185,66,59,64.86,15,8,female,65 185,66,59,64.86,15,9,female,64 185,66,59,64.86,15,10,female,63 185,66,59,64.86,15,11,female,62 185,66,59,64.86,15,12,female,61 185,66,59,64.86,15,13,female,60 185,66,59,64.86,15,14,female,58 185,66,59,64.86,15,15,female,57 186,65,67,68.68,4,1,male,66.5 186,65,67,68.68,4,2,male,66 186,65,67,68.68,4,3,male,66 186,65,67,68.68,4,4,female,65 187,65,67,68.68,1,1,female,63 188,65,66,68.14,4,1,male,63 188,65,66,68.14,4,2,female,63 188,65,66,68.14,4,3,female,63 188,65,66,68.14,4,4,female,60 189,65,66,68.14,5,1,male,67 189,65,66,68.14,5,2,male,66 189,65,66,68.14,5,3,male,65 189,65,66,68.14,5,4,female,65 189,65,66,68.14,5,5,female,61 190,65,65,67.6,9,1,male,69 190,65,65,67.6,9,2,male,68 190,65,65,67.6,9,3,male,68 190,65,65,67.6,9,4,female,65 190,65,65,67.6,9,5,female,65 190,65,65,67.6,9,6,female,62 190,65,65,67.6,9,7,female,62 190,65,65,67.6,9,8,female,61 190,65,65,67.6,9,9,female,59 191,65,65.5,67.87,2,1,male,70.7 191,65,65.5,67.87,2,2,female,65.5 192,65,65,67.6,6,1,male,69.2 192,65,65,67.6,6,2,male,69 192,65,65,67.6,6,3,male,68 192,65,65,67.6,6,4,male,67.7 192,65,65,67.6,6,5,female,64.5 192,65,65,67.6,6,6,female,60.5 193,65,64,67.06,6,1,male,67 193,65,64,67.06,6,2,male,67 193,65,64,67.06,6,3,female,64 193,65,64,67.06,6,4,female,64 193,65,64,67.06,6,5,female,62.5 193,65,64,67.06,6,6,female,60.5 194,65,63,66.52,2,1,male,70 194,65,63,66.52,2,2,female,63 195,65,63,66.52,3,1,male,66 195,65,63,66.52,3,2,male,66 195,65,63,66.52,3,3,female,63 196,65.5,63,66.77,4,1,male,71 196,65.5,63,66.77,4,2,male,71 196,65.5,63,66.77,4,3,male,69 196,65.5,63,66.77,4,4,female,63.5 197,65.5,60,65.15,5,1,male,68 197,65.5,60,65.15,5,2,male,68 197,65.5,60,65.15,5,3,male,67 197,65.5,60,65.15,5,4,male,67 197,65.5,60,65.15,5,5,female,62 198,64,64,66.56,7,1,male,71.5 198,64,64,66.56,7,2,male,68 198,64,64,66.56,7,3,female,65.5 198,64,64,66.56,7,4,female,64 198,64,64,66.56,7,5,female,62 198,64,64,66.56,7,6,female,62 198,64,64,66.56,7,7,female,61 199,64,64,66.56,7,1,male,70.5 199,64,64,66.56,7,2,male,68 199,64,64,66.56,7,3,female,67 199,64,64,66.56,7,4,female,65 199,64,64,66.56,7,5,female,64 199,64,64,66.56,7,6,female,64 199,64,64,66.56,7,7,female,60 200,64,63,66.02,1,1,male,64.5 201,64,60,64.4,2,1,male,66 201,64,60,64.4,2,2,female,60 202,63,63.5,65.79,2,1,female,68.5 202,63,63.5,65.79,2,2,female,63.5 203,62,66,66.64,3,1,male,64 203,62,66,66.64,3,2,female,62 203,62,66,66.64,3,3,female,61 204,62.5,63,65.27,2,1,male,66.5 204,62.5,63,65.27,2,2,female,57 \ No newline at end of file diff --git a/lectures/lec23/data/hybrid.csv b/lectures/lec23/data/hybrid.csv new file mode 100644 index 0000000..0c7b136 --- /dev/null +++ b/lectures/lec23/data/hybrid.csv @@ -0,0 +1,154 @@ +vehicle,year,price,acceleration,mpg,class +Prius (1st Gen),1997,24509.74,7.46,41.26,Compact +Tino,2000,35354.97,8.2,54.1,Compact +Prius (2nd Gen),2000,26832.25,7.97,45.23,Compact +Insight,2000,18936.41,9.52,53.0,Two Seater +Civic (1st Gen),2001,25833.38,7.04,47.04,Compact +Insight,2001,19036.71,9.52,53.0,Two Seater +Insight,2002,19137.01,9.71,53.0,Two Seater +Alphard,2003,38084.77,8.33,40.46,Minivan +Insight,2003,19137.01,9.52,53.0,Two Seater +Civic,2003,14071.92,8.62,41.0,Compact +Escape,2004,36676.1,10.32,31.99,SUV +Insight,2004,19237.31,9.35,52.0,Two Seater +Prius,2004,20355.64,9.9,46.0,Midsize +Silverado 15 2WD,2004,30089.64,9.09,17.0,Pickup Truck +Lexus RX400h,2005,58521.14,12.76,28.23,SUV +Civic (2nd Gen),2005,26354.44,7.63,39.99,Compact +Highlander,2005,29186.21,12.76,29.4,SUV +Insight,2005,19387.76,9.71,52.0,Two Seater +Civic,2005,18236.33,8.26,41.0,Compact +Escape 2WD,2005,19322.56,9.52,29.0,SUV +Accord,2005,16343.69,14.93,28.0,Midsize +Silverado 15 2WD,2005,32647.26,11.11,17.0,Pickup Truck +Mercury Mariner,2006,34772.4,8.98,32.93,SUV +Camry,2006,29853.25,11.28,33.64,Midsize +Lexus GS450h,2006,64547.56,18.65,33.4,Midsize +Estima,2006,36012.7,9.26,47.04,Minivan +Altima,2006,29524.75,13.29,32.93,Midsize +Chevrolet Tahoe,2007,42924.35,10.91,22.35,SUV +Kluger,2007,46229.48,12.76,25.87,SUV +Lexus LS600h/hL,2007,118543.6,17.54,21.0,Midsize +Tribute,2007,24823.83,11.28,31.75,SUV +GMC Yukon,2007,57094.81,12.28,21.78,SUV +Aura,2007,22110.87,10.87,27.0,Midsize +Vue,2007,22938.33,10.75,26.0,SUV +Silverado 15 2WD,2007,34653.23,11.49,17.0,Pickup Truck +Crown,2008,62290.38,8.7,37.16,Midsize +Cadillac Escalade,2008,78932.81,9.09,22.35,SUV +F3DM,2008,23744.06,9.52,30.11,Midsize +Altima,2008,18675.63,13.7,34.0,Midsize +A5 BSG,2009,11849.43,7.87,35.28,Midsize +Lexus RX450h,2009,46233.36,13.47,31.99,SUV +ML450 Blue HV,2009,60519.83,12.6,23.99,SUV +Prius (3rd Gen),2009,24641.18,9.6,47.98,Compact +S400 Long,2009,96208.93,13.89,26.34,Large +Mercury Milan,2009,30522.57,11.55,40.69,Midsize +Lexus HS250h,2009,38478.15,11.55,54.1,Compact +Avante/Elantra LPI,2009,21872.71,10.21,41.87,Compact +ActiveHybrid X6,2009,97237.9,17.96,18.82,SUV +SAI,2009,39172.44,11.55,54.1,Midsize +Malibu,2009,24768.79,9.09,29.0,Midsize +Vue,2009,26408.67,13.7,28.0,SUV +Aspen HEV,2009,44903.77,13.51,21.0,SUV +Durango,2009,41033.24,8.33,21.0,SUV +Auris HSD,2010,35787.29,8.85,68.21,Compact +CR-Z,2010,21435.54,9.24,37.0,Two Seater +F3DM PHEV,2010,23124.59,9.24,30.15,Midsize +Touareg,2010,64198.95,15.38,28.7,SUV +Audi Q5,2010,37510.86,14.08,33.64,SUV +Jeep Patriot,2010,17045.06,12.05,29.4,SUV +Besturn B50 ,2010,14586.61,7.14,31.28,Midsize +ActiveHybrid 7,2010,104300.43,20.41,22.11,Large +Lincoln MKZ,2010,37036.64,11.15,37.63,Midsize +Fit/Jazz,2010,16911.85,8.26,30.0,Compact +Sonata,2010,28287.66,14.7,37.0,Midsize +Cayenne S,2010,73183.47,14.71,26.11,SUV +Insight,2010,19859.16,9.17,41.0,Compact +Fuga Infiniti M35H,2010,70157.02,18.65,33.64,Midsize +Chevrolet Volt,2010,42924.35,10.78,35.0,Compact +Tribute 4WD,2010,27968.32,12.35,29.0,SUV +Fusion FWD,2010,28033.51,11.49,39.0,Midsize +HS 250h,2010,34753.53,11.76,35.0,Compact +Mariner FWD,2010,30194.95,11.63,32.0,SUV +RX 450h,2010,42812.54,13.89,30.0,SUV +ML450 4natic,2010,55164.33,12.99,22.0,SUV +Silverado 15 2WD,2010,38454.56,11.76,22.0,Pickup Truck +S400,2010,88212.78,12.99,21.0,Large +Aqua,2011,22850.87,9.35,50.0,Compact +Lexus CT200h,2011,30082.16,9.71,42.0,Compact +Civic (3rd Gen),2011,24999.59,9.6,44.36,Compact +Prius alpha (V),2011,30588.35,10.0,72.92,Midsize +3008,2011,45101.54,11.36,61.16,Compact +Fit Shuttle,2011,16394.36,7.52,58.8,Minivan +Buick Regal,2011,27948.93,12.05,25.99,Midsize +Prius V,2011,27272.28,9.51,32.93,Midsize +Freed/Freed Spike,2011,27972.07,6.29,50.81,Minivan +Optima K5,2011,26549.16,10.54,36.0,Midsize +Escape FWD,2011,30661.34,12.35,32.0,SUV +Insight,2011,18254.38,9.52,41.0,Compact +MKZ FWD,2011,34748.52,11.49,39.0,Midsize +CR-Z,2011,19402.8,12.2,37.0,Two Seater +Sonata,2011,25872.07,11.9,36.0,Midsize +Camry,2011,27130.82,13.89,33.0,Midsize +Tribute 2WD,2011,26213.09,12.5,32.0,SUV +Cayenne S,2011,67902.28,18.52,21.0,SUV +Touareg,2011,50149.39,16.13,21.0,SUV +ActiveHybrid 7i,2011,102605.66,18.18,20.0,Midsize +Prius C,2012,19006.62,9.35,50.0,Compact +Prius PHV,2012,32095.61,8.82,50.0,Midsize +Ampera,2012,31739.55,11.11,37.0,Compact +ActiveHybrid 5,2012,62180.23,16.67,26.0,Midsize +Lexus GS450h,2012,59126.14,16.95,31.0,Midsize +Insight,2012,18555.28,9.42,42.0,Compact +Chevrolet Volt,2012,39261.96,11.11,37.0,Compact +Camry LE,2012,26067.66,13.16,41.0,Midsize +MKZ FWD,2012,34858.84,11.49,39.0,Midsize +M35h,2012,53860.45,19.23,29.0,Midsize +LaCrosse,2012,30049.52,11.36,29.0,Midsize +ActiveHybrid 5,2012,61132.11,17.54,26.0,Midsize +Panamera S,2012,95283.85,17.54,25.0,Large +Yukon 1500,2012,52626.77,13.5,21.0,SUV +Prius C,2013,19080.0,8.7,50.0,Compact +Jetta,2013,24995.0,12.66,45.0,Compact +Civic,2013,24360.0,10.2,44.0,Compact +Prius,2013,24200.0,10.2,50.0,Midsize +Fusion FWD,2013,27200.0,11.72,47.0,Midsize +C-Max FWD,2013,25200.0,12.35,43.0,Large +Insight,2013,18600.0,11.76,42.0,Compact +Camry LE,2013,26140.0,13.51,41.0,Midsize +Camry LXLE,2013,27670.0,13.33,40.0,Midsize +Sonata,2013,25650.0,11.76,38.0,Midsize +Optima,2013,25900.0,11.63,38.0,Midsize +Sonata Limited,2013,30550.0,11.76,37.0,Midsize +Optima EX,2013,31950.0,11.36,37.0,Midsize +Malibu,2013,24985.0,11.49,29.0,Midsize +LaCrosse,2013,31660.0,11.36,29.0,Midsize +Regal,2013,29015.0,12.2,29.0,Midsize +RX 450h,2013,46310.0,12.99,30.0,SUV +Highlander 4WD,2013,40170.0,13.89,28.0,SUV +Q5,2013,50900.0,14.71,26.0,SUV +Cayenne S,2013,69850.0,16.39,21.0,SUV +Touareg,2013,62575.0,16.13,21.0,SUV +Escalade 2WD,2013,74425.0,11.63,21.0,SUV +Tahoe 2WD,2013,53620.0,11.9,21.0,SUV +Yukon 1500,2013,54145.0,11.88,21.0,SUV +Yukon 1500,2013,61960.0,13.33,21.0,SUV +MKZ FWD,2013,35925.0,14.03,45.0,Midsize +CT 200h,2013,32050.0,10.31,42.0,Compact +ES 300h,2013,39250.0,12.35,40.0,Midsize +ILX,2013,28900.0,9.26,38.0,Compact +ActiveHybrid 3,2013,49650.0,14.93,28.0,Compact +Silverado 15 2WD,2013,41135.0,12.35,21.0,Pickup Truck +Sierra 15 2WD,2013,41555.0,10.0,21.0,Pickup Truck +GS 450h,2013,59450.0,16.67,31.0,Midsize +M35h,2013,54750.0,19.61,29.0,Midsize +E400,2013,55800.0,14.93,26.0,Midsize +ActiveHybrid 5,2013,61400.0,12.99,26.0,Midsize +ActiveHybrid 7L,2013,84300.0,18.18,25.0,Large +Panamera S,2013,96150.0,18.52,25.0,Large +S400,2013,92350.0,13.89,21.0,Large +Prius Plug-in,2013,32000.0,9.17,50.0,Midsize +C-Max Energi Plug-in,2013,32950.0,11.76,43.0,Midsize +Fusion Energi 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file mode 100644 index 0000000..af654e9 --- /dev/null +++ b/lectures/lec23/lec23.ipynb @@ -0,0 +1,1414 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "skip" + } + }, + "outputs": [], + "source": [ + "# Set up packages for lecture. Don't worry about understanding this code, but\n", + "# make sure to run it if you're following along.\n", + "import numpy as np\n", + "import babypandas as bpd\n", + "import pandas as pd\n", + "from matplotlib_inline.backend_inline import set_matplotlib_formats\n", + "import matplotlib.pyplot as plt\n", + "from matplotlib.animation import FuncAnimation, PillowWriter\n", + "from scipy import stats\n", + "set_matplotlib_formats(\"svg\")\n", + "plt.style.use('ggplot')\n", + "\n", + "np.set_printoptions(threshold=20, precision=2, suppress=True)\n", + "pd.set_option(\"display.max_rows\", 7)\n", + "pd.set_option(\"display.max_columns\", 8)\n", + "pd.set_option(\"display.precision\", 2)\n", + "\n", + "# Animations\n", + "from IPython.display import display, HTML, IFrame, clear_output\n", + "import ipywidgets as widgets\n", + "\n", + "import warnings\n", + "warnings.filterwarnings('ignore')\n", + "\n", + "# Demonstration code\n", + "def r_scatter(r):\n", + " \"Generate a scatter plot with a correlation approximately r\"\n", + " x = np.random.normal(0, 1, 1000)\n", + " z = np.random.normal(0, 1, 1000)\n", + " y = r * x + (np.sqrt(1 - r ** 2)) * z\n", + " plt.scatter(x, y)\n", + " plt.xlim(-4, 4)\n", + " plt.ylim(-4, 4)\n", + " \n", + "def show_scatter_grid():\n", + " plt.subplots(1, 4, figsize=(10, 2))\n", + " for i, r in enumerate([-1, -2/3, -1/3, 0]):\n", + " plt.subplot(1, 4, i+1)\n", + " r_scatter(r)\n", + " plt.title(f'r = {np.round(r, 2)}')\n", + " plt.show()\n", + " plt.subplots(1, 4, figsize=(10, 2))\n", + " for i, r in enumerate([1, 2/3, 1/3]):\n", + " plt.subplot(1, 4, i+1)\n", + " r_scatter(r)\n", + " plt.title(f'r = {np.round(r, 2)}')\n", + " plt.subplot(1, 4, 4)\n", + " plt.axis('off')\n", + " plt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "# Lecture 23 – Correlation\n", + "\n", + "## DSC 10, Winter 2023" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Announcements\n", + "\n", + "- Homework 6 is due **tomorrow at 11:59pm**.\n", + "- Lab 7 is due on **Saturday 3/11 at 11:59pm**.\n", + "- The Final Project is due on **Tuesday 3/14 at 11:59pm**." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Agenda\n", + "\n", + "- Recap: Statistical inference.\n", + "- Association.\n", + "- Correlation.\n", + "- Regression." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Recap: Statistical inference" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Four big ideas in statistical inference\n", + "\n", + "Every statistical test and simulation we've run in the second half of the class is related to one of the following four ideas. To solidify your understanding of what we've done, it's a good idea to review past lectures and assignments and see how what we did in each section relates to one of these four ideas." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "source": [ + "- To test whether a sample came from a known population distribution, use \"standard\" hypothesis testing. " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "source": [ + "- To test whether two samples came from the same unknown population distribution, use permutation testing." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "source": [ + "- To estimate a population parameter given a single sample, construct a confidence interval using bootstrapping (for most statistics) or the CLT (for the sample mean)." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "source": [ + "- To test whether a population parameter is equal to a particular value, $x$, construct a confidence interval using bootstrapping (for most statistics) or the CLT (for the sample mean), and check whether $x$ is in the interval." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Association" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Prediction\n", + "\n", + "- Suppose we have a dataset with at least two numerical variables." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We're interested in **predicting** one variable based on another:\n", + " - Given my education level, what is my income?\n", + " - Given my height, how tall will my kid be as an adult?\n", + " - Given my age, how many countries have I visited?" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To do this effectively, we need to first observe a pattern between the two numerical variables." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- To see if a pattern exists, we'll need to draw a scatter plot." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Association\n", + "\n", + "- An **association** is any relationship or link between two variables in a **scatter plot**. Associations can be linear or non-linear.🔗.\"" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If two variables have a positive association ↗️, then as one variable increases, the other tends to increase.\n", + "- If two variables have a negative association ↘️, then as one variable increases, the other tends to decrease." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If two variables are associated, then we can predict the value of one variable based on the value of the other." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Hybrid cars 🚗" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "hybrid = bpd.read_csv('data/hybrid.csv')\n", + "hybrid" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### `'acceleration'` and `'price'`\n", + "\n", + "Is there an association between these two variables? If so, what kind?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "hybrid.plot(kind='scatter', x='acceleration', y='price', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### `'mpg'` and `'price'`\n", + "\n", + "Is there an association between these two variables? If so, what kind?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "hybrid.plot(kind='scatter', x='mpg', y='price', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "**Observations:**\n", + "- There is an association – cars with better fuel economy tended to be cheaper.\n", + " - Why do we think that is? 🤔\n", + "- The association looks more curved than linear. \n", + " - It may roughly follow $y \\approx \\frac{1}{x}$.\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Linear changes in units\n", + "- A linear change in units doesn't change the shape of the plot, it only changes the scale of the plot.\n", + " - Linear change means adding or subtracting a constant, and multiplying or dividing by a constant." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- In other words, instead of plotting price in _dollars_ and fuel economy in _miles per gallon_, we can plot price in _Yen (🇯🇵)_ and fuel economy in _kilometers per liter_ and the plot would look the same, just with different axes:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid.assign(\n", + " km_per_liter=hybrid.get('mpg') * 0.425144,\n", + " yen=hybrid.get('price') * 140.34 \n", + ").plot(kind='scatter', x='km_per_liter', y='yen', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Converting columns to standard units\n", + "- Recall: Suppose $x$ is a numerical variable, and $x_i$ is one value of that variable. To convert $x_i$ to standard units,\n", + "$$x_{i \\: \\text{(su)}} = \\frac{x_i - \\text{mean of $x$}}{\\text{SD of $x$}}$$" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Converting columns to standard units makes different scatter plots comparable, by putting the $x$ and $y$ axes on the **same scale**.\n", + " - Both axes measure the number of standard deviations above the mean." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Converting columns to standard units doesn't change shape of the scatter plot, because the conversion is linear." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def standard_units(any_numbers):\n", + " \"Convert any array of numbers to standard units.\"\n", + " any_numbers = np.array(any_numbers)\n", + " return (any_numbers - any_numbers.mean()) / np.std(any_numbers)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def standardize(df):\n", + " \"\"\"Return a DataFrame in which all columns of df are converted to standard units.\"\"\"\n", + " df_su = bpd.DataFrame()\n", + " for column in df.columns:\n", + " df_su = df_su.assign(**{column + ' (su)': standard_units(df.get(column))})\n", + " return df_su" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Standard units for hybrid cars\n", + "For a given pair of variables:\n", + "- Which cars are average in both variables?\n", + "- Which cars are well above or well below average in both variables?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "hybrid_su = standardize(hybrid.get(['price', 'acceleration', 'mpg'])).assign(vehicle=hybrid.get('vehicle'))\n", + "hybrid_su" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### `'acceleration'` and `'price'`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false, + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='acceleration (su)', y='price (su)', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Which cars have `'acceleration'`s and `'price'`s that are more than 2 SDs above average?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su[(hybrid_su.get('acceleration (su)') > 2) &\n", + " (hybrid_su.get('price (su)') > 2)]" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### `'mpg'` and `'price'`" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false, + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='mpg (su)', y='price (su)', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "source": [ + "Which cars have close to average `'mpg'`s and close to average `'price'`s?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "hybrid_su[(hybrid_su.get('mpg (su)') <= 0.3) &\n", + " (hybrid_su.get('mpg (su)') >= -0.3) &\n", + " (hybrid_su.get('price (su)') <= 0.3) &\n", + " (hybrid_su.get('price (su)') >= -0.3)]" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Observation on associations in standard units\n", + "- If two variables are positively associated ↗️,\n", + " - their high, positive values in standard units are typically seen together, and\n", + " - their low, negative values are typically seen together as well." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If two variables are negatively associated ↘️,\n", + " - high, positive values of one are typically coupled with low, negative values of the other.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If two variables aren't associated, there should be no such pattern." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "When there is a positive association, most data points fall in the lower left and upper right quadrants." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='acceleration (su)', y='price (su)', figsize=(10, 5))\n", + "plt.axvline(0, color='black');\n", + "plt.axhline(0, color='black');" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "When there is a negative association, most data points fall in the upper left and lower right quadrants." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='mpg (su)', y='price (su)', figsize=(10, 5))\n", + "plt.axvline(0, color='black');\n", + "plt.axhline(0, color='black');" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Correlation" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Definition: Correlation coefficient\n", + "\n", + "The correlation coefficient $r$ of two variables $x$ and $y$ is defined as the \n", + "- **average** value of the \n", + "- **product** of $x$ and $y$\n", + "- when both are measured in **standard units**." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "If `x` and `y` are two Series or arrays, \n", + "```py\n", + "r = (x_su * y_su).mean()\n", + "```\n", + "where `x_su` and `y_su` are `x` and `y` converted to standard units." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Let's calculate $r$ for `'acceleration'` and `'price'`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "hybrid_su" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "r_acc_price = (hybrid_su.get('acceleration (su)') * hybrid_su.get('price (su)')).mean()\n", + "r_acc_price" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='acceleration (su)', y='price (su)', figsize=(10, 5))\n", + "plt.axvline(0, color='black');\n", + "plt.axhline(0, color='black');" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Note that the correlation is positive, and most data points fall in the lower left and upper right quadrants!" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "Let's now calculate $r$ for `'mpg'` and `'price'`." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "hybrid_su" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "-" + } + }, + "outputs": [], + "source": [ + "r_mpg_price = (hybrid_su.get('mpg (su)') * hybrid_su.get('price (su)')).mean()\n", + "r_mpg_price" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "hybrid_su.plot(kind='scatter', x='mpg (su)', y='price (su)', figsize=(10, 5));\n", + "plt.axvline(0, color='black');\n", + "plt.axhline(0, color='black');" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "Note that the correlation is negative, and most data points fall in the upper left and lower right quadrants!" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### The correlation coefficient, $r$\n", + "\n", + "- $r$ measures how clustered points are around a straight line – **it measures linear association**.\n", + " - If two variables are correlated, it means they are linearly associated.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- $r$ is always between $-1$ and $1$.\n", + " - If $r = 1$, the scatter plot is a line of slope 1.\n", + " - If $r = -1$, the scatter plot is a line of slope -1.\n", + " - If $r = 0$, there is no linear association (_uncorrelated_)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "show_scatter_grid()" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- $r$ is computed based on standard units.\n", + " - The correlation between price in _dollars_ and fuel economy in _miles per gallon_ is the same as the correlation between price in _Yen_ and fuel economy in _kilometers per liter_." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- $r$ quantifies the **predictive power** of the relationship between two variables. \n", + " - If $r$ is close to $1$ or $-1$ we can predict one variable from the other quite accurately.\n", + " - If $r$ is close to $0$, we cannot make good predictions." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Concept Check ✅ – Answer at [cc.dsc10.com](http://cc.dsc10.com) \n", + "\n", + "Which of the following does the scatter plot below show?\n", + "\n", + "- A. Association and correlation\n", + "- B. Association but not correlation\n", + "- C. Correlation but not association\n", + "- D. Neither association nor correlation" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x2 = bpd.DataFrame().assign(\n", + " x=np.arange(-6, 6.1, 0.5), \n", + " y=np.arange(-6, 6.1, 0.5) ** 2\n", + ")\n", + "x2.plot(kind='scatter', x='x', y='y', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "

Click here to see the answer after trying it yourself.\n", + "\n", + "B. Association but not correlation\n", + " \n", + "Since there is a pattern in the scatter plot of $x$ and $y$, there is an association between $x$ and $y$. However, correlation refers to linear association, and there is no linear association between $x$ and $y$. The relationship between $x$ and $y$ is actually $y = x^2$. Even though the association between $x$ and $y$ is very strong, the association cannot be described by a linear function because as $x$ increases, $y$ first decreases, and then increases. \n", + " \n", + "The correlation ($r$) between $x$ and $y$ is 0 – try to calculate it yourself!\n", + "\n", + "
" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Regression" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Example: Predicting heights 👪 📏\n", + "\n", + "The data below was collected in the late 1800s by [Francis Galton](https://en.wikipedia.org/wiki/Francis_Galton).\n", + "- He was a eugenicist and proponent of scientific racism, which is why he collected this data.\n", + "- Today, we understand that eugenics is immoral, and that there is no scientific evidence or any other justification for racism.\n", + "- Galton is credited with discovering regression using this data." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "galton = bpd.read_csv('data/galton.csv')\n", + "galton" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Mothers and sons 👵👨\n", + "\n", + "Let's just consider the relationship between mothers' heights and their adult sons' heights." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "male_children = galton[galton.get('gender') == 'male']\n", + "mom_son = bpd.DataFrame().assign(mom = male_children.get('mother'), \n", + " son = male_children.get('childHeight'))\n", + "mom_son" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "mom_son.plot(kind='scatter', x='mom', y='son', figsize=(10, 5));" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Predicting a son's height based on his mother's height\n", + "\n", + "- The scatter plot demonstrates a positive association between a mother's height (`'mom'`) and her son's height (`'son'`). \n", + "- Let's quantify how *linear* that association is by computing the correlation between `'mom'` and `'son'`. \n", + "- First, we standardize the data." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "mom_son_su = standardize(mom_son)\n", + "mom_son_su.plot(kind='scatter', x='mom (su)', y='son (su)', figsize=(10, 5));" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": true + }, + "outputs": [], + "source": [ + "r_mom_son = (mom_son_su.get('mom (su)') * mom_son_su.get('son (su)')).mean()\n", + "r_mom_son" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Many possible ways to make predictions\n", + "\n", + "- We want a _simple_ strategy, or rule, for predicting a son's height. \n", + "- The simplest possible prediction strategy just predicts the same value for every son's height, regardless of his mother's height. \n", + "- Some such predictions are better than others." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def constant_prediction(prediction):\n", + " mom_son_su.plot(kind='scatter', x='mom (su)', y='son (su)', title=f'Predicting a height of {prediction} SUs for all sons', figsize=(10, 5));\n", + " plt.axhline(prediction, color='orange', lw=4);\n", + " plt.xlim(-3, 3)\n", + " plt.show()\n", + "\n", + "prediction = widgets.FloatSlider(value=-3, min=-3,max=3,step=0.5, description='prediction')\n", + "ui = widgets.HBox([prediction])\n", + "out = widgets.interactive_output(constant_prediction, {'prediction': prediction})\n", + "display(ui, out)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Which of these predictions is the best? \n", + " - It depends on what we mean by \"best,\" but a natural choice is the rule that predicts 0 standard units, because this corresponds to the mean height of all sons." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "mom_son_su.plot(kind='scatter', x='mom (su)', y='son (su)', title='A good prediction is the mean height of sons (0 SUs)', figsize=(10, 5));\n", + "plt.axhline(0, color='orange', lw=4);\n", + "plt.xlim(-3, 3);" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Better predictions\n", + "\n", + "- Since there is an association between a son's height and his mother's height, we can make better predictions by allowing our predictions to vary with the mother's height. \n", + "- The simplest way to do this uses a line to make predictions.\n", + "- As before, some lines are better than others." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "def linear_prediction(slope):\n", + " x = np.linspace(-3, 3)\n", + " y = x * slope\n", + " mom_son_su.plot(kind='scatter', x='mom (su)', y='son (su)', figsize=(10, 5));\n", + " plt.plot(x, y, color='orange', lw=4)\n", + " plt.xlim(-3, 3)\n", + " plt.title(r\"Predicting sons' heights using $\\mathrm{son}_{\\mathrm{(su)}}$ = \" + str(np.round(slope, 2)) + r\"$ \\cdot \\mathrm{mother}_{\\mathrm{(su)}}$\")\n", + " plt.show()\n", + "\n", + "slope = widgets.FloatSlider(value=0, min=-1,max=1,step=1/6, description='slope')\n", + "ui = widgets.HBox([slope])\n", + "out = widgets.interactive_output(linear_prediction, {'slope': slope})\n", + "display(ui, out)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- Which of these lines is the best? \n", + " - Again, it depends what we mean by \"best,\" but a good choice is the line that goes through the origin and has a slope of $r$.\n", + " - This line is called the **regression line**, and we'll see next time that it is the \"best\" line for making predictions in a certain sense." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "x = np.linspace(-3, 3)\n", + "y = x * r_mom_son\n", + "mom_son_su.plot(kind='scatter', x='mom (su)', y='son (su)', title=r'A good line goes through the origin and has slope $r$', figsize=(10, 5));\n", + "plt.plot(x, y, color='orange', label='regression line', lw=4)\n", + "plt.xlim(-3, 3)\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### The regression line\n", + "\n", + "- The regression line is the line through $(0,0)$ with slope $r$, when both variables are measured in **standard units**.\n", + "\n", + "
\n", + "\n", + "- We use the regression line to make predictions!" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Making predictions in standard units\n", + "\n", + "
\n", + "\n", + "- If $r = 0.32$, and the given $x$ is $2$ in standard units, then the prediction for $y$ is $0.64$ standard units.\n", + " - The regression line predicts that a mother whose height is $2$ SDs above average has a son whose height is $0.64$ SDs above average." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- If $r = 0.32$, and the given $x$ is $-1$ in standard units, then the prediction for $y$ is $-0.32$ standard units." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- We always _predict_ that a son will be somewhat **closer to average** in height than his mother.\n", + " - This is a consequence of the slope $r$ having magnitude less than 1.\n", + " - This effect is called **regression to the mean**." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "fragment" + } + }, + "source": [ + "- The regression line passes through the origin $(0, 0)$ in standard units. This means that, no matter what $r$ is, **for an average $x$ value, we predict an average $y$ value**." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Making predictions in original units\n", + "\n", + "Of course, we'd like to be able to predict a son's height in inches, not just in standard units. Given a mother's height in inches, here's how we'll predict her son's height in inches:\n", + "1. Convert the mother's height from inches to standard units.\n", + "\n", + "$$x_{i \\: \\text{(su)}} = \\frac{x_i - \\text{mean of $x$}}{\\text{SD of $x$}}$$\n", + "\n", + "2. Multiply by the correlation coefficient to predict the son's height in standard units.\n", + "\n", + "$$\\text{predicted } y_{i \\: \\text{(su)}} = r \\cdot x_{i \\: \\text{(su)}}$$\n", + "\n", + "3. Convert the son's predicted height from standard units back to inches.\n", + "\n", + "$$\\text{predicted } y_i = \\text{predicted } y_{i \\: \\text{(su)}} \\cdot \\text{SD of $y$} + \\text{mean of $y$}$$" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "mom_mean = mom_son.get('mom').mean()\n", + "mom_sd = np.std(mom_son.get('mom'))\n", + "son_mean = mom_son.get('son').mean()\n", + "son_sd = np.std(mom_son.get('son'))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [], + "source": [ + "def predict_with_r(mom):\n", + " \"\"\"Return a prediction for the height of a son whose mother has height mom, \n", + " using linear regression.\n", + " \"\"\"\n", + " mom_su = (mom - mom_mean) / mom_sd\n", + " son_su = r_mom_son * mom_su\n", + " return son_su * son_sd + son_mean" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "predict_with_r(68)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "predict_with_r(60)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "outputs": [], + "source": [ + "preds = mom_son.assign(\n", + " predicted_height=mom_son.get('mom').apply(predict_with_r)\n", + ")\n", + "ax = preds.plot(kind='scatter', x='mom', y='son', title='Regression line predictions, in original units', figsize=(10, 5), label='original data')\n", + "preds.plot(kind='line', x='mom', y='predicted_height', ax=ax, color='orange', label='regression line', lw=4);\n", + "plt.legend();" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Concept Check ✅ – Answer at [cc.dsc10.com](http://cc.dsc10.com) \n", + "\n", + "A course has a midterm (mean 80, standard deviation 15) and a really hard final (mean 50, standard deviation 12).\n", + "\n", + "If the scatter plot comparing midterm & final scores for students looks linearly associated with correlation 0.75, then what is the predicted final exam score for a student who received a 90 on the midterm?\n", + "\n", + "- A. 54\n", + "- B. 56\n", + "- C. 58\n", + "- D. 60\n", + "- E. 62" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "## Summary, next time" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Summary\n", + "\n", + "- The correlation coefficient, $r$, measures the linear association between two variables $x$ and $y$.\n", + " - It ranges between -1 and 1.\n", + "- When both variables are measured in standard units, the regression line is the straight line passing through $(0, 0)$ with slope $r$. We can use it to make predictions for a $y$ value (e.g. son's height) given an $x$ value (e.g. mother's height)." + ] + }, + { + "cell_type": "markdown", + "metadata": { + "slideshow": { + "slide_type": "slide" + } + }, + "source": [ + "### Next time\n", + "\n", + "More on regression, including:\n", + " - What is the equation of the regression line in original units (e.g. inches)?\n", + " - In what sense is the regression line the \"best\" line for making predictions?" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.13" + }, + "livereveal": { + "scroll": true, + "transition": "none" + }, + "rise": { + "enable_chalkboard": true + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/lectures/lec23/rise.css b/lectures/lec23/rise.css new file mode 100644 index 0000000..e88c46b --- /dev/null +++ b/lectures/lec23/rise.css @@ -0,0 +1,55 @@ +/* +copy this file as rise.css adjacent to a lecture notebook to auto-include +*/ + +.reveal, +.reveal h1, +.reveal h2, +.reveal h3, +.reveal h4, +.reveal h5, +.reveal h6 { + /* reset slideshow fonts to system default */ + font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Helvetica, + Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'; +} + +/* make table styles more similar to pandas */ +.reveal table th, +.reveal table td { + border: none; + padding: 0.5em; +} + +.rise-enabled .reveal pre { + margin: 0; +} + +/* +RISE sets table font-size to 75%, but this makes pandas tutor tables super +small +*/ +table.wstRoot { + font-size: 18px !important; +} + +.wstRoot td.rowLabel, +.wstRoot td.rowLabelLevelName, +.wstRoot .columnLabel, +.wstRoot .columnLabelLevelName { + font-size: 20px !important; +} + +/* using .menti in a cell lets us put the menti code side by side */ +.menti { + display: flex; + gap: 1rem; +} + +.menti > div { + flex-grow: 1; +} + +.menti > div:last-child { + max-width: 30%; +}