{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Plots\n",
"using Statistics\n",
"using StatsBase\n",
"using PyCall\n",
"using DataFrames\n",
"using GLM\n",
"using Tables\n",
"using XLSX\n",
"using MLBase\n",
"using RDatasets\n",
"using LsqFit"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
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"source": [
"xvals = repeat(1:0.5:10, inner=2)\n",
"yvals = 3 .+ xvals .+ 2 .* rand(length(xvals)) .-1\n",
"scatter(xvals, yvals, color=:black, leg=false)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"find_best_fit (generic function with 1 method)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function find_best_fit(xvals,yvals)\n",
" meanx = mean(xvals)\n",
" meany = mean(yvals)\n",
" stdx = std(xvals)\n",
" stdy = std(yvals)\n",
" r = cor(xvals,yvals)\n",
" a = r*stdy/stdx\n",
" b = meany - a*meanx\n",
" return a,b\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"38-element Vector{Float64}:\n",
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" ⋮\n",
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],
"source": [
"a,b = find_best_fit(xvals,yvals)\n",
"ynew = a .* xvals .+ b"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"np = pyimport(\"numpy\");"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 0.261024 seconds (1.26 M allocations: 75.000 MiB, 5.50% gc time, 0.44% compilation time)\n"
]
},
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},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xdata = xvals\n",
"ydata = yvals\n",
"@time myfit = np.polyfit(xdata, ydata, 1);\n",
"ynew2 = collect(xdata) .* myfit[1] .+ myfit[2];\n",
"scatter(xvals,yvals)\n",
"plot!(xvals,ynew)\n",
"plot!(xvals,ynew2)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
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]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = DataFrame(X=xdata, Y=ydata)\n",
"ols = lm(@formula(Y ~ X), data)\n",
"plot!(xdata,predict(ols))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now let's get some real data. We will use housing information from zillow, check out the file `zillow_data_download_april2020.xlsx` for a quick look of what the data looks like. Our goal will be to build a linear regression model between the number of houses listed vs the number of houses sold in a few states. Fitting these models can serve as a key real estate indicator."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"XLSXFile(\"zillow_data_download_april2020.xlsx\") containing 4 Worksheets\n",
" sheetname size range \n",
"-------------------------------------------------\n",
" MonthlyListings_City 8348x91 A1:CM8348 \n",
" Sale_counts_city 28760x148 A1:ER28760 \n",
"Sales_median_price_c… 3767x148 A1:ER3767 \n",
" meta 1x1 A1:A1 \n"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# play around with data for a bit\n",
"R = XLSX.readxlsx(\"data/zillow_data_download_april2020.xlsx\")"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"data-frame\"><thead><tr><th></th><th>SizeRank</th><th>RegionID</th><th>RegionName</th><th>RegionType</th><th>StateName</th><th>2013-01</th><th>2013-02</th><th>2013-03</th></tr><tr><th></th><th>Any</th><th>Any</th><th>Any</th><th>Any</th><th>Any</th><th>Any</th><th>Any</th><th>Any</th></tr></thead><tbody><p>8,347 rows × 91 columns (omitted printing of 83 columns)</p><tr><th>1</th><td>1</td><td>6181</td><td>New York</td><td>City</td><td>NY</td><td>28904</td><td>28155</td><td>30596</td></tr><tr><th>2</th><td>2</td><td>12447</td><td>Los Angeles</td><td>City</td><td>CA</td><td>6431</td><td>6613</td><td>6467</td></tr><tr><th>3</th><td>3</td><td>39051</td><td>Houston</td><td>City</td><td>TX</td><td>11696</td><td>11737</td><td>12291</td></tr><tr><th>4</th><td>4</td><td>17426</td><td>Chicago</td><td>City</td><td>IL</td><td>9523</td><td>9609</td><td>9772</td></tr><tr><th>5</th><td>5</td><td>6915</td><td>San Antonio</td><td>City</td><td>TX</td><td>7223</td><td>7170</td><td>7438</td></tr><tr><th>6</th><td>6</td><td>13271</td><td>Philadelphia</td><td>City</td><td>PA</td><td>7505</td><td>7266</td><td>7578</td></tr><tr><th>7</th><td>7</td><td>40326</td><td>Phoenix</td><td>City</td><td>AZ</td><td>6232</td><td>5720</td><td>5872</td></tr><tr><th>8</th><td>8</td><td>18959</td><td>Las Vegas</td><td>City</td><td>NV</td><td>7027</td><td>6218</td><td>6199</td></tr><tr><th>9</th><td>9</td><td>54296</td><td>San Diego</td><td>City</td><td>CA</td><td>4121</td><td>3844</td><td>3932</td></tr><tr><th>10</th><td>10</td><td>38128</td><td>Dallas</td><td>City</td><td>TX</td><td>4332</td><td>4395</td><td>4568</td></tr><tr><th>11</th><td>11</td><td>10221</td><td>Austin</td><td>City</td><td>TX</td><td>2345</td><td>2230</td><td>2301</td></tr><tr><th>12</th><td>12</td><td>33839</td><td>San Jose</td><td>City</td><td>CA</td><td><em>missing</em></td><td><em>missing</em></td><td><em>missing</em></td></tr><tr><th>13</th><td>13</td><td>25290</td><td>Jacksonville</td><td>City</td><td>FL</td><td>4517</td><td>4495</td><td>4481</td></tr><tr><th>14</th><td>14</td><td>24043</td><td>Charlotte</td><td>City</td><td>NC</td><td>4550</td><td>4451</td><td>4812</td></tr><tr><th>15</th><td>15</td><td>18172</td><td>Fort Worth</td><td>City</td><td>TX</td><td>3709</td><td>3738</td><td>3703</td></tr><tr><th>16</th><td>16</td><td>7481</td><td>Tucson</td><td>City</td><td>AZ</td><td>4422</td><td>4355</td><td>4376</td></tr><tr><th>17</th><td>17</td><td>10920</td><td>Columbus</td><td>City</td><td>OH</td><td>3873</td><td>3849</td><td>4099</td></tr><tr><th>18</th><td>18</td><td>12455</td><td>Louisville</td><td>City</td><td>KY</td><td>3517</td><td>3355</td><td>3567</td></tr><tr><th>19</th><td>19</td><td>13121</td><td>Orlando</td><td>City</td><td>FL</td><td>2821</td><td>2781</td><td>2711</td></tr><tr><th>20</th><td>20</td><td>17933</td><td>El Paso</td><td>City</td><td>TX</td><td>3255</td><td>3287</td><td>3405</td></tr><tr><th>21</th><td>21</td><td>17762</td><td>Detroit</td><td>City</td><td>MI</td><td>3550</td><td>3413</td><td>3358</td></tr><tr><th>22</th><td>22</td><td>11093</td><td>Denver</td><td>City</td><td>CO</td><td>2486</td><td>2404</td><td>2588</td></tr><tr><th>23</th><td>23</td><td>16037</td><td>Seattle</td><td>City</td><td>WA</td><td>1514</td><td>1569</td><td>1709</td></tr><tr><th>24</th><td>24</td><td>32811</td><td>Memphis</td><td>City</td><td>TN</td><td>3024</td><td>2867</td><td>2986</td></tr><tr><th>25</th><td>25</td><td>44269</td><td>Boston</td><td>City</td><td>MA</td><td>1193</td><td>1287</td><td>1570</td></tr><tr><th>26</th><td>26</td><td>41568</td><td>Washington</td><td>City</td><td>DC</td><td>1362</td><td>1339</td><td>1418</td></tr><tr><th>27</th><td>27</td><td>13373</td><td>Portland</td><td>City</td><td>OR</td><td>2133</td><td>2158</td><td>2220</td></tr><tr><th>28</th><td>28</td><td>6118</td><td>Nashville</td><td>City</td><td>TN</td><td>4285</td><td>4280</td><td>4448</td></tr><tr><th>29</th><td>29</td><td>20288</td><td>Sacramento</td><td>City</td><td>CA</td><td>905</td><td>938</td><td>990</td></tr><tr><th>30</th><td>30</td><td>3523</td><td>Baltimore</td><td>City</td><td>MD</td><td>2907</td><td>2894</td><td>3026</td></tr><tr><th>⋮</th><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>"
],
"text/latex": [
"\\begin{tabular}{r|ccccccccc}\n",
"\t& SizeRank & RegionID & RegionName & RegionType & StateName & 2013-01 & 2013-02 & 2013-03 & \\\\\n",
"\t\\hline\n",
"\t& Any & Any & Any & Any & Any & Any & Any & Any & \\\\\n",
"\t\\hline\n",
"\t1 & 1 & 6181 & New York & City & NY & 28904 & 28155 & 30596 & $\\dots$ \\\\\n",
"\t2 & 2 & 12447 & Los Angeles & City & CA & 6431 & 6613 & 6467 & $\\dots$ \\\\\n",
"\t3 & 3 & 39051 & Houston & City & TX & 11696 & 11737 & 12291 & $\\dots$ \\\\\n",
"\t4 & 4 & 17426 & Chicago & City & IL & 9523 & 9609 & 9772 & $\\dots$ \\\\\n",
"\t5 & 5 & 6915 & San Antonio & City & TX & 7223 & 7170 & 7438 & $\\dots$ \\\\\n",
"\t6 & 6 & 13271 & Philadelphia & City & PA & 7505 & 7266 & 7578 & $\\dots$ \\\\\n",
"\t7 & 7 & 40326 & Phoenix & City & AZ & 6232 & 5720 & 5872 & $\\dots$ \\\\\n",
"\t8 & 8 & 18959 & Las Vegas & City & NV & 7027 & 6218 & 6199 & $\\dots$ \\\\\n",
"\t9 & 9 & 54296 & San Diego & City & CA & 4121 & 3844 & 3932 & $\\dots$ \\\\\n",
"\t10 & 10 & 38128 & Dallas & City & TX & 4332 & 4395 & 4568 & $\\dots$ \\\\\n",
"\t11 & 11 & 10221 & Austin & City & TX & 2345 & 2230 & 2301 & $\\dots$ \\\\\n",
"\t12 & 12 & 33839 & San Jose & City & CA & \\emph{missing} & \\emph{missing} & \\emph{missing} & $\\dots$ \\\\\n",
"\t13 & 13 & 25290 & Jacksonville & City & FL & 4517 & 4495 & 4481 & $\\dots$ \\\\\n",
"\t14 & 14 & 24043 & Charlotte & City & NC & 4550 & 4451 & 4812 & $\\dots$ \\\\\n",
"\t15 & 15 & 18172 & Fort Worth & City & TX & 3709 & 3738 & 3703 & $\\dots$ \\\\\n",
"\t16 & 16 & 7481 & Tucson & City & AZ & 4422 & 4355 & 4376 & $\\dots$ \\\\\n",
"\t17 & 17 & 10920 & Columbus & City & OH & 3873 & 3849 & 4099 & $\\dots$ \\\\\n",
"\t18 & 18 & 12455 & Louisville & City & KY & 3517 & 3355 & 3567 & $\\dots$ \\\\\n",
"\t19 & 19 & 13121 & Orlando & City & FL & 2821 & 2781 & 2711 & $\\dots$ \\\\\n",
"\t20 & 20 & 17933 & El Paso & City & TX & 3255 & 3287 & 3405 & $\\dots$ \\\\\n",
"\t21 & 21 & 17762 & Detroit & City & MI & 3550 & 3413 & 3358 & $\\dots$ \\\\\n",
"\t22 & 22 & 11093 & Denver & City & CO & 2486 & 2404 & 2588 & $\\dots$ \\\\\n",
"\t23 & 23 & 16037 & Seattle & City & WA & 1514 & 1569 & 1709 & $\\dots$ \\\\\n",
"\t24 & 24 & 32811 & Memphis & City & TN & 3024 & 2867 & 2986 & $\\dots$ \\\\\n",
"\t25 & 25 & 44269 & Boston & City & MA & 1193 & 1287 & 1570 & $\\dots$ \\\\\n",
"\t26 & 26 & 41568 & Washington & City & DC & 1362 & 1339 & 1418 & $\\dots$ \\\\\n",
"\t27 & 27 & 13373 & Portland & City & OR & 2133 & 2158 & 2220 & $\\dots$ \\\\\n",
"\t28 & 28 & 6118 & Nashville & City & TN & 4285 & 4280 & 4448 & $\\dots$ \\\\\n",
"\t29 & 29 & 20288 & Sacramento & City & CA & 905 & 938 & 990 & $\\dots$ \\\\\n",
"\t30 & 30 & 3523 & Baltimore & City & MD & 2907 & 2894 & 3026 & $\\dots$ \\\\\n",
"\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ & \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m8347×91 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m SizeRank \u001b[0m\u001b[1m RegionID \u001b[0m\u001b[1m RegionName \u001b[0m\u001b[1m RegionType \u001b[0m\u001b[1m StateName \u001b[0m\u001b[1m 2013-\u001b[0m ⋯\n",
"\u001b[1m \u001b[0m│\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m ⋯\n",
"──────┼─────────────────────────────────────────────────────────────────────────\n",
" 1 │ 1 6181 New York City NY 28904 ⋯\n",
" 2 │ 2 12447 Los Angeles City CA 6431\n",
" 3 │ 3 39051 Houston City TX 11696\n",
" 4 │ 4 17426 Chicago City IL 9523\n",
" 5 │ 5 6915 San Antonio City TX 7223 ⋯\n",
" 6 │ 6 13271 Philadelphia City PA 7505\n",
" 7 │ 7 40326 Phoenix City AZ 6232\n",
" 8 │ 8 18959 Las Vegas City NV 7027\n",
" 9 │ 9 54296 San Diego City CA 4121 ⋯\n",
" 10 │ 10 38128 Dallas City TX 4332\n",
" 11 │ 11 10221 Austin City TX 2345\n",
" ⋮ │ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱\n",
" 8338 │ 7335 15309 Evans Mills City NY \u001b[90m missi\u001b[0m\n",
" 8339 │ 7335 15829 Point Roberts City WA 24 ⋯\n",
" 8340 │ 7335 25413 Kingston City GA \u001b[90m missi\u001b[0m\n",
" 8341 │ 7335 28637 Everglades City FL 26\n",
" 8342 │ 7335 29921 Ennis City MT 32\n",
" 8343 │ 7335 32151 Ingleside on the Bay City TX \u001b[90m missi\u001b[0m ⋯\n",
" 8344 │ 7335 38337 El Paso City IL \u001b[90m missi\u001b[0m\n",
" 8345 │ 7335 46631 Mount Charleston City NV \u001b[90m missi\u001b[0m\n",
" 8346 │ 7335 52665 Jamaica Beach City TX 58\n",
" 8347 │ 7335 54635 Surfside Beach City TX 72 ⋯\n",
"\u001b[36m 86 columns and 8326 rows omitted\u001b[0m"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sale_counts = R[\"Sale_counts_city\"][:]\n",
"df_sale_counts = DataFrame(sale_counts[2:end,:],Symbol.(sale_counts[1,:]))\n",
"\n",
"monthly_listings = R[\"MonthlyListings_City\"][:]\n",
"df_monthly_listings = DataFrame(monthly_listings[2:end,:],Symbol.(monthly_listings[1,:]))"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"data-frame\"><thead><tr><th></th><th>RegionID</th><th>sales</th></tr><tr><th></th><th>Any</th><th>Any</th></tr></thead><tbody><p>28,759 rows × 2 columns</p><tr><th>1</th><td>6181</td><td>4054</td></tr><tr><th>2</th><td>12447</td><td>1522</td></tr><tr><th>3</th><td>39051</td><td>2682</td></tr><tr><th>4</th><td>17426</td><td>2100</td></tr><tr><th>5</th><td>6915</td><td>1626</td></tr><tr><th>6</th><td>13271</td><td>1620</td></tr><tr><th>7</th><td>40326</td><td>2325</td></tr><tr><th>8</th><td>18959</td><td>2547</td></tr><tr><th>9</th><td>54296</td><td><em>missing</em></td></tr><tr><th>10</th><td>38128</td><td>1292</td></tr><tr><th>11</th><td>10221</td><td>1023</td></tr><tr><th>12</th><td>33839</td><td>439</td></tr><tr><th>13</th><td>25290</td><td><em>missing</em></td></tr><tr><th>14</th><td>32149</td><td>1443</td></tr><tr><th>15</th><td>20330</td><td>291</td></tr><tr><th>16</th><td>24043</td><td>1439</td></tr><tr><th>17</th><td>18172</td><td>1059</td></tr><tr><th>18</th><td>7481</td><td>870</td></tr><tr><th>19</th><td>10920</td><td>895</td></tr><tr><th>20</th><td>12455</td><td>972</td></tr><tr><th>21</th><td>13121</td><td>726</td></tr><tr><th>22</th><td>17933</td><td>465</td></tr><tr><th>23</th><td>17762</td><td><em>missing</em></td></tr><tr><th>24</th><td>11093</td><td>1198</td></tr><tr><th>25</th><td>16037</td><td>482</td></tr><tr><th>26</th><td>32811</td><td>898</td></tr><tr><th>27</th><td>44269</td><td>147</td></tr><tr><th>28</th><td>41568</td><td>845</td></tr><tr><th>29</th><td>13373</td><td>642</td></tr><tr><th>30</th><td>6118</td><td>1107</td></tr><tr><th>⋮</th><td>⋮</td><td>⋮</td></tr></tbody></table>"
],
"text/latex": [
"\\begin{tabular}{r|cc}\n",
"\t& RegionID & sales\\\\\n",
"\t\\hline\n",
"\t& Any & Any\\\\\n",
"\t\\hline\n",
"\t1 & 6181 & 4054 \\\\\n",
"\t2 & 12447 & 1522 \\\\\n",
"\t3 & 39051 & 2682 \\\\\n",
"\t4 & 17426 & 2100 \\\\\n",
"\t5 & 6915 & 1626 \\\\\n",
"\t6 & 13271 & 1620 \\\\\n",
"\t7 & 40326 & 2325 \\\\\n",
"\t8 & 18959 & 2547 \\\\\n",
"\t9 & 54296 & \\emph{missing} \\\\\n",
"\t10 & 38128 & 1292 \\\\\n",
"\t11 & 10221 & 1023 \\\\\n",
"\t12 & 33839 & 439 \\\\\n",
"\t13 & 25290 & \\emph{missing} \\\\\n",
"\t14 & 32149 & 1443 \\\\\n",
"\t15 & 20330 & 291 \\\\\n",
"\t16 & 24043 & 1439 \\\\\n",
"\t17 & 18172 & 1059 \\\\\n",
"\t18 & 7481 & 870 \\\\\n",
"\t19 & 10920 & 895 \\\\\n",
"\t20 & 12455 & 972 \\\\\n",
"\t21 & 13121 & 726 \\\\\n",
"\t22 & 17933 & 465 \\\\\n",
"\t23 & 17762 & \\emph{missing} \\\\\n",
"\t24 & 11093 & 1198 \\\\\n",
"\t25 & 16037 & 482 \\\\\n",
"\t26 & 32811 & 898 \\\\\n",
"\t27 & 44269 & 147 \\\\\n",
"\t28 & 41568 & 845 \\\\\n",
"\t29 & 13373 & 642 \\\\\n",
"\t30 & 6118 & 1107 \\\\\n",
"\t$\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"\u001b[1m28759×2 DataFrame\u001b[0m\n",
"\u001b[1m Row \u001b[0m│\u001b[1m RegionID \u001b[0m\u001b[1m sales \u001b[0m\n",
"\u001b[1m \u001b[0m│\u001b[90m Any \u001b[0m\u001b[90m Any \u001b[0m\n",
"───────┼───────────────────\n",
" 1 │ 6181 4054\n",
" 2 │ 12447 1522\n",
" 3 │ 39051 2682\n",
" 4 │ 17426 2100\n",
" 5 │ 6915 1626\n",
" 6 │ 13271 1620\n",
" 7 │ 40326 2325\n",
" 8 │ 18959 2547\n",
" 9 │ 54296 \u001b[90m missing \u001b[0m\n",
" 10 │ 38128 1292\n",
" 11 │ 10221 1023\n",
" ⋮ │ ⋮ ⋮\n",
" 28750 │ 396952 \u001b[90m missing \u001b[0m\n",
" 28751 │ 397236 \u001b[90m missing \u001b[0m\n",
" 28752 │ 398030 \u001b[90m missing \u001b[0m\n",
" 28753 │ 398104 \u001b[90m missing \u001b[0m\n",
" 28754 │ 398357 \u001b[90m missing \u001b[0m\n",
" 28755 │ 398712 \u001b[90m missing \u001b[0m\n",
" 28756 │ 398716 2\n",
" 28757 │ 399081 0\n",
" 28758 │ 737789 \u001b[90m missing \u001b[0m\n",
" 28759 │ 760882 0\n",
"\u001b[36m 28738 rows omitted\u001b[0m"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"monthly_listings_2020_02 = df_monthly_listings[!,[1,2,3,4,5,end]]\n",
"rename!(monthly_listings_2020_02, Symbol(\"2020-02\") .=> Symbol(\"listings\"))\n",
"\n",
"sale_counts_2020_02 = df_sale_counts[!,[1,end]]\n",
"rename!(sale_counts_2020_02, Symbol(\"2020-02\") .=> Symbol(\"sales\"))"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"ename": "LoadError",
"evalue": "UndefVarError: Feb2020data not defined",
"output_type": "error",
"traceback": [
"UndefVarError: Feb2020data not defined",
"",
"Stacktrace:",
" [1] top-level scope",
" @ :0",
" [2] eval",
" @ ./boot.jl:360 [inlined]",
" [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
" @ Base ./loading.jl:1094"
]
}
],
"source": [
"Feb2020data"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10-element Vector{Plots.Plot}:\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef\n",
" #undef"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Feb2020data = innerjoin(monthly_listings_2020_02,sale_counts_2020_02,on=:RegionID) #, type=\"outer\")\n",
"dropmissing!(Feb2020data)\n",
"sales = Feb2020data[!,:sales]\n",
"# prices = Feb2020data[!,:price]\n",
"counts = Feb2020data[!,:listings]\n",
"using DataStructures\n",
"states = Feb2020data[!,:StateName]\n",
"C = counter(states)\n",
"C.map\n",
"countvals = values(C.map)\n",
"topstates = sortperm(collect(countvals),rev=true)[1:10]\n",
"states_of_interest = collect(keys(C.map))[topstates]\n",
"all_plots = Array{Plots.Plot}(undef,10)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(si, coef(ols)) = (\"CA\", [0.29595375979515304])\n",
"(si, coef(ols)) = (\"FL\", [0.16693529057344902])\n",
"(si, coef(ols)) = (\"IL\", [0.22291703276169347])\n",
"(si, coef(ols)) = (\"NJ\", [0.25556391864492345])\n",
"(si, coef(ols)) = (\"MI\", [0.2925436820147375])\n",
"(si, coef(ols)) = (\"TX\", [0.24732835944836878])\n",
"(si, coef(ols)) = (\"PA\", [0.310066636803275])\n",
"(si, coef(ols)) = (\"OH\", [0.4569949754409864])\n",
"(si, coef(ols)) = (\"NC\", [0.4800747758857476])\n",
"(si, coef(ols)) = (\"NY\", [0.19151984869233082])\n"
]
},
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}
],
"source": [
"all_plots = Array{Plots.Plot}(undef,10)\n",
"for (i,si) in enumerate(states_of_interest)\n",
" curids = findall(Feb2020data[!,:StateName].==si)\n",
" data = DataFrame(X=float.(counts[curids]), Y=float.(sales[curids]))\n",
" ols = GLM.lm(@formula(Y ~ 0 + X), data) \n",
" all_plots[i] = scatter(counts[curids],sales[curids],markersize=2,\n",
" xlim=(0,500),ylim=(0,500),color=i,aspect_ratio=:equal,\n",
" legend=false,title=si)\n",
" @show si,coef(ols)\n",
" plot!(counts[curids],predict(ols),color=:black)\n",
"end\n",
"plot(all_plots...,layout=(2,5),size=(900,300))"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(si, coef(ols)) = (\"CA\", [6.803086464628309, 0.28798374289950357])\n",
"(si, coef(ols)) = (\"FL\", [19.126255819233386, 0.15510798334559428])\n",
"(si, coef(ols)) = (\"IL\", [0.6626439876883514, 0.2226325765807212])\n",
"(si, coef(ols)) = (\"NJ\", [-1.9635911642225732, 0.2650303535602978])\n",
"(si, coef(ols)) = (\"MI\", [-0.9181942041703473, 0.29768705032459525])\n",
"(si, coef(ols)) = (\"TX\", [5.241220767319292, 0.24566620049636262])\n",
"(si, coef(ols)) = (\"PA\", [-0.21362448579952723, 0.3101955410309725])\n",
"(si, coef(ols)) = (\"OH\", [-6.026958739166818, 0.47909280478486926])\n",
"(si, coef(ols)) = (\"NC\", [-24.60003985327959, 0.5302793903364814])\n",
"(si, coef(ols)) = (\"NY\", [-0.20048638474403055, 0.191535092167981])\n"
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}
],
"source": [
"all_plots = Array{Plots.Plot}(undef,10)\n",
"for (i,si) in enumerate(states_of_interest)\n",
" curids = findall(Feb2020data[!,:StateName].==si)\n",
" data = DataFrame(X=float.(counts[curids]), Y=float.(sales[curids]))\n",
" ols = GLM.lm(@formula(Y ~ X), data) \n",
" all_plots[i] = scatter(counts[curids],sales[curids],markersize=2,\n",
" xlim=(0,500),ylim=(0,500),color=i,aspect_ratio=:equal,\n",
" legend=false,title=si)\n",
" @show si,coef(ols)\n",
" plot!(counts[curids],predict(ols),color=:black)\n",
"end\n",
"plot(all_plots...,layout=(2,5),size=(900,300))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(si, coef(ols)) = (\"CA\", [0.29595375979515304])\n",
"(si, coef(ols)) = (\"FL\", [0.16693529057344902])\n",
"(si, coef(ols)) = (\"IL\", [0.22291703276169347])\n",
"(si, coef(ols)) = (\"NJ\", [0.25556391864492345])\n",
"(si, coef(ols)) = (\"MI\", [0.2925436820147375])\n",
"(si, coef(ols)) = (\"TX\", [0.24732835944836878])\n",
"(si, coef(ols)) = (\"PA\", [0.310066636803275])\n",
"(si, coef(ols)) = (\"OH\", [0.4569949754409864])\n",
"(si, coef(ols)) = (\"NC\", [0.4800747758857476])\n",
"(si, coef(ols)) = (\"NY\", [0.19151984869233082])\n"
]
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},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot()\n",
"for (i,si) in enumerate(states_of_interest)\n",
" curids = findall(Feb2020data[!,:StateName].==si)\n",
" data = DataFrame(X=float.(counts[curids]), Y=float.(sales[curids]))\n",
" ols = GLM.lm(@formula(Y ~ 0 + X), data) \n",
" scatter!(counts[curids],sales[curids],markersize=2,\n",
" xlim=(0,500),ylim=(0,500),color=i,aspect_ratio=:equal,\n",
" legend=false,marker=(3,3,stroke(0)),alpha=0.2)\n",
" if si == \"NC\" || si == \"CA\" || si == \"FL\"\n",
" annotate!([(500-20,10+coef(ols)[1]*500,text(si,10))])\n",
" end\n",
" @show si,coef(ols)\n",
" plot!(counts[curids],predict(ols),color=i,linewidth=2)\n",
"end\n",
"# plot(all_plots...,layout=(2,5),size=(900,300))\n",
"xlabel!(\"listings\")\n",
"ylabel!(\"sales\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---- \n",
"### 🟠 Logistic regression\n",
"So far, we have shown several ways to solve the linear regression problem in Julia. Here, we will first start with a motivating example of when you would want to use logistic regression. Let's assume that our predictor vector is binary (`0` or `1`), let's fit a linear regression model."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
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]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = DataFrame(X=[1,2,3,4,5,6,7], Y=[1,0,1,1,1,1,1])\n",
"linear_reg = lm(@formula(Y ~ X), data)\n",
"scatter(data[!,:X],data[!,:Y],legend=false,size=(300,200))\n",
"plot!(1:7,predict(linear_reg))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What this plot quickly shows is that linear regression may end up predicting values outside the `[0,1]` interval. For an example like this, we will use logistic regression. Interestingly, a generalized linear model (https://en.wikipedia.org/wiki/Generalized_linear_model) unifies concepts like linear regression and logistic regression, and the `GLM` package allows you to apply either of these regressions easily by specifying the `distribution family` and the `link` function. \n",
"\n",
"To apply logistic regression via the `GLM` package, you can readily use the `Binomial()` family and the `LogitLink()` link function. \n",
"\n",
"Let's load some data and take a look at one example."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"data-frame\"><thead><tr><th></th><th>Sex</th><th>BWt</th><th>HWt</th></tr><tr><th></th><th>Cat…</th><th>Float64</th><th>Float64</th></tr></thead><tbody><p>144 rows × 3 columns</p><tr><th>1</th><td>F</td><td>2.0</td><td>7.0</td></tr><tr><th>2</th><td>F</td><td>2.0</td><td>7.4</td></tr><tr><th>3</th><td>F</td><td>2.0</td><td>9.5</td></tr><tr><th>4</th><td>F</td><td>2.1</td><td>7.2</td></tr><tr><th>5</th><td>F</td><td>2.1</td><td>7.3</td></tr><tr><th>6</th><td>F</td><td>2.1</td><td>7.6</td></tr><tr><th>7</th><td>F</td><td>2.1</td><td>8.1</td></tr><tr><th>8</th><td>F</td><td>2.1</td><td>8.2</td></tr><tr><th>9</th><td>F</td><td>2.1</td><td>8.3</td></tr><tr><th>10</th><td>F</td><td>2.1</td><td>8.5</td></tr><tr><th>11</th><td>F</td><td>2.1</td><td>8.7</td></tr><tr><th>12</th><td>F</td><td>2.1</td><td>9.8</td></tr><tr><th>13</th><td>F</td><td>2.2</td><td>7.1</td></tr><tr><th>14</th><td>F</td><td>2.2</td><td>8.7</td></tr><tr><th>15</th><td>F</td><td>2.2</td><td>9.1</td></tr><tr><th>16</th><td>F</td><td>2.2</td><td>9.7</td></tr><tr><th>17</th><td>F</td><td>2.2</td><td>10.9</td></tr><tr><th>18</th><td>F</td><td>2.2</td><td>11.0</td></tr><tr><th>19</th><td>F</td><td>2.3</td><td>7.3</td></tr><tr><th>20</th><td>F</td><td>2.3</td><td>7.9</td></tr><tr><th>21</th><td>F</td><td>2.3</td><td>8.4</td></tr><tr><th>22</th><td>F</td><td>2.3</td><td>9.0</td></tr><tr><th>23</th><td>F</td><td>2.3</td><td>9.0</td></tr><tr><th>24</th><td>F</td><td>2.3</td><td>9.5</td></tr><tr><th>25</th><td>F</td><td>2.3</td><td>9.6</td></tr><tr><th>26</th><td>F</td><td>2.3</td><td>9.7</td></tr><tr><th>27</th><td>F</td><td>2.3</td><td>10.1</td></tr><tr><th>28</th><td>F</td><td>2.3</td><td>10.1</td></tr><tr><th>29</th><td>F</td><td>2.3</td><td>10.6</td></tr><tr><th>30</th><td>F</td><td>2.3</td><td>11.2</td></tr><tr><th>⋮</th><td>⋮</td><td>⋮</td><td>⋮</td></tr></tbody></table>"
],
"text/latex": [
"\\begin{tabular}{r|ccc}\n",
"\t& Sex & BWt & HWt\\\\\n",
"\t\\hline\n",
"\t& Cat… & Float64 & Float64\\\\\n",
"\t\\hline\n",
"\t1 & F & 2.0 & 7.0 \\\\\n",
"\t2 & F & 2.0 & 7.4 \\\\\n",
"\t3 & F & 2.0 & 9.5 \\\\\n",
"\t4 & F & 2.1 & 7.2 \\\\\n",
"\t5 & F & 2.1 & 7.3 \\\\\n",
"\t6 & F & 2.1 & 7.6 \\\\\n",
"\t7 & F & 2.1 & 8.1 \\\\\n",
"\t8 & F & 2.1 & 8.2 \\\\\n",
"\t9 & F & 2.1 & 8.3 \\\\\n",
"\t10 & F & 2.1 & 8.5 \\\\\n",
"\t11 & F & 2.1 & 8.7 \\\\\n",
"\t12 & F & 2.1 & 9.8 \\\\\n",
"\t13 & F & 2.2 & 7.1 \\\\\n",
"\t14 & F & 2.2 & 8.7 \\\\\n",
"\t15 & F & 2.2 & 9.1 \\\\\n",
"\t16 & F & 2.2 & 9.7 \\\\\n",
"\t17 & F & 2.2 & 10.9 \\\\\n",
"\t18 & F & 2.2 & 11.0 \\\\\n",
"\t19 & F & 2.3 & 7.3 \\\\\n",
"\t20 & F & 2.3 & 7.9 \\\\\n",
"\t21 & F & 2.3 & 8.4 \\\\\n",
"\t22 & F & 2.3 & 9.0 \\\\\n",
"\t23 & F & 2.3 & 9.0 \\\\\n",
"\t24 & F & 2.3 & 9.5 \\\\\n",
"\t25 & F & 2.3 & 9.6 \\\\\n",
"\t26 & F & 2.3 & 9.7 \\\\\n",
"\t27 & F & 2.3 & 10.1 \\\\\n",
"\t28 & F & 2.3 & 10.1 \\\\\n",
"\t29 & F & 2.3 & 10.6 \\\\\n",
"\t30 & F & 2.3 & 11.2 \\\\\n",
"\t$\\dots$ & $\\dots$ & $\\dots$ & $\\dots$ \\\\\n",
"\\end{tabular}\n"
],
"text/plain": [
"144×3 DataFrame\n",
"│ Row │ Sex │ BWt │ HWt │\n",
"│ │ \u001b[90mCat…\u001b[39m │ \u001b[90mFloat64\u001b[39m │ \u001b[90mFloat64\u001b[39m │\n",
"├─────┼──────┼─────────┼─────────┤\n",
"│ 1 │ F │ 2.0 │ 7.0 │\n",
"│ 2 │ F │ 2.0 │ 7.4 │\n",
"│ 3 │ F │ 2.0 │ 9.5 │\n",
"│ 4 │ F │ 2.1 │ 7.2 │\n",
"│ 5 │ F │ 2.1 │ 7.3 │\n",
"│ 6 │ F │ 2.1 │ 7.6 │\n",
"│ 7 │ F │ 2.1 │ 8.1 │\n",
"│ 8 │ F │ 2.1 │ 8.2 │\n",
"│ 9 │ F │ 2.1 │ 8.3 │\n",
"│ 10 │ F │ 2.1 │ 8.5 │\n",
"⋮\n",
"│ 134 │ M │ 3.5 │ 15.7 │\n",
"│ 135 │ M │ 3.5 │ 17.2 │\n",
"│ 136 │ M │ 3.6 │ 11.8 │\n",
"│ 137 │ M │ 3.6 │ 13.3 │\n",
"│ 138 │ M │ 3.6 │ 14.8 │\n",
"│ 139 │ M │ 3.6 │ 15.0 │\n",
"│ 140 │ M │ 3.7 │ 11.0 │\n",
"│ 141 │ M │ 3.8 │ 14.8 │\n",
"│ 142 │ M │ 3.8 │ 16.8 │\n",
"│ 143 │ M │ 3.9 │ 14.4 │\n",
"│ 144 │ M │ 3.9 │ 20.5 │"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# we will load this data from RDatasets\n",
"cats = dataset(\"MASS\", \"cats\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We will map the sex of each cat to a binary 0/1 value."
]
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"execution_count": 21,
"metadata": {},
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}
],
"source": [
"lmap = labelmap(cats[!,:Sex])\n",
"ci = labelencode(lmap, cats[!,:Sex])\n",
"scatter(cats[!,:BWt],cats[!,:HWt],color=ci,legend=false)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"LabelMap (with 2 labels):\n",
"[1] F\n",
"[2] M\n"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lmap"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Females (color 1) seem to be more present in the lower left corner and Males (color 2) seem to be present in the top right corner. Let's run a logistic regression model on this data."
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
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},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = DataFrame(X=cats[!,:HWt], Y=ci.-1)\n",
"probit = glm(@formula(Y ~ X), data, Binomial(), LogitLink())\n",
"scatter(data[!,:X],data[!,:Y],label=\"ground truth gender\",color=6)\n",
"scatter!(data[!,:X],predict(probit),label=\"predicted gender\",color=7)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As you can see, contrary to the linear regression case, the predicted values do not go beyond 1."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"-----\n",
"### 🟠 Non linear regression\n",
"Finally, sometimes you may have a set of points and the goal is to fit a non-linear function (maybe a quadratic function, a cubic function, an exponential function...). The way we would solve such a problem is by minimizing the least square error between the fitted function and the observations we have. We will use the package `LsqFit` for this task. Note that this problem is usually modeled as a numerical optimizaiton problem.\n",
"\n",
"We will first set up our data."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
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"source": [
"xvals = 0:0.05:10\n",
"yvals = 1*exp.(-xvals*2) + 2*sin.(0.8*pi*xvals) + 0.15 * randn(length(xvals));\n",
"scatter(xvals,yvals,legend=false)"
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"Then, we set up the model with `model(x,p)`. The vector `p` is what to be estimated given a set of values `x`."
]
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"cell_type": "code",
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"data": {
"text/plain": [
"LsqFit.LsqFitResult{Vector{Float64}, Vector{Float64}, Matrix{Float64}, Vector{Float64}}([1.0415140237037126, 2.1693486431737714, 2.0181247909313385], [0.02752895291159496, -0.07389276157679525, -0.011588736447885939, 0.07005887221187113, 0.13545559451970735, 0.20377674694992653, -0.3816295524295559, 0.055818537181232086, -0.24105254601566406, 0.03344957136175175 … 0.17882293221686463, 0.27231195003464026, 0.42863922028744716, -0.1488432806781279, 0.11540485560759861, 0.09837732624408746, -0.24814190503014721, 0.08351719361346754, -0.0945998664033223, -0.028474437883129682], [1.0000000000063065 0.0 0.0; 0.8972081144921082 -0.046722741661822303 0.12533323356031575; … ; 4.2248335930741175e-10 -4.377880403761785e-9 -0.12533323356258697; 3.7902478458698786e-10 -3.947381787803067e-9 0.0], true, Float64[])"
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"source": [
"@. model(x, p) = p[1]*exp(-x*p[2]) + p[3]*sin(0.8*pi*x)\n",
"p0 = [0.5, 0.5, 0.5]\n",
"myfit = curve_fit(model, xvals, yvals, p0)"
]
},
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"metadata": {},
"source": [
"⚠️ A note about `curve_fit`: this function can take multiple other inputs, for instance the Jacobian of what you are trying to fit. We don't dive into these details here, but be sure to check out the `LsqFit` package to see what other things you can can pass to create a better fit.\n",
"\n",
"Also note that julia has multiple packages that allow you to create Jacobians so you don't have to write them yourself. Two such packages are `FiniteDifferences` or `ForwardDiff`.\n",
"\n",
"↪️ Back to our example. We are ready now to plot the curve we have generated"
]
},
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},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"p = myfit.param\n",
"findyvals = p[1]*exp.(-xvals*p[2]) + p[3]*sin.(0.8*pi*xvals)\n",
"scatter(xvals,yvals,legend=false)\n",
"plot!(xvals,findyvals)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Just for fun... Let's try to fit a linear function"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
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]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@. model(x, p) = p[1]*x\n",
"myfit = curve_fit(model, xvals, yvals, [0.5])\n",
"p = myfit.param\n",
"findyvals = p[1]*xvals\n",
"scatter(xvals,yvals,legend=false)\n",
"plot!(xvals,findyvals)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Finally...\n",
"After finishing this notebook, you should be able to:\n",
"- [ ] run a linear regression model\n",
"- [ ] use the GLM package to pass functions and probability distributions to solve your special regression problem\n",
"- [ ] use GLM to solve a logistic regression problem\n",
"- [ ] fit a nonlinear regression to your data using the LsqFit package\n",
"- [ ] use the LsqFit package to fit a linear function too"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 🥳 One cool finding\n",
"\n",
"One metric in real estate is to find the number of houses being sold out of the number of houses on the market. We collect multiple data points from multiple states and fit a linear model to these states. It turns out that North Carolina has the highest sold/listed ratios. Florida is one of the least, and California is somewhere in between.\n",
"\n",
"<img src=\"data/0701.png\" width=\"400\">\n"
]
}
],
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