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326 lines
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326 lines
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HTML
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<!-- Mirrored from www.mathsisfun.com/data/least-squares-regression.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:31:59 GMT -->
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<title>Least Squares Regression</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 class="center">Least Squares Regression</h1>
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<h2>Line of Best Fit</h2>
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<p>Imagine you have some points, and want to have a <b>line</b> that best fits them like this:</p>
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<p class="center"><img src="images/scatter-ice-cream1a.svg" alt="scatter plot ice cream vs temp with line" style="max-width:100%" /><br>
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</p>
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<p>We can place the line "by eye": try to have the line as close as possible to all points, and a similar number of points above and below the line.</p>
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<p>But for better accuracy let's see how to calculate the line using <b>Least Squares Regression</b>.</p>
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<h2>The Line</h2>
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<p>Our aim is to calculate the values <b>m</b> (slope) and <b>b</b> (y-intercept) in the <a href="../equation_of_line.html">equation of a line</a> :</p>
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<div class="center80">
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<div align="center"><span class="largest">y = mx + b</span></div>
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</div>
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<p>Where: </p>
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<ul>
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<li><b class="larger">y</b> = how far up</li>
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<li><b class="larger">x</b> = how far along</li>
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<li><b class="larger">m</b> = <a href="../geometry/slope.html">Slope</a> or <a href="../gradient.html">Gradient</a> (how steep the line is)</li>
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<li><b class="larger">b</b> = the <a href="../y_intercept.html">Y Intercept</a> (where the line crosses the Y axis)</li>
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</ul>
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<h2>Steps</h2>
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<p>To find the line of best fit for <b>N</b> points:</p>
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<div class="def">
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<p><b>Step 1</b>: For each (x,y) point calculate x<sup>2</sup> and xy</p>
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<p><b>Step 2</b>: Sum all x, y, x<sup>2</sup> and xy, which gives us Σx, Σy, Σx<sup>2</sup> and Σxy (<a href="../algebra/sigma-notation.html">Σ means "sum up"</a>)</p>
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<p><b>Step 3</b>: Calculate Slope <b>m</b>:</p>
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<p class="center larger"><b>m</b> = <span class="intbl"> <em>N Σ(xy) − Σx Σy</em> <strong>N Σ(x<sup>2</sup>) − (Σx)<sup>2</sup></strong></span></p>
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<p>(N is the number of points.)</p>
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<p><b>Step 4</b>: Calculate Intercept <b>b</b>:</p>
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<p class="center larger"><b>b</b> = <span class="intbl"> <em>Σy − m Σx</em> <strong>N</strong></span></p>
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<p><b>Step 5</b>: Assemble the equation of a line</p>
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<p class="center large">y = mx + b</p>
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<p>Done! </p>
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</div>
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<h2>Example </h2>
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<p>Let's have an example to see how to do it!</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../combinatorics/images/ice-cream.jpg" width="100" height="133" alt="ice cream"></p>
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<h3>Example: Sam found how many <b>hours of sunshine</b> vs how many <b>ice creams</b> were sold at the shop from Monday to Friday:</h3>
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<div class="simple">
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<table border="0" align="center">
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<tr align="right">
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<th width="100">"x"<br>
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Hours of Sunshine</th>
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<th width="100">"y"<br>
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Ice Creams Sold</th>
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</tr>
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<tr>
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<td align="right">2</td>
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<td align="right">4</td>
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</tr>
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<tr>
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<td align="right">3</td>
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<td align="right">5</td>
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</tr>
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<tr>
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<td align="right">5</td>
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<td align="right">7</td>
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</tr>
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<tr>
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<td align="right">7</td>
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<td align="right">10</td>
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</tr>
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<tr>
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<td align="right">9</td>
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<td align="right">15</td>
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</tr>
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</table></div>
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<p>Let us find the best <b>m</b> (slope) and <b>b</b> (y-intercept) that suits that data</p>
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<p class="center larger">y = mx + b</p>
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<p> </p>
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<p><b>Step 1</b>: For each (x,y) calculate x<sup>2</sup> and xy:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="right">
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<th width="70">x</th>
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<th width="70">y</th>
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<th width="70">x<sup>2</sup></th>
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<th width="70">xy</th>
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</tr>
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<tr>
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<td width="70" align="right">2</td>
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<td width="70" align="right">4</td>
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<td width="70" align="right">4</td>
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<td width="70" align="right">8</td>
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</tr>
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<tr>
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<td width="70" align="right">3</td>
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<td width="70" align="right">5</td>
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<td width="70" align="right">9</td>
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<td width="70" align="right">15</td>
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</tr>
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<tr>
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<td width="70" align="right">5</td>
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<td width="70" align="right">7</td>
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<td width="70" align="right">25</td>
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<td width="70" align="right">35</td>
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</tr>
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<tr>
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<td width="70" align="right">7</td>
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<td width="70" align="right">10</td>
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<td width="70" align="right">49</td>
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<td width="70" align="right">70</td>
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</tr>
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<tr>
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<td width="70" align="right">9</td>
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<td width="70" align="right">15</td>
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<td width="70" align="right">81</td>
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<td width="70" align="right">135</td>
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</tr>
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</table>
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</div>
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<p><b>Step 2</b>: Sum x, y, x<sup>2</sup> and xy (gives us Σx, Σy, Σx<sup>2</sup> and Σxy):</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="right">
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<th width="90">x</th>
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<th width="90">y</th>
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<th width="90">x<sup>2</sup></th>
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<th width="90">xy</th>
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</tr>
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<tr>
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<td align="right">2</td>
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<td align="right">4</td>
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<td align="right">4</td>
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<td align="right">8</td>
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</tr>
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<tr>
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<td align="right">3</td>
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<td align="right">5</td>
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<td align="right">9</td>
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<td align="right">15</td>
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</tr>
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<tr>
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<td align="right">5</td>
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<td align="right">7</td>
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<td align="right">25</td>
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<td align="right">35</td>
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</tr>
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<tr>
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<td align="right">7</td>
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<td align="right">10</td>
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<td align="right">49</td>
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<td align="right">70</td>
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</tr>
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<tr>
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<td align="right">9</td>
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<td align="right">15</td>
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<td align="right">81</td>
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<td align="right">135</td>
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</tr>
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<tr align="right">
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<td><b>Σx: 26</b></td>
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<td><b>Σy: 41</b></td>
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<td><b>Σx<sup>2</sup>: 168</b></td>
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<td><b>Σxy: 263</b></td>
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</tr>
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</table>
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</div>
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<p class="center">Also <b>N</b> (number of data values) = <b>5 </b></p>
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<p><b>Step 3</b>: Calculate Slope <b>m</b>:</p>
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<p class="center larger"><b>m</b> = <span class="intbl"><em>N Σ(xy) − Σx Σy</em>
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<strong>N Σ(x<sup>2</sup>) − (Σx)<sup>2</sup></strong></span></p>
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<p class="center larger"> = <span class="intbl"> <em>5 x 263 − 26 x 41</em> <strong>5 x 168 − 26<sup>2</sup></strong></span></p>
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<p class="center larger"> = <span class="intbl"> <em>1315 − 1066</em> <strong>840 − 676</strong></span></p>
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<p class="center larger"> = <span class="intbl"> <em>249</em> <strong>164</strong></span> = 1.5183...</p>
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<p><b>Step 4</b>: Calculate Intercept <b>b</b>:</p>
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<p class="center larger"><b>b</b> = <span class="intbl">
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<em>Σy − m Σx</em>
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<strong>N</strong>
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</span></p>
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<p class="center larger">= <span class="intbl">
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<em>41 − 1.5183 x 26</em>
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<strong>5</strong>
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</span></p>
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<p class="center larger">= 0.3049...</p>
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<p><b>Step 5</b>: Assemble the equation of a line:</p>
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<p class="center large">y = mx + b</p>
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<p class="center large">y = 1.518x + 0.305</p>
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<p>Let's see how it works out:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr align="right">
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<th width="70">x</th>
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<th width="70">y</th>
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<th>y = 1.518x + 0.305</th>
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<th width="70">error</th>
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</tr>
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<tr>
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<td width="70" align="right">2</td>
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<td width="70" align="right">4</td>
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<td align="right">3.34</td>
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<td align="right">−0.66</td>
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</tr>
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<tr>
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<td width="70" align="right">3</td>
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<td width="70" align="right">5</td>
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<td align="right">4.86</td>
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<td align="right">−0.14</td>
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</tr>
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<tr>
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<td width="70" align="right">5</td>
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<td width="70" align="right">7</td>
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<td align="right">7.89</td>
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<td align="right">0.89</td>
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</tr>
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<tr>
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<td width="70" align="right">7</td>
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<td width="70" align="right">10</td>
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<td align="right">10.93</td>
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<td align="right">0.93</td>
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</tr>
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<tr>
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<td width="70" align="right">9</td>
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<td width="70" align="right">15</td>
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<td align="right">13.97</td>
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<td align="right">−1.03</td>
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</tr>
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</table>
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</div>
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<p>Here are the (x,y) points and the line <b>y = 1.518x + 0.305</b> on a graph:</p>
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<p class="center"><img src="images/least-squares-example.gif" width="235" height="330" alt="least squares example graph"></p>
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<p class="center">Nice fit!</p>
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<p>Sam hears the weather forecast which says "we expect 8 hours of sun tomorrow", so he uses the above equation to estimate that he will sell</p>
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<p class="center large">y = 1.518 x 8 + 0.305 = 12.45 Ice Creams</p>
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<p>Sam makes fresh waffle cone mixture for 14 ice creams just in case. Yum.</p>
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</div>
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<h2>How does it work?</h2>
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<p>It works by making the total of the <b>square of the errors</b> as small as possible (that is why it is called "least squares"):</p>
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<p class="center"><img src="images/least-squares2.svg" alt="least squares example graph" ><br>
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The straight line minimizes the sum of squared errors</p>
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<p>So, when we square each of those errors and add them all up, the total is as small as possible. </p>
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<p>You can <b>imagine</b> (but not accurately) each data point connected to a straight bar by springs:</p>
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<p class="center"><img src="images/least-squares-springs.svg" alt="least squares example graph" ><br>
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Boing! </p>
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<h2>Outliers</h2>
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<p>Be careful! Least squares is sensitive to <a href="outliers.html">outliers</a>. A strange value will pull the line towards it.</p>
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<h2>Use the App</h2>
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<p>Have a play with the <a href="least-squares-calculator.html">Least Squares Calculator</a></p>
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<h2>Not Just For Lines</h2>
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<p>This <b>idea</b> can be used in many other areas, not just lines.</p>
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<p class="center"><img src="images/least-squares-circle.svg" alt="least squares circle graph" ><br>
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A "circle of best fit"</p>
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<p>But the formulas (and the steps taken) will be very different!</p>
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<p> </p>
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<p> </p>
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<div class="related">
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<a href="scatter-xy-plots.html">Scatter (x,y) Graphs</a> <a href="../equation_of_line.html">Equation of a Line</a> <a href="least-squares-calculator.html">Least Squares Calculator</a> <a href="index.html">Data Index</a>
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