Ml regression in MATLAB

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class="--welcome"> <div class="--welcome-body">  <div class="--title"> <div class="--body"> <div class="nav-breadcrumb-container"> <div> <div class="breadcrumb-nav"> <a href="/matlab"> MATLAB<sup>®</sup> </a> > <a href="/matlab/#ai-ml">Artificial Intelligence and Machine Learning</a> > <span>ML Regression</span> </div> </div> <div class="nav-breadcrumb-right"> <div class="--fork"> <a id="forklink" href= "https://github.com/plotly/graphing-library-docs/edit/master/_posts/matlab/md/2021-08-04-ml-regression.md" > <div class="icon"> <svg style="width:20px;height:20px" viewbox="0 0 24 24"> <path fill="#000000" d="M2.6,10.59L8.38,4.8L10.07,6.5C9.83,7.35 10.22,8.28 11,8.73V14.27C10.4,14.61 10,15.26 10,16A2,2 0 0,0 12,18A2,2 0 0,0 14,16C14,15.26 13.6,14.61 13,14.27V9.41L15.07,11.5C15,11.65 15,11.82 15,12A2,2 0 0,0 17,14A2,2 0 0,0 19,12A2,2 0 0,0 17,10C16.82,10 16.65,10 16.5,10.07L13.93,7.5C14.19,6.57 13.71,5.55 12.78,5.16C12.35,5 11.9,4.96 11.5,5.07L9.8,3.38L10.59,2.6C11.37,1.81 12.63,1.81 13.41,2.6L21.4,10.59C22.19,11.37 22.19,12.63 21.4,13.41L13.41,21.4C12.63,22.19 11.37,22.19 10.59,21.4L2.6,13.41C1.81,12.63 1.81,11.37 2.6,10.59Z"> </path> </svg> </div> <span>Suggest an edit to this page</span> </a> </div> </div> </div> <h1> ML Regression in MATLAB<sup>®</sup> </h1> <p>How to make ML Regression plots in MATLAB<sup>®</sup> with Plotly. </p> <br>  </div> </div> </div> </header>  <section class="tutorial-content"> <h2>Simple Linear Regression</h2> <p>This example shows how to perform simple linear regression using the accidents dataset. The example also shows you how to calculate the coefficient of determination R<sup>2</sup> to evaluate the regressions. The accidents dataset contains data for fatal traffic accidents in U.S. states.</p> <p>Linear regression models the relation between a dependent, or response, variable y and one or more independent, or predictor, variables x<sub>1</sub>,...,x<sub>n</sub>. Simple linear regression considers only one independent variable using the relation</p> <ul> <li><code>y=β<sub>0</sub>+β<sub>1</sub>x+ϵ,</code></li> </ul> <p>where β<sub>0</sub> is the y-intercept, β<sub>1</sub> is the slope (or regression coefficient), and ϵ is the error term. This can be simplified to <code>Y=XB</code></p> <p>From the dataset accidents, load accident data in y and state population data in x. Find the linear regression relation y=β<sub>1</sub>x between the accidents in a state and the population of a state using the \ operator. The \ operator performs a least-squares regression.</p> <div class="highlight"><pre><code class="language-matlab" data-lang="matlab"><span class="nb">load</span> <span class="n">accidents</span> <span class="n">x</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">14</span><span class="p">);</span> <span class="c1">%Population of states</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">4</span><span class="p">);</span> <span class="c1">%Accidents per state</span> <span class="nb">format</span> <span class="n">long</span> <span class="n">b1</span> <span class="o">=</span> <span class="n">x</span><span class="p">\</span><span class="n">y</span> </code></pre></div> <pre class="code-output"> b1 = 1.372716735564871e-04 </pre> <p>b1 is the slope or regression coefficient. The linear relation is y=β<sub>1</sub>x=0.0001372x.</p> <p>Calculate the accidents per state yCalc from x using the relation. Visualize the regression by plotting the actual values y and the calculated values yCalc.</p> <div class="highlight"><pre><code class="language-matlab" data-lang="matlab"><span class="nb">load</span> <span class="n">accidents</span> <span class="n">x</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">14</span><span class="p">);</span> <span class="c1">%Population of states</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">4</span><span class="p">);</span> <span class="c1">%Accidents per state</span> <span class="nb">format</span> <span class="n">long</span> <span class="n">b1</span> <span class="o">=</span> <span class="n">x</span><span class="p">\</span><span class="n">y</span><span class="p">;</span> <span class="n">yCalc1</span> <span class="o">=</span> <span class="n">b1</span><span class="o">*</span><span class="n">x</span><span class="p">;</span> <span class="nb">scatter</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">)</span> <span class="nb">hold</span> <span class="n">on</span> <span class="nb">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">yCalc1</span><span class="p">)</span> <span class="nb">xlabel</span><span class="p">(</span><span class="s1">'Population of state'</span><span class="p">)</span> <span class="nb">ylabel</span><span class="p">(</span><span class="s1">'Fatal traffic accidents per state'</span><span class="p">)</span> <span class="nb">title</span><span class="p">(</span><span class="s1">'Linear Regression Relation Between Accidents & Population'</span><span class="p">)</span> <span class="nb">grid</span> <span class="n">on</span> <span class="n">fig2plotly</span><span class="p">(</span><span class="nb">gcf</span><span class="p">);</span> </code></pre></div> <div id="plot-4707876" class="plotly-graph-div js-plotly-plot"></div> <script type = "text/javascript" > require(["plotly"], function(Plotly) { window.PLOTLYENV = window.PLOTLYENV || {}; if (document.getElementById("plot-4707876" )) { Plotly.newPlot("plot-4707876", {"data": [{"type": "scatter", "xaxis": "x1", "yaxis": "y1", "mode": "markers", "visible": true, "name": "", "x": [493782, 572059, 608827, 626932, 642200, 754844, 783600, 902195, 1048319, 1211537, 1235786, 1274923, 1293953, 1711263, 1808344, 1819046, 1998257, 2233169, 2673400, 2688418, 2844658, 2926324, 3405565, 3421399, 3450654, 4012012, 4041769, 4301261, 4447100, 4468976, 4919479, 5130632, 5296486, 5363675, 5595211, 5689283, 5894121, 6080485, 6349097, 7078515, 8049313, 8186453, 8414350, 9938444, 11353140, 12281054, 12419293, 15982378, 18976457, 20851820, 33871648], "y": [164, 43, 98, 101, 100, 197, 134, 229, 83, 142, 171, 194, 260, 254, 411, 521, 395, 296, 704, 461, 900, 390, 291, 456, 774, 1046, 964, 665, 1154, 904, 567, 1150, 643, 792, 1130, 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{"side": "left", "zeroline": false, "autorange": false, "linecolor": "rgb(38.250000,38.250000,38.250000)", "linewidth": 1, "exponentformat": "none", "tickfont": {"size": 10, "family": "Arial, sans-serif", "color": "rgb(38.250000,38.250000,38.250000)"}, "ticklen": 6.51, "tickcolor": "rgb(38.250000,38.250000,38.250000)", "tickwidth": 1, "tickangle": -0, "ticks": "inside", "showgrid": true, "gridwidth": 1, "gridcolor": "rgba(38.250000,,38.250000,38.250000,0.150000)", "type": "linear", "showticklabels": true, "tickmode": "array", "tickvals": [0, 500, 1000, 1500, 2000, 2500, 3000, 3500, 4000, 4500, 5000], "range": [0, 5000], "mirror": false, "ticktext": ["0", "500", "1000", "1500", "2000", "2500", "3000", "3500", "4000", "4500", "5000"], "title": "Fatal traffic accidents per state", "titlefont": {"color": "rgb(38.250000,38.250000,38.250000)", "size": 11, "family": "Arial, sans-serif"}, "showline": true, "domain": [0.11, 0.925], "anchor": "x1"}}, "frames": []} , {"responsive": true} ).then(function() { var gd = document.getElementById("plot-4707876"); var x = new MutationObserver(function(mutations, observer) { { var display = window.getComputedStyle(gd).display; if (!display || display === 'none') { { console.log([gd, 'removed!']); Plotly.purge(gd); observer.disconnect(); } } } }); // Listen for the removal of the full notebook cells var notebookContainer = gd.closest('#notebook-container'); if (notebookContainer) { { x.observe(notebookContainer, { childList: true }); } } // Listen for the clearing of the current output cell var outputEl = gd.closest('.output'); if (outputEl) { { x.observe(outputEl, { childList: true }); } } }) }; }); </script> <p>Improve the fit by including a y-intercept β<sub>0</sub> in your model as y=β<sub>0</sub>+β<sub>1</sub>x. Calculate β<sub>0</sub> by padding x with a column of ones and using the \ operator.</p> <div class="highlight"><pre><code class="language-matlab" data-lang="matlab"><span class="nb">load</span> <span class="n">accidents</span> <span class="n">x</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">14</span><span class="p">);</span> <span class="c1">%Population of states</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">4</span><span class="p">);</span> <span class="c1">%Accidents per state</span> <span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="nb">ones</span><span class="p">(</span><span class="nb">length</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mi">1</span><span class="p">)</span> <span class="n">x</span><span class="p">];</span> <span class="n">b</span> <span class="o">=</span> <span class="n">X</span><span class="p">\</span><span class="n">y</span> </code></pre></div> <pre class="code-output"> b = 1.0e+02 * 1.427120171726538 0.000001256394274 </pre> <p>This result represents the relation y=β<sub>0</sub>+β<sub>1</sub>x=142.7120+0.0001256x.</p> <p>Visualize the relation by plotting it on the same figure.</p> <div class="highlight"><pre><code class="language-matlab" data-lang="matlab"><span class="nb">load</span> <span class="n">accidents</span><span class="p">;</span> <span class="n">x</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">14</span><span class="p">);</span> <span class="c1">%Population of states</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">4</span><span class="p">);</span> <span class="c1">%Accidents per state</span> <span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="nb">ones</span><span class="p">(</span><span class="nb">length</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mi">1</span><span class="p">)</span> <span class="n">x</span><span class="p">];</span> <span class="n">b</span> <span class="o">=</span> <span class="n">X</span><span class="p">\</span><span class="n">y</span><span class="p">;</span> <span class="n">yCalc2</span> <span class="o">=</span> <span class="n">X</span><span class="o">*</span><span class="n">b</span><span class="p">;</span> <span class="nb">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">yCalc2</span><span class="p">,</span><span class="s1">'--'</span><span class="p">);</span> <span class="nb">legend</span><span class="p">({</span><span class="s1">'Data'</span><span class="p">},</span><span class="s1">'Location'</span><span class="p">,</span><span class="s1">'best'</span><span class="p">);</span> <span class="n">fig2plotly</span><span class="p">(</span><span class="nb">gcf</span><span class="p">);</span> </code></pre></div> <div id="plot-4954201" class="plotly-graph-div js-plotly-plot"></div> <script type = "text/javascript" > require(["plotly"], function(Plotly) { window.PLOTLYENV = window.PLOTLYENV || {}; if (document.getElementById("plot-4954201" )) { Plotly.newPlot("plot-4954201", {"data": [{"type": "scatter", "xaxis": "x1", "yaxis": "y1", "visible": true, "name": "Data", "mode": "lines", "x": [493782, 572059, 608827, 626932, 642200, 754844, 783600, 902195, 1048319, 1211537, 1235786, 1274923, 1293953, 1711263, 1808344, 1819046, 1998257, 2233169, 2673400, 2688418, 2844658, 2926324, 3405565, 3421399, 3450654, 4012012, 4041769, 4301261, 4447100, 4468976, 4919479, 5130632, 5296486, 5363675, 5595211, 5689283, 5894121, 6080485, 6349097, 7078515, 8049313, 8186453, 8414350, 9938444, 11353140, 12281054, 12419293, 15982378, 18976457, 20851820, 33871648], "y": 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"family": "Arial, sans-serif", "color": "rgb(38.250000,38.250000,38.250000)"}, "ticklen": 6.51, "tickcolor": "rgb(38.250000,38.250000,38.250000)", "tickwidth": 1, "tickangle": -0, "ticks": "inside", "showgrid": false, "gridcolor": "rgba(38.250000,,38.250000,38.250000,0.150000)", "type": "linear", "showticklabels": true, "tickmode": "array", "tickvals": [0, 500, 1000, 1500, 2000, 2500, 3000, 3500, 4000, 4500], "range": [0, 4500], "mirror": "ticks", "ticktext": ["0", "500", "1000", "1500", "2000", "2500", "3000", "3500", "4000", "4500"], "titlefont": {"color": "rgb(38.250000,38.250000,38.250000)", "size": 11, "family": "Arial, sans-serif"}, "showline": true, "domain": [0.11, 0.925], "anchor": "x1"}, "legend": {"x": 0.15125, "xref": "paper", "xanchor": "left", "y": 0.874606227106227, "yref": "paper", "yanchor": "bottom", "traceorder": "normal", "borderwidth": 0.5, "bordercolor": "rgb(38.25,38.25,38.25)", "bgcolor": "rgb(255,255,255)", "font": {"size": 9, "family": "Arial, sans-serif", "color": "rgb(0,0,0_)"}}}, "frames": []} , {"responsive": true} ).then(function() { var gd = document.getElementById("plot-4954201"); var x = new MutationObserver(function(mutations, observer) { { var display = window.getComputedStyle(gd).display; if (!display || display === 'none') { { console.log([gd, 'removed!']); Plotly.purge(gd); observer.disconnect(); } } } }); // Listen for the removal of the full notebook cells var notebookContainer = gd.closest('#notebook-container'); if (notebookContainer) { { x.observe(notebookContainer, { childList: true }); } } // Listen for the clearing of the current output cell var outputEl = gd.closest('.output'); if (outputEl) { { x.observe(outputEl, { childList: true }); } } }) }; }); </script> <p>If with to plot the data alongside the slope, you can do it in the following way.</p> <div class="highlight"><pre><code class="language-matlab" data-lang="matlab"><span class="nb">load</span> <span class="n">accidents</span> <span class="n">x</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">14</span><span class="p">);</span> <span class="c1">%Population of states</span> <span class="n">y</span> <span class="o">=</span> <span class="n">hwydata</span><span class="p">(:,</span><span class="mi">4</span><span class="p">);</span> <span class="c1">%Accidents per state</span> <span class="n">X</span> <span class="o">=</span> <span class="p">[</span><span class="nb">ones</span><span class="p">(</span><span class="nb">length</span><span class="p">(</span><span class="n">x</span><span class="p">),</span><span class="mi">1</span><span class="p">)</span> <span class="n">x</span><span class="p">];</span> <span class="n">b</span> <span class="o">=</span> <span class="n">X</span><span class="p">\</span><span class="n">y</span><span class="p">;</span> <span class="n">yCalc2</span> <span class="o">=</span> <span class="n">X</span><span class="o">*</span><span class="n">b</span><span class="p">;</span> <span class="nb">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">yCalc2</span><span class="p">,</span><span class="s1">'--'</span><span class="p">);</span> <span class="nb">legend</span><span class="p">({</span><span class="s1">'Data'</span><span class="p">},</span><span class="s1">'Location'</span><span class="p">,</span><span class="s1">'best'</span><span class="p">);</span> <span class="nb">hold</span> <span class="n">on</span> <span class="nb">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span><span class="s1">'o'</span><span class="p">);</span> <span class="n">fig2plotly</span><span class="p">(</span><span class="nb">gcf</span><span class="p">);</span> </code></pre></div> <div id="plot-5130942" class="plotly-graph-div js-plotly-plot"></div> <script type = "text/javascript" > require(["plotly"], function(Plotly) { window.PLOTLYENV = 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Ml regression in MATLAB