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<h1>Ridge Regression</h1>
<blockquote>
<p>Ridge Regression is a commonly used technique to address the problem of multi-collinearity. The effectiveness of the application is however debatable.</p>
</blockquote>
<h2>Introduction</h2>
<p>Let us see a use case of the application of Ridge regression on the <code>longley</code> dataset. We will try to predict the <code>GNP.deflator</code> using <code>lm()</code> with the rest of the variables as predictors. This model and results will be compared with the model created using ridge regression.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span> (car) <span class="co"># for VIF</span>
<span class="kw">library</span> (ridge)
<span class="kw">data</span>(longley, <span class="dt">package=</span><span class="st">"datasets"</span>) <span class="co"># initialize data</span>
<span class="kw">head</span> (longley, <span class="dv">4</span>) <span class="co"># show top 4 rows of data</span>
<span class="co">#> GNP.deflator GNP Unemployed Armed.Forces Population Year Employed</span>
<span class="co">#> 1947 83.0 234.289 235.6 159.0 107.608 1947 60.323</span>
<span class="co">#> 1948 88.5 259.426 232.5 145.6 108.632 1948 61.122</span>
<span class="co">#> 1949 88.2 258.054 368.2 161.6 109.773 1949 60.171</span>
<span class="co">#> 1950 89.5 284.599 335.1 165.0 110.929 1950 61.187</span></code></pre></div>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">inputData <-<span class="st"> </span><span class="kw">data.frame</span> (longley) <span class="co"># plug in your data here</span>
<span class="kw">colnames</span>(inputData)[<span class="dv">1</span>] <-<span class="st"> "response"</span> <span class="co"># rename response var</span></code></pre></div>
<h2>Calculate Correlations</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">XVars <-<span class="st"> </span>inputData[, -<span class="dv">1</span>] <span class="co"># X variables</span>
<span class="kw">round</span>(<span class="kw">cor</span>(XVars), <span class="dv">2</span>) <span class="co"># Correlation Test</span>
<span class="co">#> GNP Unemployed Armed.Forces Population Year Employed</span>
<span class="co">#> GNP 1.00 0.60 0.45 0.99 1.00 0.98</span>
<span class="co">#> Unemployed 0.60 1.00 -0.18 0.69 0.67 0.50</span>
<span class="co">#> Armed.Forces 0.45 -0.18 1.00 0.36 0.42 0.46</span>
<span class="co">#> Population 0.99 0.69 0.36 1.00 0.99 0.96</span>
<span class="co">#> Year 1.00 0.67 0.42 0.99 1.00 0.97</span>
<span class="co">#> Employed 0.98 0.50 0.46 0.96 0.97 1.00</span></code></pre></div>
<h2>Prepare Training And Test Data</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">set.seed</span>(<span class="dv">100</span>) <span class="co"># set seed to replicate results</span>
trainingIndex <-<span class="st"> </span><span class="kw">sample</span>(<span class="dv">1</span>:<span class="kw">nrow</span>(inputData), <span class="fl">0.8</span>*<span class="kw">nrow</span>(inputData)) <span class="co"># indices for 80% training data</span>
trainingData <-<span class="st"> </span>inputData[trainingIndex, ] <span class="co"># training data</span>
testData <-<span class="st"> </span>inputData[-trainingIndex, ] <span class="co"># test data</span></code></pre></div>
<h2>Predict Using Linear Regression</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">lmMod <-<span class="st"> </span><span class="kw">lm</span>(response ~<span class="st"> </span>., trainingData) <span class="co"># the linear reg model</span>
<span class="kw">summary</span> (lmMod) <span class="co"># get summary</span>
<span class="kw">vif</span>(lmMod) <span class="co"># get VIF</span>
<span class="co">#> VIF</span>
<span class="co">#> GNP Unemployed Armed.Forces Population Year Employed </span>
<span class="co">#> 1523.74714 93.07635 10.74587 350.58472 2175.29221 182.93609</span>
<span class="co">#> Coefficients:</span>
<span class="co">#> (Intercept) GNP Unemployed Armed.Forces Population Year Employed</span>
<span class="co">#> 7652.25192 0.39214 0.06462 0.01573 -2.33550 -3.83113 0.53060</span></code></pre></div>
<p>There is significant multi-collinearity between GNP & Year and Population & Employed, with negative coefficients in ‘population’ and ‘Employed’. These variables may not contribute much to explain the dependent variable, nevertheless, lets see what this model predicts.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">predicted <-<span class="st"> </span><span class="kw">predict</span> (lmMod, testData) <span class="co"># predict on test data</span>
compare <-<span class="st"> </span><span class="kw">cbind</span> (<span class="dt">actual=</span>testData$response, predicted) <span class="co"># combine actual and predicted</span>
<span class="co">#> actual predicted</span>
<span class="co">#> 1949 88.2 88.45501</span>
<span class="co">#> 1953 99.0 96.67492</span>
<span class="co">#> 1957 108.4 106.59672</span>
<span class="co">#> 1959 112.6 113.31106</span>
<span class="kw">mean</span> (<span class="kw">apply</span>(compare, <span class="dv">1</span>, min)/<span class="kw">apply</span>(compare, <span class="dv">1</span>, max)) <span class="co"># calculate accuracy</span>
<span class="co">#> 98.76%</span></code></pre></div>
<h2>Apply Ridge Regression On Same Data</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">linRidgeMod <-<span class="st"> </span><span class="kw">linearRidge</span>(response ~<span class="st"> </span>., <span class="dt">data =</span> trainingData) <span class="co"># the ridge regression model</span>
<span class="co">#> No more Negative Coefficients!</span>
<span class="co">#> (Intercept) GNP Unemployed Armed.Forces Population Year Employed</span>
<span class="co">#> -1.015385e+03 3.715498e-02 1.328002e-02 1.707769e-02 1.294903e-01 5.318930e-01 5.976266e-01</span>
predicted <-<span class="st"> </span><span class="kw">predict</span>(linRidgeMod, testData) <span class="co"># predict on test data</span>
compare <-<span class="st"> </span><span class="kw">cbind</span> (<span class="dt">actual=</span>testData$response, predicted) <span class="co"># combine</span>
<span class="co">#> actual predicted</span>
<span class="co">#> 1949 88.2 88.68584</span>
<span class="co">#> 1953 99.0 99.26104</span>
<span class="co">#> 1957 108.4 106.99370</span>
<span class="co">#> 1959 112.6 110.95450</span>
<span class="kw">mean</span> (<span class="kw">apply</span>(compare, <span class="dv">1</span>, min)/<span class="kw">apply</span>(compare, <span class="dv">1</span>, max)) <span class="co"># calculate accuracy</span>
<span class="co">#> 99.10%</span></code></pre></div>
<p>Clearly, in this case, ridge regression is successful in improving the accuracy by a minor but significant fraction.</p>
<h2>Predicting With A Re-calibrated Linear Model</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">newlmMod <-<span class="st"> </span><span class="kw">lm</span>(response ~<span class="st"> </span>., trainingData[, -<span class="kw">c</span>(<span class="dv">2</span>, <span class="dv">5</span>, <span class="dv">6</span>)]) <span class="co"># without "GNP", "Population" & "Year"</span>
<span class="kw">summary</span> (newlmMod) <span class="co"># get summary</span>
<span class="kw">vif</span>(newlmMod) <span class="co"># get VIF</span>
<span class="co">#> Coefficients:</span>
<span class="co">#> (Intercept) Unemployed Armed.Forces Employed </span>
<span class="co">#> -62.19771 0.03248 0.02714 2.24039 </span>
<span class="co">#> VIF</span>
<span class="co">#> Unemployed Armed.Forces Employed </span>
<span class="co">#> 2.124153 1.452648 2.592474</span>
predicted <-<span class="st"> </span><span class="kw">predict</span>(newlmMod, testData) <span class="co"># predict on test data</span>
compare <-<span class="st"> </span><span class="kw">cbind</span> (<span class="dt">actual=</span>testData$response, predicted) <span class="co"># for comparison</span>
<span class="kw">mean</span> (<span class="kw">apply</span>(compare, <span class="dv">1</span>, min)/<span class="kw">apply</span>(compare, <span class="dv">1</span>, max)) <span class="co"># calculate accuracy</span>
<span class="co">#> 99.21%</span></code></pre></div>
<p>The re-calibrated linear model yields better accuracy when the multicollinearity is taken care of. This analysis may not be sufficient to draw conclusions about the effectiveness of ridge regression. The intention, however, is to open up considerations for new modeling options for problem solving.</p>
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