Pca visualization in MATLAB

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observations for 4 variables. Find the principal components for the ingredients data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald coeff = pca(ingredients) </code></pre></div> <pre class="code-output"> coeff = Columns 1 through 3 -0.067799985695474 -0.646018286568728 0.567314540990512 -0.678516235418647 -0.019993340484099 -0.543969276583817 0.029020832106229 0.755309622491133 0.403553469172668 0.730873909451461 -0.108480477171676 -0.468397518388289 Column 4 0.506179559977705 0.493268092159297 0.515567418476836 0.484416225289198 </pre> The rows of <code>coeff</code> contain the coefficients for the four ingredient variables, and its columns correspond to four principal components. <h2>PCA in the Presence of Missing Data</h2> Find the principal component coefficients when there are missing values in a data set. Load the sample data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 </code></pre></div> Data matrix <code>X</code> has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg. The variables bore and stroke are missing four values in rows 56 to 59, and the variables horsepower and peak-rpm are missing two values in rows 131 and 132. Perform principal component analysis. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 coeff = pca(X(:,3:15)); </code></pre></div> By default, <code>pca</code> performs the action specified by the <code>'Rows','complete'</code> name-value pair argument. This option removes the observations with <code>NaN</code> values before calculation. Rows of <code>NaN</code>s are reinserted into <code>score</code> and <code>tsquared</code> at the corresponding locations, namely rows 56 to 59, 131, and 132. Use <code>'pairwise'</code> to perform the principal component analysis. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 coeff = pca(X(:,3:15),'Rows','pairwise'); </code></pre></div> In this case, <code>pca</code> computes the (i,j) element of the covariance matrix using the rows with no <code>NaN</code> values in the columns i or j of <code>X</code>. Note that the resulting covariance matrix might not be positive definite. This option applies when the algorithm <code>pca</code> uses is eigenvalue decomposition. When you don’t specify the algorithm, as in this example, <code>pca</code> sets it to <code>'eig'</code>. If you require <code>'svd'</code> as the algorithm, with the <code>'pairwise'</code> option, then <code>pca</code> returns a warning message, sets the algorithm to <code>'eig'</code> and continues. If you use the <code>'Rows','all'</code> name-value pair argument, <code>pca</code> terminates because this option assumes there are no missing values in the data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 coeff = pca(X(:,3:15),'Rows','all'); </code></pre></div> <pre class="code-output"> ...ERROR... Raw data contains NaN missing value while 'Rows' option is set to 'all'. Consider using 'complete' or 'pairwise' option, or using 'als' algorithm instead. </pre> <h2>Weighted PCA</h2> Use the inverse variable variances as weights while performing the principal components analysis. Load the sample data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald </code></pre></div> Perform the principal component analysis using the inverse of variances of the ingredients as variable weights. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance'); </code></pre></div> Note that the coefficient matrix, <code>wcoeff</code>, is not orthonormal. Calculate the orthonormal coefficient matrix. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance'); coefforth = inv(diag(std(ingredients)))* wcoeff </code></pre></div> <pre class="code-output"> coefforth = Columns 1 through 3 -0.475955172748972 0.508979384806409 -0.675500187964285 -0.563870242191993 -0.413931487136986 0.314420442819292 0.394066533909305 -0.604969078471438 -0.637691091806566 0.547931191260862 0.451235109330017 0.195420962611708 Column 4 0.241052184051094 0.641756074427214 0.268466110294533 0.676734019481283 </pre> Check orthonormality of the new coefficient matrix, <code>coefforth</code>. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [wcoeff,~,latent,~,explained] = pca(ingredients,'VariableWeights','variance'); coefforth = inv(diag(std(ingredients)))* wcoeff; coefforth*coefforth' </code></pre></div> <pre class="code-output"> ans = Columns 1 through 3 1.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 1.000000000000000 0.000000000000000 -0.000000000000000 0.000000000000000 1.000000000000000 -0.000000000000000 0.000000000000000 -0.000000000000000 Column 4 -0.000000000000000 0.000000000000000 -0.000000000000000 1.000000000000000 </pre> <h2>PCA Using ALS for Missing Data</h2> Find the principal components using the alternating least squares (ALS) algorithm when there are missing values in the data. Load the sample data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald </code></pre></div> The ingredients data has 13 observations for 4 variables. Perform principal component analysis using the ALS algorithm and display the component coefficients. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); coeff </code></pre></div> <pre class="code-output"> coeff = Columns 1 through 3 -0.067799985695474 -0.646018286568728 0.567314540990512 -0.678516235418647 -0.019993340484099 -0.543969276583817 0.029020832106229 0.755309622491133 0.403553469172668 0.730873909451461 -0.108480477171676 -0.468397518388289 Column 4 0.506179559977705 0.493268092159297 0.515567418476836 0.484416225289198 </pre> Introduce missing values randomly. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN </code></pre></div> <pre class="code-output"> y = 7 26 6 NaN 1 29 15 52 NaN NaN 8 20 11 31 NaN 47 7 52 6 33 NaN 55 NaN NaN NaN 71 NaN 6 1 31 NaN 44 2 NaN NaN 22 21 47 4 26 NaN 40 23 34 11 66 9 NaN 10 68 8 12 </pre> Approximately 30% of the data has missing values now, indicated by <code>NaN</code>. Perform principal component analysis using the ALS algorithm and display the component coefficients. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als'); coeff1 </code></pre></div> <pre class="code-output"> coeff1 = Columns 1 through 3 -0.036212338541055 0.821521833463707 -0.525180807034516 -0.683141604556494 -0.099847237932408 0.182836485390674 0.016929757180232 0.557510085824824 0.821489828858883 0.729191057256728 -0.065687977768361 0.126136436504135 Column 4 0.219033475985740 0.699942066753560 -0.118534166411284 0.669369173908958 </pre> Display the estimated mean. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als'); mu1 </code></pre></div> <pre class="code-output"> mu1 = Columns 1 through 3 8.995597151700496 47.908800869772193 9.045133815910839 Column 4 28.551458070546648 </pre> Reconstruct the observed data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als'); t = score1*coeff1' + repmat(mu1,13,1) </code></pre></div> <pre class="code-output"> t = Columns 1 through 3 6.999999999999997 25.999999999999972 5.999999999999996 0.999999999999986 28.999999999999968 15.000000000000011 10.781877383662085 53.023021922864480 7.999999999999999 11.000000000000002 30.999999999999996 13.549987432049472 6.999999999999996 52.000000000000000 6.000000000000012 10.481832029571297 55.000000000000000 7.832799509066273 3.098161701207570 71.000000000000014 11.949103806476206 0.999999999999980 30.999999999999968 -0.516112356202619 1.999999999999988 53.791389384174103 5.770961215451544 21.000000000000014 47.000000000000000 3.999999999999998 21.580891857665534 39.999999999999986 23.000000000000014 10.999999999999996 66.000000000000014 9.000000000000002 9.999999999999998 68.000000000000000 8.000000000000009 Column 4 51.524967145094379 52.000000000000021 19.999999999999996 47.000000000000014 32.999999999999950 17.936171158236188 5.999999999999915 44.000000000000000 22.000000000000000 25.999999999999989 33.999999999999993 5.707816613776032 11.999999999999943 </pre> The ALS algorithm estimates the missing values in the data. Another way to compare the results is to find the angle between the two spaces spanned by the coefficient vectors. Find the angle between the coefficients found for complete data and data with missing values using ALS. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff1,score1,latent,tsquared,explained,mu1] = pca(y,'algorithm','als'); t = score1*coeff1' + repmat(mu1,13,1); subspace(coeff,coeff1) </code></pre></div> <pre class="code-output"> ans = 6.872983007538972e-16 </pre> This is a small value. It indicates that the results if you use <code>pca</code> with <code>'Rows','complete'</code> name-value pair argument when there is no missing data and if you use <code>pca</code> with <code>'algorithm','als'</code> name-value pair argument when there is missing data are close to each other. Perform the principal component analysis using <code>'Rows','complete'</code> name-value pair argument and display the component coefficients. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete'); coeff2 </code></pre></div> <pre class="code-output"> coeff2 = -0.205420225953307 0.858688026698708 0.049181247828161 -0.669364063599413 -0.371998098041251 0.551021387596589 0.147356319504436 -0.351257041216125 -0.518714827525170 0.698598880784301 -0.029845918549515 0.651837067815822 </pre> In this case, <code>pca</code> removes the rows with missing values, and <code>y</code> has only four rows with no missing values. <code>pca</code> returns only three principal components. You cannot use the <code>'Rows','pairwise'</code> option because the covariance matrix is not positive semidefinite and <code>pca</code> returns an error message. Find the angle between the coefficients found for complete data and data with missing values using listwise deletion (when <code>'Rows','complete'</code>). <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete'); subspace(coeff(:,1:3),coeff2) </code></pre></div> <pre class="code-output"> ans = 0.357607357590711 </pre> The angle between the two spaces is substantially larger. This indicates that these two results are different. Display the estimated mean. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete'); mu2 </code></pre></div> <pre class="code-output"> mu2 = Columns 1 through 3 7.888888888888889 46.909090909090907 9.875000000000000 Column 4 29.600000000000001 </pre> In this case, the mean is just the sample mean of <code>y</code>. Reconstruct the observed data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared,explained] = pca(ingredients); y = ingredients; rng('default'); % for reproducibility ix = random('unif',0,1,size(y))<0.30; y(ix) = NaN; [coeff2,score2,latent,tsquared,explained,mu2] = pca(y,'Rows','complete'); score2*coeff2' </code></pre></div> <pre class="code-output"> ans = Columns 1 through 3 NaN NaN NaN -7.516185143683813 -18.354534728448066 4.096758018165185 NaN NaN NaN NaN NaN NaN -0.564429969715416 5.321307756733086 -3.343158339022808 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN 12.831514933168062 -0.107632488812891 -6.333304231731043 NaN NaN NaN NaN NaN NaN 1.467986895710465 20.634225817831361 -2.929186618770285 Column 4 NaN 22.005631390194360 NaN NaN 3.603980833482452 NaN NaN NaN NaN -3.775776525301393 NaN NaN -18.004319332087810 </pre> This shows that deleting rows containing <code>NaN</code> values does not work as well as the ALS algorithm. Using ALS is better when the data has too many missing values. <h2>Principal Component Coefficients, Scores, and Variances</h2> Find the coefficients, scores, and variances of the principal components. Load the sample data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald </code></pre></div> The ingredients data has 13 observations for 4 variables. Find the principal component coefficients, scores, and variances of the components for the ingredients data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent] = pca(ingredients); </code></pre></div> Each column of <code>score</code> corresponds to one principal component. The vector, <code>latent</code>, stores the variances of the four principal components. Reconstruct the centered ingredients data. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent] = pca(ingredients); Xcentered = score*coeff' </code></pre></div> <pre class="code-output"> Xcentered = Columns 1 through 3 -0.461538461538459 -22.153846153846157 -5.769230769230774 -6.461538461538464 -19.153846153846153 3.230769230769226 3.538461538461541 7.846153846153848 -3.769230769230770 3.538461538461538 -17.153846153846143 -3.769230769230777 -0.461538461538459 3.846153846153846 -5.769230769230769 3.538461538461541 6.846153846153848 -2.769230769230769 -4.461538461538460 22.846153846153818 5.230769230769237 -6.461538461538467 -17.153846153846153 10.230769230769228 -5.461538461538463 5.846153846153832 6.230769230769233 13.538461538461540 -1.153846153846135 -7.769230769230773 -6.461538461538466 -8.153846153846160 11.230769230769232 3.538461538461542 17.846153846153840 -2.769230769230766 2.538461538461542 19.846153846153840 -3.769230769230764 Column 4 30.000000000000021 22.000000000000021 -10.000000000000000 17.000000000000000 3.000000000000000 -8.000000000000005 -23.999999999999996 14.000000000000007 -7.999999999999992 -4.000000000000015 4.000000000000006 -18.000000000000007 -18.000000000000000 </pre> The new data in <code>Xcentered</code> is the original ingredients data centered by subtracting the column means from corresponding columns. Visualize both the orthonormal principal component coefficients for each variable and the principal component scores for each observation in a single plot. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent] = pca(ingredients); Xcentered = score*coeff'; biplot(coeff(:,1:2),'scores',score(:,1:2),'varlabels',{'v_1','v_2','v_3','v_4'}); fig2plotly(gcf); </code></pre></div> <div id="plot-9997773" class="plotly-graph-div js-plotly-plot"></div> <script type = "text/javascript" > require(["plotly"], function(Plotly) { window.PLOTLYENV = window.PLOTLYENV || {}; if (document.getElementById("plot-9997773" )) { Plotly.newPlot("plot-9997773", {"data": [{"type": "scatter", "xaxis": "x1", "yaxis": "y1", "visible": true, "name": "", "mode": "lines", "x": [0, -0.0677999856954741, null, 0, -0.678516235418647, null, 0, 0.029020832106229, null, 0, 0.730873909451461, null], "y": [0, -0.646018286568728, null, 0, -0.0199933404840993, null, 0, 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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> All four variables are represented in this biplot by a vector, and the direction and length of the vector indicate how each variable contributes to the two principal components in the plot. For example, the first principal component, which is on the horizontal axis, has positive coefficients for the third and fourth variables. Therefore, vectors v3 and v4 are directed into the right half of the plot. The largest coefficient in the first principal component is the fourth, corresponding to the variable v4. The second principal component, which is on the vertical axis, has negative coefficients for the variables v1, v2, and v4, and a positive coefficient for the variable v3. This 2-D biplot also includes a point for each of the 13 observations, with coordinates indicating the score of each observation for the two principal components in the plot. For example, points near the left edge of the plot have the lowest scores for the first principal component. The points are scaled with respect to the maximum score value and maximum coefficient length, so only their relative locations can be determined from the plot. <h2>T-Squared Statistic</h2> Find the Hotelling’s T-squared statistic values. Load the sample data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald </code></pre></div> The ingredients data has 13 observations for 4 variables. Perform the principal component analysis and request the T-squared values. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared] = pca(ingredients); tsquared </code></pre></div> <pre class="code-output"> tsquared = 5.680340841490951 3.075837035211966 6.000232793912613 2.619763232047952 3.368139445336655 0.566796674771214 3.481840731756962 3.979397901620163 2.608586242833568 7.481756329327637 4.183022234152593 2.232718720505654 2.721567817032093 </pre> Request only the first two principal components and compute the T-squared values in the reduced space of requested principal components. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2); tsquared </code></pre></div> <pre class="code-output"> tsquared = 5.680340841490951 3.075837035211966 6.000232793912613 2.619763232047952 3.368139445336655 0.566796674771214 3.481840731756962 3.979397901620163 2.608586242833568 7.481756329327637 4.183022234152593 2.232718720505654 2.721567817032093 </pre> Note that even when you specify a reduced component space, <code>pca</code> computes the T-squared values in the full space, using all four components. The T-squared value in the reduced space corresponds to the Mahalanobis distance in the reduced space. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2); tsqreduced = mahal(score,score) </code></pre></div> <pre class="code-output"> tsqreduced = 3.317920809184487 2.007906801224291 0.587427300774747 1.738161538935849 0.295525889978777 0.422793560910442 3.245670063829735 2.691429716874080 1.361859279884830 2.990295498144076 2.437109148953179 1.378818611291408 1.525081780014097 </pre> Calculate the T-squared values in the discarded space by taking the difference of the T-squared values in the full space and Mahalanobis distance in the reduced space. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load hald [coeff,score,latent,tsquared] = pca(ingredients,'NumComponents',2); tsqreduced = mahal(score,score); tsqdiscarded = tsquared - tsqreduced </code></pre></div> <pre class="code-output"> tsqdiscarded = 2.362420032306464 1.067930233987675 5.412805493137867 0.881601693112104 3.072613555357878 0.144003113860771 0.236170667927227 1.287968184746083 1.246726962948738 4.491460831183561 1.745913085199414 0.853900109214246 1.196486037017995 </pre> <h2>Percent Variability Explained by Principal Components</h2> Find the percent variability explained by the principal components. Show the data representation in the principal components space. Load the sample data set. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 </code></pre></div> Data matrix <code>X</code> has 13 continuous variables in columns 3 to 15: wheel-base, length, width, height, curb-weight, engine-size, bore, stroke, compression-ratio, horsepower, peak-rpm, city-mpg, and highway-mpg. Find the percent variability explained by principal components of these variables. <div class="highlight"><pre><code class="language-matlab" data-lang="matlab">load imports-85 [coeff,score,latent,tsquared,explained] = pca(X(:,3:15)); explained </code></pre></div> <pre class="code-output"> explained = 64.342913356114238 35.448392662688399 0.154993220067715 0.037863222808984 0.007807396888358 0.004771746794250 0.001304050511812 0.001056438605476 0.000529182750646 0.000186613509121 0.000158017147436 0.000017269286049 0.000006822827515 </pre> The first three components explain 99.95% of all variability. 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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> The data shows the largest variability along the first principal component axis. This is the largest possible variance among all possible choices of the first axis. The variability along the second principal component axis is the largest among all possible remaining choices of the second axis. The third principal component axis has the third largest variability, which is significantly smaller than the variability along the second principal component axis. The fourth through thirteenth principal component axes are not worth inspecting, because they explain only 0.05% of all variability in the data. To skip any of the outputs, you can use <code>~</code> instead in the corresponding element. 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Pca visualization in MATLAB