{"id":1015,"date":"2023-09-04T19:54:15","date_gmt":"2023-09-04T19:54:15","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=1015"},"modified":"2023-09-05T12:56:18","modified_gmt":"2023-09-05T12:56:18","slug":"how-principal-are-your-components","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2023\/09\/04\/how-principal-are-your-components\/","title":{"rendered":"How Principal Are Your Components?"},"content":{"rendered":"\n
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This post is an offshoot from Chapter 1 of Probably Overthinking It<\/a><\/em>, which is available for pre-order now<\/a>!<\/p>\n<\/blockquote>\n\n\n\n

In a previous post I explored the correlations between measurements in the ANSUR-II dataset<\/a>, which includes 93 measurements from a sample of U.S. military personnel. I found that measurements of the head were weakly correlated with measurements from other parts of the body \u2013 and in particular the protrusion of the ears is almost entirely uncorrelated with anything else.<\/p>\n\n\n\n

A friend of mine, and co-developer of the Modeling and Simulation<\/a> class I taught at Olin, asked whether I had tried running principal component analysis<\/a> (PCA). I had not, but now I have. Let\u2019s look at the results.<\/p>\n\n\n\n

Click here to run this notebook on Colab<\/a>. <\/p>\n\n\n\n

The ANSUR data is available from The OPEN Design Lab<\/a>. <\/p>\n\n\n\n

Explained Variance<\/h2>\n\n\n\n

Here\u2019s a visualization of explained variance versus number of components. <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

With one component, we can capture 44% of the variation in the measurements. With two components, we\u2019re up to 62%. After that, the gains are smaller (as we expect), but with 10 measurements, we get up to 78%.<\/p>\n\n\n\n

Loadings<\/h2>\n\n\n\n

Looking at the loadings<\/a>, we can see which measurements contribute the most to each of the components, so we can get a sense of which characteristics each component captures. <\/p>\n\n\n\n

I won\u2019t explain all of the measurements, but if there are any you are curious about, you can look them up in The Measurer\u2019s Handbook<\/em><\/a>, which includes details on \u201csampling strategy and measuring techniques\u201d as well as descriptions and diagrams of the landmarks and measurements between them.<\/p>\n\n\n\n

Principal Component 1:\n0.135 \t suprasternaleheight\n0.134 \t cervicaleheight\n0.134 \t buttockkneelength\n0.134 \t acromialheight\n0.133 \t kneeheightsitting\n\nPrincipal Component 2:\n0.166 \t waistcircumference\n-0.163 \t poplitealheight\n0.163 \t abdominalextensiondepthsitting\n0.161 \t waistdepth\n0.159 \t buttockdepth\n\nPrincipal Component 3:\n0.338 \t elbowrestheight\n0.31 \t eyeheightsitting\n0.307 \t sittingheight\n0.228 \t waistfrontlengthsitting\n-0.225 \t heelbreadth\n\nPrincipal Component 4:\n0.247 \t balloffootcircumference\n0.232 \t bimalleolarbreadth\n0.22 \t footbreadthhorizontal\n0.218 \t handbreadth\n0.212 \t sittingheight\n\nPrincipal Component 5:\n0.319 \t interscyeii\n0.292 \t biacromialbreadth\n0.275 \t shoulderlength\n0.273 \t interscyei\n0.184 \t shouldercircumference\n\nPrincipal Component 6:\n-0.34 \t headcircumference\n-0.321 \t headbreadth\n0.316 \t shoulderlength\n-0.277 \t tragiontopofhead\n-0.262 \t interpupillarybreadth\n\nPrincipal Component 7:\n0.374 \t crotchlengthposterioromphalion\n-0.321 \t earbreadth\n-0.298 \t earlength\n-0.284 \t waistbacklength\n0.253 \t crotchlengthomphalion\n\nPrincipal Component 8:\n0.472 \t earprotrusion\n0.346 \t earlength\n0.215 \t crotchlengthposterioromphalion\n-0.202 \t wristheight\n0.195 \t overheadfingertipreachsitting\n\nPrincipal Component 9:\n-0.299 \t tragiontopofhead\n0.294 \t crotchlengthposterioromphalion\n-0.253 \t bicristalbreadth\n-0.228 \t shoulderlength\n0.189 \t neckcircumferencebase\n\nPrincipal Component 10:\n0.406 \t earbreadth\n0.356 \t earprotrusion\n-0.269 \t waistfrontlengthsitting\n0.239 \t earlength\n-0.228 \t waistbacklength\n<\/pre>\n\n\n\n
Here’s my interpretation of the first few components.<\/p>\n\n\n\n