Statistical feature extraction A0 B0 B1 B2 · · · I38 I39 I40 J0 first step: representation of red 0 0 0 0 · · · 0 0 2 0 raw data in contingency green 0 0 0 0 · · · 0 0 0 0 matrix blue 0 0 0 0 · · · 0 0 0 0 black 0 0 0 0 · · · 18 23 21 25 rows: color terms from white 25 25 22 23 · · · 0 0 0 0 . . . . . . . . . . various languages . . . . . . . . . . . . . . . . . . . . rot 0 0 0 0 · · · 1 0 0 0 columns: Munsell chips gr¨ un 0 0 0 0 · · · 0 0 0 0 cells: number of test gelb 0 0 0 1 · · · 0 0 0 0 . . . . . . . . . . . . . . . . . . . . persons who used the . . . . . . . . . . rouge 0 0 0 0 · · · 0 0 0 0 row-term for the vert 0 0 0 0 · · · 0 0 0 0 . . . . . . . . . . column-chip . . . . . . . . . . . . . . . . . . . . further processing: divide each row by the number n of test persons using the corresponding term duplicate each row n times 42/124
Principal Component Analysis technique to reduce dimensionality of data input: set of vectors in an n -dimensional space first step: second step: rotate the coordinate system, such that choose a suitable m < n the new n coordinates are project the data on those m orthogonal to each other new coordinates where the the variations of the data data have the highest along the new coordinates variance are stochastically independent 43/124
Principal Component Analysis alternative formulation: choose an m -dimensional linear sub-manifold of your n -dimensional space project your data onto this manifold when doing so, pick your sub-manifold such that the average squared distance of the data points from the sub-manifold is minimized intuition behind this formulation: data are “actually” generated in an m -dimensional space observations are disturbed by n -dimensional noise PCA is a way to reconstruct the underlying data distribution applications: picture recognition, latent semantic analysis, statistical data analysis in general, data visualization, ... 44/124
Statistical feature extraction: PCA 0.30 0.25 first 15 principal components jointly proportion of variance explained 0.20 explain 91 . 6% of the total variance 0.15 choice of m = 15 is 0.10 determined by using “Kaiser’s stopping 0.05 rule” 0.00 principal components 45/124
Statistical feature extraction: PCA after some post-processing (“varimax” algorithm): A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 B C D E F G H I J 46/124
Projecting observed data on lower-dimensional-manifold noise removal: project observed data onto the lower-dimensional submanifold that was obtained via PCA in our case: noisy binary categories are mapped to smoothed fuzzy categories (= probability distributions over Munsell chips) some examples: 47/124
Projecting observed data on lower-dimensional-manifold 48/124
Projecting observed data on lower-dimensional-manifold 49/124
Projecting observed data on lower-dimensional-manifold 50/124
Projecting observed data on lower-dimensional-manifold 51/124
Projecting observed data on lower-dimensional-manifold 52/124
Projecting observed data on lower-dimensional-manifold 53/124
Projecting observed data on lower-dimensional-manifold 54/124
Projecting observed data on lower-dimensional-manifold 55/124
Projecting observed data on lower-dimensional-manifold 56/124
Projecting observed data on lower-dimensional-manifold 57/124
Projecting observed data on lower-dimensional-manifold 58/124
Projecting observed data on lower-dimensional-manifold 59/124
Projecting observed data on lower-dimensional-manifold 60/124
Projecting observed data on lower-dimensional-manifold 61/124
Projecting observed data on lower-dimensional-manifold 62/124
Projecting observed data on lower-dimensional-manifold 63/124
Projecting observed data on lower-dimensional-manifold 64/124
Projecting observed data on lower-dimensional-manifold 65/124
Projecting observed data on lower-dimensional-manifold 66/124
Projecting observed data on lower-dimensional-manifold 67/124
Smoothing the partitions from smoothed extensions we can recover smoothed partitions each pixel is assigned to category in which it has the highest degree of membership 68/124
Smoothed partitions of the color space 69/124
Smoothed partitions of the color space 70/124
Smoothed partitions of the color space 71/124
Smoothed partitions of the color space 72/124
Smoothed partitions of the color space 73/124
Smoothed partitions of the color space 74/124
Smoothed partitions of the color space 75/124
Smoothed partitions of the color space 76/124
Smoothed partitions of the color space 77/124
Smoothed partitions of the color space 78/124
Convexity note: so far, we only used information from the WCS the location of the 330 Munsell chips in L*a*b* space played no role so far still, apparently partition cells always form continuous clusters in L*a*b* space Hypothesis (G¨ ardenfors): extension of color terms always form convex regions of L*a*b* space 79/124
Support Vector Machines supervised learning technique smart algorithm to classify data in a high-dimensional space by a (for instance) linear boundary minimizes number of mis-classifications if the training data are not linearly separable SVM classification plot o o 3 o o o o o o o o o 2 red o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 1 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x o o 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o −1 o o o o o o o o o o o o o o o o o o o o o o o green o o o o o o o o o o o o −2 o o o o o o o o o o o −3 o o o −3 −2 −1 0 1 2 3 80/124 y
Convex partitions a binary linear classifier divides an n -dimensional space into two convex half-spaces intersection of two convex set is itself convex hence: intersection of k binary classifications leads to convex sets procedure: if a language partitions the Munsell space into m categories, train m ( m − 1) many binary SVMs, one for each pair 2 of categories in L*a*b* space leads to m convex sets (which need not split the L*a*b* space exhaustively) 81/124
Convex approximation 82/124
Convex approximation 83/124
Convex approximation 84/124
Convex approximation 85/124
Convex approximation 86/124
Convex approximation 87/124
Convex approximation 88/124
Convex approximation 89/124
Convex approximation 90/124
Convex approximation 91/124
Convex approximation on average, 93 . 7% of all Munsell chips are correctly classified by convex approximation proportion of correctly classified Munsell chips 0.95 0.90 ● ● ● ● ● ● ● 0.85 ● ● ● 0.80 92/124
Convex approximation compare to the outcome of the same procedure without PCA, and with PCA but using a random permutation of the Munsell chips 100 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 80 ● ● ● ● ● ● ● ● ● ● ● ● degree of convexity (%) ● ● 60 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 40 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 ● 1 2 3 93/124
Convex approximation choice of m = 10 is somewhat arbitrary outcome does not depend very much on this choice though 100 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 90 mean degree of convexity (%) 80 70 60 50 0 10 20 30 40 50 no. of principal components used 94/124
Implicative universals first six features correspond nicely to the six primary colors white, black, red, green, blue, yellow according to Kay et al. (1997) (and many other authors) simple system of implicative universals regarding possible partitions of the primary colors 95/124
Implicative universals I II III IV V white white red red / yellow yellow green / blue green / blue black black white white red � white / red / yellow white � red / yellow yellow red / yellow black / green / blue green green black / green / blue black / blue blue black white white red red yellow yellow green black / green / blue black / blue white white red red yellow / green yellow / green / blue blue black black white red yellow / green black / blue source: Kay et al. (1997) 96/124
Partition of the primary colors each speaker/term pair can be projected to a 15-dimensional vector primary colors correspond to first 6 entries each primary color is assigned to the term for which it has the highest value defines for each speaker a partition over the primary colors 97/124
Partition of the primary colors for instance: sample speaker (from Piraha): extracted partition: white / yellow A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 B C D red E F G H green / blue I J black supposedly impossible, but occurs 61 times in the database 98/124
Partition of primary colors most frequent partition types: 1 { white } , { red } , { yellow } , { green, blue } , { black } ( 41 . 9% ) 2 { white } , { red } , { yellow } , { green } , { blue } , { black } ( 25 . 2% ) 3 { white } , { red, yellow } , { green, blue, black } ( 6 . 3% ) 4 { white } , { red } , { yellow } , { green } , { black, blue } ( 4 . 2% ) 5 { white, yellow } , { red } , { green, blue } , { black } ( 3 . 4% ) 6 { white } , { red } , { yellow } , { green, blue, black } ( 3 . 2% ) 7 { white } , { red, yellow } , { green, blue } , { black } ( 2 . 6% ) 8 { white, yellow } , { red } , { green, blue, black } ( 2 . 0% ) 9 { white } , { red } , { yellow } , { green, blue, black } ( 1 . 6% ) 10 { white } , { red } , { green, yellow } , { blue, black } ( 1 . 2% ) 99/124
Partition of primay colors 87 . 1% of all speaker partitions obey Kay et al.’s universals the ten partitions that confirm to the universals occupy ranks 1, 2, 3, 4, 6, 7, 9, 10, 16, 18 decision what counts as an exception seems somewhat arbitrary on the basis of these counts 100/124
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