A PONDERING OF THE PASTEL PONY PIGMENTATION PUZZLE

Transcription

1 Proceedigs of the CCCCCCC The Completely Cofirmable Coferece o Colorful Cartoo Character Classificatios Dec. 10, A PONDERING OF THE PASTEL PONY PIGMENTATION PUZZLE By Hydrus Beta INTRODUCTION Sice the debut of the My Little Poy: Friedship is Magic, poy ethusiasts have desiged ad draw thousads of their ow origial characters (OCs). The coloratio of may OCs deviate from the look ad feel of those used for characters i the show. While may of these origial desigs use itetioally uique coloratio, quatitative kowledge of geuie Equestria colors may still pique the iterest of fas ad artists. The goal of this paper is to aalyze the color demographics of Equestria's poy populatio. A algorithm is preseted to radomly geerate coat ad mae colors that match these demographics, ad te such radomly-geerated characters are preseted as a demostratio ad test of the study. Figure 1. Histogram of s for Equestria s Mares COLLECTING SAMPLE DATA The "List of Poies" Wiki cotais a plethora of poy color data, orgaized by race (pegasus, uicor, earth poy, crystal poy, or alicor), geder, ad age [1]. For this study, crystal poies ad alicors were omitted. Colors were recorded from the web pages by takig large screeshots with the assistace of the Full Page Scree Capture extesio available for Google Chrome []. FreeMat was the used to sca the images alog the colums that display the coat ad mae colors ad extract the data i RGB format [3]. Uder the assumptio that hue, saturatio, ad value (brightess) are more relevat variables i desigig a character color scheme, the RGB quatities were coverted to HSV quatities before further aalysis. Before extractig the colors, the image of each poy was observed maually, ad the etry was discarded if the image appeared heavily tited or the colors otherwise biased or distorted. For example, data for Rosewood Brook, a uicor appearig i Gray Smith's memory, was discarded due to the character's colors beig artificially limited to shades of sepia. Figure. Histogram of s for Equestria s Mares ANALYSIS OF THE MARE DATA The data for mares ad for stallios were kept separate, i order to allow for a compariso of the color demographics betwee the two groups. Figures 1 through 6 show histograms of the hues, saturatios, ad values for the coats ad primary mae color of Equestria mares. Figure 3. Histogram of Brightesses for Equestria s Mares

2 flatter ad shifted right. Very high saturatio values for the maes are less commo tha lower saturatios, but are ot as rare as high coat saturatios. Coversely, brightess values are otably very high for the coats, with 90% of mares havig a coat brightess of 75 or higher ad the largest group beig i the rage. brightesses also ted to be high, although very low brightesses are also possible. The histograms for coat brightess ad mae brightess are similar i shape, but the mae brightess trails off more to the lower values. Figure 4. Histogram of s for Equestria s Mares The apparet similarity of the two hue histograms, the two saturatio histograms, ad the two brightess histograms might appear to imply that these respective pairs of quatities are strogly liked to each other i some way. However, a plot of mae hues vs coat hues reveals o obvious correlatio (figure 7). O the cotrary, they ow appear to be idepedet variables. Figure 5. Histogram of s for Equestria s Mares Figure 7. Scatterplot of s vs s for Equestria Mares Likewise, plottig every possible pair of items (coat hue vs coat saturatio, coat brightess vs mae hue, etc.) shows o immediately obvious correlatio betwee ay of the six items (although a distace covariace test does show that ot all of the variables are completely idepedet. See the sectio titled A Closer Look at Variable Idepedece, below). ANALYSIS OF THE STALLION DATA Figure 6. Histogram of Brightesses for Equestria s Mares s across the etire spectrum are preset i both the coats ad maes of Equestria mares. However, it is evidet that hues i the gree/turquoise rage from ~100 o to ~160 o are avoided for both the coats ad the maes, while a oragish yellow is the most commo hue for both. values for the coat ted o the lower side, with ~35% beig the most commo level, while the histogram for mae saturatio is Figures 8-13 show the distributios of hue, saturatio, ad brightess i the coats ad maes of the stallios. The histogram of stallio coat hues is somewhat similar to that of the mares, except that there's a sigificat dimiishmet i the rage ~60 o -340 o. Like the mares, few stallios exhibit coats ad maes i the gree/turquoise rage. The stallios' mae hues histogram is less similar to that of the mares, but agai shows a sigificatly dimiished occurrece of hues i that upper rage of ~60 o -340 o. There is also a oticeable spike aroud the rage 10 o -0 o.

3 Figure 8. Histogram of s for Equestria s Stallios Figure 11. Histogram of s for Equestria s Stallios Figure 9. Histogram of s for Equestria s Stallios Figure 1. Histogram of s for Equestria s Stallios Figure 10. Histogram of Brightesses for Equestria s Stallios The saturatio histograms are similar betwee the mares ad the stallios except that the stallios exhibit a spike i the umber of samples at the extreme low ed of both histograms (0-10% saturatio). Brightess distributios for the stallios' coats ad maes spread out further ito the lower values tha they do for the mares. Still, the brightesses ted towards the higher values, ad almost o stallio has a coat brightess Figure 13. Histogram of Brightesses for Equestria s Stallios below 30 while oly a hadful have mae brightesses below 0. The shape of the brightess histogram for the stallio maes is also slightly differet tha the other brightess histograms, beig relatively flatter from aroud 70 to 0, ad the sharply droppig off for lower values.

4 Oce agai, plottig each type of data agaist every other shows o immediately apparet correlatios. from left to right. Figure 15 below, for example, shows the cumulative histogram derived from figure 14: A PROCEDURE FOR PRODUCING PONY PIGMENTATION A algorithm is preseted below for geeratig a populatio of coat ad mae colors that matches the histograms show i figures 1-6 ad Five mare color schemes ad five stallio color schemes take directly from this algorithm are preseted for discussio. For this algorithm, it is assumed that all the types of data (hues, saturatios, ad brightesses for the mae ad for the coat) are idepedet of each other, ad ca thus be geerated idepedetly. Noe of the probability desity fuctios (pdfs) for the hues, implied by the histograms i figures 1,, 8, ad 9, closely follow ay sigle stadard pdf. While a estimatio of each pdf could coceivably be made by combiig together several Gaussia distributios, the algorithm below opts for a umerical approach that will produce umbers that follow each histogram more precisely ad ca be uiversally applied to all of the histograms. Normalized histograms ca be used as a estimatio of their respective pdfs. Histograms are ormalized by dividig every value by the total umber of samples, thus esurig that the sum of values is equal to uity without chagig the shape of the distributio. Figure 14 below shows the ormalized histogram for mare coat saturatio: Figure 15. Cumulative Normalized Histogram for Mare s Note that the last etry i the cumulative histogram is always 1, the sum of all the histogram etries. Next, we geerate a array of values, all betwee 0 ad 1, with a uiform distributio. This is easily achieved usig the built-i FreeMat fuctio rad(1,). radge = rad(1,) We ca perform logical tests o these radomly-geerated umbers to see whether they exist i a give rage. For example, we ca fid all the radomly-geerated umbers that are betwee 0.84 ad ad store their locatios i the array idexes usig the followig commad: idexes = fid(radge>0.84&radge<=0.499) Note that 0.84 ad are the values of the cumulative histogram i figure 15 for saturatios up to 0% ad saturatios up to 30%, respectively. The differece, 0.15 is the fractio of mares who had coat saturatios betwee 0% ad 30% (this is cofirmed by lookig at figure 14). Hece, the poies at these idexes should all have their coat saturatios chose to be betwee 0% ad 30%. A radom distributio i this rage ca be applied to them as follows: Figure 14. Normalized Histogram of Mare saturatios. The goal is to produce a radom populatio of poies, where is ay positive iteger we choose. Whe determiig a quatity such as coat hues, we must determie what fractio of these poies will belog to each bad of hues. Ideally, for a truly radomly-geerated populatio, the fractio of poies will ot be hard-coded, but rather allowed to be geerated probabilistically, usig the ormalized histogram as a weightig fuctio. To do this, we first create a "cumulative" histogram, which simply takes the sum of the histogram etries geerated_coat_saturatio(idexes) = 0.1*rad(1,legth(idexes))+0. This procedure ca be automated across the etire cumulative histogram, fillig i all coat saturatios. The procedure ca be applied equally as well to coat hues, coat brightesses, mae hues, etc. A sample of 100 color schemes was radomly geerated for both mares ad stallios, ad the first five of each are preseted i figure 16.

5 The, the distace covariace is defied as the square root of 1 k1 l1 ( X, Y) Oe last quatity must be foud before the test statistic is determied: A B Figure 16. Te Poy Color Schemes Geerated by the Algorithm. While ot a perfect set, all of these poies at least exhibit a appropriate rage of hues, saturatios ad brightesses that are similar their Equestria ki. Subjectively, it could be said that poies #3, #4 ad #8 appear a little oversaturated, ad ideed their coat saturatio levels are all towards the higher ed of the respective histograms (#3 is 49%, #4 is 59%, ad #8 is 58%). Number 9's coat, however, actually has the highest level of saturatio, at 6%. It could also be said that some of the poies' color combiatios could be more aesthetically pleasig. Nevertheless, a few of the poies have very decet coloratio. The author's persoal favorite is #6. A CLOSER LOOK AT VARIABLE INDEPENDENCE The algorithm i the previous sectio assumed that all the color variables could be treated as completely idepedet, because the scatter plots were spread out ad exhibited o obvious correlatio. A more cofidet measure of idepedece ca be achieved mathematically by performig a distace covariace test of idepedece [4]. Each type of data will be tested agaist every other, oe pair at a time. Let X i ad Y i (i = 1,,..., ) represet the two sets of data to be compared. The first step i performig a distace covariace test is to scale the data to esure that the test is affie-ivariat. For data ir 1, the scalig factors are simply the reciprocal of the stadard deviatios: S 1 a b k1 l1 X X 1 Fially, the test statistic is computed as k l ( X, Y) S k1 l1 Y Y Choosig a sigificace level of α, the ull hypothesis that the data are idepedet is rejected if the followig is true: ( X, Y) S 1 1 where Φ -1 ( ) is the iverse stadard ormal cumulative distributio fuctio. Tables 1 ad summarize the results of all the idepedece tests for mares ad for stallios, at a sigificace level of 5%. Table 1. Covariace Idepedece Test Results for Mares. "Id" idicates that the two variables are idepedet, while "Dep" idicates that the data failed the distace covariace test ad are therefore ot idepedet. k l X* = X/σ(X) Y* = Y/σ(Y) Before the distace covariace ca be determied, the followig quatities must be computed for X: a X k X l a k a l1 1 a 1 l a k1 a 1 a k1 l1 A a a k a l a Similarly, b, bk, b l, b,adb are computed for Y.

6 Table. Covariace Idepedece Test Results for Stallios. "Id" idicates that the two variables are idepedet, while "Dep" idicates that the data failed the distace covariace test ad are therefore ot idepedet. icluded i the wiki ad would have to be measured ad recorded maually. A quick prelimiary observatio suggests that the character outlies retai a similar hue to the part that they are outliig, but have slightly higher saturatio ad lower brightess. A more complete study would be desirable to cofirm ad quatify this relatioship across all hues, saturatios, ad brightess levels. Aother way to expad the curret research would be to iclude crystal poies i the data set. Note that, due to the shiftig brightess ad saturatio levels across their coats ad maes while i "crystally" form, crystal poy data may eed to be subjected to closer scrutiy. A study of this special texturig of a crystal poy's colors would make a ice complemet to this research. CONCLUSIONS These results show that there are may pairs of variables that have some level of depedecy o each other. The actual relatioships are yet to be determied, ad could be very subtle. For the pairs of variables that passed the distace covariace test for idepedece, it ca be said with a good degree of cofidece that those variables are idepedet of each other. DATA BIAS, RESEARCH CHALLENGES, AND ROOM FOR IMPROVEMENT There are a couple of biases i the data that are worth metioig. First, ote that all of the hue histograms have a peculiar spike at a hue of 0 o. This is because whe the saturatio of a color is 0%, hue becomes irrelevat ad defaults to 0 o. Obviously, this meas that the algorithm will produce more red poies tha are preset i a true Equestria populatio. Such a bias might be filtered out by redistributig the hues of all the grey poies. Secodly, although the list of poies was maually scrutiized before data extractio, mildlytited images may still be preset i the data. A commo tit that is applied for ighttime scees makes the poies appear bluer, so it is suspected that the hue histograms are slightly biased towards blue. Distace covariace tests were performed to see whether the data types could be show mathematically to be idepedet. A umber of data pairs passed the test, but ufortuately, may of the pairigs did ot. That is, there is likely some relatioship (though evidetly weak) betwee those data pairs. A thorough uderstadig of all the relatioships is lackig ad may be very difficult to ail dow. There are obviously may ways to expad the curret study ad refie its results. Eye color data ca be easily icorporated ito the study, as it is icluded alog with the coat ad mae data i the List of Poies wiki. Other colors, such as secodary mae colors ad character outlie colors, would require additioal work to icorporate sice these colors are ot The histograms reveal the shapes of the probability desity fuctios for the hue, saturatio, ad brightess of Equestria poies' coats ad maes. The rage of hues from ~100 o to ~160 o (gree-turquoise) is curiously avoided for all poies. It has bee suggested by The L-Trai that these hues might be avoided sice they are similar to those used for the grass ad tree leaves that are commo to the settig of Poyville [5]. A amusig implicatio of this is that camouflage has played a egative evolutioary impact o poykid. Additioally, the rage of hues from ~60 o to ~340 o is dimiished for the stallios. Perhaps this rage was avoided for the male characters because it icludes the various hues of pik, a color traditioally associated with femiiity. A computer algorithm, makig a couple of simplifyig assumptios, ca be desiged to produce poy colors i a appropriate rage to match the demographics of Equestria's poy populatio. The less-tha-perfect aesthetics of the color schemes idicates that the algorithm's assumptios may be a little flawed, ad distace covariace tests ideed idicate that several pairs of variables are ot truly idepedet. A closer study of the relatioships betwee the variables could lead to a improved algorithm. Still, the colors produced by this simple desig are ot very far off. REFERENCES [1] "List of poies," accessed Oct. 11, 015, [] Coles, P., "Full Page Scree Capture," 015, retrieved from [3] Basu, S. "FreeMat," stable release versio 4., Jue 30, 013, [4] Szekey, G., Rizzo, M., ad Bakirov, N, "Measurig ad Testig Depedece by Correlatio of Distaces," The Aals of Statistics, Vol 35, No 6 (007): [5] The L-Trai, electroic message to author, October 13, 015

7 APPENDIX A: EXTRACTED DATA FROM THE LIST OF PONIES WIKI Mare Data

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9 Stallio Data:

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11 APPENDIX B: 00 GENERATED PONY COLORS Mares Stallios