Smogon Community Correlation Analysis of Pokemon Usages
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 May 17th, 2009, 11:41:27 AM #1 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta Correlation Analysis of Pokemon Usages Here is a further statistic that I like to crank out from time to time. This time it regards looking for correlation in Pokemon usages. What is correlation, though? Two Pokemon's usages are said to correlate if the usage increase or decrease of one of them affects the usage increase or decrease of the other. There are two types of correlation: direct correlation and inverse correlation. Two Pokemon's usages are said to exhibit direct correlation if, when the usage of the first Pokemon increases, the usage of the second one increases as well by roughly the same margin, and vice-versa. Two Pokemon's usages are said to exhibit inverse correlation if, when the usage of the first Pokemon increases, the usage of the second one decreases by roughly the same margin, and vice-versa. In statistics, there is an important measure of correlation called Pearson's product-moment correlation coefficient. I didn't use this method of correlation, however; I used a simpler method. What I did was simply to find the difference of (i.e. subtract) the increase or decrease of a Pokemon's usage in the last 6 months from the increase or decrease in usage of the other Pokemon. If this subtraction is quite small, the Pokemon would exhibit usage correlation. I analysed the top 100 Pokemon used in the Standard metagame and the top 50 Pokemon used in the Uber metagame. The reason why the UU metagame was not accounted for is that we don't have enough statistics for its metagame yet to formulate a good correlation analysis out of it. Here are the results. Standard Metagame Correlation: Code: ```Abomasnow Direct: Froslass, Moltres Inverse: Rotom-H Absol: Direct: None Inverse: None Aerodactyl: Direct: None Inverse: Crobat, Forretress, Skarmory Alakazam Direct: Mismagius Inverse: Umbreon Ambipom Direct: None Inverse: None Arcanine Direct: Honchkrow, Registeel, Sceptile Inverse: Dugtrio, Gardevoir, Magmortar Azelf Direct: Weavile Inverse: None Azumarill Direct: Blastoise Inverse: None Blastoise Direct: Azumarill, Porygon2 Inverse: None Blaziken Direct: Rampardos Inverse: None Blissey Direct: None Inverse: None Breloom Direct: None Inverse: None Bronzong Direct: Heatran, Suicune Inverse: None Cacturne Direct: Froslass Inverse: Crobat Celebi: Direct: None Inverse: None Charizard Direct: None Inverse: None Claydol Direct: None Inverse: None Clefable Direct: Walrein Inverse: None Cresselia Direct: None Inverse: None Crobat Direct: None Inverse: Aerodactyl, Cacturne, Electrode, Kabutops Donphan Direct: None Inverse: None Dragonite Direct: None Inverse: None Dugtrio Direct: Magmortar Inverse: Arcanine, Staraptor Dusknoir Direct: None Inverse: None Electivire: Direct: None Inverse: None Electrode Direct: Kabutops Inverse: Crobat Empoleon Direct: Jolteon Inverse: Rhyperior, Rotom-W Espeon Direct: None Inverse: None Flygon Direct: None Inverse: None Forretress Direct: Skarmory Inverse: Aerodactyl Froslass Direct: Abomasnow, Cacturne Inverse: None Gallade Direct: None Inverse: None Gardevoir Direct: Magmortar Inverse: Arcanine Gengar Direct: Skarmory Inverse: None Glaceon Direct: None Inverse: None Gliscor Direct: Ninjask Inverse: None Gyarados Direct: Starmie Inverse: None Heatran Direct: Bronzong Inverse: Infernape Heracross Direct: None Inverse: None Hippowdon Direct: None Inverse: None Hitmontop Direct: None Inverse: None Honchkrow Direct: Arcanine, Registeel, Sceptile, Typhlosion Inverse: None Houndoom Direct: None Inverse: None Infernape Direct: None Inverse: Heatran Jirachi Direct: None Inverse: None Jolteon Direct: Empoleon Inverse: Rotom-W, Zapdos Kabutops Direct: Electrode Inverse: Crobat Kingdra Direct: None Inverse: None Lanturn Direct: None Inverse: None Latias Direct: None Inverse: None Lucario Direct: None Inverse: Magnezone Ludicolo Direct: Tangrowth Inverse: Porygon-Z, Roserade Machamp Direct: None Inverse: None Magmortar Direct: Dugtrio, Gardevoir, Medicham Inverse: Arcanine Magnezone Direct: None Inverse: Lucario Mamoswine Direct: None Inverse: None Medicham Direct: Magmortar, Spiritomb Inverse: None Metagross Direct: None Inverse: None Milotic Direct: None Inverse: None Mismagius Direct: Alakazam Inverse: Umbreon Moltres Direct: Abomasnow Inverse: None Ninjask Direct: Gliscor, Smeargle Inverse: None Porygon2 Direct: Blastoise, Sceptile Inverse: None Porygon-Z Direct: None Inverse: Ludicolo Raikou Direct: Rampardos Inverse: None Rampardos Direct: Blaziken, Raikou Inverse: None Registeel Direct: Arcanine, Honchkrow Inverse: None Rhyperior Direct: None Inverse: Empoleon Roserade Direct: None Inverse: Ludicolo, Rotom-C Rotom-C Direct: None Inverse: Roserade Rotom-H Direct: None Inverse: Abomasnow Rotom-W Direct: None Inverse: Empoleon, Jolteon Salamence Direct: None Inverse: None Sceptile Direct: Arcanine, Honchkrow, Porygon2 Inverse: None Scizor Direct: None Inverse: None Shaymin Direct: None Inverse: Snorlax, Vaporeon Shuckle Direct: Walrein Inverse: None Skarmory Direct: Forretress, Gengar Inverse: Aerodactyl Slowbro Direct: None Inverse: None Smeargle Direct: Ninjask Inverse: None Snorlax Direct: None Inverse: Shaymin Spiritomb Direct: Medicham Inverse: None Staraptor Direct: None Inverse: Dugtrio Starmie Direct: Gyarados Inverse: None Suicune Direct: Bronzong Inverse: None Swampert Direct: None Inverse: None Tangrowth Direct: Ludicolo Inverse: None Tentacruel Direct: None Inverse: None Togekiss Direct: None Inverse: None Torterra Direct: None Inverse: None Typhlosion Direct: Honchkrow Inverse: None Tyranitar Direct: None Inverse: None Umbreon Direct: None Inverse: Alakazam, Mismagius, Weezing Uxie Direct: None Inverse: None Vaporeon Direct: None Inverse: Shaymin Walrein Direct: Shuckle, Clefable Inverse: None Weavile Direct: Azelf Inverse: None Weezing Direct: None Inverse: Umbreon Yanmega Direct: None Inverse: None Zapdos Direct: None Inverse: Jolteon``` Uber Metagame Correlation: Code: ```Blissey Direct: None Inverse: None Bronzong Direct: None Inverse: None Celebi Direct: None Inverse: None Darkrai Direct: None Inverse: None Deoxys-S Direct: Groudon Inverse: None Deoxys-A Direct: None Inverse: None Deoxys-D Direct: Gengar Inverse: None Dialga Direct: None Inverse: None Forretress Direct: None Inverse: None Garchomp Direct: None Inverse: None Gengar Direct: Deoxys-D Inverse: None Giratina Direct: None Inverse: None Giratina-O Direct: Weavile Inverse: Metagross Groudon Direct: Deoxys-S Inverse: None Gyarados Direct: Manaphy Inverse: None Heatran Direct: None Inverse: None Heracross Direct: None Inverse: None Ho-oh Direct: Quagsire Inverse: Primeape, Toxicroak Infernape Direct: Manaphy Inverse: None Jirachi Direct: None Inverse: None Jumpluff Direct: None Inverse: None Kingdra Direct: None Inverse: None Kyogre Direct: None Inverse: None Latias Direct: Lugia Inverse: None Latios Direct: None Inverse: None Lucario Direct: None Inverse: None Ludicolo Direct: None Inverse: None Lugia Direct: Latias Inverse: None Manaphy Direct: Gyarados, Infernape Inverse: Primeape, Smeargle Metagross Direct: None Inverse: Giratina-O Mew Direct: None Inverse: None Mewtwo Direct: None Inverse: None Palkia Direct: None Inverse: None Primeape Direct: Smeargle Inverse: Ho-oh, Manaphy Quagsire Direct: Ho-oh Inverse: None Rayquaza Direct: None Inverse: None Registeel Direct: None Inverse: None Roserade Direct: None Inverse: None Salamence Direct: None Inverse: None Scizor Direct: None Inverse: None Shaymin-S Direct: None Inverse: None Shedinja Direct: None Inverse: None Skarmory Direct: None Inverse: None Smeargle Direct: Primeape Inverse: Manaphy Starmie Direct: None Inverse: None Toxicroak Direct: None Inverse: Ho-oh Tyranitar Direct: None Inverse: None Weavile Direct: Giratina-O Inverse: None Wobbuffet Direct: None Inverse: None``` Finally, here are some graphs that confirm a few of the above correlations graphically: Heatran and Bronzong exhibit direct correlation in the Standard metagame. Empoleon and Rhyperior exhibit inverse correlation in the Standard metagame. Jolteon and Zapdos exhibit inverse correlation in the Standard metagame. Manaphy and Infernape exhibit direct correlation in the Uber metagame. Deoxys-D and Gengar exhibit direct correlation in the Uber metagame. Metagross and Giratina-O exhibit inverse correlation in the Uber metagame. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
 May 17th, 2009, 12:09:03 PM #2 Seymor   Join Date: Apr 2009 Posts: 15 Graphs could use axis labels.
 May 17th, 2009, 12:09:39 PM #3 eric the espeon maybe I just misunderstood   Join Date: Aug 2007 Posts: 3,695 Could you show how strong or otherwise the correlations are? This could allow your system to detect more slight correlations without exaggerating them. Very cool stats, its interesting to see that because of the massive amount of sample data you can see correlations between some rare Pokemon than would only actually encounter each other in a small % of matches (like Umbreon and Zam). It also seems that a lot of the Pokemon with noticeable correlations are leads, could you possibly do a lead correlation analysis? __________________ For people who like storing things: The Box Reading and LC? LCF, LC Guide, LC Analyses Good channels: #littlecup, #C&C, #1v1, others And for SCMS editors: SCMS group ete on IRC. Goodbye Smogon. Good luck, was fun while it lasted.
 May 17th, 2009, 12:19:39 PM #4 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta About axis labels: the vertical axis is percentage usage, while the horizontal axis is: 1 - Nov, 2 - Dec, 3 - Jan, 4 - Feb, 5 - Mar, 6 - Apr The correlation formula indicates how much they correlate as a number. The nearer the number is to zero, the better the correlation is. I decided not to show any numbers so that I don't confuse the new Smogon user. I could provide the formulae though. I might do a lead correlation analysis, yes - however, not in the near future. It didn't take me long to implement this (on Excel), but it did take me a good deal of time to perfect the formulae. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
 May 17th, 2009, 12:28:00 PM #5 eric the espeon maybe I just misunderstood   Join Date: Aug 2007 Posts: 3,695 Then how about a "simple" table for the less math inclined (what you have done so far), as well as one with the correlation displayed as a number for those who want more in depth info (and more sensitive to slight correlations)? __________________ For people who like storing things: The Box Reading and LC? LCF, LC Guide, LC Analyses Good channels: #littlecup, #C&C, #1v1, others And for SCMS editors: SCMS group ete on IRC. Goodbye Smogon. Good luck, was fun while it lasted.
 May 18th, 2009, 4:17:27 AM #6 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta eric, I could just as well post the 25,000 numbers (that's twenty five thousand) that signify the correlation of each Pokemon with each other. There will basically be two 100x100 tables for OU, and two 50x50 tables for Ubers. I don't know if people _really_ want to browse through a 100x100 table of numbers, which is why I didn't post it (and why Excel is excellent for doing this). I guess that if people want to see it, I'd just upload the Excel sheet for people to download if they feel like. I'll do that after I return home from work, though. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
 May 18th, 2009, 8:23:31 AM #7 eric the espeon maybe I just misunderstood   Join Date: Aug 2007 Posts: 3,695 mm, that would be over the top (though I suppose some people may like it) but a more sensitive readout than given in the OP would be nice. I mean, looking at the top 50 Ubers a vast majority of them are: Direct: None Inverse: None And the same can be said for a good portion of OU, when probably most of them have some weaker correlation with at least a few Pokemon that has not shown up. Making the tables less sensitive could mean that statistical noise creates some odd results, but so long as you can see how strong the correlation is you can judge for yourself which to pay the most attention to. __________________ For people who like storing things: The Box Reading and LC? LCF, LC Guide, LC Analyses Good channels: #littlecup, #C&C, #1v1, others And for SCMS editors: SCMS group ete on IRC. Goodbye Smogon. Good luck, was fun while it lasted.
 May 18th, 2009, 8:43:50 AM #8 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta That's why I might as well provide the correlation for every Pokemon with every Pokemon... because I don't know how much am I going to slacken the criterion to allow two Pokemon to show correlation. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
 May 18th, 2009, 11:06:44 AM #9 eric the espeon maybe I just misunderstood   Join Date: Aug 2007 Posts: 3,695 The problem with just having the entire table could be that most Pokemon have a level of correlation with each other that is not separable from the statistical noise. The ideal situation would be to have a table of all correlations that you can set a "threshold" for notable correlation yourself, and it shows you all the correlations stronger than that value. But having you pick a single (more lenient) value and including the strength of correlation would be awesome. Also, why so few comments? This seems like a pretty interesting set of data already for OU/Ubers players. __________________ For people who like storing things: The Box Reading and LC? LCF, LC Guide, LC Analyses Good channels: #littlecup, #C&C, #1v1, others And for SCMS editors: SCMS group ete on IRC. Goodbye Smogon. Good luck, was fun while it lasted.
 May 18th, 2009, 11:23:48 AM #10 Arzamo     Join Date: Aug 2007 Posts: 317 #vark I find it strange that none of these are exhibiting the predator-prey relationships that occur in real-life ecology. If you have the time, X-Act, maybe you could account for some time delay for some of the obvious ones? The statistics are interesting, but they make me want to form possibly nonexistent reasons for the correlations. The only reason I can right now come up with for the inverse correlations without time delay is that two Pokémon fill the same niche in the metagame, and that explanation only works for a few, e.g. Jolteon and Zapdos, Weezing and Umbreon, Electrode/Aerodactyl/Crobat. And even those only very roughly fill the same niche. __________________ P: 2577-9630-3448; Pt: 5284-7637-0795 | Trade Thread (dead, PM me for a trade for something) Naive Azelf, 27/31/31/26/22/31, HP Ground 70 (RNG'ed, not listed in thread, PM me to trade)
 May 18th, 2009, 4:26:49 PM #11 zarator Don't cry, little one     Join Date: Mar 2008 Posts: 3,461 Italy It's all very interesting, although I think that - as every statistic should remember^^ - correlation is not causation. For example, there's little to no concept link (counter/teampartner and so on) between Blaziken and Rampardos IMO
 May 18th, 2009, 9:59:12 PM #12 Arzamo     Join Date: Aug 2007 Posts: 317 #vark Even though correlation is not necessarily causation, a true correlation always has some explanation (at least in the vast majority of cases). It might be the predator-prey relationship I discussed in my previous post, which is a direct causation, or it may just be a third, unknown Pokémon or set of Pokémon causing the correlation. The confounding factor - the third Pokémon - could also explain correlations between two Pokémon that fill the same niche. __________________ P: 2577-9630-3448; Pt: 5284-7637-0795 | Trade Thread (dead, PM me for a trade for something) Naive Azelf, 27/31/31/26/22/31, HP Ground 70 (RNG'ed, not listed in thread, PM me to trade)
 May 19th, 2009, 3:15:26 AM #13 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta Okay, I'm going to post the criterion I used for two Pokemon to show 'enough' correlation. Say Pokemon A has percentage usages a_1, a_2, a_3, ..., a_n from months 1, 2, 3, ..., n in chronological order, while Pokemon B has percentage usages b_1, b_2, b_3, ..., b_n. My definitions for direct and inverse correlation were: Code: ```Direct Correlation = abs((a_1 + b_2) - (a_2 + b_1)) + abs((a_2 + b_3) - (a_3 + b_2)) + ... + abs((a_(n-1) + b_n) - (a_n + b_(n-1))) Inverse Correlation = abs((a_2 + b_2) - (a_1 + b_1)) + abs((a_3 + b_3) - (a_2 + b_2)) + ... + abs((a_n + b_n) - (a_(n-1) + b_(n-1))) where abs(x) is the unsigned value of x``` The nearer the correlation is to 0, the more correlation is exhibited. A value of 0 shows perfect correlation. The average change between a month usage and the next, m_a, for Pokemon A is defined as follows: Code: `m_a = (abs(a_2 - a_1) + abs(a_3 - a_2) + ... + abs(a_n - a_(n-1))) / (n-1)` Finally, Pokemon A was said to show 'enough' correlation with Pokemon B if Code: `[Direct/Inverse] Strong Correlation < max(m_a / 2, m_b / 2)` Now what I could do to slacken the above definition is Code: `[Direct/Inverse] Weak Correlation < max(m_a, m_b)` I'll try this and see what happens. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
 May 19th, 2009, 5:03:18 PM #14 Imran     Join Date: May 2008 Posts: 1,290 33 / 38 / 28 X-Act, you say that you did not provide a PMCC value. Obviously I do not want to pile more onto your plate, would it be too hard to generate a list of "correlation partners" for each Pokemon? If we could just find the |PMCC| values of each Pokemon with respect to the top 30 or 40 Pokemon in each tier, and then order them largest to smallest (obviously we would want to display the parity of each PMCC value in the list), this would give us an easier way to quantify how well each Pokemon correlates with another, and I believe would be very useful for analysis writing and suspect nominations. From what I remember, PMCC is just a function of three "varience type" calculations, but I do not know how complicated it would be to write a script that could calculate these and order them for you. It might be worth looking into, it would at the very least, mean that we wouldn't necessarily need this arbitary threshold value, and the PMCC is easy to interpret (being simply a value where |x|<1) __________________ Possibly the most interesting thread on Smogon.Bored of having to type coherantly? it just feels nice to be able to whack that thing lol FRIGGIN IDIOTS OF A NATIONS If you are where I am, I'll give you my crsid and we can go to a pub or something :) I make music, please take a listen if you are that way inclined. Here is my Soundcloud and my Youtube.
 May 20th, 2009, 1:52:29 AM #15 X-Act np: Biffy Clyro - Shock Shock     Join Date: Feb 2006 Posts: 4,679 Malta First of all, PMCC is not simpler to interpret than my method. For PMCC, the nearer the number is to 1 or -1, the stronger the correlation, but where are you going to take the cut-off? Is it +/-0.9? +/-0.8? Secondly, I did apply PMCC to the Pokemon usages, and found that the correlation generated wasn't as good as I wanted it to be. Basically, my idea of perfect correlation is: if a Pokemon increased by p% usage from one month to the next, the other Pokemon usage should increase or decrease by that same amount. I found that PMCC doesn't cater for this enough, so I discarded it. __________________ http://users.smogon.com/X-Act For all your Pokemon needs (and more!) including: the Defensive EVs applet, the Probabilities of Breeding IVs in Pokemon applet, and the Ratings of Pokemon Base Stats applet (now Version 2.0!). And also the IV to PID applet!
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