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Hajime Nanjo, Nobuhiro Shimizu, Taku Yamanaka 2017-12-28 Kuno

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Hajime Nanjo, Nobuhiro Shimizu, Taku Yamanaka 2017-12-28 Kuno How many days would be holidays if we respect the birthdays of all the past emperors? Hajime Nanjo, Nobuhiro Shimizu, Taku Yamanaka 2017-12-28 Kuno and Yamanaka Groups End-of-the Year Presentation Introduction • 125 emperors in Japan in the past • Birthdays of 3 emperors are holidays • Meiji Emperor : Nov. 3, Culture Day • Showa Emperor : Apr. 29, Showa Day • Current Emperor : Dec. 23, The Emperor’s Birthday 2 Question • If we make the birthdays of all the past 125 emperors as holidays, how many “Emperor’s birthday” holidays will we have? • N = 365 (#days/year) • M = 125 (#emperors) • n : #holidays 3 1. Brute-force method • Split emperors into multiple days • Example: 7 emperors . Day A Day A B Day A B Day A B C D (7) (1+6) (2+5) (1+2+2+2) 1 day 2 days 24 days 4 days Example: #cases for the splitting pattern • 365C1 : pick 1 day Day A B C D • × 7C1 : pick 1 emperor (1+2+2+2) × 365-1C3 : pick 3 days 4 days • • × 7-1C2x3 : pick 2x3 emperors • ×(2x3)! : Line up emperors • × 1/(2!)3 : Remove double-counts = 1,833,153,121,800 5 #holidays 1 2 3 4 5 6 7 partitions (7) (1+6) (1+1+5) (1+1+1+4) (1+1+1+1+3) (1+1+1+1+1+2) (1+1+1+1+1+1+1) (2+5) (1+2+4) (1+1+2+3) (1+1+1+2+2) (3+4) (1+3+3) (1+2+2+2) Partition Number (分割数) = 15 (2+2+3) #cases 365 837018 1.45E+10 6.11E+12 8.82E+14 4.76E+16 8.15E+17 × 1 / 365 7 Probability 1 7 #holidays 6 Problem of Partition No. 109 • It is LARGE! • Took 100 min for M=90. • Expect 80 hours for M=125 Partition Number 100 0 N 100 7 2. Monte Carlo Method • Monte Carlo method can be used to calculate the expected number of holidays M • M emperors are uniformly assigned to 365 boxes. Jan. Dec. 1st 31st … hol hol hol … hol ß Number of holidays is immediately obtained. 8 N histories More realistic distribution Japanese government provides the statistic information of fraction of birth-month. † The assignment of birth days is weighted according to this fraction. fraction 1 3 5 7 9 11 The real distribution is NOT uniform at all! Jan … Sep Dec † http://www.mhlw.go.jp/toukei/saikin/hw/jinkou/tokusyu/syussyo-4/syussyo1-2.html N histories Expected number of holidays decreases by only 0.03 days (43 minutes). à Relax! Still we can take sufficient number of holidays even in this realistic situation. 3. Recurrence Formula となる。ただし、差し引く日数の最大値は n 1 と M のうち小さい方であるため、 − Q(M | n) : The number of cases for distributing M emperors • min(M,n 1) to n specific days − Mall in 1 day all in 2 days Q(M n)=M n nCi Q(M i) (2) Q(M|n)=n nC−1 Q(M 1) nC2 Q(M 2) | | − i=1| − | ··· min(M,n 1)! − = nM C Q(M i) と表せる。 − n i | i=1 N n M X 次に、１年 日のうち、• Probability日に that M emperors人が入る確率は fit in n days out of N days in a year Q(M n) P (M,N n)= C | (3) | N n N M である。したがって、漸化式の式• Used Python (infinite(2) を用いて #digits for iintegers!)M について Q(M i) を求めておけば、 12 ≤ | 祝日の日数の確率分布 P (M,N n) が計算できる。 | 3.2 漸化式の計算 Q(M i) は整数であるので正確に計算できるが、N M = 365125 10320 という大きな | ≃ 数を扱う。そこで、整数の桁数に制限のない Python を用いて計算した。P (M,N n) の | 計算では実数で割り算するために、割る数と割られる数 (整数) を、Python で表せる実数 O(10308) の範囲に収まるまで 10100(整数) で約分した。 図 2 に、漸化式で求めた天皇誕生日の日数の確率分布 P (125, 365 n) を示す。期待値は | 106.0 日であり、分布の RMS は 3.5 日である。この結果は、＊＊＊＊＊の方法で計算して 求めた結果と一致する。 図 2: 漸化式を用いた、天皇誕生日の日数の確率分布。 3 Result of recurrence formula 0.1 M = 125 emperors <n> = 106.0 days RMS = 3.5 days Probability 0 0 100 n (#holidays)13 Result of recurrence formula 10-11 Probability 10-186 0 100 n (#holidays)14 4. Simple calculation 0 1 2 3 i N-1 Day 0 × × ⚪ × … × N(365) patterns/row 1 × × × × … × 2 × × × × … × j × × × × … × M-1 M Pworkday =(1 1/N ) − M Emperor ID Pholiday =1 (1 1/N ) 15 − − Result of simple calculation • Binomial distribution : N,Pholiday → P(n) • Mean = N x Pholiday = 365 x (1-(1-1/365)125 )=106 days • RMS = √ NPholiday(1-Pholiday) = 8.7 days 16 <n> Comparisons • Recurrence formula and the simple formula agree up to 12 digits. But why n C Q(M n)/N M = N 1 (1 1/N )M ? · N n | − − n • MC is consistentX with the recurrence formula⇥ ⇤ 300 <n> formula MC 0.04 MC-formula 0 M 17 1000 RMS Comparisons • RMS of the simple formula does not agree formula 8 MC + simple RMS 0 0 1000 M 18 Number of holidays 25000 M=50 Mean:46.7837 M=125 RMS :1.63511 20000 M=200 Mean:105.953 M=300 RMS :3.4784 M=500 15000 Mean:154.141 Number of holidays RMS :4.70842 25000 Mean:204.701 10000 M=50 Mean:46.7837 RMS :5.6498 M=125 RMS :1.63511 Mean:272.409 20000 M=200 5000 Mean:105.953 RMS :6.0882 M=300 RMS :3.4784 M=500 15000 0 Mean:154.141 0 50 100 150 200 250 300 350 RMS :4.70842 Mean:204.701 10000 図 4: 実際の誕生月の割合を考慮して，シミュレーションした祝日の数の分布。赤 RMS :5.6498 (M = 50)， Mean:272.409 黒 (M = 125) ，マゼンタ (M = 200)，青 (M = 300)，緑 (M = 500) 5000 RMS :6.0882 認識できる。 0 0 50 100 150 200 250 300 350 4 漸化式を用いた方法 図 4: 実際の誕生月の割合を考慮して，シミュレーションした祝日の数の分布。赤 (M = 50)， 4.1 漸化式の導出黒 (M = 125) ，マゼンタ (M = 200)，青 (M = 300)，緑 (M = 500) 次に、漸化式を用いた方法を考える。Q(M n) を、ある特定の n 日に M 人を分配する | 場合の数とする。認識できる。M 人それぞれが n 通り選ぶ選び方は nM 通りあるが、これは M 人全員 が 1 日に集中してしまう場合や、2 日間に集中してしまう場合などを含む。したがってそ れらの寄与を差し引くと、 4 漸化式を用いた方法 Q(M n)=nM C Q(M 1) C Q(M 2) (1) | − n 1 | − n 2 | − ··· 4.1 漸化式の導出 となる。ただし、差し引く日数の最大値は n 1 と M のうち小さい方であるため、 次に、漸化式を用いた方法を考える。− Q(M n) を、ある特定の n 日に M 人を分配する Conclusionmin(M,n 1) | − M 場合の数とする。M 人それぞれがM n 通り選ぶ選び方は n 通りあるが、これは M 人全員 Q(M n)=n nCMi Q(M i) (2) が 1 日に集中してしまう場合や、• #holidays| = N −12 日間に集中してしまう場合などを含む。したがってそ(1 1/N ) | − i=1− ! れらの寄与を差し引くと、 106.0⇥ ± 3.5 days (for ⇤125 emperors) と表せる。 この Q(M n) を用いると、１年ProbabilityQ(M n)=N distribution日のうち、nM functionnC日にQ(M M= 人が入る確率は1) C Q(M 2) (1) | • | − n 1 | − n 2 | − ··· Q(M n) となる。ただし、差し引く日数の最大値はP (M,N n)= C n |1 と M のうち小さい方であるため、(3) | N n N−M min(M,n 1) − である。したがって、漸化式の式 (2) を用いて QM(M i) を i =1から M まで順に求めてお Q(M n)=n | nCi Q(M i) (2) けば、祝日の日数の確率分布 P (M,N| n) を計算できる。− | | !i=1 と表せる。 • RMS needs the recurrence5 formula or MC この Q(M n) を用いると、１年 N 日のうち、n 日に M 人が入る確率は | 19 Q(M n) P (M,N n)= C | (3) | N n N M である。したがって、漸化式の式 (2) を用いて Q(M i) を i =1から M まで順に求めてお | けば、祝日の日数の確率分布 P (M,N n) を計算できる。 | 5 Conclusion • Depending on how you approach/think, the problem can be • extremely difficult and time consuming, or • extremely simple and quick • Monte Carlo is easy and robust • Writeup 「全ての天皇の誕生日を祝日にすると何日休みにな るか」is available on the program page 20 Happy Holidays! 21 Backup 22 View this page in: English Translate Turn off for: Japanese Check with data Options ▼ View this page in: English Translate Turn off for: Japanese Options ▼ Web member registration Web member registration Select Language AKB48 Group ID Password Select Language Chinese Japanese AKB48 Group ID Password Member Schedule Goods Ticket AKB48 Theater Discography Official mobile Fan club Handshake event Blog Chinese Japanese TOP > Member Display all members Narrow your search Input a member name Team A Team K 入山 杏奈 大家 志津香 小嶋 菜月 佐々木 優佳里 Team B Anna Iriyama Shizuka Oya Natsuki Kojima Yukari Sasaki 1995.12.03 1991.12.28 1995.03.08 1995.08.28 Member Schedule Goods Ticket AKB48Team 4Theater AKB48 Team A AKB48 TeamDiscography A AKB48 Team A AKB48 Team A MOfficial = 138 mobile members Fan http://oshitan.com/akb48/club Handshake event Blog Team 8 • 研究生 Date of birth 白間 美瑠 田北 香世子 谷口 めぐ 中西 智代梨 Miru Shiroma Kayoko Takita Megu Taniguchi Chiyori Nakanishi n = 114 different birthdays 1997.10.14 1997.02.13 1998.11.12 1995.05.12 • AKB48 Team A / AKB48 Team A AKB48 Team A AKB48 Team A 「レッツゴー研究生！」公演 NMB48 Team M 「レッツゴー研究生！」公演 2017年12月24日(日) TOP > Member How to apply for tickets • expected n = 115.0 ± 3.7 樋渡 結依 宮崎 美穂 宮脇 咲良 横山 由依 Yui Hiwatashi Miho Miyazaki Sakura Miyawaki Yui Yokoyama 2000.04.30 1993.07.30 1998.03.19 1992.12.08 AKB48 Team A AKB48 Team A AKB48 Team A / AKB48 Team A HKT48 Team K IV Display all members 市川 愛美 久保 怜音 兒玉 遥 篠崎 彩奈 Narrow your search Manami Ichikawa Satone Kubo Haruka Kodama Ayana Shinozaki 1999.08.26 2003.11.20 1996.09.19 1996.01.08 AKB48 Team K AKB48 Team K AKB48 Team K / AKB48 Team K HKT48 Team H Input a member name https://www.akb48.co.jp/about/members/ 23 Team A Team K 入山 杏奈 大家 志津香 小嶋 菜月 佐々木 優佳里 Team B Anna Iriyama Shizuka Oya Natsuki Kojima Yukari Sasaki 1995.12.03 1991.12.28 1995.03.08 1995.08.28 Team 4 AKB48 Team A AKB48 Team A AKB48 Team A AKB48 Team A Team 8 研究生 Date of birth 白間 美瑠 田北 香世子 谷口 めぐ 中西 智代梨 Miru Shiroma Kayoko Takita Megu Taniguchi Chiyori Nakanishi 1997.10.14 1997.02.13 1998.11.12 1995.05.12 AKB48 Team A / AKB48 Team A AKB48 Team A AKB48 Team A 「レッツゴー研究生！」公演「レッツゴー研究生！」公演 NMB48 Team M 「レッツゴー研究生！」公演「レッツゴー研究生！」公演 2017年12月24日(日) How to apply for tickets 樋渡 結依 宮崎 美穂 宮脇 咲良 横山 由依 Yui Hiwatashi Miho Miyazaki Sakura Miyawaki Yui Yokoyama 2000.04.30 1993.07.30 1998.03.19 1992.12.08 AKB48 Team A AKB48 Team A AKB48 Team A / AKB48 Team A HKT48 Team K IV 市川 愛美 久保 怜音 兒玉 遥 篠崎 彩奈 Manami Ichikawa Satone Kubo Haruka Kodama Ayana Shinozaki 1999.08.26 2003.11.20 1996.09.19 1996.01.08 AKB48 Team K AKB48 Team K AKB48 Team K / AKB48 Team K HKT48 Team H Check with data sumo, basketball, boxing, volleyball politicians 400 wrestling actors/actresses movie directors baseball players medical scholars tennis players jurists translators boat racers jockeys https://www.tanjoubi.org/birth/ golf players Takarazuka http://zukaholic.3album.net/student/index/S_birthday/ 100 AKB48 n (days) NMB48, SKE48 HelloPro http://oshitan.com HKT48 40 Nogizaka46 102 M 103 104 24 Differences between data and calculations RMS 5 0 -5 n (data - expected) 102 M 103 104 25 5.
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