r - 使用tsclean从R中的多个时间序列中删除异常值(Remove outliers from multiple timeseries in R using tsclean)

Question

I would like to clean multiple time series of outliers in R .

(我想清除R多个时间序列离群值。)

The forecast::tsclean() function works well for univariate time series data (a single time series of observations) but tsclean() does not work for multivariate timeseries (many different timeseries of observations over a given period).

(forecast::tsclean()函数适用于单变量时间序列数据（单个时间序列的观测值），但tsclean()不适用于多元时间序列（给定时间段内许多不同的观测时间序列）。)

I would prefer to do this following the tidyverse as I am using tsibbles .

(我更喜欢在tidyverse按照我使用的tsibbles 。)

I am aware of the feasts package which makes working with multivariate time series easier, but I do not believe it has adapted the tsclean function.

(我知道feasts包使使用多元时间序列更容易，但是我不相信它已经适应了tsclean函数。)

I provide an example where I attempt two solutions, which both fail, which highlights my problem:

(我提供了一个示例，其中我尝试了两个都失败的解决方案，这突出了我的问题：)

• Attempt 1 naively applies the tsclean function to the tsibble which fails.

  (Attempt 1天真地将tsclean函数应用于失败的tsibble 。)

• Attempt 2 tries to exploit that tsclean works on univariate series, by applying the group_by_key() function in order to deal with each of the many time series separately.

  (Attempt 2尝试通过应用group_by_key()函数来利用tsclean在单变量序列上工作，以便分别处理多个时间序列中的每个时间序列。)

library(fable)
#> Loading required package: fabletools
library(tidyverse)

df <- structure(list(date = structure(c(
  11413, 11504, 11596, 11688,
  11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509,
  12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330,
  13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153,
  14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975,
  15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796,
  15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617,
  16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440,
  17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778,
  11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600,
  12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422,
  13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245,
  14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065,
  15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887,
  15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709,
  16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532,
  17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869,
  11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692,
  12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514,
  13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335,
  14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156,
  15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979,
  16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801,
  16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622,
  17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961,
  12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784,
  12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604,
  13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426,
  14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248,
  15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071,
  16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892,
  16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713,
  17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961, 12053,
  12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784, 12874,
  12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604, 13695,
  13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426, 14518,
  14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248, 15340,
  15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071, 16161,
  16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892, 16983,
  17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713, 17805,
  11413, 11504, 11596, 11688, 11778, 11869, 11961, 12053, 12143,
  12234, 12326, 12418, 12509, 12600, 12692, 12784, 12874, 12965,
  13057, 13149, 13239, 13330, 13422, 13514, 13604, 13695, 13787,
  13879, 13970, 14061, 14153, 14245, 14335, 14426, 14518, 14610,
  14700, 14791, 14883, 14975, 15065, 15156, 15248, 15340, 15431,
  15522, 15614, 15706, 15796, 15887, 15979, 16071, 16161, 16252,
  16344, 16436, 16526, 16617, 16709, 16801, 16892, 16983, 17075,
  17167, 17257, 17348, 17440, 17532, 17622, 17713, 17805, 11413,
  11504, 11596, 11688, 11778, 11869, 11961, 12053, 12143, 12234,
  12326, 12418, 12509, 12600, 12692, 12784, 12874, 12965, 13057,
  13149, 13239, 13330, 13422, 13514, 13604, 13695, 13787, 13879,
  13970, 14061, 14153, 14245, 14335, 14426, 14518, 14610, 14700,
  14791, 14883, 14975, 15065, 15156, 15248, 15340, 15431, 15522,
  15614, 15706, 15796, 15887, 15979, 16071, 16161, 16252, 16344,
  16436, 16526, 16617, 16709, 16801, 16892, 16983, 17075, 17167,
  17257, 17348, 17440, 17532, 17622, 17713, 17805, 11413, 11504,
  11596, 11688, 11778, 11869, 11961, 12053, 12143, 12234, 12326,
  12418, 12509, 12600, 12692, 12784, 12874, 12965, 13057, 13149,
  13239, 13330, 13422, 13514, 13604, 13695, 13787, 13879, 13970,
  14061, 14153, 14245, 14335, 14426, 14518, 14610, 14700, 14791,
  14883, 14975, 15065, 15156, 15248, 15340, 15431, 15522, 15614,
  15706, 15796, 15887, 15979, 16071, 16161, 16252, 16344, 16436,
  16526, 16617, 16709, 16801, 16892, 16983, 17075, 17167, 17257,
  17348, 17440, 17532, 17622, 17713, 17805, 11413, 11504, 11596,
  11688, 11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418,
  12509, 12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239,
  13330, 13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061,
  14153, 14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883,
  14975, 15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706,
  15796, 15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526,
  16617, 16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348,
  17440, 17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688,
  11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509,
  12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330,
  13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153,
  14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975,
  15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796,
  15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617,
  16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440,
  17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778,
  11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600,
  12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422,
  13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245,
  14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065,
  15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887,
  15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709,
  16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532,
  17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869,
  11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692,
  12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514,
  13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335,
  14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156,
  15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979,
  16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801,
  16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622,
  17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961,
  12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784,
  12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604,
  13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426,
  14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248,
  15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071,
  16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892,
  16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713,
  17805
), class = c("yearquarter", "Date")), Series = c(
  "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "10

Answer 1

Here is my solution, which could be improved in various dimensions to keep it tidy .

(这是我的解决方案，可以在各个方面进行改进，以使其保持整洁 。)

It requires a lot of manual tinkering:

(它需要大量的手动修改：)

• I convert the tsibble to a ts object and specify dates manually for the ts .

  (我将tsibble转换为ts对象，并手动为ts指定日期。)

  This seems inefficient.

  (这似乎效率低下。)

• I specify the columns manually df_unclean[, 2:14] in order to apply tsclean .

  (为了应用tsclean df_unclean[, 2:14]我手动指定了df_unclean[, 2:14]列。)

library(fable)
#> Loading required package: fabletools
library(tidyverse)

df <- structure(list(date = structure(c(
  11413, 11504, 11596, 11688,
  11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509,
  12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330,
  13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153,
  14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975,
  15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796,
  15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617,
  16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440,
  17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778,
  11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600,
  12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422,
  13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245,
  14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065,
  15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887,
  15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709,
  16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532,
  17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869,
  11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692,
  12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514,
  13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335,
  14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156,
  15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979,
  16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801,
  16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622,
  17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961,
  12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784,
  12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604,
  13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426,
  14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248,
  15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071,
  16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892,
  16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713,
  17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961, 12053,
  12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784, 12874,
  12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604, 13695,
  13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426, 14518,
  14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248, 15340,
  15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071, 16161,
  16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892, 16983,
  17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713, 17805,
  11413, 11504, 11596, 11688, 11778, 11869, 11961, 12053, 12143,
  12234, 12326, 12418, 12509, 12600, 12692, 12784, 12874, 12965,
  13057, 13149, 13239, 13330, 13422, 13514, 13604, 13695, 13787,
  13879, 13970, 14061, 14153, 14245, 14335, 14426, 14518, 14610,
  14700, 14791, 14883, 14975, 15065, 15156, 15248, 15340, 15431,
  15522, 15614, 15706, 15796, 15887, 15979, 16071, 16161, 16252,
  16344, 16436, 16526, 16617, 16709, 16801, 16892, 16983, 17075,
  17167, 17257, 17348, 17440, 17532, 17622, 17713, 17805, 11413,
  11504, 11596, 11688, 11778, 11869, 11961, 12053, 12143, 12234,
  12326, 12418, 12509, 12600, 12692, 12784, 12874, 12965, 13057,
  13149, 13239, 13330, 13422, 13514, 13604, 13695, 13787, 13879,
  13970, 14061, 14153, 14245, 14335, 14426, 14518, 14610, 14700,
  14791, 14883, 14975, 15065, 15156, 15248, 15340, 15431, 15522,
  15614, 15706, 15796, 15887, 15979, 16071, 16161, 16252, 16344,
  16436, 16526, 16617, 16709, 16801, 16892, 16983, 17075, 17167,
  17257, 17348, 17440, 17532, 17622, 17713, 17805, 11413, 11504,
  11596, 11688, 11778, 11869, 11961, 12053, 12143, 12234, 12326,
  12418, 12509, 12600, 12692, 12784, 12874, 12965, 13057, 13149,
  13239, 13330, 13422, 13514, 13604, 13695, 13787, 13879, 13970,
  14061, 14153, 14245, 14335, 14426, 14518, 14610, 14700, 14791,
  14883, 14975, 15065, 15156, 15248, 15340, 15431, 15522, 15614,
  15706, 15796, 15887, 15979, 16071, 16161, 16252, 16344, 16436,
  16526, 16617, 16709, 16801, 16892, 16983, 17075, 17167, 17257,
  17348, 17440, 17532, 17622, 17713, 17805, 11413, 11504, 11596,
  11688, 11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418,
  12509, 12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239,
  13330, 13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061,
  14153, 14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883,
  14975, 15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706,
  15796, 15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526,
  16617, 16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348,
  17440, 17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688,
  11778, 11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509,
  12600, 12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330,
  13422, 13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153,
  14245, 14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975,
  15065, 15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796,
  15887, 15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617,
  16709, 16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440,
  17532, 17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778,
  11869, 11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600,
  12692, 12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422,
  13514, 13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245,
  14335, 14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065,
  15156, 15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887,
  15979, 16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709,
  16801, 16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532,
  17622, 17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869,
  11961, 12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692,
  12784, 12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514,
  13604, 13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335,
  14426, 14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156,
  15248, 15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979,
  16071, 16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801,
  16892, 16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622,
  17713, 17805, 11413, 11504, 11596, 11688, 11778, 11869, 11961,
  12053, 12143, 12234, 12326, 12418, 12509, 12600, 12692, 12784,
  12874, 12965, 13057, 13149, 13239, 13330, 13422, 13514, 13604,
  13695, 13787, 13879, 13970, 14061, 14153, 14245, 14335, 14426,
  14518, 14610, 14700, 14791, 14883, 14975, 15065, 15156, 15248,
  15340, 15431, 15522, 15614, 15706, 15796, 15887, 15979, 16071,
  16161, 16252, 16344, 16436, 16526, 16617, 16709, 16801, 16892,
  16983, 17075, 17167, 17257, 17348, 17440, 17532, 17622, 17713,
  17805
), class = c("yearquarter", "Date")), Series = c(
  "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "10", "10", "10", "10", "10", "10", "10", "10",
  "10", "10", "10", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "101", "101", "101", "101", "101", "101", "101", "101",
  "101", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1011", "1011", "1011", "1011", "1011", "1011", "1011", "1011",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1012",
  "1012", "1012", "1012", "1012", "1012", "1012", "1012", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "1013", "1013",
  "1013", "1013", "1013", "1013", "1013", "1013", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "102", "102", "102",
  "102", "102", "102", "102", "102", "102", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1021", "1021", "1021", "1021",
  "1021", "1021", "1021", "1021", "1022", "1022", "1022", "1022",
  "1022", "1022", "1022", "1022", "1022", "1022", "1022", "1022",
  "1022", "1022", "1022", "1022", "1022", "1022", "1022", "1022",
  "1022", "1022", "1022", "1022", "1022", "1022", "1022", "1022",
  "1022", "1022", "1022", "1022", "1022", "1022", "1022", "1022",
  "1022", "1022", "1022", "1022", "1022", "1022", "102