Quels problèmes de remodelage peuvent fondre/ne pas résoudre en une seule étape?

Question

reshape2 est un paquet qui permet un puissant ensemble de transformations de données, grâce à son approche de fusion/coulée en deux parties. Cependant, comme tous les outils, il incorpore des hypothèses qui limitent les cas qu'il peut gérer.

Quel problème de reconfiguration de données reshape2 peut pas gérer dans sa forme actuelle?

La réponse idéale comprendra:

• Une description du type de cas d'utilisation où cette forme de données se trouve généralement
• données exemples
• Code pour accomplir la transformation (de préférence en utilisant autant de la transformation avec reshape2 que possible)

Exemple

Les données "larges" sont courantes dans les applications de panneau.

melt.wide <- function(data, id.vars, new.names, sep=".", variable.name="variable", ...) { 
    # Guess number of variables currently wide 
    colnames(data) <- sub(paste0(sep,"$"), "", colnames(data)) 
    wide.vars <- colnames(data)[grep(sep, colnames(data))] 
    n.wide <- str_count(wide.vars, sep) 
    stopifnot(length(new.names)==unique(n.wide)) 
    # Melt 
    data.melt <- melt(data,id.vars=id.vars,measure.vars=wide.vars,...) 
    new <- stack.list(str_split(data.melt$variable,sep)) 
    colnames(new) <- c(variable.name,new.names) 
    data.melt <- subset(data.melt,select=c(-variable)) 
    cbind(data.melt,new) 
} 

choice.vars <- colnames(res)[grep("_",colnames(res))] 
melt.wide(subset(res,select=c("WorkerId",choice.vars)), id.vars="WorkerId", new.names=c("set","option"), sep="_") 

La nouvelle fonction retourne un objet fondu qui peut alors être *cast.

où les données sont:

so <- structure(list(WorkerId = c(12L, 13L, 27L, 25L, 30L, 8L), pio_1_1 = structure(c(2L, 
1L, 2L, 1L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    pio_1_2 = structure(c(1L, 2L, 2L, 2L, 1L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), pio_1_3 = structure(c(1L, 1L, 
    1L, 1L, 2L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    pio_1_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    pio_2_1 = structure(c(1L, 2L, 2L, 1L, 1L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), pio_2_2 = structure(c(1L, 1L, 
    1L, 2L, 1L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    pio_2_3 = structure(c(2L, 2L, 2L, 2L, 2L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), pio_2_4 = structure(c(1L, 1L, 
    1L, 1L, 1L, 1L), .Label = "No", class = "factor"), pio_3_1 = structure(c(2L, 
    2L, 2L, 2L, 2L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    pio_3_2 = structure(c(2L, 1L, 1L, 1L, 2L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), pio_3_3 = structure(c(2L, 1L, 
    2L, 1L, 1L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    pio_3_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    pio_4_1 = structure(c(2L, 1L, 2L, 2L, 1L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), pio_4_2 = structure(c(2L, 2L, 
    2L, 1L, 2L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    pio_4_3 = structure(c(1L, 2L, 1L, 1L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), pio_4_4 = structure(c(1L, 1L, 
    1L, 1L, 1L, 1L), .Label = "No", class = "factor"), caremgmt_1_1 = structure(c(2L, 
    2L, 1L, 2L, 1L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_1_2 = structure(c(1L, 2L, 2L, 2L, 1L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_1_3 = structure(c(1L, 
    1L, 1L, 1L, 2L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_1_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    caremgmt_2_1 = structure(c(2L, 2L, 2L, 2L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_2_2 = structure(c(1L, 
    2L, 1L, 2L, 1L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_2_3 = structure(c(2L, 1L, 2L, 1L, 1L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_2_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), caremgmt_3_1 = structure(c(2L, 
    1L, 2L, 1L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_3_2 = structure(c(2L, 1L, 2L, 2L, 2L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_3_3 = structure(c(2L, 
    2L, 2L, 2L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_3_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    caremgmt_4_1 = structure(c(1L, 1L, 2L, 1L, 2L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_4_2 = structure(c(2L, 
    2L, 2L, 2L, 1L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    caremgmt_4_3 = structure(c(1L, 1L, 1L, 1L, 1L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), caremgmt_4_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), prev_1_1 = structure(c(1L, 
    1L, 2L, 1L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    prev_1_2 = structure(c(1L, 2L, 1L, 2L, 1L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), prev_1_3 = structure(c(2L, 1L, 
    1L, 2L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    prev_1_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    prev_2_1 = structure(c(1L, 1L, 2L, 1L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), prev_2_2 = structure(c(2L, 2L, 
    1L, 2L, 1L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    prev_2_3 = structure(c(1L, 2L, 1L, 1L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), prev_2_4 = structure(c(1L, 1L, 
    1L, 1L, 1L, 1L), .Label = "No", class = "factor"), prev_3_1 = structure(c(1L, 
    2L, 1L, 1L, 2L, 1L), .Label = c("No", "Yes"), class = "factor"), 
    prev_3_2 = structure(c(1L, 1L, 2L, 1L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), prev_3_3 = structure(c(2L, 2L, 
    1L, 2L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    prev_3_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = "No", class = "factor"), 
    prev_4_1 = structure(c(1L, 2L, 2L, 1L, 2L, 2L), .Label = c("No", 
    "Yes"), class = "factor"), prev_4_2 = structure(c(1L, 1L, 
    2L, 1L, 2L, 2L), .Label = c("No", "Yes"), class = "factor"), 
    prev_4_3 = structure(c(1L, 1L, 1L, 2L, 2L, 1L), .Label = c("No", 
    "Yes"), class = "factor"), prev_4_4 = structure(c(1L, 1L, 
    1L, 1L, 1L, 1L), .Label = "No", class = "factor"), price_1_1 = structure(c(30L, 
    12L, 1L, 16L, 28L, 17L), .Label = c("$2,500", "$2,504", "$2,507", 
    "$2,509", "$2,512", "$2,513", "$2,515", "$2,526", "$2,547", 
    "$2,548", "$2,578", "$2,588", "$2,594", "$2,605", "$2,607", 
    "$2,617", "$2,618", "$2,622", "$2,635", "$2,649", "$2,670", 
    "$2,672", "$2,679", "$2,681", "$2,698", "$2,704", "$2,721", 
    "$2,782", "$2,851", "$2,884", "$2,919", "$2,925", "$2,935", 
    "$3,022"), class = "factor"), price_1_2 = structure(c(1L, 
    19L, 5L, 17L, 7L, 1L), .Label = c("$2,500", "$2,501", "$2,502", 
    "$2,504", "$2,513", "$2,515", "$2,517", "$2,532", "$2,535", 
    "$2,558", "$2,564", "$2,571", "$2,575", "$2,578", "$2,608", 
    "$2,633", "$2,634", "$2,675", "$2,678", "$2,687", "$2,730", 
    "$2,806", "$2,827", "$2,848", "$2,891", "$2,901", "$2,923", 
    "$2,933", "$2,937", "$2,958", "$2,987"), class = "factor"), 
    price_1_3 = structure(c(11L, 1L, 1L, 8L, 19L, 14L), .Label = c("$2,500", 
    "$2,504", "$2,507", "$2,513", "$2,516", "$2,518", "$2,564", 
    "$2,579", "$2,580", "$2,583", "$2,584", "$2,592", "$2,604", 
    "$2,608", "$2,639", "$2,643", "$2,646", "$2,665", "$2,667", 
    "$2,695", "$2,698", "$2,709", "$2,710", "$2,713", "$2,714", 
    "$2,750", "$2,757", "$2,876", "$2,978", "$2,984", "$3,024", 
    "$3,059"), class = "factor"), price_1_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "$2,500", class = "factor"), 
    price_2_1 = structure(c(27L, 32L, 19L, 22L, 4L, 26L), .Label = c("$2,500", 
    "$2,504", "$2,505", "$2,510", "$2,511", "$2,512", "$2,515", 
    "$2,517", "$2,518", "$2,529", "$2,533", "$2,537", "$2,551", 
    "$2,553", "$2,574", "$2,593", "$2,600", "$2,605", "$2,608", 
    "$2,612", "$2,613", "$2,618", "$2,639", "$2,657", "$2,714", 
    "$2,730", "$2,747", "$2,764", "$2,771", "$2,773", "$2,813", 
    "$2,859", "$2,901", "$3,019", "$3,037"), class = "factor"), 
    price_2_2 = structure(c(12L, 2L, 1L, 27L, 1L, 7L), .Label = c("$2,500", 
    "$2,502", "$2,510", "$2,514", "$2,515", "$2,516", "$2,517", 
    "$2,518", "$2,520", "$2,521", "$2,523", "$2,536", "$2,544", 
    "$2,575", "$2,583", "$2,592", "$2,602", "$2,624", "$2,644", 
    "$2,652", "$2,662", "$2,677", "$2,720", "$2,761", "$2,765", 
    "$2,770", "$2,772", "$2,835", "$2,873", "$2,911", "$2,950", 
    "$2,962"), class = "factor"), price_2_3 = structure(c(32L, 
    1L, 8L, 33L, 29L, 11L), .Label = c("$2,500", "$2,506", "$2,507", 
    "$2,510", "$2,511", "$2,512", "$2,515", "$2,517", "$2,527", 
    "$2,528", "$2,540", "$2,554", "$2,562", "$2,565", "$2,568", 
    "$2,581", "$2,597", "$2,611", "$2,616", "$2,631", "$2,652", 
    "$2,663", "$2,671", "$2,672", "$2,685", "$2,727", "$2,731", 
    "$2,742", "$2,771", "$2,778", "$2,781", "$2,970", "$2,984", 
    "$2,986", "$3,030"), class = "factor"), price_2_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "$2,500", class = "factor"), 
    price_3_1 = structure(c(24L, 1L, 28L, 7L, 18L, 21L), .Label = c("$2,500", 
    "$2,501", "$2,503", "$2,505", "$2,509", "$2,512", "$2,535", 
    "$2,537", "$2,542", "$2,553", "$2,556", "$2,560", "$2,561", 
    "$2,574", "$2,584", "$2,618", "$2,624", "$2,629", "$2,637", 
    "$2,664", "$2,761", "$2,840", "$2,875", "$2,883", "$2,891", 
    "$2,933", "$2,953", "$2,978", "$3,039", "$3,043", "$3,067" 
    ), class = "factor"), price_3_2 = structure(c(3L, 1L, 5L, 
    19L, 25L, 9L), .Label = c("$2,500", "$2,501", "$2,503", "$2,504", 
    "$2,512", "$2,517", "$2,540", "$2,543", "$2,546", "$2,560", 
    "$2,567", "$2,573", "$2,586", "$2,592", "$2,594", "$2,603", 
    "$2,604", "$2,606", "$2,628", "$2,633", "$2,635", "$2,693", 
    "$2,696", "$2,714", "$2,734", "$2,739", "$2,770", "$2,791", 
    "$2,797", "$2,936", "$2,967", "$3,021", "$3,024"), class = "factor"), 
    price_3_3 = structure(c(26L, 7L, 5L, 32L, 10L, 24L), .Label = c("$2,500", 
    "$2,501", "$2,502", "$2,505", "$2,506", "$2,507", "$2,508", 
    "$2,509", "$2,512", "$2,515", "$2,519", "$2,547", "$2,556", 
    "$2,574", "$2,587", "$2,592", "$2,608", "$2,616", "$2,621", 
    "$2,635", "$2,638", "$2,667", "$2,671", "$2,688", "$2,694", 
    "$2,700", "$2,717", "$2,759", "$2,809", "$2,864", "$2,891", 
    "$2,912", "$3,011", "$3,012"), class = "factor"), price_3_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "$2,500", class = "factor"), 
    price_4_1 = structure(c(29L, 13L, 16L, 24L, 33L, 19L), .Label = c("$2,500", 
    "$2,505", "$2,506", "$2,508", "$2,511", "$2,525", "$2,549", 
    "$2,562", "$2,577", "$2,582", "$2,586", "$2,591", "$2,621", 
    "$2,636", "$2,654", "$2,670", "$2,722", "$2,726", "$2,733", 
    "$2,744", "$2,745", "$2,755", "$2,768", "$2,805", "$2,817", 
    "$2,827", "$2,835", "$2,888", "$2,925", "$2,959", "$3,001", 
    "$3,027", "$3,061", "$3,071"), class = "factor"), price_4_2 = structure(c(33L, 
    31L, 21L, 16L, 25L, 13L), .Label = c("$2,500", "$2,502", 
    "$2,503", "$2,505", "$2,506", "$2,511", "$2,513", "$2,516", 
    "$2,529", "$2,539", "$2,547", "$2,554", "$2,557", "$2,562", 
    "$2,567", "$2,579", "$2,581", "$2,583", "$2,585", "$2,591", 
    "$2,612", "$2,629", "$2,640", "$2,670", "$2,695", "$2,726", 
    "$2,737", "$2,788", "$2,790", "$2,798", "$2,852", "$3,031", 
    "$3,063"), class = "factor"), price_4_3 = structure(c(4L, 
    30L, 4L, 19L, 1L, 27L), .Label = c("$2,500", "$2,504", "$2,507", 
    "$2,509", "$2,511", "$2,512", "$2,514", "$2,516", "$2,543", 
    "$2,552", "$2,562", "$2,575", "$2,578", "$2,581", "$2,594", 
    "$2,614", "$2,615", "$2,617", "$2,636", "$2,640", "$2,641", 
    "$2,652", "$2,749", "$2,755", "$2,805", "$2,812", "$2,867", 
    "$2,906", "$2,910", "$2,917", "$2,924", "$2,927", "$2,961", 
    "$3,028", "$3,053", "$3,054"), class = "factor"), price_4_4 = structure(c(1L, 
    1L, 1L, 1L, 1L, 1L), .Label = "$2,500", class = "factor"), 
    plan_1_1 = structure(c(2L, 2L, 2L, 1L, 1L, 2L), .Label = c("", 
    "X"), class = "factor"), plan_1_2 = structure(c(1L, 1L, 1L, 
    2L, 1L, 1L), .Label = c("", "X"), class = "factor"), plan_1_3 = structure(c(1L, 
    1L, 1L, 1L, 2L, 1L), .Label = c("", "X"), class = "factor"), 
    plan_1_4 = structure(c(1L, 1L, 1L, 1L, 1L, 1L), .Label = c("", 
    "X"), class = "factor"), plan_2_1 = structure(c(1L, 2L, 1L, 
    2L, 2L, 2L), .Label = c("", "X"), class = "factor"), plan_2_2 = structure(c(1L, 
    1L, 2L, 1L, 1L, 1L), .Label = c("", "X"), class = "factor"), 
    plan_2_3 = structure(c(2L, 1L, 1L, 1L, 2L, 1L), .Label = c("", 
    "X"), class = "factor"), plan_2_4 = structure(c(1L, 1L, 1L, 
    1L, 1L, 1L), .Label = c("", "X"), class = "factor"), plan_3_1 = structure(c(1L, 
    2L, 1L, 1L, 2L, 1L), .Label = c("", "X"), class = "factor"), 
    plan_3_2 = structure(c(1L, 1L, 1L, 2L, 1L, 1L), .Label = c("", 
    "X"), class = "factor"), plan_3_3 = structure(c(2L, 1L, 1L, 
    1L, 1L, 2L), .Label = c("", "X"), class = "factor"), plan_3_4 = structure(c(1L, 
    1L, 2L, 1L, 1L, 1L), .Label = c("", "X"), class = "factor"), 
    plan_4_1 = structure(c(2L, 2L, 1L, 1L, 1L, 1L), .Label = c("", 
    "X"), class = "factor"), plan_4_2 = structure(c(2L, 1L, 1L, 
    2L, 1L, 1L), .Label = c("", "X"), class = "factor"), plan_4_3 = structure(c(1L, 
    1L, 1L, 1L, 2L, 2L), .Label = c("", "X"), class = "factor"), 
    plan_4_4 = structure(c(1L, 1L, 2L, 1L, 1L, 1L), .Label = c("", 
    "X"), class = "factor")), .Names = c("WorkerId", "pio_1_1", 
"pio_1_2", "pio_1_3", "pio_1_4", "pio_2_1", "pio_2_2", "pio_2_3", 
"pio_2_4", "pio_3_1", "pio_3_2", "pio_3_3", "pio_3_4", "pio_4_1", 
"pio_4_2", "pio_4_3", "pio_4_4", "caremgmt_1_1", "caremgmt_1_2", 
"caremgmt_1_3", "caremgmt_1_4", "caremgmt_2_1", "caremgmt_2_2", 
"caremgmt_2_3", "caremgmt_2_4", "caremgmt_3_1", "caremgmt_3_2", 
"caremgmt_3_3", "caremgmt_3_4", "caremgmt_4_1", "caremgmt_4_2", 
"caremgmt_4_3", "caremgmt_4_4", "prev_1_1", "prev_1_2", "prev_1_3", 
"prev_1_4", "prev_2_1", "prev_2_2", "prev_2_3", "prev_2_4", "prev_3_1", 
"prev_3_2", "prev_3_3", "prev_3_4", "prev_4_1", "prev_4_2", "prev_4_3", 
"prev_4_4", "price_1_1", "price_1_2", "price_1_3", "price_1_4", 
"price_2_1", "price_2_2", "price_2_3", "price_2_4", "price_3_1", 
"price_3_2", "price_3_3", "price_3_4", "price_4_1", "price_4_2", 
"price_4_3", "price_4_4", "plan_1_1", "plan_1_2", "plan_1_3", 
"plan_1_4", "plan_2_1", "plan_2_2", "plan_2_3", "plan_2_4", "plan_3_1", 
"plan_3_2", "plan_3_3", "plan_3_4", "plan_4_1", "plan_4_2", "plan_4_3", 
"plan_4_4"), row.names = c(NA, 6L), class = "data.frame") 

Answer 1

... presque un an plus tard ...

Cela est venu à l'esprit l'autre jour, et je soupçonne que ce furtif est ce que vous avez essayé pour montrer dans votre exemple, mais malheureusement, votre code d'exemple ne fonctionne pas! Parfois, les choses vont un peu trop loin pour moi lorsque je rends mes données "longues". Parfois, même si ce n'est pas nécessairement ce que l'on appelle "données propres", je préfère avoir un "semi-long" data.frame. Ceci est facilement réalisé en utilisant reshape de base R, mais nécessite quelques étapes supplémentaires avec le paquet "reshape2", comme démontré ci-dessous:

Condition préalable: données d'échantillon.

set.seed(1) 
myDf <- data.frame(
    ID.1 = sample(letters[1:5], 5, replace = TRUE), 
    ID.2 = 1:5, 
    V.1 = sample(10:14, 5, replace = TRUE), 
    V.2 = sample(5:9, 5, replace = TRUE), 
    V.3 = sample(3:14, 5, replace = TRUE), 
    W.1 = sample(LETTERS, 5, replace = TRUE), 
    W.2 = sample(LETTERS, 5, replace = TRUE), 
    W.3 = sample(LETTERS, 5, replace = TRUE) 
) 
myDf 
# ID.1 ID.2 V.1 V.2 V.3 W.1 W.2 W.3 
# 1 b 1 14 6 8 Y K M 
# 2 b 2 14 5 11 F A P 
# 3 c 3 13 8 14 Q J M 
# 4 e 4 13 6 7 D W E 
# 5 b 5 10 8 12 G I V 

La sortie "semi-longue" que je recherche. Facilement réalisé avec reshape de base R.

reshape(myDf, direction = "long", idvar=1:2, varying = 3:ncol(myDf)) 
#  ID.1 ID.2 time V W 
# b.1.1 b 1 1 14 Y 
# b.2.1 b 2 1 14 F 
# c.3.1 c 3 1 13 Q 
# e.4.1 e 4 1 13 D 
# b.5.1 b 5 1 10 G 
# b.1.2 b 1 2 6 K 
# b.2.2 b 2 2 5 A 
# c.3.2 c 3 2 8 J 
# e.4.2 e 4 2 6 W 
# b.5.2 b 5 2 8 I 
# b.1.3 b 1 3 8 M 
# b.2.3 b 2 3 11 P 
# c.3.3 c 3 3 14 M 
# e.4.3 e 4 3 7 E 
# b.5.3 b 5 3 12 V 

melt est très bien si vous vouliez l'équivalent de stack, d'autant plus que les rejets stack toutes factor variables, ce qui est frustrant quand read.table et par défaut à la famille stringsAsFactors = TRUE. (Vous pouvez le faire fonctionner, mais vous devez convertir les colonnes pertinentes en character avant de pouvoir utiliser stack). Mais, ce n'est pas ce que je cherche, notamment à cause de la façon dont il a géré la colonne "variable".

library(reshape2) 
myDfL <- melt(myDf, id.vars=1:2) 
head(myDfL) 
# ID.1 ID.2 variable value 
# 1 b 1  V.1 14 
# 2 b 2  V.1 14 
# 3 c 3  V.1 13 
# 4 e 4  V.1 13 
# 5 b 5  V.1 10 
# 6 b 1  V.2  6 

Pour résoudre ce problème, il faut d'abord diviser la colonne « variable », puis utilisez dcast pour obtenir le même format de sortie que vous obtiendrez de reshape.

myDfL <- cbind(myDfL, colsplit(myDfL$variable, "\\.", names=c("var", "time"))) 
dcast(myDfL, ID.1 + ID.2 + time ~ var, value.var="value") 
# ID.1 ID.2 time V W 
# 1  b 1 1 14 Y 
# 2  b 1 2 6 K 
# 3  b 1 3 8 M 
# 4  b 2 1 14 F 
# 5  b 2 2 5 A 
# 6  b 2 3 11 P 
# 7  b 5 1 10 G 
# 8  b 5 2 8 I 
# 9  b 5 3 12 V 
# 10 c 3 1 13 Q 
# 11 c 3 2 8 J 
# 12 c 3 3 14 M 
# 13 e 4 1 13 D 
# 14 e 4 2 6 W 
# 15 e 4 3 7 E