Data

To illustrate this problem I’m using the dataset flights from Hadley Wickham’s package nycflights13:

# install.packages("nycflights13")
library(nycflights13)
data("flights")

Per the documentation, flights contains data on all flights in 2013 that departed NYC via JFK, LaGuardia, or Newark. The variables we’re interested in here are month, dest, and arr_delay.

library(dplyr)
flights <- flights %>%
  select(month, dest, arr_delay)

bicluster() requires data to be in table format, which is what we will do here using spread() from tidyr after using dplyr to summarize the data since we are analyzing average arrival delay.

library(tidyr)
flights <- flights %>%
  group_by(month, dest) %>%
  summarise(mean_arr_delay = mean(arr_delay, na.rm = TRUE)) %>%
  spread(dest, mean_arr_delay) %>% 
  as.data.frame()

Now we name the rows using the text version of month followed by removing month and converting the data to a matrix.

rownames(flights) <- month.name[flights$month]
flights <- as.matrix(flights[, -1])
flights[1:5, 1:5]

##                ABQ      ACK      ALB ANC       ATL
## January         NA       NA 35.17460  NA  4.152047
## February        NA       NA 17.38889  NA  5.174092
## March           NA       NA 17.16667  NA  7.029286
## April    12.222222       NA 18.00000  NA 11.724280
## May      -6.516129 3.904762 10.19643  NA  8.187036

Biclustering

Now that our data is in correct form we can bicluster it.

We first need to determine how many groups of months (r) and how many groups of destination airports (c) to define. Since rows correspond to months, in this analysis, setting r = 4 may not be a bad idea: create a group for each season/quarter of the year. For the columns (airports) the number of groups is a bit more ambiguous, just like with k-means clustering. Since each continent of the world may take a different amount of time and may have different policies for flights from the US, let’s set c = 6.

Now we’re ready to bicluster. I’ll put the code first and give brief descriptions to selected arguments. This will mostly be a repeat of the documentation for biclustermd().

library(biclustermd)
bc <- biclustermd(data = flights, col_clusters = 6, row_clusters = 4)
bc

## 
##  Data has 1260 values, 11.75% of which are missing
##  11 Iterations
##  Initial SSE = 198251; Final SSE = 83107, a 58.1% reduction
##  Rand similarity used; Indices: Columns (P) = 1, Rows (Q) = 1

biclustermd outputs a list of class biclustermd with useful information. Execute ?biclustermd to view the output in detail.

This list is used to make plots and training data, for example.

Plotting

To plot SSE by iteration, use autoplot.biclustermd_sse(), which outputs points by default:

library(ggplot2)
?autoplot.biclustermd_sse
autoplot(bc$SSE)

Next, cluster similarity measures. If a similarity measure for rows equals one, that means that the row shuffling from the last iteration and second to last iteration are identical. If zero, the two shufflings have nothing in common. The same goes for column similarities. autoplot.biclustermd_sim() plots the RIs by iteration.

?autoplot.biclustermd_sim
autoplot(bc$Similarities)

autoplot.biclustermd() makes visual analysis of biclustering results easy by making a heat map of the clustered data. The algorithm does use randomness, so my results may look different from yours.

?autoplot.biclustermd
autoplot(bc) +
  scale_fill_viridis_c(na.value = "white") +
  labs(x = "Destination Airport", y = "Month", fill = "Average Delay")

Often times it is helpful to run the data through an S-shaped function before plotting. This is easy with autoplot.biclustermd(): set transform_colors = TRUE and specify by what constant you’d like to scale your data by (with c) before running it through a standard normal CDF:

autoplot(bc, transform_colors = TRUE, c = 1/10) +
  scale_fill_viridis_c(na.value = "white", limits = c(0, 1)) +
  labs(x = "Destination Airport", y = "Month", fill = "Average Delay")

To make the results really stand out we can reorder the row and column groups with the reorder argument:

autoplot(bc, transform_colors = TRUE, c = 1/10, reorder = TRUE) +
  scale_fill_viridis_c(na.value = "white", limits = c(0, 1)) +
  labs(x = "Destination Airport", y = "Month", fill = "Average Delay")

Minimization of SSE for a set of parameters

Since the algorithm uses purposeful randomness, it is recommend that analysts run the biclustering multiple times and keep the one with the smallest SSE. The function rep_biclustermd() does exactly that. Arguments to rep_biclustermd() are the same as those for biclustermd() with 3 additional arguments: 1. nrep: the number of times to repeat the biclustering. 2. parallel: a logical indicating if parallelization should be used or not. By default this is FALSE. 3. ncores: the number of cores to parallelize over. Ignored if parallel = FALSE.

Below we run biclustering 100 times without using parallelization.

repeated_bc <- rep_biclustermd(flights, nrep = 100, col_clusters = 6, row_clusters = 4)
repeated_bc

## $best_bc
## 
##  Data has 1260 values, 11.75% of which are missing
##  9 Iterations
##  Initial SSE = 182584; Final SSE = 79965, a 56.2% reduction
##  Rand similarity used; Indices: Columns (P) = 1, Rows (Q) = 1
## 
## $rep_sse
##   [1] 87317.68 80467.22 83597.98 86136.64 81709.14 83904.91 83311.03 88874.16
##   [9] 89290.83 83149.45 82599.95 80478.32 83678.68 83208.12 81135.95 91052.73
##  [17] 82141.15 83522.20 89346.45 82906.23 81622.75 87459.93 84022.41 82663.20
##  [25] 81764.85 85712.65 84977.54 87980.69 84836.74 82764.63 82802.40 84977.54
##  [33] 86799.56 85883.12 91002.27 82644.91 83678.68 85452.28 83445.97 87348.08
##  [41] 84046.88 91160.01 82632.77 85446.45 80001.85 86097.43 81684.58 80513.17
##  [49] 84976.47 82154.32 90865.93 83374.63 81793.79 82501.96 81518.55 83708.10
##  [57] 81608.50 83491.11 83880.60 86524.91 83930.70 83509.52 84795.75 81634.99
##  [65] 85679.59 82813.54 82480.60 85690.52 92757.65 83792.49 82373.11 83067.79
##  [73] 87981.39 91026.90 80895.88 84260.72 87962.42 83230.33 84000.47 84285.40
##  [81] 85877.01 86654.26 84354.56 83276.25 88562.70 85711.85 83773.49 82563.32
##  [89] 90507.00 83695.70 84847.23 81600.47 80388.86 85149.75 79964.57 81466.87
##  [97] 88661.07 88450.06 84058.06 90328.87
## 
## $runtime
##    user  system elapsed 
##  11.632   0.000  11.632

1. repeated_bc$best_bc returns the best biclustering, and has the same structure as the output of biclustermd().
2. repeated_bc$rep_sse is the SSE for each repeat, in order.
3. repeated_bc$runtime provides the CPU runtime for the user and system as well as elapsed time in seconds.

The 100 repeats take approximately 20 seconds to complete. Parallelization is quicker given sufficient complexity. However, as nrep gets large, parallelization gets slow, since all nrep objects have to be written to memory versus the minimum SSE object when processing serially. These remarks are current as of July 17, 2019.

repeated_bc$runtime

##    user  system elapsed 
##  11.632   0.000  11.632

We can plot the best SSE at each repeat, that is, we can take the cumulative minimum of rep_sse and plot:

plot(cummin(repeated_bc$rep_sse), type = 'o', ylab = 'Cumulative Minimum', xlab = 'Repeat Number')

Predicting average delays

We’ll end this tutorial by making a training dataset from the clustered data using gather.biclustermd(), which gives the name of the row and column the data point comes from as well as the row and column group it belongs to.

training <- gather(repeated_bc$best_bc)
training %>% head()

##   row_name col_name row_cluster col_cluster bicluster_no    value
## 1  January      ABQ           1           1            1       NA
## 2    March      ABQ           1           1            1       NA
## 3    April      ABQ           1           1            1 12.22222
## 4  January      ACK           1           1            1       NA
## 5    March      ACK           1           1            1       NA
## 6    April      ACK           1           1            1       NA

Now we’ll plot cell (1, 4) and fit a linear model to it.

autoplot(repeated_bc$best_bc, row_clusts = 1, col_clusts = 4) +
  scale_fill_viridis_c(na.value = "white") +
  labs(x = "Destination Airport", y = "Month", fill = "Average Delay")

model <- training %>% 
  filter(row_cluster == 1, col_cluster == 4) %>% 
  lm(value ~ row_name + col_name, data = .)
summary(model)

## 
## Call:
## lm(formula = value ~ row_name + col_name, data = .)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -10.946  -2.760  -0.887   3.195  13.647 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)   
## (Intercept)       10.313      3.190   3.233  0.00239 **
## row_nameJanuary   -4.229      1.613  -2.623  0.01211 * 
## row_nameMarch     -3.803      1.586  -2.398  0.02098 * 
## col_nameBDL        5.069      4.317   1.174  0.24696   
## col_nameBHM        8.352      4.317   1.934  0.05982 . 
## col_nameBNA        4.837      4.317   1.120  0.26897   
## col_nameBUR       -5.389      4.317  -1.248  0.21888   
## col_nameBWI        5.064      4.317   1.173  0.24745   
## col_nameCAK        7.040      4.317   1.631  0.11045   
## col_nameCHO       -2.038      4.849  -0.420  0.67636   
## col_nameCLE       -2.152      4.317  -0.498  0.62076   
## col_nameCRW        7.710      4.317   1.786  0.08134 . 
## col_nameCVG       11.662      4.317   2.701  0.00992 **
## col_nameIAD        6.931      4.317   1.605  0.11591   
## col_nameJAC        1.554      4.849   0.320  0.75027   
## col_nameJAX        4.635      4.317   1.073  0.28919   
## col_nameMDW        4.756      4.317   1.102  0.27692   
## col_nameMEM        3.738      4.317   0.866  0.39154   
## col_nameMKE        4.194      4.317   0.971  0.33688   
## col_nameORF        1.575      4.317   0.365  0.71701   
## col_namePSE       -3.899      4.317  -0.903  0.37168   
## col_nameROC        4.809      4.317   1.114  0.27164   
## col_nameSAT       -8.490      4.317  -1.967  0.05587 . 
## col_nameSMF        1.877      4.317   0.435  0.66591   
## col_nameSTL        4.539      4.317   1.051  0.29913   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.288 on 42 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.5917, Adjusted R-squared:  0.3584 
## F-statistic: 2.536 on 24 and 42 DF,  p-value: 0.004026

sqrt(mean(resid(model) ^ 2))

## [1] 4.18657

If you have questions you can email me at [email protected]. In the case you’ve found an error please open up an issue.