Preamble

Report for the first assignment of the Reproducible Research course of the Coursera/JHSPH Data Science Specialization.

The source files are posted on GitHub.

Preliminaries

Libraries needed for data processing and plotting:

library("ggplot2")
library("plyr")
library("reshape2")
library("xtable")

INTRODUCTION

It is now possible to collect a large amount of data about personal movement using activity monitoring devices such as a Fitbit, Nike Fuelband, or Jawbone Up.

These type of devices are part of the “quantified self” movement – a group of enthusiasts who take measurements about themselves regularly to improve their health, to find patterns in their behavior, or because they are tech geeks. But these data remain under-utilized both because the raw data are hard to obtain and there is a lack of statistical methods and software for processing and interpreting the data.

The Data

This assignment makes use of data from a personal activity monitoring device.
This device collects data at 5 minute intervals through out the day. The data consists of two months of data from an anonymous individual collected during the months of October and November, 2012 and include the number of steps taken in 5 minute intervals each day.

Outline

DATA LOADING AND PREPROCESSING
- New Variables
- Checks on NA and valid steps entries
QUESTIONS
- MEAN TOTAL NUMBER OF STEPS TAKEN PER DAY
- THE AVERAGE DAILY ACTIVITY PATTERN
  - Time-series plot of average activity
  - Maximum average number of steps
- IMPUTING MISSING VALUES
  - Some thoughts about missing data
  - What do the NA in this dataset look like? How many are there and how are they distributed?
  - An imputing strategy
  - Average activity pattern by day of the week
- DIFFERENCES IN ACTIVITY PATTERNS BETWEEN WEEKDAYS AND WEEKENDS
  - Average activity by day type (weekday vs. weekend)
APPENDIX
- R Session Info

LOADING AND PREPROCESSING THE DATA

The data were downloaded from the course web site:

Dataset: Activity monitoring data [52K]

main <- read.csv("data/activity.csv.gz")

The dataset comprises 17568 observations of the following variables:

steps: Number of steps taking in a 5-minute interval (missing values are coded as NA)
date: The date on which the measurement was taken in YYYY-MM-DD format
interval: Identifier for the 5-minute interval in which measurement was taken

New Variables

We added a few new variables related to time:

fullTime: combines date and interval (reformatted).
hour: interval is converted to a HH:MM format, of class POSIXct (because of ggplot).
dayName: weekdays() are extracted from fullTime.
dayFactor: additional variable with day names saved as ordered factor.
dayFlag: a 2-level factor to more easily recognize week [0] and week end [1] days.
chunk: a 4-level factor marking four 6-hours periods during a day.

main$fullTime <- strptime(paste(main$date,sprintf("%04d",main$interval),sep=" "), "%F %H%M")
main$hour <- as.POSIXct(strptime(sprintf("%04d",main$interval), "%H%M"))
main$dayName <- weekdays(main$fullTime)
day.names <- c("Monday","Tuesday","Wednesday","Thursday","Friday","Saturday","Sunday")
main$dayFactor <- factor(main$dayName, levels=day.names, ordered=TRUE)
main$dayFlag <- factor(ifelse(as.integer(main$dayFactor)<=5,0,1),labels=c("weekday","weekend"))

chunkTimes <- strptime(c("00:00:00","06:00:00","12:00:00","18:00:00","23:59:59"), "%H:%M:%S")
main$chunk <- cut(main$hour, breaks=chunkTimes, labels=c("00to06","06to12","12to18","18to24"))

The resulting data frame has the following structure:

str(main)
# 'data.frame': 17568 obs. of  9 variables:
#  $ steps    : int  NA NA NA NA NA NA NA NA NA NA ...
#  $ date     : Factor w/ 61 levels "2012-10-01","2012-10-02",..: 1 1 1 1 1 1 1 1 1 1 ...
#  $ interval : int  0 5 10 15 20 25 30 35 40 45 ...
#  $ fullTime : POSIXlt, format: "2012-10-01 00:00:00" "2012-10-01 00:05:00" "2012-10-01 00:10:00" ...
#  $ hour     : POSIXct, format: "2015-08-03 00:00:00" "2015-08-03 00:05:00" "2015-08-03 00:10:00" ...
#  $ dayName  : chr  "Monday" "Monday" "Monday" "Monday" ...
#  $ dayFactor: Ord.factor w/ 7 levels "Monday"<"Tuesday"<..: 1 1 1 1 1 1 1 1 1 1 ...
#  $ dayFlag  : Factor w/ 2 levels "weekday","weekend": 1 1 1 1 1 1 1 1 1 1 ...
#  $ chunk    : Factor w/ 4 levels "00to06","06to12",..: 1 1 1 1 1 1 1 1 1 1 ...

Checks on `NA` and valid `steps` entries

A few basic checks of the data completeness by day, creating a data frame with:

nNA: count of NA.
nGood.All: valid entries, i.e. non-NA.
nGood.Not0: valid entries different from 0.
nSteps: total number of steps.

… summarized by day.

daily.stats <- ddply(main, .(date), summarize, 
                     nNA = sum(is.na(steps)), 
                     nGood.All = sum(!is.na(steps)),
                     nGood.Not0 = sum(!is.na(steps) & steps>0), 
                     nSteps = sum(steps))

table(daily.stats$nNA)
# 
#   0 288 
#  53   8

The table of the number of NA in each day reveals that all missing values are concentrated in 8 days, for which in fact there are no data (i.e. 288 NA).
The other 53 days have complete data, 288 entries each, although the majority of them are zeros.

The following plot illustrates this latter point: it show the distribution of the number of intervals with steps>0 in a day.
The red bin comprises the days without any valid value (8). The distribution is centered at around 80, which is about 28% of the intervals in each day (288).

hist(daily.stats$nGood.Not0, xlim=c(-10,160), breaks=seq(-10,305,by=10), 
     col=c("red",rep("forestgreen",20)), cex.main=0.85, 
     main="Distribution of the daily number of 'good values' of 'steps' (!NA & >0)", 
     xlab="Number of 'good' intervals")

If we split each day in four time intervals (the chunk factor) the summary of good and non-zero values not surprisingly shows a very low fraction for the night period and higher fractions for the morning and afternoon period.

tmp.df <- ddply(main, .(chunk), summarize, 
                Ntot = length(steps[!is.na(steps)]), 
                Ndata.gt.0 = sum(!is.na(steps) & steps>0), 
                fraction = sprintf("%6.2f %%",sum(!is.na(steps) & steps>0)/length(steps[!is.na(steps)])*100.))

chunk	N.tot	N.good	pct.good
00to06	3816	148	3.88 %
06to12	3816	1642	43.03 %
12to18	3816	1377	36.08 %
18to24	3816	1083	28.38 %

byChunk <- subset(ddply(main, .(date, chunk), summarize, Ndata=sum(!is.na(steps) & steps>0)), Ndata>0)

ggplot(byChunk, aes(x=Ndata)) + theme_bw() + theme(legend.position="none") + 
           geom_histogram(aes(fill=chunk), colour="grey50", breaks=seq(-4,54,by=4), right=TRUE) +
           ggtitle("Distribution by 'day chunk' of the number of 'good values' of steps (!NA & >0)") + 
           xlab("Number of 'good' intervals") + 
           ylab("Number of days") + 
           facet_wrap( ~ chunk, nrow = 1)

QUESTIONS

Mean Total Number of Steps Taken per Day

The daily.stats data frame contains all the pieces of information needed to address this question.
The values in the nSteps column are the number of steps summed over each day, and we can get the statistics with the summary() command.
The mean and median are 10770 and 10760 respectively.
Here is the full summary, followed by the distribution.

summary(daily.stats$nSteps)
#    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
#      41    8841   10760   10770   13290   21190       8

hist(daily.stats$nSteps, breaks = seq(0, 30000, by = 2500), col = "orange", 
     main = "Distribution of Total Daily Steps", 
     xlab = "Total Daily Steps", 
     ylab = "N_{days}")

The Average Daily Activity Pattern

For convenience in further processing and to work with ggplot we reshape the data frame to long format by melting it.
With the na.rm=TRUE option the NA are excluded (i.e. left implicit), which in this case is acceptable because they can easily be reconstructed on the basis of the continuity of the date+interval variable.

main.m2 <- melt(main[,c("steps","date","hour","dayFactor","dayFlag")], na.rm = TRUE, 
                id = c("date","hour","dayFactor","dayFlag"), 
                measured = c("steps"), 
                value.name = "steps")

To plot the number of steps as a function of time during the day, averaged over days, we prepare the data using the ddply() function to summarize the data frame by hour.

We decided to include in the summarizing interval entries with steps=0, assuming that the device counting steps was accurate enough that all non-NA values are reliable. Please note that NA are already excluded from the molten data frame used here.
For each hour we compute mean, standard deviation and number of data contributing to that bin in the time-series.
The summarizing operation returns NaN for the mean and standard deviation if the number of contributions is 0 or <2, respectively. We reset those values to NA.

ts.ByHour <- ddply(main.m2, .(hour), summarize, Avrg=mean(steps), Sd=sd(steps), N=length(steps))
ts.ByHour$Avrg[ts.ByHour$N==0] <- NA
ts.ByHour$Sd[ts.ByHour$N<2] <- NA

Time-series plot of average activity

The next two plots show the activity averaged over all days.
The two plots are identical, except that on the second one we show \(\langle steps\rangle/\sqrt{N_{data}}\) errorbars.

gm.v1 <- ggplot(ts.ByHour, aes(hour, Avrg)) + theme_bw() + 
            geom_line(lty = 1, col = "red2") + 
            labs(title = "Activity (steps/5min) Averaged over all days") + 
            labs(x = "Time during the day") + 
            labs(y = "Mean number of steps") 

gm.v1

gm.v1 + geom_errorbar(aes(ymin = ts.ByHour$Avrg-ts.ByHour$Sd/sqrt(ts.ByHour$N),
                          ymax = ts.ByHour$Avrg+ts.ByHour$Sd/sqrt(ts.ByHour$N)), 
                      color = "red2", alpha = "0.5")

Maximum average number of steps

maxHr <- format(ts.ByHour[which.max(x=ts.ByHour$Avrg),"hour"], "%H:%M")
maxN  <- ts.ByHour[which.max(x=ts.ByHour$Avrg),"Avrg"]

The maximum value of the number of steps per interval averaged over the full period is 206.17, occurring at 08:35:

ts.ByHour[which.max(x=ts.ByHour$Avrg),c("hour","Avrg")]
#                    hour     Avrg
# 104 2015-08-03 08:35:00 206.1698

Imputing Missing Values

Some thoughts about missing data

Two important points to address when deciding how to handle missing data are to understand why they are missing, what is the goal of the analysis, and what would be the impact of the missing data in that context.

I can imagine that the application of some methods requires complete datasets, and this may force to impute the missing data. This may be an issue with the dataset at hand, if one wanted to perform some kinds of time-series analyses cutting data across dates, because as we will see in a moment missing data are such that a few whole days are missing.
On the other hand, if one wanted to study patterns within individual days, e.g. to compare week and weekend days, morning and afternoons, etc, the fact that effectively the missing data are missing days may not be so pernicious, certainly as long as we can assume that the data are missing at random.

It is also worth considering that imputing data may introduce a bias in the dataset, and in turn bias the results of the analysis.

I do not have much experience with imputing data, as in my field of research this is rarely (if ever) done, and broadly speaking an important reason for this is that in the end by imputing data we are not adding any new information to the dataset (although naively this may sometimes seem to be the case), and it may be preferable and more robust to deal with missing data by other means/methods. One of the few exceptions to this “rule” is the application of time-series analysis that simply do not work with gapped data.

What do the `NA` in this dataset look like?
How many are there and how are they distributed?

First questions are how many there are and how they are distributed.

Total.NA <- sum(daily.stats$nNA)

BadDays <- as.character(daily.stats$date[daily.stats$nNA > 0])

Ntot <- tapply(main$steps, main$dayFactor, function(x){length(x)/288}, simplify=TRUE)
Nbad <- tapply(main$steps, main$dayFactor, function(x){length(x[is.na(x)])/288}, simplify=TRUE)

The total number of NA values is 2304, and as noted in the previous section they are all concentrated in 8 days for which in fact there are no valid values for steps, effectively making the issue of missing data one of missing days.

This is the list of these bad days:

BadDays
2012-10-01	Monday
2012-10-08	Monday
2012-11-01	Thursday
2012-11-04	Sunday
2012-11-09	Friday
2012-11-10	Saturday
2012-11-14	Wednesday
2012-11-30	Friday

They are fairly evenly distributed over weekdays considering the total number of occurrences for each of the week:

	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Ntot	9	9	9	9	9	8	8
Nbad	2	0	1	1	2	1	1

df <- data.frame(daily.stats[,c(1,2)], flag = ifelse(daily.stats$nNA == 0,1,0), 
                 dayF = factor(weekdays(as.Date(daily.stats$date)), 
                 levels = day.names, 
                 ordered = TRUE))
mat <- matrix(c(df$flag,-1,-1),nrow=7)

image(x=1:7, y=1:9, mat[,ncol(mat):1],
      col=c("grey80", "#DD4444", "#55CC55"), 
      xaxt = "n", yaxt = "n", ylab = "", xlab = "", 
      main = "Matrix plot of 'good' and 'bad' days", cex.main = 0.9)

grid(nx = 7, ny = 9, col = "white")
axis( 1, at = 1:7, labels=c("M","Tu","W","Th","F","S","Su"), las=1, padj=-1, tcl=0 )
axis( 2, at = 1:9, labels=paste("wk",9:1,sep=" "), las=1, tcl=0, hadj=0.7 )

An imputing strategy

Because the missing data

are concentrated in a few days,
constitute the totality of the entries for those days, and
these missing days are evenly distributed over the days of the week,

and because of the limited scope of the analysis, it would seem reasonable to consider a very conservative approach towards imputing missing data, such as the three following choices:

Not impute the missing values, simply exclude the missing days from the analysis.
Replace the missing values with a representative value, which could for instance be the average steps computed over an appropriately defined sample (e.g. the same day-of-the-week).
Replace the missing values by sampling from the distribution of values for similar days, for instance the same day of the week (e.g. Wednesday for a Wednesday).

One important caveat with the latter option is that while it is certainly doable, our sampling is not particularly robust given that we have only a few values for each hour for each day of the week (at most 9).

Because it would seem lazy to adopt the first approach, let’s take a slightly closer look at the properties of the activity to formulate a plausible plan to implement the second strategy.

Average activity pattern by day of the week

The first thing we can do is to look at the activity data split up by day of the week, making time-series plot averaged by day. First step is summarizing the data set by dayFactor and hour and then plot the seven time-series.

# summarizing by weekday and hour
ts.ByDay.ByHour <- ddply(main.m2, .(dayFactor, hour), summarize, 
                         Avrg=mean(steps), 
                         Sd=sd(steps), 
                         N=length(steps))

str(ts.ByDay.ByHour)
# 'data.frame': 2016 obs. of  5 variables:
#  $ dayFactor: Ord.factor w/ 7 levels "Monday"<"Tuesday"<..: 1 1 1 1 1 1 1 1 1 1 ...
#  $ hour     : POSIXct, format: "2015-08-03 00:00:00" "2015-08-03 00:05:00" "2015-08-03 00:10:00" ...
#  $ Avrg     : num  1.43 0 0 0 0 ...
#  $ Sd       : num  3.78 0 0 0 0 ...
#  $ N        : int  7 7 7 7 7 7 7 7 7 7 ...

gm.ByDay.ByHour <- ggplot(ts.ByDay.ByHour, aes(hour, Avrg))
gm.ByDay.ByHour + theme_bw() + theme(legend.position = "none") +
                geom_line(aes(color = dayFactor), lty = 1) +
                labs(title = "activity averaged over time, by day") + 
                labs(x = "hour of the day") + 
                labs(y = "number of steps / 5 minutes") + 
                facet_wrap( ~ dayFactor, ncol = 1)

Visual “inspection” seems to suggest that week days may actually be split in two groups Monday/Tuesday/Wednesday and Thursday/Friday instead of being combined all together. The averaged Saturday is fairly similar to Friday, sort of in between that and Sunday.

Let’s see the average activity for the three groups

G1 = Monday/Tuesday/Wednesday
G2 = Thursday/Friday and
G3 = Saturday/Sunday.

main$dayGroup <- main$dayFactor
levels(main$dayGroup) <- c("G1","G1","G1","G2","G2","G3","G3")
main.m2 <- melt(main[, c("steps","date","hour","dayFactor","dayFlag","dayGroup")],
                id = c("date","hour","dayFactor","dayFlag","dayGroup"), 
                measured = c("steps"), value.name="steps", na.rm=TRUE)

ts.ByGroup.ByHour <- ddply(main.m2, .(dayGroup, hour), summarize, 
                           Avrg = mean(steps), 
                           Sd = sd(steps), 
                           N = length(steps))

ggplot(ts.ByGroup.ByHour, aes(hour, Avrg)) + theme_bw() + theme(legend.position="none") +
    geom_line(aes(color=dayGroup),lty=1) +
    labs(title="activity averaged over time, by type of day") + 
    labs(x="hour of the day") + labs(y="number of steps / 5 minutes") + 
    facet_wrap( ~ dayGroup, ncol=1)

Imputation by group

The plan is then to fill-in the missing days with the average of activity (i.e. steps) of the days of the same group G1|G2|G3.
First we create a vector with the sequence of dayGroup strings for the missing days.

bd <- subset(main, date %in% BadDays, select=c("date","hour","dayGroup"))
bd.sequence <- as.character(ddply(bd, .(date, dayGroup), summarize, l = length(dayGroup))$dayGroup)
bd.sequence
# [1] "G1" "G1" "G2" "G3" "G2" "G3" "G1" "G2"

We prepare templates of the activity for each dayGroup, as numeric vectors, and define a function to get them by string.

template.G1 <- as.vector(subset(ts.ByGroup.ByHour, dayGroup=="G1", select=c("Avrg"))$Avrg)
template.G2 <- as.vector(subset(ts.ByGroup.ByHour, dayGroup=="G2", select=c("Avrg"))$Avrg)
template.G3 <- as.vector(subset(ts.ByGroup.ByHour, dayGroup=="G3", select=c("Avrg"))$Avrg)

get.template <- function(x) { name <- paste("template", x, sep = "."); v <- get(name); return(v)}

Next we fill a new vector with a sequence of “imputed values”:

newvec <- vector(mode="numeric", length=Total.NA)
for(i in 1:length(bd.sequence)) { 
    i1<-288*(i-1)+1; i2<-i*288; newvec[i1:i2]<-get.template(bd.sequence[i]); 
}

length(newvec)
# [1] 2304
summary(newvec[newvec>0])
#     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#   0.1667   7.4000  29.3800  44.8700  67.4300 273.7000

Finally we duplicate the steps data in the data frame and replace the subset of NA with values from the the newly vector:

main$stepsNew <- main$steps
main$stepsNew[is.na(main$steps)] <- newvec

Let’s put together daily stats for the new stepsNew variable:

daily.stats.new <- ddply(main, .(date), summarize, 
                     nNA = sum(is.na(stepsNew)), 
                     nGood.All = sum(!is.na(stepsNew)),
                     nGood.Not0 = sum(!is.na(stepsNew) & steps>0), 
                     nSteps = sum(stepsNew))

And compare summaries with the original dataset:

summary(daily.stats$nSteps)
#    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
#      41    8841   10760   10770   13290   21190       8

summary(daily.stats.new$nSteps)
#    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#      41    9819   10570   10760   12810   21190

par(mfrow=c(1,2))
hist(daily.stats.new$nSteps, breaks=seq(0,30000,by=2500), col="orange", 
     xlim = c(0, 25000), 
     ylim = c(0, 25),
     main = "New Distribution of Total Daily Steps", 
     xlab = "Total Daily Steps", 
     ylab = "N_{days}")

hist(daily.stats$nSteps, breaks=seq(0,30000,by=2500), col="orange", 
     xlim=c(0,25000), ylim=c(0,25),
     main="Distribution of Total Daily Steps", xlab="Total Daily Steps", ylab="N_{days}")

par(mfrow=c(1,1))

There distributions exhibit a modest difference but they are in line with what one would have expected with this approach for imputing missing values, namely that the distribution of total daily steps has become more “concentrated” because we added average days values for the 8 missing days. The same effect is noticeable in the summaries, with the 1st and 3rd quantiles shifted slightly towards the mean.

Differences in Activity Patterns Between Weekdays and Weekends

Average activity by day type (weekday vs. weekend)

We commented above about a possible dichotomy in the activity patterns of week days, that seem to exhibit two kinds of behaviors.

Beside that, there exist noticeable differences between week and weekend days, although they are more marked between Monday/Tuesday/Wednesday and weekend days (see plot above).

Here we show the activity time-series for the averaged 5 week days and the weekend days.

ts.ByFlag.ByHour <- ddply(main.m2, .(dayFlag, hour), summarize, 
                          Avrg = mean(steps), 
                          Sd = sd(steps), 
                          N = length(steps))

gm.ByFlag.ByHour <- ggplot(ts.ByFlag.ByHour, aes(hour, Avrg))
gm.ByFlag.ByHour + theme_bw() + theme(legend.position = "none") +
    geom_line(aes(color = dayFlag), lty = 1) +
    labs(title="Activity averaged over time, by type of day") + 
    labs(x = "Hour of the day") + 
    labs(y = "Number of steps / 5 minutes") + 
    facet_wrap( ~ dayFlag, ncol = 1)

APPENDIX

[Back to the Top]

R Session Info

sessionInfo()
# R version 3.1.3 (2015-03-09)
# Platform: x86_64-pc-linux-gnu (64-bit)
# Running under: Ubuntu 14.04.2 LTS
# 
# locale:
#  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C               LC_TIME=en_US.UTF-8       
#  [4] LC_COLLATE=C               LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
#  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                  LC_ADDRESS=C              
# [10] LC_TELEPHONE=C             LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# other attached packages:
# [1] xtable_1.7-4   reshape2_1.4.1 plyr_1.8.3     ggplot2_1.0.1  knitr_1.10.5  
# 
# loaded via a namespace (and not attached):
#  [1] MASS_7.3-41      Rcpp_0.11.6      colorspace_1.2-6 digest_0.6.8     evaluate_0.7    
#  [6] formatR_1.2      grid_3.1.3       gtable_0.1.2     highr_0.5        htmltools_0.2.6 
# [11] labeling_0.3     magrittr_1.5     munsell_0.4.2    proto_0.3-10     rmarkdown_0.7   
# [16] scales_0.2.4     stringi_0.5-5    stringr_1.0.0    tools_3.1.3      yaml_2.1.13

Human Activity: Steps History and Patterns

Giovanni Fossati

Preamble

Preliminaries

INTRODUCTION

The Data

Outline

LOADING AND PREPROCESSING THE DATA

New Variables

Checks on `NA` and valid `steps` entries

QUESTIONS

Mean Total Number of Steps Taken per Day

The Average Daily Activity Pattern

Time-series plot of average activity

Maximum average number of steps

Imputing Missing Values

Some thoughts about missing data

What do the `NA` in this dataset look like?
How many are there and how are they distributed?

An imputing strategy

Average activity pattern by day of the week

Imputation by group

Differences in Activity Patterns Between Weekdays and Weekends

Average activity by day type (weekday vs. weekend)

APPENDIX

R Session Info

Human Activity: Steps History and Patterns

Giovanni Fossati

Preamble

Preliminaries

INTRODUCTION

The Data

Outline

LOADING AND PREPROCESSING THE DATA

New Variables

Checks on NA and valid steps entries

QUESTIONS

Mean Total Number of Steps Taken per Day

The Average Daily Activity Pattern

Time-series plot of average activity

Maximum average number of steps

Imputing Missing Values

Some thoughts about missing data

What do the NA in this dataset look like? How many are there and how are they distributed?

An imputing strategy

Average activity pattern by day of the week

Imputation by group

Differences in Activity Patterns Between Weekdays and Weekends

Average activity by day type (weekday vs. weekend)

APPENDIX

R Session Info

Checks on `NA` and valid `steps` entries

What do the `NA` in this dataset look like?
How many are there and how are they distributed?