This is the first post I’ve written in a while. I have been somewhat radio silent on social media, but I’m jumping back in.

Now, I work with brain images, which can have millions of elements (referred to as voxels). Many of these elements are zero (for background). We want to calculate basic statistics on the data usually and I wanted to describe how you can speed up operations or reduce memory requirements if you want to calculate many statistics on a *large* vector with integer values by using summary tables.

## Why to use Tables

Tables are relatively computationally expensive to calculate. They must operate over the entire vector, find the unique values, and bin the data into these values. Let be the length of the vector. For integer vectors (i.e. whole number), the number of unique values is much less than . Therefore, the table is stored much more efficiently than the entire vector.

### Tables are sufficient statistics

You can think of the frequencies and bins as summary statistics for the entire distribution of the data. I will not discuss a formal proof here, but you can easily re-create the entire vector using the table (see `epitools::expand.table`

for a function to do this), and thus the table is a sufficient (but not likely a minimal) statistic.

As a sufficient statistic, we can create any statistic that we’d like relatively easy. Now, `R`

has very efficient functions for many statistics, such as the median and quantiles, so it may not make sense why we’d want to rewrite some of these functions using tables.

I can think of 2 reasons: 1) you want to calculate *many* statistics on the data and don’t want to pass the vector in multiple times, and 2) you want to preprocess the data to summarize the data into tables to only use these in memory versus the entire vector.

Here are some examples when this question has been asked on stackoverflow: 1, 2 and the R list-serv: 1. What we’re going to do is show some basic operations on tables to get summary statistics and show they agree.

## R Implementation

Let’s make a large vector:

set.seed(20150301) vec = sample(-10:100, size= 1e7, replace = TRUE)

### Quantile function for tables

I implemented a quantile function for tables (of only type 1). The code takes in a table, creates the cumulative sum, extracts the unique values of the table, then computes and returns the quantiles.

quantile.table = function(tab, probs = c(0, 0.25, 0.5, 0.75, 1)){ n = sum(tab) #### get CDF cs = cumsum(tab) ### get values (x) uvals = unique(as.numeric(names(tab))) # can add different types of quantile, but using default m = 0 qs = sapply(probs, function(prob){ np = n * prob j = floor(np) + m g = np + m - j # type == 1 gamma = as.numeric(g != 0) cs <= j quant = uvals[min(which(cs >= j))] return(quant) }) dig <- max(2L, getOption("digits")) names(qs) <- paste0(if (length(probs) < 100) formatC(100 * probs, format = "fg", width = 1, digits = dig) else format(100 * probs, trim = TRUE, digits = dig), "%") return(qs) }

### Quantile Benchmarks

Let’s benchmark the quantile functions: 1) creating the table and then getting the quantiles, 2) creating an empircal CDF function then creating the quantiles, 3) creating the quantiles on the original data.

library(microbenchmark) options(microbenchmark.unit='relative') qtab = function(vec){ tab = table(vec) quantile.table(tab) } qcdf = function(vec){ cdf = ecdf(vec) quantile(cdf, type=1) } # quantile(vec, type = 1) microbenchmark(qtab(vec), qcdf(vec), quantile(vec, type = 1), times = 10L)

Unit: relative expr min lq mean median uq qtab(vec) 12.495569 12.052644 9.109178 11.589662 7.499691 qcdf(vec) 5.407606 5.802752 4.375459 5.553492 3.708795 quantile(vec, type = 1) 1.000000 1.000000 1.000000 1.000000 1.000000 max neval cld 5.481202 10 c 2.653728 10 b 1.000000 10 a

### More realistic benchmarks

Not surprisingly, simply running `quantile`

on the vector beats the other 2 methods, **by far**. So computational speed may not be beneficial for using a table. But if tables or CDFs are already created in a previous processing step, we should compare that procedure:

options(microbenchmark.unit="relative") tab = table(vec) cdf = ecdf(vec) all.equal(quantile.table(tab), quantile(cdf, type=1))

[1] TRUE

all.equal(quantile.table(tab), quantile(vec, type=1))

[1] TRUE

microbenchmark(quantile.table(tab), quantile(cdf, type=1), quantile(vec, type = 1), times = 10L)

Unit: relative expr min lq mean median uq quantile.table(tab) 1.000 1.000 1.0000 1.000 1.0000 quantile(cdf, type = 1) 774.885 1016.172 596.3217 1144.063 868.8105 quantile(vec, type = 1) 1029.696 1122.550 653.2146 1199.143 910.3743 max neval cld 1.0000 10 a 198.1590 10 b 206.5936 10 b

As we can see, if you had already computed tables, then you get the same quantiles as performing the operation on the vector, and also **much** faster results. Using `quantile`

on a `ecdf`

object is not much better, which mainly is due to the fact that the `quantile`

function remakes the factor and then calculate quantiles:

stats:::quantile.ecdf

function (x, ...) quantile(evalq(rep.int(x, diff(c(0, round(nobs * y)))), environment(x)), ...) <bytecode: 0x107493e28> <environment: namespace:stats>

### Median for tables

Above we show the `quantile.table`

function, so the median function is trivial where `probs = 0.5`

:

median.table = function(tab){ quantile.table(tab, probs = 0.5) }

## Mean of a table

Other functions can be used to calculate statstics on the table, such as the mean:

mean.table = function(tab){ uvals = unique(as.numeric(names(tab))) sum(uvals * tab)/sum(tab) } mean.table(tab)

[1] 44.98991

mean(tab)

[1] 44.98991

mean(cdf)

Warning in mean.default(cdf): argument is not numeric or logical: returning NA

[1] NA

As we see, we can simply use `mean`

and do not need to define a new function for tables.

mean(vec)

[1] 44.98991

all.equal(mean(tab), mean(vec))

[1] TRUE

### Subsetting tables

One problem with using `mean`

vs. `mean.table`

is when you subset the table or perform an operation that causes it to lose the attribute of the class of `table`

. For example, let’s say I want to estimate the mean of the data for values :

mean(vec[vec > 0])

[1] 50.50371

over0 = tab[as.numeric(names(tab)) > 0] mean(over0)

[1] 90065.98

mean.table(over0)

[1] 50.50371

class(over0)

[1] "array"

We see that after subsetting, `over0`

is an `array`

and not a `table`

, so `mean`

computes the mean using the `array`

method, treating the frequences as data and the estimated mean is not correct. `mean.table`

calculates the correct value, as it does not depend on the class of `tab`

. Another way to circumvent this is to reassign a class of `table`

to `over0`

:

class(over0) = "table" mean(over0)

[1] 50.50371

This process requires the user to know what the class is of the object passed to `mean`

, and may not be correct if the user changes the class of the object.

### Aside on NA values

Let’s see what happens when there are `NA`

s in the vector. We’ll put in 20 `NA`

values:

navec = vec navec[sample(length(navec), 20)] = NA natab = table(navec, useNA="ifany") nacdf = ecdf(navec) mean(navec)

[1] NA

mean(natab)

[1] NA

# mean(nacdf)

We see that if we `table`

the data with `NA`

being a category, then any operation that returns `NA`

if `NA`

are present will return `NA`

. For example, if we do a table on the data with the `table`

option `useNA="always"`

, then the mean will be `NA`

even though no `NA`

are present in the original vector. Also, `ecdf`

objects do not keep track of `NA`

values after they are computed.

tab2 = table(vec, useNA="always") mean(tab2)

[1] NA

nonatab = table(navec, useNA="no") mean(nonatab)

[1] 44.98993

mean(navec, na.rm=TRUE)

[1] 44.98993

If you are using tables for statistics, the equivalent of `na.rm=FALSE`

is `table(..., useNA="ifany")`

and `na.rm=TRUE`

is `table(..., useNA="no")`

. We also see that an object of `ecdf`

do not ever show NAs. Although we said tables are sufficient statistics, that may not be entirely correct if depending on how you make the table when the data have missing data.

### Mean benchmark

Let’s benchmark the mean function, assuming we have pre-computed the table:

options(microbenchmark.unit="relative") microbenchmark(mean(tab), mean(vec), times = 10L)

Unit: relative expr min lq mean median uq max neval cld mean(tab) 1.0000 1.0000 1.0000 1.0000 1.0000 1.00000 10 a mean(vec) 374.0648 132.3851 111.2533 104.7355 112.7517 75.21185 10 b

Again, if we have the table pre-computed, then estimating means is much faster using the table.

## Getting standard deviation

The `mean`

example may be misleading when we try `sd`

on the table:

sd(vec)

[1] 32.04476

sd(tab)

[1] 302.4951

This are not even remotely close. This is because `sd`

is operating on the table as if it were a `vector`

and not a frequency table.

Note, we cannot calculate `sd`

from the `ecdf`

object:

sd(cdf)

Error in as.double(x): cannot coerce type 'closure' to vector of type 'double'

## SD and Variance for frequency table

We will create a function to run `sd`

on a table:

var.table = function(tab){ m = mean(tab) uvals = unique(as.numeric(names(tab))) n = sum(tab) sq = (uvals - m)^2 ## sum of squared terms var = sum(sq * tab) / (n-1) return(var) } sd.table = function(tab){ sqrt(var.table(tab)) } sd.table(tab)

[1] 32.04476

We create the mean, get the squared differences, and sum these up (`sum(sq * tab)`

) , divide by `n-1`

to get the variance and the `sd`

is the square root of the variance.

### Benchmarking SD

Let’s similarly benchmark the data for `sd`

:

options(microbenchmark.unit="relative") microbenchmark(sd.table(tab), sd(vec), times = 10L)

Unit: relative expr min lq mean median uq max neval sd.table(tab) 1.0000 1.0000 1.000 1.000 1.0000 1.0000 10 sd(vec) 851.8676 952.7785 847.225 1142.225 732.3427 736.2757 10 cld a b

## Mode of distribution

Another statistic we may want for tabular data is the mode. We can simply find the maximum frequency in the table. The `multiple`

option returns multiple values if there is a tie for the maximum frequency.

mode.table = function(tab, multiple = TRUE){ uvals = unique(as.numeric(names(tab))) ind = which.max(tab) if (multiple){ ind = which(tab == max(tab)) } uvals[ind] } mode.table(tab)

[1] 36

## Memory of each object

We wish to simply show the memory profile for using a `table`

verus the entire vector:

format(object.size(vec), "Kb")

[1] "39062.5 Kb"

format(object.size(tab), "Kb")

[1] "7.3 Kb"

round(as.numeric(object.size(vec) / object.size(tab)))

[1] 5348

We see that the table **much** smaller than the vector. Therefore, computing and storing summary tables for integer data can be much more efficient.

# Conclusion

Tables are computationally expensive. If tables are pre-computed for integer data, however, then statistics can be calculated quickly and accurately, even if `NA`

s are present. These tables are also much smaller in memory so that they can be stored with less space. This may be an important thing to think about computing and storage of large vectors in the future.