fromo

Fast Robust Moments -- Pick Three!

Fast, numerically robust, higher order moments in R, computed via Rcpp, mostly as an exercise to learn Rcpp. Supports computation on vectors and matrices, and Monoidal append (and unappend) of moments. Computations are via the Welford-Terriberry algorithm, as described by Bennett et al.

-- Steven E. Pav, shabbychef@gmail.com

Installation

This package can be installed from CRAN, via drat, or from github:

# via CRAN:
install.packages("fromo")
# via drat:
if (require(drat)) {
    drat:::add("shabbychef")
    install.packages("fromo")
}
# get snapshot from github (may be buggy)
if (require(devtools)) {
    install_github("shabbychef/fromo")
}

Basic Usage

Currently the package functionality can be divided into the following:

• Functions which reduce a vector to an array of moments.
• Functions which take a vector to a matrix of the running moments.
• Functions which transform a vector to some normalized form, like a centered, rescaled, z-scored sample, or a summarized form, like the running Sharpe or t-stat.
• Functions for computing the covariance of a vector robustly.
• Object representations of moments with join and unjoin methods.

Summary moments

A function which computes, say, the kurtosis, typically also computes the mean and standard deviation, and has performed enough computation to easily return the skew. However, the default functions in R for higher order moments discard these lower order moments. So, for example, if you wish to compute Merten's form for the standard error of the Sharpe ratio, you have to call separate functions to compute the kurtosis, skew, standard deviation, and mean.

The summary functions in fromo return all the moments up to some order, namely the functions sd3, skew4, and kurt5. The latter of these, kurt5 returns an array of length 5 containing the excess kurtosis, the skewness, the standard deviation, the mean, and the observation count. (The number in the function name denotes the length of the output.) Along the same lines, there are summarizing functions that compute centered moments, standardized moments, and 'raw' cumulants:

• cent_moments: return a k+1-vector of the kth centered moment, the k-1th, all the way down to the 2nd (the variance), then the mean and the observation count.
• std_moments: return a k+1-vector of the kth standardized moment, the k-1th, all the way down to the 3rd, then the standard deviation, the mean, and the observation count.
• cent_cumulants: computes the centered cumulants (yes, this is redundant, but they are not standardized). return a k+1-vector of the kth raw cumulant, the k-1th, all the way down to the second, then the mean, and the observation count.
• std_cumulants: computes the standardized (and, of course, centered) cumulants. return a k+1-vector of the kth standardized cumulant, all the way down to the third, then the variance, the mean, and the observation count.

library(fromo)
set.seed(12345)
x <- rnorm(1000, mean = 10, sd = 2)
show(cent_moments(x, max_order = 4, na_rm = TRUE))

## [1]   47.276   -0.047    3.986   10.092 1000.000

show(std_moments(x, max_order = 4, na_rm = TRUE))

## [1]  3.0e+00 -5.9e-03  2.0e+00  1.0e+01  1.0e+03

show(cent_cumulants(x, max_order = 4, na_rm = TRUE))

## [1]   -0.388   -0.047    3.986   10.092 1000.000

show(std_cumulants(x, max_order = 4, na_rm = TRUE))

## [1] -2.4e-02 -5.9e-03  4.0e+00  1.0e+01  1.0e+03

Speed

In theory these operations should be just as fast as the default functions, but faster than calling multiple default functions. Here is a speed comparison of the basic moment computations:

library(fromo)
library(moments)
library(microbenchmark)

set.seed(1234)
x <- rnorm(1000)

dumbk <- function(x) {
    c(kurtosis(x) - 3, skewness(x), sd(x), mean(x), 
        length(x))
}

microbenchmark(kurt5(x), skew4(x), sd3(x), dumbk(x), 
    dumbk(x), kurtosis(x), skewness(x), sd(x), mean(x))

## Unit: microseconds
##         expr   min    lq  mean median  uq   max neval cld
##     kurt5(x) 138.1 140.5 152.3  142.8 151   326   100   a
##     skew4(x)  76.6  79.0 374.5   80.2  83 29119   100   a
##       sd3(x)   9.6  10.6  14.0   11.5  12   161   100   a
##     dumbk(x) 192.8 207.9 272.6  216.4 229  9540   200   a
##  kurtosis(x)  85.7  90.6  99.5   93.2 102   217   100   a
##  skewness(x)  85.8  90.8  96.2   92.5  95   172   100   a
##        sd(x)  15.4  17.8  21.6   18.9  20    72   100   a
##      mean(x)   3.9   4.5   5.4    4.7   5    19   100   a

x <- rnorm(1e+07, mean = 1e+12)

microbenchmark(kurt5(x), skew4(x), sd3(x), dumbk(x), 
    kurtosis(x), skewness(x), sd(x), mean(x), times = 10L)

## Unit: milliseconds
##         expr  min   lq mean median   uq  max neval  cld
##     kurt5(x) 1470 1481 1515   1501 1534 1593    10   c 
##     skew4(x)  808  813  869    834  870 1069    10  b  
##       sd3(x)   75   75   78     76   81   87    10 a   
##     dumbk(x) 1830 1852 1924   1873 1918 2328    10    d
##  kurtosis(x)  906  909  947    917  945 1138    10  b  
##  skewness(x)  864  872  938    912  954 1184    10  b  
##        sd(x)   52   52   54     52   54   64    10 a   
##      mean(x)   19   19   20     20   20   21    10 a

Weight! Weight!

Many of the methods now support the computation of weighted moments. There are a few options around weights: whether to check them for negative values, whether to normalize them to unit mean.

library(fromo)
library(moments)
library(microbenchmark)

set.seed(987)
x <- rnorm(1000)
w <- runif(length(x))

# no weights:
show(cent_moments(x, max_order = 4, na_rm = TRUE))

## [1] 2.9e+00 1.2e-02 1.0e+00 1.0e-02 1.0e+03

# with weights:
show(cent_moments(x, max_order = 4, wts = w, na_rm = TRUE))

## [1] 3.1e+00 4.1e-02 1.0e+00 1.3e-02 1.0e+03

# if you turn off weight normalization, the last
# element is sum(wts):
show(cent_moments(x, max_order = 4, wts = w, na_rm = TRUE, 
    normalize_wts = FALSE))

## [1]   3.072   0.041   1.001   0.013 493.941

# let's compare for speed!
x <- rnorm(1e+07)
w <- runif(length(x))

slow_sd <- function(x, w) {
    n0 <- length(x)
    mu <- weighted.mean(x, w = w)
    sg <- sqrt(sum(w * (x - mu)^2)/(n0 - 1))
    c(sg, mu, n0)
}
microbenchmark(sd3(x, wts = w), slow_sd(x, w))

## Unit: milliseconds
##             expr min  lq mean median  uq max neval cld
##  sd3(x, wts = w) 104 107  111    110 115 139   100  a 
##    slow_sd(x, w) 261 278  310    297 318 483   100   b

Monoid mumbo-jumbo

The as.centsums object performs the summary (centralized) moment computation, and stores the centralized sums. There is a print method that shows raw, centralized, and standardized moments of the ingested data. This object supports concatenation and unconcatenation. These should satisfy 'monoidal homomorphism', meaning that concatenation and taking moments commute with each other. So if you have two vectors, x1 and x2, the following should be equal: c(as.centsums(x1,4),as.centsums(x2,4)) and as.centsums(c(x1,x2),4). Moreover, the following should also be equal: as.centsums(c(x1,x2),4) %-% as.centsums(x2,4)) and as.centsums(x1,4). This is a small step of the way towards fast machine learning methods (along the lines of Mike Izbicki's Hlearn library).

Some demo code:

set.seed(12345)
x1 <- runif(100)
x2 <- rnorm(100, mean = 1)
max_ord <- 6L

obj1 <- as.centsums(x1, max_ord)
# display:
show(obj1)

##           class: centsums 
##     raw moments: 100 0.0051 0.09 -0.00092 0.014 -0.00043 0.0027 
## central moments: 0 0.09 -0.0023 0.014 -0.00079 0.0027 
##     std moments: 0 1 -0.086 1.8 -0.33 3.8

# join them together
obj1 <- as.centsums(x1, max_ord)
obj2 <- as.centsums(x2, max_ord)
obj3 <- as.centsums(c(x1, x2), max_ord)
alt3 <- c(obj1, obj2)
# it commutes!
stopifnot(max(abs(sums(obj3) - sums(alt3))) < 1e-07)
# unjoin them, with this one weird operator:
alt2 <- obj3 %-% obj1
alt1 <- obj3 %-% obj2
stopifnot(max(abs(sums(obj2) - sums(alt2))) < 1e-07)
stopifnot(max(abs(sums(obj1) - sums(alt1))) < 1e-07)

We also have 'raw' join and unjoin methods, not nicely wrapped:

set.seed(123)
x1 <- rnorm(1000, mean = 1)
x2 <- rnorm(1000, mean = 1)
max_ord <- 6L
rs1 <- cent_sums(x1, max_ord)
rs2 <- cent_sums(x2, max_ord)
rs3 <- cent_sums(c(x1, x2), max_ord)
rs3alt <- join_cent_sums(rs1, rs2)
stopifnot(max(abs(rs3 - rs3alt)) < 1e-07)

rs1alt <- unjoin_cent_sums(rs3, rs2)
rs2alt <- unjoin_cent_sums(rs3, rs1)
stopifnot(max(abs(rs1 - rs1alt)) < 1e-07)
stopifnot(max(abs(rs2 - rs2alt)) < 1e-07)

For multivariate input

There is also code for computing co-sums and co-moments, though as of this writing only up to order 2. Some demo code for the monoidal stuff here:

set.seed(54321)
x1 <- matrix(rnorm(100 * 4), ncol = 4)
x2 <- matrix(rnorm(100 * 4), ncol = 4)

max_ord <- 2L
obj1 <- as.centcosums(x1, max_ord, na.omit = TRUE)
# display:
show(obj1)

## An object of class "centcosums"
## Slot "cosums":
##          [,1]    [,2]   [,3]     [,4]    [,5]
## [1,] 100.0000  -0.093  0.045  -0.0046   0.046
## [2,]  -0.0934 111.012  4.941 -16.4822   6.660
## [3,]   0.0450   4.941 71.230   0.8505   5.501
## [4,]  -0.0046 -16.482  0.850 117.3456  13.738
## [5,]   0.0463   6.660  5.501  13.7379 100.781
## 
## Slot "order":
## [1] 2

# join them together
obj1 <- as.centcosums(x1, max_ord)
obj2 <- as.centcosums(x2, max_ord)
obj3 <- as.centcosums(rbind(x1, x2), max_ord)
alt3 <- c(obj1, obj2)
# it commutes!
stopifnot(max(abs(cosums(obj3) - cosums(alt3))) < 1e-07)
# unjoin them, with this one weird operator:
alt2 <- obj3 %-% obj1
alt1 <- obj3 %-% obj2
stopifnot(max(abs(cosums(obj2) - cosums(alt2))) < 1e-07)
stopifnot(max(abs(cosums(obj1) - cosums(alt1))) < 1e-07)

Running moments

Since an online algorithm is used, we can compute cumulative running moments. Moreover, we can remove observations, and thus compute moments over a fixed length lookback window. The code checks for negative even moments caused by roundoff, and restarts the computation to correct; periodic recomputation can be forced by an input parameter.

A demonstration:

library(fromo)
library(moments)
library(microbenchmark)

set.seed(1234)
x <- rnorm(20)

k5 <- running_kurt5(x, window = 10L)
colnames(k5) <- c("excess_kurtosis", "skew", "stdev", 
    "mean", "nobs")
k5

##       excess_kurtosis  skew stdev   mean nobs
##  [1,]             NaN   NaN   NaN -1.207    1
##  [2,]             NaN   NaN  1.05 -0.465    2
##  [3,]             NaN -0.34  1.16  0.052    3
##  [4,]          -1.520 -0.13  1.53 -0.548    4
##  [5,]          -1.254 -0.50  1.39 -0.352    5
##  [6,]          -0.860 -0.79  1.30 -0.209    6
##  [7,]          -0.714 -0.70  1.19 -0.261    7
##  [8,]          -0.525 -0.64  1.11 -0.297    8
##  [9,]          -0.331 -0.58  1.04 -0.327    9
## [10,]          -0.331 -0.42  1.00 -0.383   10
## [11,]           0.262 -0.65  0.95 -0.310   10
## [12,]           0.017 -0.30  0.95 -0.438   10
## [13,]           0.699 -0.61  0.79 -0.624   10
## [14,]          -0.939  0.69  0.53 -0.383   10
## [15,]          -0.296  0.99  0.64 -0.330   10
## [16,]           1.078  1.33  0.57 -0.391   10
## [17,]           1.069  1.32  0.57 -0.385   10
## [18,]           0.868  1.29  0.60 -0.421   10
## [19,]           0.799  1.31  0.61 -0.449   10
## [20,]           1.193  1.50  1.07 -0.118   10

# trust but verify
alt5 <- sapply(seq_along(x), function(iii) {
    rowi <- max(1, iii - 10 + 1)
    kurtosis(x[rowi:iii]) - 3
}, simplify = TRUE)

cbind(alt5, k5[, 1])

##         alt5       
##  [1,]    NaN    NaN
##  [2,] -2.000    NaN
##  [3,] -1.500    NaN
##  [4,] -1.520 -1.520
##  [5,] -1.254 -1.254
##  [6,] -0.860 -0.860
##  [7,] -0.714 -0.714
##  [8,] -0.525 -0.525
##  [9,] -0.331 -0.331
## [10,] -0.331 -0.331
## [11,]  0.262  0.262
## [12,]  0.017  0.017
## [13,]  0.699  0.699
## [14,] -0.939 -0.939
## [15,] -0.296 -0.296
## [16,]  1.078  1.078
## [17,]  1.069  1.069
## [18,]  0.868  0.868
## [19,]  0.799  0.799
## [20,]  1.193  1.193

See also

If you like rolling computations, do also check out the following packages (I believe they are all on CRAN):

• RollingWindow, available only from github.
• RcppRoll.
• roll.

Of these three, it seems that RollingWindow implements the optimal algorithm of reusing computations, while the other two packages gain efficiency from parallelization and implementation in C++.

Running 'scale' operations

Through template magic, the same code was modified to perform running centering, scaling, z-scoring and so on:

library(fromo)
library(moments)
library(microbenchmark)

set.seed(1234)
x <- rnorm(20)

xz <- running_zscored(x, window = 10L)

# trust but verify
altz <- sapply(seq_along(x), function(iii) {
    rowi <- max(1, iii - 10 + 1)
    (x[iii] - mean(x[rowi:iii]))/sd(x[rowi:iii])
}, simplify = TRUE)

cbind(xz, altz)

##              altz
##  [1,]   NaN    NA
##  [2,]  0.71  0.71
##  [3,]  0.89  0.89
##  [4,] -1.18 -1.18
##  [5,]  0.56  0.56
##  [6,]  0.55  0.55
##  [7,] -0.26 -0.26
##  [8,] -0.23 -0.23
##  [9,] -0.23 -0.23
## [10,] -0.51 -0.51
## [11,] -0.17 -0.17
## [12,] -0.59 -0.59
## [13,] -0.19 -0.19
## [14,]  0.84  0.84
## [15,]  2.02  2.02
## [16,]  0.49  0.49
## [17,] -0.22 -0.22
## [18,] -0.82 -0.82
## [19,] -0.64 -0.64
## [20,]  2.37  2.37

A list of the available running functions:

• running_centered : from the current value, subtract the mean over the trailing window.
• running_scaled: divide the current value by the standard deviation over the trailing window.
• running_zscored: from the current value, subtract the mean then divide by the standard deviation over the trailing window.
• running_sharpe: divide the mean by the standard deviation over the trailing window. There is a boolean flag to also compute and return the Mertens' form of the standard error of the Sharpe ratio over the trailing window in the second column.
• running_tstat: compute the t-stat over the trailing window.
• running_cumulants: computes cumulants over the trailing window.
• running_apx_quantiles: computes approximate quantiles over the trailing window based on the cumulants and the Cornish-Fisher approximation.
• running_apx_median: uses running_apx_quantiles to give the approximate median over the trailing window.

Lookahead

The functions running_centered, running_scaled and running_zscored take an optional lookahead parameter that allows you to peek ahead (or behind if negative) to the computed moments for comparing against the current value. These are not supported for running_sharpe or running_tstat because they do not have an idea of the 'current value'.

Here is an example of using the lookahead to z-score some data, compared to a purely time-safe lookback. Around a timestamp of 1000, you can see the difference in outcomes from the two methods:

set.seed(1235)
z <- rnorm(1500, mean = 0, sd = 0.09)
x <- exp(cumsum(z)) - 1

xz_look <- running_zscored(x, window = 301, lookahead = 150)
xz_safe <- running_zscored(x, window = 301, lookahead = 0)
df <- data.frame(timestamp = seq_along(x), raw = x, 
    lookahead = xz_look, lookback = xz_safe)

library(tidyr)
gdf <- gather(df, key = "smoothing", value = "x", -timestamp)

library(ggplot2)
ph <- ggplot(gdf, aes(x = timestamp, y = x, group = smoothing, 
    colour = smoothing)) + geom_line()
print(ph)

Time-Based Running Computations

The standard running moments computations listed above work on a running window of a fixed number of observations. However, sometimes one needs to compute running moments over a different kind of window. The most common form of this is over time-based windows. For example, the following computations:

• Compute the total sales over the past six months, as of every day.
• Compute the volatility of an asset's daily returns, over a yearly window, computed at the end of every trading month.

These are now supported in fromo via the t_running class of functions, which are like the running functions, but accept also the 'times' at which the input are marked, and optionally also the times at which one will 'look back' to perform the computations. The times can be computed implicitly as the cumulative sum of given (non-negative) time deltas.

Here is an example of computing the volatility of daily 'returns' of the Fama French Market factor, based on a one year window, computed at month ends:

# devtools::install_github('shabbychef/aqfb_data')
library(aqfb.data)
library(fromo)
# daily 'returns' of Fama French 4 factors
data(dff4)
# compute month end dates:
library(lubridate)
mo_ends <- unique(lubridate::ceiling_date(index(dff4), 
    "month") %m-% days(1))
res <- t_running_sd3(dff4$Mkt, time = index(dff4), 
    window = 365.25, min_df = 180, lb_time = mo_ends)
df <- cbind(data.frame(mo_ends), data.frame(res))
colnames(df) <- c("date", "sd", "mean", "num_days")
knitr::kable(tail(df), row.names = FALSE)

date	sd	mean	num_days
2018-07-31	0.79	0.07	253
2018-08-31	0.78	0.08	253
2018-09-30	0.79	0.07	251
2018-10-31	0.89	0.03	253
2018-11-30	0.95	0.03	253
2018-12-31	1.09	-0.01	251

And the plot of the time series:

library(ggplot2)
library(scales)
ph <- df %>% ggplot(aes(date, 0.01 * sd)) + geom_line() + 
    geom_point(alpha = 0.1) + scale_y_continuous(labels = scales::percent) + 
    labs(x = "lookback date", y = "standard deviation of percent returns", 
        title = "rolling 1 year volatility of daily Mkt factor returns, computed monthly")
print(ph)

---

Efficiency

We make every attempt to balance numerical robustness, computational efficiency and memory usage. As a bit of strawman-bashing, here we microbenchmark the running Z-score computation against the naive algorithm:

library(fromo)
library(moments)
library(microbenchmark)

set.seed(4422)
x <- rnorm(10000)

dumb_zscore <- function(x, window) {
    altz <- sapply(seq_along(x), function(iii) {
        rowi <- max(1, iii - window + 1)
        xrang <- x[rowi:iii]
        (x[iii] - mean(xrang))/sd(xrang)
    }, simplify = TRUE)
}

val1 <- running_zscored(x, 250)
val2 <- dumb_zscore(x, 250)
stopifnot(max(abs(val1 - val2), na.rm = TRUE) <= 1e-14)

microbenchmark(running_zscored(x, 250), dumb_zscore(x, 
    250))

## Unit: microseconds
##                     expr    min     lq   mean median     uq    max neval cld
##  running_zscored(x, 250)    340    359    397    387    415    576   100  a 
##      dumb_zscore(x, 250) 233681 256483 276580 267985 277785 398526   100   b

Timing against the roll package

More seriously, here we compare the running_sd3 function, which computes the standard deviation, mean and number of elements with the roll_sd and roll_mean functions from the roll package.

# dare I?
library(fromo)
library(microbenchmark)
library(roll)

set.seed(4422)
x <- rnorm(1e+05)
xm <- matrix(x)

v1 <- running_sd3(xm, 250)
rsd <- roll::roll_sd(xm, 250)
rmu <- roll::roll_mean(xm, 250)
# compute error on the 1000th row:
stopifnot(max(abs(v1[1000, ] - c(rsd[1000], rmu[1000], 
    250))) < 1e-14)
# now timings:
microbenchmark(running_sd3(xm, 250), roll::roll_mean(xm, 
    250), roll::roll_sd(xm, 250))

## Unit: milliseconds
##                      expr  min   lq mean median uq  max neval cld
##      running_sd3(xm, 250)  3.3  3.5  3.9    3.7  4  5.8   100 a  
##  roll::roll_mean(xm, 250) 13.0 13.1 14.0   13.4 14 24.2   100  b 
##    roll::roll_sd(xm, 250) 34.8 35.2 37.2   36.1 38 50.0   100   c

OK, that's not a fair comparison: roll_mean is optimized to work columwise on a matrix. Let's unbash this strawman. I create a function using fromo::running_sd3 to compute a running mean or running standard deviation columnwise on a matrix, then compare that to roll_mean and roll_sd:

library(fromo)
library(microbenchmark)
library(roll)

set.seed(4422)
xm <- matrix(rnorm(4e+05), ncol = 100)
fromo_sd <- function(x, wins) {
    apply(x, 2, function(xc) {
        running_sd3(xc, wins)[, 1]
    })
}
fromo_mu <- function(x, wins) {
    apply(x, 2, function(xc) {
        running_sd3(xc, wins)[, 2]
    })
}
wins <- 1000
v1 <- fromo_sd(xm, wins)
rsd <- roll::roll_sd(xm, wins, min_obs = 3)

v2 <- fromo_mu(xm, wins)
rmu <- roll::roll_mean(xm, wins)
# compute error on the 2000th row:
stopifnot(max(abs(v1[2000, ] - rsd[2000, ])) < 1e-14)
stopifnot(max(abs(v2[2000, ] - rmu[2000, ])) < 1e-14)

# now timings: note fromo_mu and fromo_sd do
# exactly the same work, so only time one of them
microbenchmark(fromo_sd(xm, wins), roll::roll_mean(xm, 
    wins), roll::roll_sd(xm, wins), times = 50L)

## Unit: milliseconds
##                       expr  min   lq mean median   uq max neval cld
##         fromo_sd(xm, wins) 35.0 36.6 45.7   37.3 38.3 134    50   b
##  roll::roll_mean(xm, wins)  1.3  1.4  5.0    2.1  2.4  58    50  a 
##    roll::roll_sd(xm, wins)  3.7  3.7  5.4    4.4  4.9  56    50  a

I suspect, however, that roll_mean is literally recomputing moments over the entire window for every cell of the output, instead of reusing computations, which fromo mostly does:

library(roll)
library(microbenchmark)
set.seed(91823)
xm <- matrix(rnorm(2e+05), ncol = 10)
fromo_mu <- function(x, wins, ...) {
    apply(x, 2, function(xc) {
        running_sd3(xc, wins, ...)[, 2]
    })
}

microbenchmark(roll::roll_mean(xm, 10, min_obs = 3), 
    roll::roll_mean(xm, 100, min_obs = 3), roll::roll_mean(xm, 
        1000, min_obs = 3), roll::roll_mean(xm, 10000, 
        min_obs = 3), fromo_mu(xm, 10, min_df = 3), 
    fromo_mu(xm, 100, min_df = 3), fromo_mu(xm, 1000, 
        min_df = 3), fromo_mu(xm, 10000, min_df = 3), 
    times = 100L)

## Unit: milliseconds
##                                     expr   min    lq  mean median    uq    max neval   cld
##     roll::roll_mean(xm, 10, min_obs = 3)   1.5   1.7   1.9    1.8   1.9    3.4   100 a    
##    roll::roll_mean(xm, 100, min_obs = 3)  10.5  10.8  11.4   11.1  11.8   15.7   100 a    
##   roll::roll_mean(xm, 1000, min_obs = 3)  95.9  99.4 105.6  102.2 107.4  166.9   100    d 
##  roll::roll_mean(xm, 10000, min_obs = 3) 738.0 770.1 803.5  781.6 803.1 1086.6   100     e
##             fromo_mu(xm, 10, min_df = 3)   6.2   6.8   7.6    7.3   8.4   12.5   100 a    
##            fromo_mu(xm, 100, min_df = 3)   7.7   8.1  10.0    8.6   9.8   94.7   100 a    
##           fromo_mu(xm, 1000, min_df = 3)  20.3  21.6  23.2   22.6  23.9   34.5   100  b   
##          fromo_mu(xm, 10000, min_df = 3)  81.7  84.5  90.3   87.6  94.3  130.6   100   c

The runtime for operations from roll grow with the window size. The equivalent operations from fromo also consume more time for longer windows. In theory they would be invariant with respect to window size, but I coded them to 'restart' the computation periodically for improved accuracy. The user has control over how often this happens, in order to balance speed and accuracy. Here I set that parameter very large to show that runtimes need not grow with window size:

library(fromo)
library(microbenchmark)
set.seed(91823)
xm <- matrix(rnorm(2e+05), ncol = 10)
fromo_mu <- function(x, wins, ...) {
    apply(x, 2, function(xc) {
        running_sd3(xc, wins, ...)[, 2]
    })
}
rp <- 1L + nrow(xm)

microbenchmark(fromo_mu(xm, 10, min_df = 3, restart_period = rp), 
    fromo_mu(xm, 100, min_df = 3, restart_period = rp), 
    fromo_mu(xm, 1000, min_df = 3, restart_period = rp), 
    fromo_mu(xm, 10000, min_df = 3, restart_period = rp), 
    times = 100L)

## Unit: milliseconds
##                                                  expr min  lq mean median  uq max neval cld
##     fromo_mu(xm, 10, min_df = 3, restart_period = rp) 6.1 6.6  7.3    6.9 7.9  10   100   a
##    fromo_mu(xm, 100, min_df = 3, restart_period = rp) 6.3 6.8  7.4    7.1 7.8  11   100   a
##   fromo_mu(xm, 1000, min_df = 3, restart_period = rp) 6.1 6.7  7.4    7.1 7.8  12   100   a
##  fromo_mu(xm, 10000, min_df = 3, restart_period = rp) 6.3 6.8  7.5    7.1 8.2  13   100   a

Here are some more benchmarks, also against the rollingWindow package, for running sums:

library(microbenchmark)
library(fromo)
library(RollingWindow)
library(roll)

set.seed(12345)
x <- rnorm(10000)
xm <- matrix(x)
wins <- 1000

# run fun on each wins sized window...
silly_fun <- function(x, wins, fun, ...) {
    xout <- rep(NA, length(x))
    for (iii in seq_along(x)) {
        xout[iii] <- fun(x[max(1, iii - wins + 1):iii], 
            ...)
    }
    xout
}
vals <- list(running_sum(x, wins, na_rm = FALSE), RollingWindow::RollingSum(x, 
    wins, na_method = "ignore"), roll::roll_sum(xm, 
    wins), silly_fun(x, wins, sum, na.rm = FALSE))

# check all equal?
stopifnot(max(unlist(lapply(vals[2:length(vals)], function(av) {
    err <- vals[[1]] - av
    max(abs(err[wins:length(err)]), na.rm = TRUE)
}))) < 1e-12)

# benchmark it
microbenchmark(running_sum(x, wins, na_rm = FALSE), 
    RollingWindow::RollingSum(x, wins), running_sum(x, 
        wins, na_rm = TRUE), RollingWindow::RollingSum(x, 
        wins, na_method = "ignore"), roll::roll_sum(xm, 
        wins))

## Unit: microseconds
##                                                      expr  min   lq mean median   uq  max neval  cld
##                       running_sum(x, wins, na_rm = FALSE)   70   73   89     79  105  197   100 a   
##                        RollingWindow::RollingSum(x, wins)  108  116  146    129  165  329   100  b  
##                        running_sum(x, wins, na_rm = TRUE)  101  105  138    109  133 1918   100 ab  
##  RollingWindow::RollingSum(x, wins, na_method = "ignore")  353  369  415    403  434  697   100   c 
##                                  roll::roll_sum(xm, wins) 4153 4205 4309   4236 4338 5570   100    d

And running means:

library(microbenchmark)
library(fromo)
library(RollingWindow)
library(roll)

set.seed(12345)
x <- rnorm(10000)
xm <- matrix(x)
wins <- 1000

vals <- list(running_mean(x, wins, na_rm = FALSE), 
    RollingWindow::RollingMean(x, wins, na_method = "ignore"), 
    roll::roll_mean(xm, wins), silly_fun(x, wins, mean, 
        na.rm = FALSE))

# check all equal?
stopifnot(max(unlist(lapply(vals[2:length(vals)], function(av) {
    err <- vals[[1]] - av
    max(abs(err[wins:length(err)]), na.rm = TRUE)
}))) < 1e-12)

# benchmark it:
microbenchmark(running_mean(x, wins, na_rm = FALSE, 
    restart_period = 1e+05), RollingWindow::RollingMean(x, 
    wins), running_mean(x, wins, na_rm = TRUE, restart_period = 1e+05), 
    RollingWindow::RollingMean(x, wins, na_method = "ignore"), 
    roll::roll_mean(xm, wins))

## Unit: microseconds
##                                                          expr  min   lq mean median   uq  max neval  cld
##  running_mean(x, wins, na_rm = FALSE, restart_period = 1e+05)   71   78  101     96  115  225   100 a   
##                           RollingWindow::RollingMean(x, wins)  133  167  230    218  268  466   100  b  
##   running_mean(x, wins, na_rm = TRUE, restart_period = 1e+05)  102  111  165    137  164 2271   100 ab  
##     RollingWindow::RollingMean(x, wins, na_method = "ignore")  376  451  570    534  669 1170   100   c 
##                                     roll::roll_mean(xm, wins) 5014 5260 5667   5530 5952 7535   100    d