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, [email protected]

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),
    kurtosis(x), skewness(x), sd(x), mean(x))

## Unit: microseconds
##         expr  min    lq  mean median    uq  max neval cld
##     kurt5(x) 80.5  81.4  87.0     82  84.1  162   100 a  
##     skew4(x) 69.2  70.0  73.9     71  71.9  117   100 a  
##       sd3(x) 12.4  13.1  13.8     13  13.8   35   100  b 
##     dumbk(x) 98.1  99.6 142.2    101 108.4 3322   100   c
##  kurtosis(x) 40.3  41.3  46.1     42  43.5  215   100 ab 
##  skewness(x) 39.6  41.1  42.8     42  42.7   72   100 ab 
##        sd(x)  9.6  10.5  11.4     11  11.4   24   100  b 
##      mean(x)  4.2   4.7   5.1      5   5.3   15   100  b

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) 822 831  873    859  927  943    10 a     
##     skew4(x) 701 715  756    752  783  825    10  b    
##       sd3(x) 105 106  113    108  114  137    10   c   
##     dumbk(x) 893 900  982   1009 1035 1079    10    d  
##  kurtosis(x) 419 420  483    470  508  648    10     e 
##  skewness(x) 414 421  468    470  489  571    10     e 
##        sd(x)  38  38   42     42   44   47    10      f
##      mean(x)  19  19   20     19   20   22    10      f

# clean up
rm(x)

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) 120 125  133    130 136 180   100  a 
##    slow_sd(x, w) 235 277  318    294 329 575   100   b

# clean up
rm(x, w)

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 adjustment 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)

plot of chunk toy_zscore

Running Bivariate Computations

The package now supports operations on a running window of two aligned input series, $x$ and $y$, and can compute the correlation, covariance, and OLS regression on the two over a running windo. The following are supported:

• running_correlation: the correlation over the trailing window.
• running_covariance: the covariance of the two series.
• running_covariance3: the full variance-covariance of the two series, returning matrix with three columns.
• running_regression_intercept: the intercept of the OLS regression fit over the trailing window.
• running_regression_slope: the slope of the OLS regression fit over the trailing window.
• running_regression_fit: a matrix of the intercept and slope of the OLS regression fit over the trailing window.
• running_regression_diagnostics: a matrix of the intercept and slope of the OLS regression fit, the regression standard error and the standard errors of the intercept and slope, over the trailing window.

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.

The t_running functions now also include the bivariate computations t_running_correlation, t_running_covariance, t_running_covariance3, t_running_regression_intercept, t_running_regression_slope, t_running_regression_fit, t_running_regression_diagnostics.

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)

plot of chunk trun_testing

Now consider the running correlation of the "SMB" factor against the "Mkt" factor over time:

rho <- t_running_correlation(x = dff4$Mkt, y = dff4$SMB,
    time = index(dff4), window = 365.25, min_df = 180,
    lb_time = mo_ends)
library(ggplot2)
library(scales)
ph <- cbind(data.frame(mo_ends), data.frame(rho)) %>%
    setNames(c("date", "rho")) %>%
    ggplot(aes(date, rho)) + geom_line() + geom_point(alpha = 0.1) +
    geom_hline(yintercept = 0, linetype = 2, alpha = 0.5) +
    labs(x = "lookback date", y = expression(rho),
        title = "Rolling 1 year correlation of daily SMB and Mkt factor returns, computed monthly")
print(ph)

plot of chunk trun_corr_testing

---

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)    427    454    519    479    565    840   100  a 
##      dumb_zscore(x, 250) 173408 195446 219268 209673 227987 425059   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: microseconds
##                      expr  min   lq mean median   uq   max neval cld
##      running_sd3(xm, 250) 5455 5638 5955   5753 5842 18177   100 a  
##  roll::roll_mean(xm, 250)  818 1025 1089   1052 1097  1931   100  b 
##    roll::roll_sd(xm, 250) 3327 3432 3661   3508 3601 16591   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) 43.6 44.4 47.7   46.8 48.3  61    50 a  
##  roll::roll_mean(xm, wins)  1.3  1.4  2.0    1.5  2.4   5    50  b 
##    roll::roll_sd(xm, wins)  3.5  3.6  4.6    3.9  4.7  13    50   c

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: microseconds
##                                     expr    min     lq   mean median     uq    max neval  cld
##     roll::roll_mean(xm, 10, min_obs = 3)    823    866   1121    909   1032   5444   100 a   
##    roll::roll_mean(xm, 100, min_obs = 3)    823    886   1030    941    990   4239   100 a   
##   roll::roll_mean(xm, 1000, min_obs = 3)    823    869   1085    924    970   7718   100 a   
##  roll::roll_mean(xm, 10000, min_obs = 3)    813    873   1041    909    947   4255   100 a   
##             fromo_mu(xm, 10, min_df = 3)   6746   6924   7961   7011   7633  20672   100  b  
##            fromo_mu(xm, 100, min_df = 3)   8553   8763   9938   8932  11541  16519   100  b  
##           fromo_mu(xm, 1000, min_df = 3)  25729  26368  27494  26680  28197  41629   100   c 
##          fromo_mu(xm, 10000, min_df = 3) 107917 109834 113658 111135 112992 274406   100    d

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) 6.3 6.7 10.2   11.1 11  19   100  a 
##    fromo_mu(xm, 100, min_df = 3, restart_period = rp) 6.2 6.6  9.6   10.9 11  15   100  ab
##   fromo_mu(xm, 1000, min_df = 3, restart_period = rp) 6.3 6.5  9.2    8.4 11  16   100   b
##  fromo_mu(xm, 10000, min_df = 3, restart_period = rp) 5.6 5.9  8.9   10.3 11  15   100   b

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)  99 101  116    105 131 166   100 a  
##                        RollingWindow::RollingSum(x, wins) 119 124  195    169 245 391   100  b 
##                        running_sum(x, wins, na_rm = TRUE)  98 100  118    107 131 250   100 a  
##  RollingWindow::RollingSum(x, wins, na_method = "ignore") 262 270  408    366 499 796   100   c
##                                  roll::roll_sum(xm, wins)  78  93  131    116 150 331   100 a

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)  96  99  113    102 125 188   100 a  
##                           RollingWindow::RollingMean(x, wins) 121 125  189    170 241 383   100  b 
##   running_mean(x, wins, na_rm = TRUE, restart_period = 1e+05)  96  99  114    102 126 206   100 a  
##     RollingWindow::RollingMean(x, wins, na_method = "ignore") 260 268  373    371 465 618   100   c
##                                     roll::roll_mean(xm, wins)  93 101  132    123 150 301   100 a