# How to find cumulative variance or standard deviation in R

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I have a column `X` in a data frame, for which I need to find the cumulative standard deviation.

``X  Cumulative.SD 1     - 4    2.12 5    2.08 6    2.16 9    2.91 ``

the basic idea

``x <- c(1, 4, 5, 6, 9)  ## cumulative sample size n <- seq_along(x)  ## cumulative mean m <- cumsum(x) / n #[1] 1.000000 2.500000 3.333333 4.000000 5.000000  ## cumulative squared mean m2 <- cumsum(x * x) / n #[1]  1.0  8.5 14.0 19.5 31.8  ## cumulative variance v <- (m2 - m * m) * (n / (n - 1)) #[1]      NaN 4.500000 4.333333 4.666667 8.500000  ## cumulative standard deviation s <- sqrt(v) #[1]      NaN 2.121320 2.081666 2.160247 2.915476 ``

utility functions

``cummean <- function (x) cumsum(x) / seq_along(x)  cumvar <- function (x, sd = FALSE) {   x <- x - x[sample.int(length(x), 1)]  ## see Remark 2 below   n <- seq_along(x)   v <- (cumsum(x ^ 2) - cumsum(x) ^ 2 / n) / (n - 1)   if (sd) v <- sqrt(v)   v   }  ## Rcpp version: `cumvar_cpp` library(Rcpp)  cppFunction('NumericVector cumvar_cpp(NumericVector x, bool sd) {   int n = x.size();   NumericVector v(n);   srand(time(NULL));   double pivot = x[rand() % n];   double *xx = &x[0], *xx_end = &x[n], *vv = &v[0];   int i = 0; double xi, sum2 = 0.0, sum = 0.0, vi;   for (; xx < xx_end; xx++, vv++, i++) {     xi = *xx - pivot;     sum += xi; sum2 += xi * xi;     vi = (sum2 - (sum * sum) / (i + 1)) / i;     if (sd) vi = sqrt(vi);     *vv = vi;     }   return v;   }') ``

speed

`cumvar` and `cumvar_cpp` are fast for two reasons:

1. They are "vectorized";
2. They have `O(n)` complexity rather than `O(n ^ 2)` complexity.

They are increasingly faster than simple rolling computations for increasingly longer vectors.

``x <- rnorm(1e+3) library(microbenchmark) library(TTR) microbenchmark("zheyuan" = cumvar(x, TRUE),                "zheyuan_cpp" = cumvar_cpp(x, TRUE),                "Rich" = vapply(seq_along(x), function(i) sd(x[1:i]), 1),                "akrun" = runSD(x, n = 1, cumulative = TRUE)) #Unit: microseconds #        expr       min         lq        mean     median         uq        max #     zheyuan   101.261   105.2505   118.85214   121.0040   128.5925    157.702 # zheyuan_cpp    69.749    72.7190    81.38878    82.2335    84.2820    213.193 #        Rich 74595.329 75201.9420 77533.38803 75814.6945 77465.9945 136099.832 #       akrun  4329.892  4436.0145  4710.82440  4669.8380  4715.6035   6908.231  x <- rnorm(1e+4) microbenchmark("zheyuan" = cumvar(x, TRUE),                "zheyuan_cpp" = cumvar_cpp(x, TRUE),                "akrun" = runSD(x, n = 1, cumulative = TRUE)) #Unit: microseconds #        expr        min         lq        mean      median         uq #     zheyuan    842.844    863.676    997.9585    880.2245    968.077 # zheyuan_cpp    618.823    632.254    709.1971    639.2990    657.366 #       akrun 147279.410 148200.370 150839.8161 149599.6170 151981.069  x <- rnorm(1e+5) microbenchmark("zheyuan" = cumvar(x, TRUE),                "zheyuan_cpp" = cumvar_cpp(x, TRUE),                "akrun" = runSD(x, n = 1, cumulative = TRUE),                times = 10) #Unit: milliseconds #        expr          min           lq         mean       median           uq #     zheyuan     8.446502     8.657557    22.531637     9.431389    11.082594 # zheyuan_cpp     6.189955     6.305053     6.698292     6.365656     6.812374 #       akrun 14477.847050 14559.844609 14787.200820 14755.526655 15021.524429 ``

Remark 1

I googled "cumulative variance R" and found a small package `cumstats`. It has a `cumvar` function but is written with `sapply` (like Rich Scriven's answer), so there is no need for me to experiment it.

Remark 2

Thanks for Benjamin Christoffersen's professional elaboration. I've added to my original `cumvar` the following line to enhance numerical stability.

``x <- x - x[sample.int(length(x), 1)] ``

Then it is returning correct values compared with Ben's `roll_var`.

``## using Ben's example set.seed(99858398) x <- rnorm(1e2, mean = 1e8) all.equal(cumvar_cpp(x, FALSE), base::c(roll_var(x))) #[1] TRUE ``

A speed comparison for computing cumulative variance:

``x <- rnorm(1e+6) microbenchmark("zheyuan" = cumvar(x, TRUE),                "zheyuan_cpp" = cumvar_cpp(x, FALSE),                "Ben_cpp" = roll_var(x),                times = 20) #Unit: milliseconds #        expr      min       lq     mean   median       uq       max neval #     zheyuan 85.47676 87.36403 91.05656 89.64444 93.99912 102.04964    20 # zheyuan_cpp 42.27983 42.41443 44.29919 42.65548 46.43293  51.24379    20 #     Ben_cpp 46.99105 47.12178 49.48072 47.76016 50.44587  60.11491    20  x <- rnorm(1e+7) microbenchmark("zheyuan" = cumvar(x, TRUE),                "zheyuan_cpp" = cumvar_cpp(x, FALSE),                "Ben_cpp" = roll_var(x),                times = 10) #Unit: milliseconds #        expr       min        lq      mean    median        uq       max neval #     zheyuan 1171.3624 1215.8870 1278.3862 1266.9576 1330.6168 1486.7895    10 # zheyuan_cpp  463.6257  473.2711  479.8156  476.8822  482.4766  512.0520    10 #     Ben_cpp  571.7481  583.2694  587.9993  584.1206  602.0050  605.1515    10 ``