Fast (Grouped, Weighted) Variance and Standard Deviation for Matrix-Like Objects
fvar_fsd.Rdfvar and fsd are generic functions that compute the (column-wise) variance and standard deviation of x, (optionally) grouped by g and/or frequency-weighted by w. The TRA argument can further be used to transform x using its (grouped, weighted) variance/sd.
Usage
fvar(x, ...)
fsd(x, ...)
# S3 method for default
fvar(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for default
fsd(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for matrix
fvar(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, drop = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for matrix
fsd(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, drop = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for data.frame
fvar(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, drop = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for data.frame
fsd(x, g = NULL, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = TRUE, drop = TRUE, stable.algo = .op[["stable.algo"]], ...)
# S3 method for grouped_df
fvar(x, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = FALSE, keep.group_vars = TRUE, keep.w = TRUE,
stub = .op[["stub"]], stable.algo = .op[["stable.algo"]], ...)
# S3 method for grouped_df
fsd(x, w = NULL, TRA = NULL, na.rm = .op[["na.rm"]],
use.g.names = FALSE, keep.group_vars = TRUE, keep.w = TRUE,
stub = .op[["stub"]], stable.algo = .op[["stable.algo"]], ...)Arguments
- x
a numeric vector, matrix, data frame or grouped data frame (class 'grouped_df').
- g
a factor,
GRPobject, atomic vector (internally converted to factor) or a list of vectors / factors (internally converted to aGRPobject) used to groupx.- w
a numeric vector of (non-negative) weights, may contain missing values.
- TRA
an integer or quoted operator indicating the transformation to perform: 0 - "na" | 1 - "fill" | 2 - "replace" | 3 - "-" | 4 - "-+" | 5 - "/" | 6 - "%" | 7 - "+" | 8 - "*" | 9 - "%%" | 10 - "-%%". See
TRA.- na.rm
logical. Skip missing values in
x. Defaults toTRUEand implemented at very little computational cost. Ifna.rm = FALSEaNAis returned when encountered.- use.g.names
logical. Make group-names and add to the result as names (default method) or row-names (matrix and data frame methods). No row-names are generated for data.table's.
- drop
matrix and data.frame method: Logical.
TRUEdrops dimensions and returns an atomic vector ifg = NULLandTRA = NULL.- keep.group_vars
grouped_df method: Logical.
FALSEremoves grouping variables after computation.- keep.w
grouped_df method: Logical. Retain summed weighting variable after computation (if contained in
grouped_df).- stub
character. If
keep.w = TRUEandstub = TRUE(default), the summed weights column is prefixed by"sum.". Users can specify a different prefix through this argument, or set it toFALSEto avoid prefixing.- stable.algo
logical.
TRUE(default) use Welford's numerically stable online algorithm.FALSEimplements a faster but numerically unstable one-pass method. See Details.- ...
arguments to be passed to or from other methods. If
TRAis used, passingset = TRUEwill transform data by reference and return the result invisibly.
Details
Welford's online algorithm used by default to compute the variance is well described here (the section Weighted incremental algorithm also shows how the weighted variance is obtained by this algorithm).
If stable.algo = FALSE, the variance is computed in one-pass as (sum(x^2)-n*mean(x)^2)/(n-1), where sum(x^2) is the sum of squares from which the expected sum of squares n*mean(x)^2 is subtracted, normalized by n-1 (Bessel's correction). This is numerically unstable if sum(x^2) and n*mean(x)^2 are large numbers very close together, which will be the case for large n, large x-values and small variances (catastrophic cancellation occurs, leading to a loss of numeric precision). Numeric precision is however still maximized through the internal use of long doubles in C++, and the fast algorithm can be up to 4-times faster compared to Welford's method.
The weighted variance is computed with frequency weights as (sum(x^2*w)-sum(w)*weighted.mean(x,w)^2)/(sum(w)-1). If na.rm = TRUE, missing values will be removed from both x and w i.e. utilizing only x[complete.cases(x,w)] and w[complete.cases(x,w)].
For further computational detail see fsum.
Value
fvar returns the (w weighted) variance of x, grouped by g, or (if TRA is used) x transformed by its (grouped, weighted) variance. fsd computes the standard deviation of x in like manor.
References
Welford, B. P. (1962). Note on a method for calculating corrected sums of squares and products. Technometrics. 4 (3): 419-420. doi:10.2307/1266577.
Examples
## default vector method
fvar(mtcars$mpg) # Simple variance (all examples also hold for fvar!)
#> [1] 36.3241
fsd(mtcars$mpg) # Simple standard deviation
#> [1] 6.026948
fsd(mtcars$mpg, w = mtcars$hp) # Weighted sd: Weighted by hp
#> [1] 5.150858
fsd(mtcars$mpg, TRA = "/") # Simple transformation: scaling (See also ?fscale)
#> [1] 3.484351 3.484351 3.783009 3.550719 3.102731 3.003178 2.372677 4.048484
#> [9] 3.783009 3.185692 2.953402 2.721112 2.870441 2.522006 1.725583 1.725583
#> [17] 2.439045 5.375855 5.044012 5.624737 3.567311 2.571783 2.522006 2.206755
#> [25] 3.185692 4.529656 4.313958 5.044012 2.621559 3.268653 2.488822 3.550719
fsd(mtcars$mpg, mtcars$cyl) # Grouped sd
#> 4 6 8
#> 4.509828 1.453567 2.560048
fsd(mtcars$mpg, mtcars$cyl, mtcars$hp) # Grouped weighted sd
#> 4 6 8
#> 4.250863 1.294689 2.390448
fsd(mtcars$mpg, mtcars$cyl, TRA = "/") # Scaling by group
#> [1] 14.447218 14.447218 5.055626 14.722403 7.304550 12.452126 5.585833
#> [8] 5.410406 5.055626 13.208885 12.245737 6.406130 6.757686 5.937388
#> [15] 4.062424 4.062424 5.742080 7.184310 6.740834 7.516917 4.767366
#> [22] 6.054574 5.937388 5.195215 7.499859 6.053446 5.765187 6.740834
#> [29] 6.171759 13.552866 5.859265 4.745192
fsd(mtcars$mpg, mtcars$cyl, mtcars$hp, "/") # Group-scaling using weighted group sds
#> [1] 16.220111 16.220111 5.363617 16.529066 7.822800 13.980191 5.982141
#> [8] 5.740011 5.363617 14.829816 13.748475 6.860638 7.237136 6.358640
#> [15] 4.350648 4.350648 6.149474 7.621982 7.151489 7.974852 5.057797
#> [22] 6.484139 6.358640 5.563810 8.031966 6.422226 6.116405 7.151489
#> [29] 6.609639 15.216009 6.274973 5.034272
## data.frame method
fsd(iris) # This works, although 'Species' is a factor variable
#> Sepal.Length Sepal.Width Petal.Length Petal.Width Species
#> 0.8280661 0.4358663 1.7652982 0.7622377 0.8192319
fsd(mtcars, drop = FALSE) # This works, all columns are numeric variables
#> mpg cyl disp hp drat wt qsec vs
#> 1 6.026948 1.785922 123.9387 68.56287 0.5346787 0.9784574 1.786943 0.5040161
#> am gear carb
#> 1 0.4989909 0.7378041 1.6152
fsd(iris[-5], iris[5]) # By Species: iris[5] is still a list, and thus passed to GRP()
#> Sepal.Length Sepal.Width Petal.Length Petal.Width
#> setosa 0.3524897 0.3790644 0.1736640 0.1053856
#> versicolor 0.5161711 0.3137983 0.4699110 0.1977527
#> virginica 0.6358796 0.3224966 0.5518947 0.2746501
fsd(iris[-5], iris[[5]]) # Same thing much faster: fsd recognizes 'Species' is a factor
#> Sepal.Length Sepal.Width Petal.Length Petal.Width
#> setosa 0.3524897 0.3790644 0.1736640 0.1053856
#> versicolor 0.5161711 0.3137983 0.4699110 0.1977527
#> virginica 0.6358796 0.3224966 0.5518947 0.2746501
head(fsd(iris[-5], iris[[5]], TRA = "/")) # Data scaled by species (see also fscale)
#> Sepal.Length Sepal.Width Petal.Length Petal.Width
#> 1 14.46851 9.233260 8.061544 1.897793
#> 2 13.90112 7.914223 8.061544 1.897793
#> 3 13.33372 8.441838 7.485720 1.897793
#> 4 13.05003 8.178031 8.637369 1.897793
#> 5 14.18481 9.497068 8.061544 1.897793
#> 6 15.31960 10.288490 9.789018 3.795585
## matrix method
m <- qM(mtcars)
fsd(m)
#> mpg cyl disp hp drat wt
#> 6.0269481 1.7859216 123.9386938 68.5628685 0.5346787 0.9784574
#> qsec vs am gear carb
#> 1.7869432 0.5040161 0.4989909 0.7378041 1.6152000
fsd(m, mtcars$cyl) # etc..
#> mpg cyl disp hp drat wt qsec vs
#> 4 4.509828 0 26.87159 20.93453 0.3654711 0.5695637 1.682445 0.3015113
#> 6 1.453567 0 41.56246 24.26049 0.4760552 0.3563455 1.706866 0.5345225
#> 8 2.560048 0 67.77132 50.97689 0.3723618 0.7594047 1.196014 0.0000000
#> am gear carb
#> 4 0.4670994 0.5393599 0.522233
#> 6 0.5345225 0.6900656 1.812654
#> 8 0.3631365 0.7262730 1.556624
## method for grouped data frames - created with dplyr::group_by or fgroup_by
mtcars |> fgroup_by(cyl,vs,am) |> fsd()
#> cyl vs am mpg disp hp drat wt qsec
#> 1 4 0 1 NA NA NA NA NA NA
#> 2 4 1 0 1.4525839 13.969371 19.65536 0.1300000 0.4075230 1.6714365
#> 3 4 1 1 4.7577005 18.802128 24.14441 0.3783926 0.4400840 0.9454628
#> 4 6 0 1 0.7505553 8.660254 37.52777 0.1616581 0.1281601 0.7687219
#> 5 6 1 0 1.6317169 44.742634 9.17878 0.5919459 0.1162164 0.8159044
#> 6 8 0 0 2.7743959 71.823494 33.35984 0.2302749 0.7683069 0.8016475
#> gear carb
#> 1 NA NA
#> 2 0.5773503 0.5773503
#> 3 0.3779645 0.5345225
#> 4 0.5773503 1.1547005
#> 5 0.5773503 1.7320508
#> 6 0.0000000 0.9003366
#> [ reached 'max' / getOption("max.print") -- omitted 1 rows ]
mtcars |> fgroup_by(cyl,vs,am) |> fsd(keep.group_vars = FALSE) # Remove grouping columns
#> mpg disp hp drat wt qsec gear
#> 1 NA NA NA NA NA NA NA
#> 2 1.4525839 13.969371 19.65536 0.1300000 0.4075230 1.67143651 0.5773503
#> 3 4.7577005 18.802128 24.14441 0.3783926 0.4400840 0.94546285 0.3779645
#> 4 0.7505553 8.660254 37.52777 0.1616581 0.1281601 0.76872188 0.5773503
#> 5 1.6317169 44.742634 9.17878 0.5919459 0.1162164 0.81590441 0.5773503
#> 6 2.7743959 71.823494 33.35984 0.2302749 0.7683069 0.80164745 0.0000000
#> 7 0.5656854 35.355339 50.20458 0.4808326 0.2828427 0.07071068 0.0000000
#> carb
#> 1 NA
#> 2 0.5773503
#> 3 0.5345225
#> 4 1.1547005
#> 5 1.7320508
#> 6 0.9003366
#> 7 2.8284271
mtcars |> fgroup_by(cyl,vs,am) |> fsd(hp) # Weighted by hp
#> cyl vs am sum.hp mpg disp drat wt qsec gear
#> 1 4 0 1 91 0.0000000 0.000000 0.0000000 0.00000000 0.0000000 0.0000000
#> 2 4 1 0 254 1.1242936 11.439070 0.1086204 0.34150141 1.4030344 0.4868090
#> 3 4 1 1 564 4.5643045 17.861603 0.3117619 0.44698787 0.9335417 0.4006210
#> 4 6 0 1 395 0.6465871 7.460621 0.1392649 0.09592878 0.6512685 0.4973747
#> 5 6 1 0 461 1.3956451 38.774265 0.5094278 0.09888402 0.7006934 0.4994102
#> 6 8 0 0 2330 2.6621685 68.845963 0.2331010 0.75554498 0.8299946 0.0000000
#> carb
#> 1 0.0000000
#> 2 0.4868090
#> 3 0.5002424
#> 4 0.9947494
#> 5 1.4982306
#> 6 0.8510728
#> [ reached 'max' / getOption("max.print") -- omitted 1 rows ]
mtcars |> fgroup_by(cyl,vs,am) |> fsd(hp, "/") # Weighted scaling transformation
#> cyl vs am hp mpg disp drat wt
#> Mazda RX4 6 0 1 110 32.478221 21.445937 28.004181 27.311929
#> Mazda RX4 Wag 6 0 1 110 32.478221 21.445937 28.004181 29.970151
#> Datsun 710 4 1 1 93 4.995285 6.046490 12.349167 5.190297
#> Hornet 4 Drive 6 1 0 110 15.333411 6.653898 6.045999 32.512836
#> Hornet Sportabout 8 0 0 175 7.024349 5.229065 13.513454 4.553005
#> Valiant 6 1 0 105 12.968913 5.802818 5.417844 34.990486
#> qsec gear carb
#> Mazda RX4 25.27376 8.042226 4.021113
#> Mazda RX4 Wag 26.13362 8.042226 4.021113
#> Datsun 710 19.93483 9.984498 1.999031
#> Hornet 4 Drive 27.74395 6.007086 0.667454
#> Hornet Sportabout 20.50616 Inf 2.349975
#> Valiant 28.85713 6.007086 0.667454
#> [ reached 'max' / getOption("max.print") -- omitted 26 rows ]
#>
#> Grouped by: cyl, vs, am [7 | 5 (3.8) 1-12]