Fast (Grouped, Weighted) Scaling and Centering of Matrix-like Objects
fscale.Rdfscale is a generic function to efficiently standardize (scale and center) data. STD is a wrapper around fscale representing the 'standardization operator', with more options than fscale when applied to matrices and data frames. Standardization can be simple or groupwise, ordinary or weighted. Arbitrary target means and standard deviations can be set, with special options for grouped scaling and centering. It is also possible to scale data without centering i.e. perform mean-preserving scaling.
Usage
fscale(x, ...)
STD(x, ...)
# Default S3 method
fscale(x, g = NULL, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# Default S3 method
STD(x, g = NULL, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'matrix'
fscale(x, g = NULL, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'matrix'
STD(x, g = NULL, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1,
stub = .op[["stub"]], ...)
# S3 method for class 'data.frame'
fscale(x, g = NULL, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'data.frame'
STD(x, by = NULL, w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]],
mean = 0, sd = 1, stub = .op[["stub"]], keep.by = TRUE, keep.w = TRUE, ...)
# Methods for indexed data / compatibility with plm:
# S3 method for class 'pseries'
fscale(x, effect = 1L, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'pseries'
STD(x, effect = 1L, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'pdata.frame'
fscale(x, effect = 1L, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1, ...)
# S3 method for class 'pdata.frame'
STD(x, effect = 1L, w = NULL, cols = is.numeric, na.rm = .op[["na.rm"]],
mean = 0, sd = 1, stub = .op[["stub"]], keep.ids = TRUE, keep.w = TRUE, ...)
# Methods for grouped data frame / compatibility with dplyr:
# S3 method for class 'grouped_df'
fscale(x, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1,
keep.group_vars = TRUE, keep.w = TRUE, ...)
# S3 method for class 'grouped_df'
STD(x, w = NULL, na.rm = .op[["na.rm"]], mean = 0, sd = 1,
stub = .op[["stub"]], keep.group_vars = TRUE, keep.w = TRUE, ...)Arguments
- x
a numeric vector, matrix, data frame, 'indexed_series' ('pseries'), 'indexed_frame' ('pdata.frame') or grouped data frame ('grouped_df').
- g
a factor,
GRPobject, or atomic vector / list of vectors (internally grouped withgroup) used to groupx.- by
STD data.frame method: Same as
g, but also allows one- or two-sided formulas i.e.~ group1orvar1 + var2 ~ group1 + group2. See Examples.- cols
STD (p)data.frame method: Select columns to scale using a function, column names, indices or a logical vector. Default: All numeric columns. Note:
colsis ignored if a two-sided formula is passed toby.- w
a numeric vector of (non-negative) weights.
STDdata frame andpdata.framemethods also allow a one-sided formula i.e.~ weightcol. Thegrouped_df(dplyr) method supports lazy-evaluation. See Examples.- na.rm
logical. Skip missing values in
xorwwhen computing means and sd's.- effect
plm methods: Select which panel identifier should be used as group-id. 1L takes the first variable in the index, 2L the second etc.. Index variables can also be called by name using a character string. More than one variable can be supplied.
- stub
character. A prefix/stub to add to the names of all transformed columns.
TRUE(default) uses"STD.",FALSEwill not rename columns.- mean
the mean to center on (default is 0). If
mean = FALSE, no centering will be performed. In that case the scaling is mean-preserving. A numeric value different from 0 (i.e.mean = 5) will be added to the data after subtracting out the mean(s), such that the data will have a mean of 5. A special option when performing grouped scaling and centering ismean = "overall.mean". In that case the overall mean of the data will be added after subtracting out group means.- sd
the standard deviation to scale the data to (default is 1). A numeric value different from 0 (i.e.
sd = 3) will scale the data to have a standard deviation of 3. A special option when performing grouped scaling issd = "within.sd". In that case the within standard deviation (= the standard deviation of the group-centered series) will be calculated and applied to each group. The results is that the variance of the data within each group is harmonized without forcing a certain variance (such as 1).- keep.by, keep.ids, keep.group_vars
data.frame, pdata.frame and grouped_df methods: Logical. Retain grouping / panel-identifier columns in the output. For
STD.data.framethis only works if grouping variables were passed in a formula.- keep.w
data.frame, pdata.frame and grouped_df methods: Logical. Retain column containing the weights in the output. Only works if
wis passed as formula / lazy-expression.- ...
arguments to be passed to or from other methods.
Details
If g = NULL, fscale by default (column-wise) subtracts the mean or weighted mean (if w is supplied) from all data points in x, and then divides this difference by the standard deviation or frequency-weighted standard deviation. The result is that all columns in x will have a (weighted) mean 0 and (weighted) standard deviation 1. Alternatively, data can be scaled to have a mean of mean and a standard deviation of sd. If mean = FALSE the data is only scaled (not centered) such that the mean of the data is preserved.
Means and standard deviations are computed using Welford's numerically stable online algorithm.
With groups supplied to g, this standardizing becomes groupwise, so that in each group (in each column) the data points will have mean mean and standard deviation sd. Naturally if mean = FALSE then each group is just scaled and the mean is preserved. For centering without scaling see fwithin.
If na.rm = FALSE and a NA or NaN is encountered, the mean and sd for that group will be NA, and all data points belonging to that group will also be NA in the output.
If na.rm = TRUE, means and sd's are computed (column-wise) on the available data points, and also the weight vector can have missing values. In that case, the weighted mean an sd are computed on (column-wise) complete.cases(x, w), and x is scaled using these statistics. Note that fscale will not insert a missing value in x if the weight for that value is missing, rather, that value will be scaled using a weighted mean and standard-deviated computed without itself! (The intention here is that a few (randomly) missing weights shouldn't break the computation when na.rm = TRUE, but it is not meant for weight vectors with many missing values. If you don't like this behavior, you should prepare your data using x[is.na(w), ] <- NA, or impute your weight vector for non-missing x).
Special options for grouped scaling are mean = "overall.mean" and sd = "within.sd". The former group-centers vectors on the overall mean of the data (see fwithin for more details) and the latter scales the data in each group to have the within-group standard deviation (= the standard deviation of the group-centered data). Thus scaling a grouped vector with options mean = "overall.mean" and sd = "within.sd" amounts to removing all differences in the mean and standard deviations between these groups. In weighted computations, mean = "overall.mean" will subtract weighted group-means from the data and add the overall weighted mean of the data, whereas sd = "within.sd" will compute the weighted within- standard deviation and apply it to each group.
Value
x standardized (mean = mean, standard deviation = sd), grouped by g/by, weighted with w. See Details.
Note
For centering without scaling see fwithin/W. For simple not mean-preserving scaling use fsd(..., TRA = "/"). To sweep pre-computed means and scale-factors out of data see TRA.
Examples
## Simple Scaling & Centering / Standardizing
head(fscale(mtcars)) # Doesn't rename columns
#> mpg cyl disp hp drat
#> Mazda RX4 0.1508848 -0.1049878 -0.57061982 -0.5350928 0.5675137
#> Mazda RX4 Wag 0.1508848 -0.1049878 -0.57061982 -0.5350928 0.5675137
#> Datsun 710 0.4495434 -1.2248578 -0.99018209 -0.7830405 0.4739996
#> Hornet 4 Drive 0.2172534 -0.1049878 0.22009369 -0.5350928 -0.9661175
#> Hornet Sportabout -0.2307345 1.0148821 1.04308123 0.4129422 -0.8351978
#> Valiant -0.3302874 -0.1049878 -0.04616698 -0.6080186 -1.5646078
#> wt qsec vs am gear
#> Mazda RX4 -0.610399567 -0.7771651 -0.8680278 1.1899014 0.4235542
#> Mazda RX4 Wag -0.349785269 -0.4637808 -0.8680278 1.1899014 0.4235542
#> Datsun 710 -0.917004624 0.4260068 1.1160357 1.1899014 0.4235542
#> Hornet 4 Drive -0.002299538 0.8904872 1.1160357 -0.8141431 -0.9318192
#> Hornet Sportabout 0.227654255 -0.4637808 -0.8680278 -0.8141431 -0.9318192
#> Valiant 0.248094592 1.3269868 1.1160357 -0.8141431 -0.9318192
#> carb
#> Mazda RX4 0.7352031
#> Mazda RX4 Wag 0.7352031
#> Datsun 710 -1.1221521
#> Hornet 4 Drive -1.1221521
#> Hornet Sportabout -0.5030337
#> Valiant -1.1221521
head(STD(mtcars)) # By default adds a prefix
#> STD.mpg STD.cyl STD.disp STD.hp STD.drat
#> Mazda RX4 0.1508848 -0.1049878 -0.57061982 -0.5350928 0.5675137
#> Mazda RX4 Wag 0.1508848 -0.1049878 -0.57061982 -0.5350928 0.5675137
#> Datsun 710 0.4495434 -1.2248578 -0.99018209 -0.7830405 0.4739996
#> Hornet 4 Drive 0.2172534 -0.1049878 0.22009369 -0.5350928 -0.9661175
#> Hornet Sportabout -0.2307345 1.0148821 1.04308123 0.4129422 -0.8351978
#> Valiant -0.3302874 -0.1049878 -0.04616698 -0.6080186 -1.5646078
#> STD.wt STD.qsec STD.vs STD.am STD.gear
#> Mazda RX4 -0.610399567 -0.7771651 -0.8680278 1.1899014 0.4235542
#> Mazda RX4 Wag -0.349785269 -0.4637808 -0.8680278 1.1899014 0.4235542
#> Datsun 710 -0.917004624 0.4260068 1.1160357 1.1899014 0.4235542
#> Hornet 4 Drive -0.002299538 0.8904872 1.1160357 -0.8141431 -0.9318192
#> Hornet Sportabout 0.227654255 -0.4637808 -0.8680278 -0.8141431 -0.9318192
#> Valiant 0.248094592 1.3269868 1.1160357 -0.8141431 -0.9318192
#> STD.carb
#> Mazda RX4 0.7352031
#> Mazda RX4 Wag 0.7352031
#> Datsun 710 -1.1221521
#> Hornet 4 Drive -1.1221521
#> Hornet Sportabout -0.5030337
#> Valiant -1.1221521
qsu(STD(mtcars)) # See that is works
#> N Mean SD Min Max
#> STD.mpg 32 0 1 -1.6079 2.2913
#> STD.cyl 32 0 1 -1.2249 1.0149
#> STD.disp 32 -0 1 -1.2879 1.9468
#> STD.hp 32 0 1 -1.381 2.7466
#> STD.drat 32 -0 1 -1.5646 2.4939
#> STD.wt 32 -0 1 -1.7418 2.2553
#> STD.qsec 32 -0 1 -1.874 2.8268
#> STD.vs 32 0 1 -0.868 1.116
#> STD.am 32 -0 1 -0.8141 1.1899
#> STD.gear 32 -0 1 -0.9318 1.7789
#> STD.carb 32 -0 1 -1.1222 3.2117
qsu(STD(mtcars, mean = 5, sd = 3)) # Assigning a mean of 5 and a standard deviation of 3
#> N Mean SD Min Max
#> STD.mpg 32 5 3 0.1764 11.8738
#> STD.cyl 32 5 3 1.3254 8.0446
#> STD.disp 32 5 3 1.1363 10.8403
#> STD.hp 32 5 3 0.8569 13.2397
#> STD.drat 32 5 3 0.3062 12.4817
#> STD.wt 32 5 3 -0.2253 11.766
#> STD.qsec 32 5 3 -0.622 13.4803
#> STD.vs 32 5 3 2.3959 8.3481
#> STD.am 32 5 3 2.5576 8.5697
#> STD.gear 32 5 3 2.2045 10.3368
#> STD.carb 32 5 3 1.6335 14.635
qsu(STD(mtcars, mean = FALSE)) # No centering: Scaling is mean-preserving
#> N Mean SD Min Max
#> STD.mpg 32 20.0906 1 18.4827 22.3819
#> STD.cyl 32 6.1875 1 4.9626 7.2024
#> STD.disp 32 230.7219 1 229.434 232.6686
#> STD.hp 32 146.6875 1 145.3065 149.4341
#> STD.drat 32 3.5966 1 2.032 6.0905
#> STD.wt 32 3.2172 1 1.4755 5.4726
#> STD.qsec 32 17.8487 1 15.9747 20.6755
#> STD.vs 32 0.4375 1 -0.4305 1.5535
#> STD.am 32 0.4062 1 -0.4079 1.5962
#> STD.gear 32 3.6875 1 2.7557 5.4664
#> STD.carb 32 2.8125 1 1.6903 6.0242
## Panel Data
head(fscale(get_vars(wlddev,9:12), wlddev$iso3c)) # Standardizing 4 series within each country
#> PCGDP LIFEEX GINI ODA
#> 1 NA -1.653181 NA -0.6498451
#> 2 NA -1.602256 NA -0.5951801
#> 3 NA -1.552023 NA -0.6517082
#> 4 NA -1.502678 NA -0.5925063
#> 5 NA -1.454122 NA -0.5649154
#> 6 NA -1.406257 NA -0.5431461
head(STD(wlddev, ~iso3c, cols = 9:12)) # Same thing using STD, id's added
#> iso3c STD.PCGDP STD.LIFEEX STD.GINI STD.ODA
#> 1 AFG NA -1.653181 NA -0.6498451
#> 2 AFG NA -1.602256 NA -0.5951801
#> 3 AFG NA -1.552023 NA -0.6517082
#> 4 AFG NA -1.502678 NA -0.5925063
#> 5 AFG NA -1.454122 NA -0.5649154
#> 6 AFG NA -1.406257 NA -0.5431461
pwcor(fscale(get_vars(wlddev,9:12), wlddev$iso3c)) # Correlaing panel series after standardizing
#> PCGDP LIFEEX GINI ODA
#> PCGDP 1 .62 -.20 .09
#> LIFEEX .62 1 -.12 .32
#> GINI -.20 -.12 1 -.09
#> ODA .09 .32 -.09 1
fmean(get_vars(wlddev, 9:12)) # This calculates the overall means
#> PCGDP LIFEEX GINI ODA
#> 1.204878e+04 6.429630e+01 3.853412e+01 4.547201e+08
fsd(fwithin(get_vars(wlddev, 9:12), wlddev$iso3c)) # This calculates the within standard deviations
#> PCGDP LIFEEX GINI ODA
#> 6.723681e+03 6.084205e+00 2.927700e+00 6.507096e+08
head(qsu(fscale(get_vars(wlddev, 9:12), # This group-centers on the overall mean and
wlddev$iso3c, # group-scales to the within standard deviation
mean = "overall.mean", sd = "within.sd"), # -> data harmonized in the first 2 moments
by = wlddev$iso3c))
#> , , PCGDP
#>
#> N Mean SD Min Max
#> ABW 32 12048.778 6723.6808 -13032.4333 19779.7812
#> AFG 18 12048.778 6723.6808 2023.3142 18822.252
#> AGO 40 12048.778 6723.6808 -1027.1433 22877.0816
#> ALB 40 12048.778 6723.6808 3212.4503 25460.2799
#> AND 50 12048.778 6723.6808 -111.8716 24171.0081
#> ARE 45 12048.778 6723.6808 3206.3558 26904.4271
#>
#> , , LIFEEX
#>
#> N Mean SD Min Max
#> ABW 60 64.2963 6.0842 49.8607 72.6147
#> AFG 60 64.2963 6.0842 54.238 73.6849
#> AGO 60 64.2963 6.0842 55.6648 77.7463
#> ALB 60 64.2963 6.0842 50.8542 74.1559
#> AND 0 0 - 0 0
#> ARE 60 64.2963 6.0842 49.6668 71.3434
#>
#> , , GINI
#>
#> N Mean SD Min Max
#> ABW 0 0 - 0 0
#> AFG 0 0 - 0 0
#>
#> [ reached getOption("max.print") -- omitted 4 row(s) and 1 matrix slice(s) ]
## Indexed data
wldi <- findex_by(wlddev, iso3c, year)
head(STD(wldi)) # Standardizing all numeric variables by country
#> iso3c year STD.decade STD.PCGDP STD.LIFEEX STD.GINI STD.ODA STD.POP
#> 1 AFG 1960 -1.448413 NA -1.653181 NA -0.6498451 -1.0754727
#> 2 AFG 1961 -1.448413 NA -1.602256 NA -0.5951801 -1.0556707
#> 3 AFG 1962 -1.448413 NA -1.552023 NA -0.6517082 -1.0347669
#> 4 AFG 1963 -1.448413 NA -1.502678 NA -0.5925063 -1.0127455
#> 5 AFG 1964 -1.448413 NA -1.454122 NA -0.5649154 -0.9895973
#> 6 AFG 1965 -1.448413 NA -1.406257 NA -0.5431461 -0.9653050
#>
#> Indexed by: iso3c [1] | year [6 (61)]
head(STD(wldi, effect = 2L)) # Standardizing all numeric variables by year
#> iso3c year STD.decade STD.PCGDP STD.LIFEEX STD.GINI STD.ODA STD.POP
#> 1 AFG 1960 NaN NA -1.755632 NA -0.185893459 -0.08682756
#> 2 AFG 1961 NaN NA -1.759208 NA -0.059419593 -0.08697252
#> 3 AFG 1962 NaN NA -1.769154 NA -0.282990408 -0.08693992
#> 4 AFG 1963 NaN NA -1.772255 NA -0.071937146 -0.08682708
#> 5 AFG 1964 NaN NA -1.774865 NA -0.010951326 -0.08661029
#> 6 AFG 1965 NaN NA -1.790502 NA 0.006003786 -0.08627312
#>
#> Indexed by: iso3c [1] | year [6 (61)]
## Weighted Standardizing
weights = abs(rnorm(nrow(wlddev)))
head(fscale(get_vars(wlddev,9:12), wlddev$iso3c, weights))
#> PCGDP LIFEEX GINI ODA
#> 1 NA -1.467564 NA -0.4994722
#> 2 NA -1.414317 NA -0.4338014
#> 3 NA -1.361793 NA -0.5017104
#> 4 NA -1.310197 NA -0.4305893
#> 5 NA -1.259427 NA -0.3974435
#> 6 NA -1.209379 NA -0.3712913
head(STD(wlddev, ~iso3c, weights, 9:12))
#> iso3c STD.PCGDP STD.LIFEEX STD.GINI STD.ODA
#> 1 AFG NA -1.467564 NA -0.4994722
#> 2 AFG NA -1.414317 NA -0.4338014
#> 3 AFG NA -1.361793 NA -0.5017104
#> 4 AFG NA -1.310197 NA -0.4305893
#> 5 AFG NA -1.259427 NA -0.3974435
#> 6 AFG NA -1.209379 NA -0.3712913
# Grouped data
wlddev |> fgroup_by(iso3c) |> fselect(PCGDP,LIFEEX) |> STD()
#> STD.PCGDP STD.LIFEEX
#> 1 NA -1.65318076
#> 2 NA -1.60225647
#> 3 NA -1.55202301
#> 4 NA -1.50267777
#> 5 NA -1.45412205
#> 6 NA -1.40625717
#> 7 NA -1.35868835
#> 8 NA -1.31092216
#> 9 NA -1.26266251
#> 10 NA -1.21361334
#> 11 NA -1.16337988
#> 12 NA -1.11196214
#> 13 NA -1.05955749
#> 14 NA -1.00606725
#> 15 NA -0.95129403
#> 16 NA -0.89504046
#> 17 NA -0.83710914
#> 18 NA -0.77740140
#> 19 NA -0.71581853
#> 20 NA -0.65255793
#> 21 NA -0.58752090
#> 22 NA -0.52051007
#> 23 NA -0.45201887
#> 24 NA -0.38224470
#> 25 NA -0.31158231
#> 26 NA -0.24042647
#> 27 NA -0.16887587
#> 28 NA -0.09732527
#> 29 NA -0.02636681
#> 30 NA 0.04370344
#> 31 NA 0.11189856
#> 32 NA 0.17782381
#> 33 NA 0.24118310
#> 34 NA 0.30187774
#> 35 NA 0.35971037
#> [ reached 'max' / getOption("max.print") -- omitted 13141 rows ]
#>
#> Grouped by: iso3c [216 | 61 (0)]
wlddev |> fgroup_by(iso3c) |> fselect(PCGDP,LIFEEX) |> STD(weights) # weighted standardizing
#> STD.PCGDP STD.LIFEEX
#> 1 NA -1.467563659
#> 2 NA -1.414317039
#> 3 NA -1.361792758
#> 4 NA -1.310197196
#> 5 NA -1.259427164
#> 6 NA -1.209379469
#> 7 NA -1.159641348
#> 8 NA -1.109696844
#> 9 NA -1.059236385
#> 10 NA -1.007950397
#> 11 NA -0.955426116
#> 12 NA -0.901663540
#> 13 NA -0.846869054
#> 14 NA -0.790939466
#> 15 NA -0.733668392
#> 16 NA -0.674849452
#> 17 NA -0.614276263
#> 18 NA -0.551845634
#> 19 NA -0.487454373
#> 20 NA -0.421308863
#> 21 NA -0.353305913
#> 22 NA -0.283239140
#> 23 NA -0.211624501
#> 24 NA -0.138668377
#> 25 NA -0.064783533
#> 26 NA 0.009617267
#> 27 NA 0.084430831
#> 28 NA 0.159244395
#> 29 NA 0.233438812
#> 30 NA 0.306704510
#> 31 NA 0.378009576
#> 32 NA 0.446941246
#> 33 NA 0.513189947
#> 34 NA 0.576652487
#> 35 NA 0.637122485
#> [ reached 'max' / getOption("max.print") -- omitted 13141 rows ]
#>
#> Grouped by: iso3c [216 | 61 (0)]
wlddev |> fgroup_by(iso3c) |> fselect(PCGDP,LIFEEX,POP) |> STD(POP) # weighting by POP ->
#> POP STD.PCGDP STD.LIFEEX
#> 1 8996973 NA -2.1727695
#> 2 9169410 NA -2.1194893
#> 3 9351441 NA -2.0669319
#> 4 9543205 NA -2.0153038
#> 5 9744781 NA -1.9645018
#> 6 9956320 NA -1.9144225
#> 7 10174836 NA -1.8646530
#> 8 10399926 NA -1.8146770
#> 9 10637063 NA -1.7641848
#> 10 10893776 NA -1.7128664
#> 11 11173642 NA -1.6603090
#> 12 11475445 NA -1.6065126
#> 13 11791215 NA -1.5516835
#> 14 12108963 NA -1.4957187
#> 15 12412950 NA -1.4384115
#> 16 12689160 NA -1.3795555
#> 17 12943093 NA -1.3189441
#> 18 13171306 NA -1.2564741
#> 19 13341198 NA -1.1920422
#> 20 13411056 NA -1.1258550
#> 21 13356511 NA -1.0578092
#> 22 13171673 NA -0.9876983
#> 23 12882528 NA -0.9160385
#> [ reached 'max' / getOption("max.print") -- omitted 13153 rows ]
#>
#> Grouped by: iso3c [216 | 61 (0)]
# ..keeps the weight column unless keep.w = FALSE