This function can be used to perform the functional boxplot of univariate or multivariate functional data.
fbplot(
Data,
Depths = "MBD",
Fvalue = 1.5,
adjust = FALSE,
display = TRUE,
xlab = NULL,
ylab = NULL,
main = NULL,
...
)
# S3 method for fData
fbplot(
Data,
Depths = "MBD",
Fvalue = 1.5,
adjust = FALSE,
display = TRUE,
xlab = NULL,
ylab = NULL,
main = NULL,
...
)
# S3 method for mfData
fbplot(
Data,
Depths = list(def = "MBD", weights = "uniform"),
Fvalue = 1.5,
adjust = FALSE,
display = TRUE,
xlab = NULL,
ylab = NULL,
main = NULL,
...
)
the univariate or multivariate functional dataset whose
functional boxplot must be determined, in form of fData
or
mfData
object.
either a vector containing the depths for each element of the dataset, or:
univariate case: a string containing the name of the method you
want to use to compute it. The default is 'MBD'
.
multivariate case: a list with elements def
, containing the
name of the depth notion to be used to compute depths (BD
or
MBD
), and weights
, containing the value of parameter
weights
to be passed to the depth function. Default is
list(def = 'MBD', weights = 'uniform')
.
In both cases the name of the functions to compute depths must be available in the caller's environment.
the value of the inflation factor \(F\), default is F =
1.5
.
either FALSE
if you would like the default value for the
inflation factor, \(F = 1.5\), to be used, or (for now only in the
univariate functional case) a list specifying the parameters required by
the adjustment:
N_trials
: the number of repetitions of the adjustment procedure
based on the simulation of a gaussian population of functional data, each
one producing an adjusted value of \(F\), which will lead to the averaged
adjusted value \(\bar{F}\). Default is 20.
trial_size
: the number of elements in the gaussian population of
functional data that will be simulated at each repetition of the adjustment
procedure. Default is 8 * Data$N
.
TPR
: the True Positive Rate of outliers, i.e. the proportion of
observations in a dataset without amplitude outliers that have to be
considered outliers. Default is 2 * pnorm(4 * qnorm(0.25))
.
F_min
: the minimum value of \(F\), defining the left boundary
for the optimization problem aimed at finding, for a given dataset of
simulated gaussian data associated to Data
, the optimal value of
\(F\). Default is 0.5.
F_max
: the maximum value of \(F\), defining the right boundary
for the optimization problem aimed at finding, for a given dataset of
simulated gaussian data associated to Data
, the optimal value of
\(F\). Default is 5.
tol
: the tolerance to be used in the optimization problem aimed at
finding, for a given dataset of simulated gaussian data associated to
Data
, the optimal value of \(F\). Default is 1e-3
.
maxiter
: the maximum number of iterations to solve the
optimization problem aimed at finding, for a given dataset of simulated
gaussian data associated to Data
, the optimal value of \(F\).
Default is 100
.
VERBOSE
: a parameter controlling the verbosity of the adjustment
process.
either a logical value indicating whether you want the functional boxplot to be displayed, or the number of the graphical device where you want the functional boxplot to be displayed.
the label to use on the x axis when displaying the functional boxplot.
the label (or list of labels for the multivariate functional case) to use on the y axis when displaying the functional boxplot.
the main title (or list of titles for the multivariate functional case) to be used when displaying the functional boxplot.
additional graphical parameters to be used in plotting functions.
Even when used in graphical way to plot the functional boxplot, the function returns a list of three elements:
Depths
: contains the depths of each element of the functional
dataset.
Fvalue
: is the value of F used to obtain the outliers.
ID_out
: contains the vector of indices of dataset elements flagged
as outliers (if any).
In the univariate functional case, when the adjustment option is
selected, the value of \(F\) is optimized for the univariate functional
dataset provided with Data
.
In practice, a number adjust$N_trials
of times a synthetic population
(of size adjust$tiral_size
with the same covariance (robustly
estimated from data) and centerline as fData
is simulated without
outliers and each time an optimized value \(F_i\) is computed so that a
given proportion (adjust$TPR
) of observations is flagged as outliers.
The final value of F
for the functional boxplot is determined as an
average of \(F_1, F_2, \dots, F_{N_{trials}}\). At each time step the
optimization problem is solved using stats::uniroot
(Brent's method).
Sun, Y., & Genton, M. G. (2012). Functional boxplots. Journal of Computational and Graphical Statistics.
Sun, Y., & Genton, M. G. (2012). Adjusted functional boxplots for spatio-temporal data visualization and outlier detection. Environmetrics, 23(1), 54-64.
# UNIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENT
set.seed(1)
N = 2 * 100 + 1
P = 2e2
grid = seq( 0, 1, length.out = P )
D = 10 * matrix( sin( 2 * pi * grid ), nrow = N, ncol = P, byrow = TRUE )
D = D + rexp(N, rate = 0.05)
# c( 0, 1 : (( N - 1 )/2), -( ( ( N - 1 ) / 2 ) : 1 ) )^4
fD = fData( grid, D )
dev.new()
oldpar <- par(mfrow = c(1, 1))
par(mfrow = c(1, 3))
plot( fD, lwd = 2, main = 'Functional dataset',
xlab = 'time', ylab = 'values' )
fbplot( fD, main = 'Functional boxplot', xlab = 'time', ylab = 'values', Fvalue = 1.5 )
#> $Depth
#> [1] 0.506666667 0.446517413 0.189054726 0.181044776 0.435621891 0.067860697
#> [7] 0.435621891 0.471194030 0.496268657 0.196965174 0.372935323 0.507263682
#> [13] 0.423830846 0.019850746 0.468457711 0.473830846 0.256666667 0.496268657
#> [19] 0.372935323 0.489502488 0.130895522 0.494726368 0.334278607 0.483383085
#> [25] 0.139502488 0.086368159 0.487562189 0.029651741 0.449950249 0.487562189
#> [31] 0.362388060 0.039353234 0.362388060 0.388009950 0.227611940 0.483383085
#> [37] 0.351442786 0.503432836 0.506218905 0.277412935 0.465621891 0.481144279
#> [43] 0.406716418 0.415472637 0.476368159 0.345820896 0.402189055 0.489502488
#> [49] 0.459651741 0.227611940 0.423830846 0.181044776 0.048955224 0.478805970
#> [55] 0.491343284 0.494726368 0.242338308 0.356965174 0.462686567 0.507263682
#> [61] 0.113383085 0.459651741 0.297263682 0.328358209 0.009950249 0.427860697
#> [67] 0.095472637 0.456517413 0.504278607 0.501442786 0.284129353 0.148009950
#> [73] 0.058457711 0.328358209 0.415472637 0.048955224 0.388009950 0.334278607
#> [79] 0.503432836 0.086368159 0.411144279 0.485522388 0.502487562 0.077164179
#> [85] 0.164726368 0.506218905 0.316218905 0.427860697 0.383084577 0.212487562
#> [91] 0.476368159 0.446517413 0.471194030 0.310000000 0.497711443 0.322338308
#> [97] 0.442985075 0.249552239 0.172935323 0.383084577 0.249552239 0.462686567
#> [103] 0.449950249 0.406716418 0.419701493 0.104477612 0.220099502 0.156417910
#> [109] 0.481144279 0.507412935 0.322338308 0.270597015 0.029651741 0.164726368
#> [115] 0.367711443 0.501442786 0.431791045 0.095472637 0.256666667 0.340099502
#> [121] 0.284129353 0.505024876 0.297263682 0.491343284 0.009950249 0.212487562
#> [127] 0.039353234 0.378059701 0.122189055 0.439353234 0.316218905 0.411144279
#> [133] 0.402189055 0.397562189 0.148009950 0.500298507 0.204776119 0.130895522
#> [139] 0.220099502 0.507014925 0.468457711 0.478805970 0.172935323 0.442985075
#> [145] 0.263681592 0.505671642 0.485522388 0.419701493 0.303681592 0.204776119
#> [151] 0.456517413 0.290746269 0.290746269 0.077164179 0.242338308 0.502487562
#> [157] 0.263681592 0.500298507 0.058457711 0.019850746 0.345820896 0.493084577
#> [163] 0.310000000 0.356965174 0.351442786 0.270597015 0.113383085 0.196965174
#> [169] 0.378059701 0.122189055 0.497711443 0.277412935 0.493084577 0.189054726
#> [175] 0.397562189 0.340099502 0.235024876 0.499054726 0.439353234 0.507014925
#> [181] 0.505671642 0.303681592 0.367711443 0.499054726 0.392835821 0.473830846
#> [187] 0.507462687 0.465621891 0.235024876 0.139502488 0.453283582 0.506666667
#> [193] 0.156417910 0.431791045 0.392835821 0.507412935 0.505024876 0.453283582
#> [199] 0.504278607 0.067860697 0.104477612
#>
#> $Fvalue
#> [1] 1.5
#>
#> $ID_outliers
#> [1] 6 14 28 53 65 73 127 154
#>
boxplot(fD$values[,1], ylim = range(fD$values), main = 'Boxplot of functional dataset at t_0 ' )
par(oldpar)
# UNIVARIATE FUNCTIONAL BOXPLOT - WITH ADJUSTMENT
set.seed( 161803 )
P = 2e2
grid = seq( 0, 1, length.out = P )
N = 1e2
# Generating a univariate synthetic gaussian dataset
Data = generate_gauss_fdata( N, centerline = sin( 2 * pi * grid ),
Cov = exp_cov_function( grid,
alpha = 0.3,
beta = 0.4 ) )
fD = fData( grid, Data )
dev.new()
# \donttest{
fbplot( fD, adjust = list( N_trials = 10,
trial_size = 5 * N,
VERBOSE = TRUE ),
xlab = 'time', ylab = 'Values',
main = 'My adjusted functional boxplot' )
#> * * * Iteration 1 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 2 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 3 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 4 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 5 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 6 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 7 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 8 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 9 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> * * * Iteration 10 / 10
#> * * * * beginning optimization
#> * * * * optimization finished.
#> $Depth
#> [1] 0.30070101 0.44452323 0.36620606 0.45850303 0.37377980 0.48135354
#> [7] 0.38181818 0.02659596 0.13369899 0.42945859 0.06214949 0.47510707
#> [13] 0.30211111 0.23325253 0.17038384 0.12049495 0.42031313 0.49717778
#> [19] 0.33230101 0.44418990 0.48692525 0.48365657 0.35160404 0.40534141
#> [25] 0.39773131 0.42371515 0.22158586 0.36634141 0.14679192 0.39269293
#> [31] 0.46537778 0.23329495 0.34899596 0.36024646 0.16920606 0.44801616
#> [37] 0.11143434 0.46201818 0.28396162 0.46948687 0.08282020 0.13537576
#> [43] 0.36103636 0.38312323 0.24703030 0.48692121 0.46313131 0.44435556
#> [49] 0.22941616 0.49802424 0.12830505 0.31013333 0.23446263 0.20488081
#> [55] 0.48862424 0.28376768 0.37487475 0.44553131 0.38107071 0.42778182
#> [61] 0.46742828 0.43395758 0.17490101 0.46710303 0.43027879 0.38315152
#> [67] 0.49235758 0.48834949 0.05709293 0.44494949 0.40256566 0.19737778
#> [73] 0.46515152 0.30163030 0.46011111 0.43029495 0.46189091 0.33044242
#> [79] 0.40226263 0.35060202 0.48137374 0.04209091 0.25347273 0.27454747
#> [85] 0.25617778 0.28562424 0.39055152 0.34064242 0.38934141 0.32964040
#> [91] 0.48462828 0.37410303 0.46637778 0.23939394 0.46542626 0.28972727
#> [97] 0.29029697 0.46380606 0.41843434 0.39590101
#>
#> $Fvalue
#> [1] 0.8693445
#>
#> $ID_outliers
#> [1] 8 11 41 42 51 69 82
#>
# }
# MULTIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENT
set.seed( 1618033 )
P = 1e2
N = 1e2
L = 2
grid = seq( 0, 1, length.out = 1e2 )
C1 = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )
C2 = exp_cov_function( grid, alpha = 0.3, beta = 0.4 )
# Generating a bivariate functional dataset of gaussian data with partially
# correlated components
Data = generate_gauss_mfdata( N, L,
centerline = matrix( sin( 2 * pi * grid ),
nrow = 2, ncol = P,
byrow = TRUE ),
correlations = rep( 0.5, 1 ),
listCov = list( C1, C2 ) )
mfD = mfData( grid, Data )
dev.new()
fbplot( mfD, Fvalue = 2.5, xlab = 'time', ylab = list( 'Values 1',
'Values 2' ),
main = list( 'First component', 'Second component' ) )
#> $Depth
#> [1] 0.43254949 0.39116364 0.35335556 0.39053131 0.08351313 0.16953131
#> [7] 0.43867677 0.20252121 0.21967273 0.45222828 0.41967879 0.25533333
#> [13] 0.42095758 0.35050101 0.38179192 0.35996364 0.29859798 0.20294747
#> [19] 0.40174545 0.46333535 0.38374949 0.39712121 0.34678384 0.31019192
#> [25] 0.48306263 0.31276768 0.35823030 0.28142222 0.31049495 0.44906869
#> [31] 0.42941010 0.38867677 0.38472323 0.44426667 0.34880000 0.38969091
#> [37] 0.45598182 0.33775758 0.45459192 0.48445051 0.31913737 0.37989091
#> [43] 0.15915354 0.34753131 0.37781414 0.27654545 0.48208081 0.15098990
#> [49] 0.36521818 0.26237778 0.44220000 0.44574949 0.24915556 0.37333131
#> [55] 0.35525455 0.28274343 0.38808283 0.11473333 0.36927071 0.16206869
#> [61] 0.33650909 0.39217778 0.39524040 0.44069091 0.04190707 0.47903636
#> [67] 0.41923030 0.18813333 0.39609697 0.43845657 0.35709697 0.49089697
#> [73] 0.39865657 0.46600606 0.39957576 0.35944444 0.19638990 0.30273333
#> [79] 0.41812323 0.27323434 0.27932929 0.29413333 0.40971717 0.20922020
#> [85] 0.29652727 0.49594141 0.38739798 0.26582020 0.11922424 0.31874747
#> [91] 0.49965859 0.33760808 0.32843232 0.38696970 0.42256364 0.18884444
#> [97] 0.46271919 0.44007475 0.36836970 0.22256364
#>
#> $Fvalue
#> [1] 2.5
#>
#> $ID_outliers
#> integer(0)
#>