latentcor

Mingze Huang, Christian L. Müller, Irina Gaynanova

2021-08-23

Introduction

R package latentcor utilizes the powerful semi-parametric latent Gaussian copula models to estimate latent correlations between mixed data types. The package allows to estimate correlations between any of continuous/binary/ternary/zero-inflated (truncated) variable types. The underlying implementation takes advantage of fast multi-linear interpolation scheme with a clever choice of grid points that give the package a small memory footprint, and allows to use the latent correlations with sub-sampling and bootstrapping.

Getting started

A simple example with two variables

First, we will generate a pair of variables with different types using a sample size \(n=100\) which will serve as example data. Here first variable will be ternary, and second variable will be continuous.

simdata = GenData(n = 100, types = c("ter", "con"))

The output of GenData is a list with 2 elements:

names(simdata)
#> [1] "X"     "plotX"
X = simdata$X
head(X, n = 6L)
#>      [,1]        [,2]
#> [1,]    0  0.20587026
#> [2,]    1 -0.04915793
#> [3,]    1 -0.65222833
#> [4,]    2  0.89411877
#> [5,]    0  0.20746957
#> [6,]    0 -0.50552757
simdata$plotX
#> NULL

Then we can estimate the latent correlation matrix based on these 2 variables using estR function.

estimate = estR(X, types = c("ter", "con"))

The output of estR is a list with several elements:

names(estimate)
#> [1] "zratios"    "K"          "R"          "Rpointwise" "plotR"
estimate$zratios
#> [[1]]
#> [1] 0.3 0.8
#> 
#> [[2]]
#> [1] NA
estimate$K
#>           X1        X2
#> X1 1.0000000 0.2359596
#> X2 0.2359596 1.0000000
estimate$Rpointwise
#>           X1        X2
#> X1 1.0000000 0.4309563
#> X2 0.4309563 1.0000000
estimate$R
#>           X1        X2
#> X1 1.0000000 0.4305253
#> X2 0.4305253 1.0000000
estimate$plotR
#> NULL

Another example with real dataset

We use the build-in dataset mtcars:

head(mtcars, n = 6L)
#>                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
#> Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
#> Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
#> Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
#> Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
#> Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
#> Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Let’s take a look at the unique values for each variable to determine the corresponding data type.

apply(mtcars, 2, table)
#> $mpg
#> 
#> 10.4 13.3 14.3 14.7   15 15.2 15.5 15.8 16.4 17.3 17.8 18.1 18.7 19.2 19.7   21 
#>    2    1    1    1    1    2    1    1    1    1    1    1    1    2    1    2 
#> 21.4 21.5 22.8 24.4   26 27.3 30.4 32.4 33.9 
#>    2    1    2    1    1    1    2    1    1 
#> 
#> $cyl
#> 
#>  4  6  8 
#> 11  7 14 
#> 
#> $disp
#> 
#>  71.1  75.7  78.7    79  95.1   108 120.1 120.3   121 140.8   145 146.7   160 
#>     1     1     1     1     1     1     1     1     1     1     1     1     2 
#> 167.6   225   258 275.8   301   304   318   350   351   360   400   440   460 
#>     2     1     1     3     1     1     1     1     1     2     1     1     1 
#>   472 
#>     1 
#> 
#> $hp
#> 
#>  52  62  65  66  91  93  95  97 105 109 110 113 123 150 175 180 205 215 230 245 
#>   1   1   1   2   1   1   1   1   1   1   3   1   2   2   3   3   1   1   1   2 
#> 264 335 
#>   1   1 
#> 
#> $drat
#> 
#> 2.76 2.93    3 3.07 3.08 3.15 3.21 3.23 3.54 3.62 3.69  3.7 3.73 3.77 3.85  3.9 
#>    2    1    1    3    2    2    1    1    1    1    1    1    1    1    1    2 
#> 3.92 4.08 4.11 4.22 4.43 4.93 
#>    3    2    1    2    1    1 
#> 
#> $wt
#> 
#> 1.513 1.615 1.835 1.935  2.14   2.2  2.32 2.465  2.62  2.77  2.78 2.875  3.15 
#>     1     1     1     1     1     1     1     1     1     1     1     1     1 
#>  3.17  3.19 3.215 3.435  3.44  3.46  3.52  3.57  3.73  3.78  3.84 3.845  4.07 
#>     1     1     1     1     3     1     1     2     1     1     1     1     1 
#>  5.25 5.345 5.424 
#>     1     1     1 
#> 
#> $qsec
#> 
#>  14.5  14.6 15.41  15.5 15.84 16.46  16.7 16.87  16.9 17.02 17.05  17.3  17.4 
#>     1     1     1     1     1     1     1     1     1     2     1     1     1 
#> 17.42  17.6 17.82 17.98    18  18.3 18.52  18.6 18.61  18.9 19.44 19.47  19.9 
#>     1     1     1     1     1     1     1     1     1     2     1     1     1 
#>    20 20.01 20.22  22.9 
#>     1     1     1     1 
#> 
#> $vs
#> 
#>  0  1 
#> 18 14 
#> 
#> $am
#> 
#>  0  1 
#> 19 13 
#> 
#> $gear
#> 
#>  3  4  5 
#> 15 12  5 
#> 
#> $carb
#> 
#>  1  2  3  4  6  8 
#>  7 10  3 10  1  1

Then we can estimate the latent correlation matrix for all variables of mtcars by using estR function.

estimate_mtcars = estR(mtcars, types = c("con", "ter", "con", "con", "con", "con", "con", "bin", "bin", "ter", "con"))
#> Using Matrix::nearPD since Minimum eigenvalue of latent correlation matrix is -0.203053866491842 smaller than 0.

The output of estR for mtcars:

names(estimate_mtcars)
#> [1] "zratios"    "K"          "R"          "Rpointwise" "plotR"
estimate_mtcars$zratios
#> [[1]]
#> [1] NA
#> 
#> [[2]]
#> [1] 0.34375 0.56250
#> 
#> [[3]]
#> [1] NA
#> 
#> [[4]]
#> [1] NA
#> 
#> [[5]]
#> [1] NA
#> 
#> [[6]]
#> [1] NA
#> 
#> [[7]]
#> [1] NA
#> 
#> [[8]]
#> [1] 0.5625
#> 
#> [[9]]
#> [1] 0.59375
#> 
#> [[10]]
#> [1] 0.46875 0.84375
#> 
#> [[11]]
#> [1] NA
estimate_mtcars$K
#>             mpg        cyl       disp         hp        drat         wt
#> mpg   1.0000000 -0.6431452 -0.7580645 -0.7278226  0.45564516 -0.7197581
#> cyl  -0.6431452  1.0000000  0.6592742  0.6310484 -0.44354839  0.5907258
#> disp -0.7580645  0.6592742  1.0000000  0.6532258 -0.48991935  0.7358871
#> hp   -0.7278226  0.6310484  0.6532258  1.0000000 -0.37298387  0.6008065
#> drat  0.4556452 -0.4435484 -0.4899194 -0.3729839  1.00000000 -0.5383065
#> wt   -0.7197581  0.5907258  0.7358871  0.6008065 -0.53830645  1.0000000
#> qsec  0.3125000 -0.3649194 -0.2983871 -0.4657258  0.03225806 -0.1411290
#> vs    0.4173387 -0.4475806 -0.4274194 -0.4435484  0.26411290 -0.3467742
#> am    0.3286290 -0.2842742 -0.3649194 -0.2116935  0.40120968 -0.4314516
#> gear  0.3427419 -0.3326613 -0.3770161 -0.2197581  0.45967742 -0.4314516
#> carb -0.4395161  0.3326613  0.3608871  0.5161290 -0.08266129  0.3245968
#>             qsec          vs          am        gear        carb
#> mpg   0.31250000  0.41733871  0.32862903  0.34274194 -0.43951613
#> cyl  -0.36491935 -0.44758065 -0.28427419 -0.33266129  0.33266129
#> disp -0.29838710 -0.42741935 -0.36491935 -0.37701613  0.36088710
#> hp   -0.46572581 -0.44354839 -0.21169355 -0.21975806  0.51612903
#> drat  0.03225806  0.26411290  0.40120968  0.45967742 -0.08266129
#> wt   -0.14112903 -0.34677419 -0.43145161 -0.43145161  0.32459677
#> qsec  1.00000000  0.46774194 -0.11895161 -0.07258065 -0.44354839
#> vs    0.46774194  1.00000000  0.08467742  0.15322581 -0.36088710
#> am   -0.11895161  0.08467742  1.00000000  0.43346774 -0.03629032
#> gear -0.07258065  0.15322581  0.43346774  1.00000000  0.06854839
#> carb -0.44354839 -0.36088710 -0.03629032  0.06854839  1.00000000
estimate_mtcars$Rpointwise
#>             mpg        cyl       disp         hp        drat         wt
#> mpg   1.0000000 -0.9990000 -0.9286530 -0.9099905  0.65616525 -0.9046652
#> cyl  -0.9990000  1.0000000  0.9990000  0.9900378 -0.77195772  0.9525997
#> disp -0.9286530  0.9990000  1.0000000  0.8552768 -0.69582182  0.9151697
#> hp   -0.9099905  0.9900378  0.8552768  1.0000000 -0.55293425  0.8097609
#> drat  0.6561652 -0.7719577 -0.6958218 -0.5529342  1.00000000 -0.7483492
#> wt   -0.9046652  0.9525997  0.9151697  0.8097609 -0.74834918  1.0000000
#> qsec  0.4713967 -0.6540431 -0.4517316 -0.6680316  0.05064917 -0.2198737
#> vs    0.8727316 -0.9623421 -0.8905658 -0.9188458  0.57685875 -0.7416377
#> am    0.7178533 -0.7124468 -0.7888268 -0.4746999  0.85723711 -0.9121559
#> gear  0.6234660 -0.6441105 -0.6786359 -0.4119442  0.80260415 -0.7617271
#> carb -0.6368382  0.6025491  0.5370028  0.7247928 -0.12947951  0.4880685
#>             qsec         vs          am       gear        carb
#> mpg   0.47139674  0.8727316  0.71785333  0.6234660 -0.63683817
#> cyl  -0.65404313 -0.9623421 -0.71244682 -0.6441105  0.60254911
#> disp -0.45173164 -0.8905658 -0.78882677 -0.6786359  0.53700280
#> hp   -0.66803158 -0.9188458 -0.47469992 -0.4119442  0.72479279
#> drat  0.05064917  0.5768588  0.85723711  0.8026041 -0.12947951
#> wt   -0.21987367 -0.7416377 -0.91215587 -0.7617271  0.48806852
#> qsec  1.00000000  0.9599123 -0.27004807 -0.1385035 -0.64170875
#> vs    0.95991229  1.0000000  0.27236999  0.4087924 -0.76863616
#> am   -0.27004807  0.2723700  1.00000000  0.9941469 -0.08284094
#> gear -0.13850346  0.4087924  0.99414687  1.0000000  0.13086286
#> carb -0.64170875 -0.7686362 -0.08284094  0.1308629  1.00000000
estimate_mtcars$R
#>             mpg        cyl       disp         hp        drat         wt
#> mpg   1.0000000 -0.9609031 -0.9415043 -0.9158428  0.66735827 -0.9184175
#> cyl  -0.9609031  1.0000000  0.9642266  0.9413892 -0.73792461  0.9100566
#> disp -0.9415043  0.9642266  1.0000000  0.8623242 -0.71394453  0.9357387
#> hp   -0.9158428  0.9413892  0.8623242  1.0000000 -0.55916149  0.8168236
#> drat  0.6673583 -0.7379246 -0.7139445 -0.5591615  1.00000000 -0.7655481
#> wt   -0.9184175  0.9100566  0.9357387  0.8168236 -0.76554810  1.0000000
#> qsec  0.4823428 -0.5822593 -0.4633043 -0.6776150  0.07314342 -0.2320168
#> vs    0.8544713 -0.9256791 -0.8492790 -0.9218190  0.53281338 -0.7091408
#> am    0.6969207 -0.6621142 -0.7398814 -0.4763248  0.82651611 -0.8702507
#> gear  0.6335341 -0.6404076 -0.7007359 -0.4148761  0.81916584 -0.7795446
#> carb -0.6389233  0.6083585  0.5440334  0.7239277 -0.13666971  0.4925960
#>             qsec         vs          am       gear        carb
#> mpg   0.48234281  0.8544713  0.69692066  0.6335341 -0.63892330
#> cyl  -0.58225925 -0.9256791 -0.66211416 -0.6404076  0.60835846
#> disp -0.46330426 -0.8492790 -0.73988144 -0.7007359  0.54403338
#> hp   -0.67761499 -0.9218190 -0.47632483 -0.4148761  0.72392768
#> drat  0.07314342  0.5328134  0.82651611  0.8191658 -0.13666971
#> wt   -0.23201675 -0.7091408 -0.87025068 -0.7795446  0.49259599
#> qsec  1.00000000  0.8394639 -0.21423820 -0.1320981 -0.65840975
#> vs    0.83946385  1.0000000  0.34069073  0.3558147 -0.72748608
#> am   -0.21423820  0.3406907  1.00000000  0.9329160 -0.06824438
#> gear -0.13209815  0.3558147  0.93291605  1.0000000  0.12065393
#> carb -0.65840975 -0.7274861 -0.06824438  0.1206539  1.00000000
estimate_mtcars$plotR
#> NULL

Mathematical framework for estimation

Latent Gaussian Copula Model for Mixed Data

latentcor utilizes the powerful semi-parametric latent Gaussian copula models to estimate latent correlations between mixed data types (continuous/binary/ternary/truncated or zero-inflated). Below we review the definitions for each type.

Definition of continuous model (Fan et al. 2017)

A random \(X\in\cal{R}^{p}\) satisfies the Gaussian copula (or nonparanormal) model if there exist monotonically increasing \(f=(f_{j})_{j=1}^{p}\) with \(Z_{j}=f_{j}(X_{j})\) satisfying \(Z\sim N_{p}(0, \Sigma)\), \(\sigma_{jj}=1\); we denote \(X\sim NPN(0, \Sigma, f)\).

X = GenData(n = 6, types = "con")$X
X
#>            [,1]
#> [1,]  0.8025116
#> [2,] -0.0168194
#> [3,] -2.2159450
#> [4,]  0.2073208
#> [5,] -0.3521075
#> [6,] -1.3127480

Definition of binary model (Fan et al. 2017)

A random \(X\in\cal{R}^{p}\) satisfies the binary latent Gaussian copula model if there exists \(W\sim NPN(0, \Sigma, f)\) such that \(X_{j}=I(W_{j}>c_{j})\), where \(I(\cdot)\) is the indicator function and \(c_{j}\) are constants.

X = GenData(n = 6, types = "bin")$X
X
#>      [,1]
#> [1,]    1
#> [2,]    0
#> [3,]    1
#> [4,]    0
#> [5,]    1
#> [6,]    0

Definition of ternary model (Quan, Booth, and Wells 2018)

A random \(X\in\cal{R}^{p}\) satisfies the ternary latent Gaussian copula model if there exists \(W\sim NPN(0, \Sigma, f)\) such that \(X_{j}=I(W_{j}>c_{j})+I(W_{j}>c'_{j})\), where \(I(\cdot)\) is the indicator function and \(c_{j}<c'_{j}\) are constants.

X = GenData(n = 6, types = "ter")$X
X
#>      [,1]
#> [1,]    1
#> [2,]    0
#> [3,]    1
#> [4,]    0
#> [5,]    2
#> [6,]    1

Definition of truncated or zero-inflated model (Yoon, Carroll, and Gaynanova 2020)

A random \(X\in\cal{R}^{p}\) satisfies the truncated latent Gaussian copula model if there exists \(W\sim NPN(0, \Sigma, f)\) such that \(X_{j}=I(W_{j}>c_{j})W_{j}\), where \(I(\cdot)\) is the indicator function and \(c_{j}\) are constants.

X = GenData(n = 6, types = "tru")$X
X
#>           [,1]
#> [1,] 0.0000000
#> [2,] 0.0000000
#> [3,] 1.1227278
#> [4,] 0.0000000
#> [5,] 1.3762351
#> [6,] 0.7223574

Mixed latent Gaussian copula model

The mixed latent Gaussian copula model jointly models \(W=(W_{1}, W_{2}, W_{3}, W_{4})\sim NPN(0, \Sigma, f)\) such that \(X_{1j}=W_{1j}\), \(X_{2j}=I(W_{2j}>c_{2j})\), \(X_{3j}=I(W_{3j}>c_{3j})+I(W_{3j}>c'_{3j})\) and \(X_{4j}=I(W_{4j}>c_{4j})W_{4j}\).

set.seed("234820")
X = GenData(n = 100, types = c("con", "bin", "ter", "tru"))$X
head(X)
#>            [,1] [,2] [,3]      [,4]
#> [1,] -0.5728663    0    1 0.0000000
#> [2,] -1.5632883    0    0 0.0000000
#> [3,]  0.4600555    1    1 0.2634213
#> [4,] -1.5186510    1    1 0.0000000
#> [5,] -1.5438165    0    1 0.0000000
#> [6,] -0.5656219    0    0 0.0000000

Moment-based estimation of \(\Sigma\) based on bridge functions

The estimation of latent correlation matrix \(\Sigma\) is achieved via the bridge function \(F\) which is defined such that \(E(\hat{\tau}_{jk})=F(\sigma_{jk})\), where \(\sigma_{jk}\) is the latent correlation between variables \(j\) and \(k\), and \(\hat{\tau}_{jk}\) is the corresponding sample Kendall’s \(\tau\).

Kendall’s \(\tau\) (\(\tau_{a}\))

Given observed \(\mathbf{x}_{j}, \mathbf{x}_{k}\in\cal{R}^{n}\),

\[ \hat{\tau}_{jk}=\hat{\tau}(\mathbf{x}_{j}, \mathbf{x}_{k})=\frac{2}{n(n-1)}\sum_{1\le i<i'\le n}sign(x_{ij}-x_{i'j})sign(x_{ik}-x_{i'k}), \] where \(n\) is the sample size.

latentcor calculates pairwise Kendall’s \(\widehat \tau\) as part of the estimation process

estimate = estR(X, types = c("con", "bin", "ter", "tru"))
K = estimate$K
K
#>           X1        X2        X3        X4
#> X1 1.0000000 0.2557576 0.2456566 0.3331313
#> X2 0.2557576 1.0000000 0.1555556 0.2339394
#> X3 0.2456566 0.1555556 1.0000000 0.2183838
#> X4 0.3331313 0.2339394 0.2183838 1.0000000

Using \(F\) and \(\widehat \tau_{jk}\), a moment-based estimator is \(\hat{\sigma}_{jk}=F^{-1}(\hat{\tau}_{jk})\) with the corresponding \(\hat{\Sigma}\) being consistent for \(\Sigma\) (Fan et al. 2017; Quan, Booth, and Wells 2018; Yoon, Carroll, and Gaynanova 2020).

The explicit form of bridge function \(F\) has been derived for all combinations of continuous(C)/binary(B)/ternary(N)/truncated(T) variable types, and we summarize the corresponding references. Each of this combinations is implemented in latentcor.

Type continuous binary ternary zero-inflated (truncated)
continuous Liu, Lafferty, and Wasserman (2009) - - -
binary Fan et al. (2017) Fan et al. (2017) - -
ternary Quan, Booth, and Wells (2018) Quan, Booth, and Wells (2018) Quan, Booth, and Wells (2018) -
zero-inflated (truncated) Yoon, Carroll, and Gaynanova (2020) Yoon, Carroll, and Gaynanova (2020) See Appendix Yoon, Carroll, and Gaynanova (2020)

Below we provide an explicit form of \(F\) for each combination.

Theorem (explicit form of bridge function) Let \(W_{1}\in\cal{R}^{p_{1}}\), \(W_{2}\in\cal{R}^{p_{2}}\), \(W_{3}\in\cal{R}^{p_{3}}\), \(W_{4}\in\cal{R}^{p_{4}}\) be such that \(W=(W_{1}, W_{2}, W_{3}, W_{4})\sim NPN(0, \Sigma, f)\) with \(p=p_{1}+p_{2}+p_{3}+p_{4}\). Let \(X=(X_{1}, X_{2}, X_{3}, X_{4})\in\cal{R}^{p}\) satisfy \(X_{j}=W_{j}\) for \(j=1,...,p_{1}\), \(X_{j}=I(W_{j}>c_{j})\) for \(j=p_{1}+1, ..., p_{1}+p_{2}\), \(X_{j}=I(W_{j}>c_{j})+I(W_{j}>c'_{j})\) for \(j=p_{1}+p_{2}+1, ..., p_{3}\) and \(X_{j}=I(W_{j}>c_{j})W_{j}\) for \(j=p_{1}+p_{2}+p_{3}+1, ..., p\) with \(\Delta_{j}=f(c_{j})\). The rank-based estimator of \(\Sigma\) based on the observed \(n\) realizations of \(X\) is the matrix \(\mathbf{\hat{R}}\) with \(\hat{r}_{jj}=1\), \(\hat{r}_{jk}=\hat{r}_{kj}=F^{-1}(\hat{\tau}_{jk})\) with block structure

\[ \mathbf{\hat{R}}=\begin{pmatrix} F_{CC}^{-1}(\hat{\tau}) & F_{CB}^{-1}(\hat{\tau}) & F_{CN}^{-1}(\hat{\tau}) & F_{CT}^{-1}(\hat{\tau})\\ F_{BC}^{-1}(\hat{\tau}) & F_{BB}^{-1}(\hat{\tau}) & F_{BN}^{-1}(\hat{\tau}) & F_{BT}^{-1}(\hat{\tau})\\ F_{NC}^{-1}(\hat{\tau}) & F_{NB}^{-1}(\hat{\tau}) & F_{NN}^{-1}(\hat{\tau}) & F_{NT}^{-1}(\hat{\tau})\\ F_{TC}^{-1}(\hat{\tau}) & F_{TB}^{-1}(\hat{\tau}) & F_{TN}^{-1}(\hat{\tau}) & F_{TT}^{-1}(\hat{\tau}) \end{pmatrix} \] \[ F(\cdot)=\begin{cases} CC:\ 2\sin^{-1}(r)/\pi \\ \\ BC: \ 4\Phi_{2}(\Delta_{j},0;r/\sqrt{2})-2\Phi(\Delta_{j}) \\ \\ BB: \ 2\{\Phi_{2}(\Delta_{j},\Delta_{k};r)-\Phi(\Delta_{j})\Phi(\Delta_{k})\} \\ \\ NC: \ 4\Phi_{2}(\Delta_{j}^{2},0;r/\sqrt{2})-2\Phi(\Delta_{j}^{2})+4\Phi_{3}(\Delta_{j}^{1},\Delta_{j}^{2},0;\Sigma_{3a}(r))-2\Phi(\Delta_{j}^{1})\Phi(\Delta_{j}^{2})\\ \\ NB: \ 2\Phi_{2}(\Delta_{j}^{2},\Delta_{k},r)\{1-\Phi(\Delta_{j}^{1})\}-2\Phi(\Delta_{j}^{2})\{\Phi(\Delta_{k})-\Phi_{2}(\Delta_{j}^{1},\Delta_{k},r)\} \\ \\ NN: \ 2\Phi_{2}(\Delta_{j}^{2},\Delta_{k}^{2};r)\Phi_{2}(-\Delta_{j}^{1},-\Delta_{k}^{1};r)-2\{\Phi(\Delta_{j}^{2})-\Phi_{2}(\Delta_{j}^{2},\Delta_{k}^{1};r)\}\{\Phi(\Delta_{k}^{2})-\Phi_{2}(\Delta_{j}^{1},\Delta_{k}^{2};r)\} \\ \\ TC: \ -2\Phi_{2}(-\Delta_{j},0;1/\sqrt{2})+4\Phi_{3}(-\Delta_{j},0,0;\Sigma_{3b}(r)) \\ \\ TB: \ 2\{1-\Phi(\Delta_{j})\}\Phi(\Delta_{k})-2\Phi_{3}(-\Delta_{j},\Delta_{k},0;\Sigma_{3c}(r))-2\Phi_{3}(-\Delta_{j},\Delta_{k},0;\Sigma_{3d}(r)) \\ \\ TN: \ -2\Phi(-\Delta_{k}^{1})\Phi(\Delta_{k}^{2}) + 2\Phi_{3}(-\Delta_{k}^{1},\Delta_{k}^{2},\Delta_{j};\Sigma_{3e}(r))+2\Phi_{4}(-\Delta_{k}^{1},\Delta_{k}^{2},-\Delta_{j},0;\Sigma_{4a}(r))+2\Phi_{4}(-\Delta_{k}^{1},\Delta_{k}^{2},-\Delta_{j},0;\Sigma_{4b}(r)) \\ \\ TT: \ -2\Phi_{4}(-\Delta_{j},-\Delta_{k},0,0;\Sigma_{4c}(r))+2\Phi_{4}(-\Delta_{j},-\Delta_{k},0,0;\Sigma_{4d}(r)) \\ \end{cases} \]

where \(\Delta_{j}=\Phi^{-1}(\pi_{0j})\), \(\Delta_{k}=\Phi^{-1}(\pi_{0k})\), \(\Delta_{j}^{1}=\Phi^{-1}(\pi_{0j})\), \(\Delta_{j}^{2}=\Phi^{-1}(\pi_{0j}+\pi_{1j})\), \(\Delta_{k}^{1}=\Phi^{-1}(\pi_{0k})\), \(\Delta_{k}^{2}=\Phi^{-1}(\pi_{0k}+\pi_{1k})\),

\[ \Sigma_{3a}(r)= \begin{pmatrix} 1 & 0 & \frac{r}{\sqrt{2}} \\ 0 & 1 & -\frac{r}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & 1 \end{pmatrix}, \;\;\; \Sigma_{3b}(r)= \begin{pmatrix} 1 & \frac{1}{\sqrt{2}} & \frac{r}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & 1 & r \\ \frac{r}{\sqrt{2}} & r & 1 \end{pmatrix}, \;\;\; \Sigma_{3c}(r)= \begin{pmatrix} 1 & -r & \frac{1}{\sqrt{2}} \\ -r & 1 & -\frac{r}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & 1 \end{pmatrix}, \]

\[ \Sigma_{3d}(r)= \begin{pmatrix} 1 & 0 & -\frac{1}{\sqrt{2}} \\ 0 & 1 & -\frac{r}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & 1 \end{pmatrix}, \;\;\; \Sigma_{3e}(r)= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & r & 1 \end{pmatrix}, \;\;\; \Sigma_{4a}(r)= \begin{pmatrix} 1 & 0 & 0 & \frac{r}{\sqrt{2}} \\ 0 & 1 & -r & \frac{r}{\sqrt{2}} \\ 0 & -r & 1 & -\frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 1 \end{pmatrix}, \]

\[ \Sigma_{4b}(r)= \begin{pmatrix} 1 & 0 & r & \frac{r}{\sqrt{2}} \\ 0 & 1 & 0 & \frac{r}{\sqrt{2}} \\ r & 0 & 1 & \frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 \end{pmatrix}, \;\;\; \Sigma_{4c}(r)= \begin{pmatrix} 1 & 0 & \frac{1}{\sqrt{2}} & -\frac{r}{\sqrt{2}} \\ 0 & 1 & -\frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & 1 & -r \\ -\frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} & -r & 1 \end{pmatrix}\;\;\text{and}\;\; \Sigma_{4d}(r)= \begin{pmatrix} 1 & r & \frac{1}{\sqrt{2}} & \frac{r}{\sqrt{2}} \\ r & 1 & \frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{r}{\sqrt{2}} & 1 & r \\ \frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} & r & 1 \end{pmatrix}. \]

Estimation methods

Given the form of bridge function \(F\), obtaining a moment-based estimation \(\widehat \sigma_{jk}\) requires inversion of \(F\). latentcor implements two methods for calcultation of the inversion.

Original method (method = "original")

Original estimation approach relies on numerical inversion of \(F\) based on solving uni-root optimization problem.

estimate = estR(X, types = c("con", "bin", "ter", "tru"), method = "original")

Algorithm for Original method

Input: \(F(r)=F(r, \mathbf{\Delta})\) - bridge function based on the type of variables \(j\), \(k\)

estimate$K
#>           X1        X2        X3        X4
#> X1 1.0000000 0.2557576 0.2456566 0.3331313
#> X2 0.2557576 1.0000000 0.1555556 0.2339394
#> X3 0.2456566 0.1555556 1.0000000 0.2183838
#> X4 0.3331313 0.2339394 0.2183838 1.0000000
estimate$zratios
#> [[1]]
#> [1] NA
#> 
#> [[2]]
#> [1] 0.5
#> 
#> [[3]]
#> [1] 0.3 0.8
#> 
#> [[4]]
#> [1] 0.5
estimate$R
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5524373 0.4476503 0.5820345
#> X2 0.5524373 1.0000000 0.4046173 0.5815691
#> X3 0.4476503 0.4046173 1.0000000 0.4649222
#> X4 0.5820345 0.5815691 0.4649222 1.0000000

Approximation method (method = "approx")

A faster approximation method is based on multi-linear interpolation of pre-computed inverse bridge function on a fixed grid of points. This is possible as the inverse bridge function is an analytic function of at most 5 parameters:

In short, d-dimensional multi-linear interpolation uses a weighted average of \(2^{d}\) neighbors to approximate the function values at the points within the d-dimensional cube of the neighbors, and to perform interpolation, latentcor takes advantage of the R package chebpol (Gaure 2019). This approximation method has been first described in (Yoon, Müller, and Gaynanova 2021) for continuous/binary/truncated cases. In latentcor, we additionally implement ternary case, and optimize the choice of grid as well as interpolation boundary for faster computations with smaller memory footprint.

estimate = estR(X, types = c("con", "bin", "ter", "tru"), method = "approx")

Algorithm for Approximation method

Input: Let \(\check{g}=h(g)\), pre-computed values \(F^{-1}(h^{-1}(\check{g}))\) on a fixed grid \(\check{g}\in\check{\cal{G}}\) based on the type of variables \(j\) and \(k\). For binary/continuous case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j})\); for binary/binary case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}, \check{\Delta}_{k})\); for truncated/continuous case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j})\); for truncated/truncated case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}, \check{\Delta}_{k})\); for ternary/continuous case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}^{1}, \check{\Delta}_{j}^{2})\); for ternary/binary case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}^{1}, \check{\Delta}_{j}^{2}, \check{\Delta}_{k})\); for ternary/truncated case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}^{1}, \check{\Delta}_{j}^{2}, \check{\Delta}_{k})\); for ternay/ternary case, \(\check{g}=(\check{\tau}_{jk}, \check{\Delta}_{j}^{1}, \check{\Delta}_{j}^{2}, \check{\Delta}_{k}^{1}, \check{\Delta}_{k}^{2})\).

To avoid interpolation in areas with high approximation errors close to the boundary, we use hybrid scheme in Step 3. For the derivation of approximate bound for BC, BB, TC, TB, TT cases see Yoon, Müller, and Gaynanova (2021). The derivation of approximate bound for NC, NB, NN, NT case is in the Appendix.

\[ \bar{\tau}_{jk}(\cdot)= \begin{cases} 2\pi_{0j}(1-\pi_{0j}) & for \; BC \; case\\ 2\min(\pi_{0j},\pi_{0k})\{1-\max(\pi_{0j}, \pi_{0k})\} & for \; BB \; case\\ 2\{\pi_{0j}(1-\pi_{0j})+\pi_{1j}(1-\pi_{0j}-\pi_{1j})\} & for \; NC \; case\\ 2\min(\pi_{0j}(1-\pi_{0j})+\pi_{1j}(1-\pi_{0j}-\pi_{1j}),\pi_{0k}(1-\pi_{0k})) & for \; NB \; case\\ 2\min(\pi_{0j}(1-\pi_{0j})+\pi_{1j}(1-\pi_{0j}-\pi_{1j}), \\ \;\;\;\;\;\;\;\;\;\;\pi_{0k}(1-\pi_{0k})+\pi_{1k}(1-\pi_{0k}-\pi_{1k})) & for \; NN \; case\\ 1-(\pi_{0j})^{2} & for \; TC \; case\\ 2\max(\pi_{0k},1-\pi_{0k})\{1-\max(\pi_{0k},1-\pi_{0k},\pi_{0j})\} & for \; TB \; case\\ 1-\{\max(\pi_{0j},\pi_{0k},\pi_{1k},1-\pi_{0k}-\pi_{1k})\}^{2} & for \; TN \; case\\ 1-\{\max(\pi_{0j},\pi_{0k})\}^{2} & for \; TT \; case\\ \end{cases} \]

By default, latentcor uses \(\mbox{ratio} = 0.9\), but this value can be modified by the user

estR(X, types = c("con", "bin", "ter", "tru"), method = "approx", ratio = 0.99)$R
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5522684 0.4472342 0.5817297
#> X2 0.5522684 1.0000000 0.4054908 0.5803080
#> X3 0.4472342 0.4054908 1.0000000 0.4563203
#> X4 0.5817297 0.5803080 0.4563203 1.0000000
estR(X, types = c("con", "bin", "ter", "tru"), method = "approx", ratio = 0.4)$R
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5524373 0.4472342 0.5820345
#> X2 0.5524373 1.0000000 0.4054908 0.5815691
#> X3 0.4472342 0.4054908 1.0000000 0.4563203
#> X4 0.5820345 0.5815691 0.4563203 1.0000000
estR(X, types = c("con", "bin", "ter", "tru"), method = "original")$R
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5524373 0.4476503 0.5820345
#> X2 0.5524373 1.0000000 0.4046173 0.5815691
#> X3 0.4476503 0.4046173 1.0000000 0.4649222
#> X4 0.5820345 0.5815691 0.4649222 1.0000000

The lower is the ratio, the closer is the approximation method to original method (with ratio = 0 being equivalent to method = "original"), but also the higher is the cost of computations.

Rescaled Grid for Interpolation

Since \(|\hat{\tau}|\le \bar{\tau}\), the grid does not need to cover the whole domain \(\tau\in[-1, 1]\). To optimize memory associated with storing the grid, we rescale \(\tau\) as follows: \(\check{\tau}_{jk}=\tau_{jk}/\bar{\tau}_{jk}\in[-1, 1]\), where \(\bar{\tau}_{jk}\) is as defined above.

In addition, for ternary variable \(j\), it always holds that \(\Delta_{j}^{2}>\Delta_{j}^{1}\) since \(\Delta_{j}^{1}=\Phi^{-1}(\pi_{0j})\) and \(\Delta_{j}^{2}=\Phi^{-1}(\pi_{0j}+\pi_{1j})\). Thus, the grid should not cover the the area corresponding to \(\Delta_{j}^{2}\ge\Delta_{j}^{1}\). We thus rescale as follows: \(\check{\Delta}_{j}^{1}=\Delta_{j}^{1}/\Delta_{j}^{2}\in[0, 1]\); \(\check{\Delta}_{j}^{2}=\Delta_{j}^{2}\in[0, 1]\).

Adjustment of pointwise-estimator for positive-definiteness

Since the estimation is performed point-wise, the resulting matrix of estimated latent correlations is not guaranteed to be positive semi-definite. For example, this could be expected when the sample size is small (and so the estimation error for each pairwise correlation is larger)

set.seed("234820")
X = GenData(n = 6, types = c("con", "bin", "ter", "tru"))$X
X
#>            [,1] [,2] [,3]      [,4]
#> [1,] -0.5182800    0    1 0.1021738
#> [2,] -1.3017092    0    0 0.0000000
#> [3,]  0.3145191    1    2 0.4213514
#> [4,] -0.6093291    0    1 1.2771610
#> [5,] -1.3175490    1    0 0.0000000
#> [6,] -0.7807245    1    1 0.0000000
out = estR(X, types = c("con", "bin", "ter", "tru"))
out$Rpointwise
#>            X1         X2        X3         X4
#> X1  1.0000000 -0.1477240 0.9990000  0.8548518
#> X2 -0.1477240  1.0000000 0.3523666 -0.5030324
#> X3  0.9990000  0.3523666 1.0000000  0.9114307
#> X4  0.8548518 -0.5030324 0.9114307  1.0000000
eigen(out$Rpointwise)$values
#> [1]  2.85954424  1.29130852  0.09944544 -0.25029820

latentcor automatically corrects the pointwise estimator to be positive definite by making two adjustments. First, if Rpointwise has smallest eigenvalue less than zero, the estR projects this matrix to the nearest positive semi-definite matrix. The user is notified of this adjustment through the message (supressed in previous code chunk), e.g.

out = estR(X, types = c("con", "bin", "ter", "tru"))
#> Using Matrix::nearPD since Minimum eigenvalue of latent correlation matrix is -0.25029819695076 smaller than 0.

Second, latentcor shrinks the adjusted matrix of correlations towards identity matrix using the parameter \(\nu\) with default value of 0.001 (nu = 0.001), so that the resulting R is strictly positive definite with the minimal eigenvalue being greater or equal to \(\nu\). That is \[ R = (1 - \nu) \widetilde R + \nu I, \] where \(\widetilde R\) is the nearest positive semi-definite matrix to Rpointwise.

out = estR(X, types = c("con", "bin", "ter", "tru"), nu = 0.001)
#> Using Matrix::nearPD since Minimum eigenvalue of latent correlation matrix is -0.25029819695076 smaller than 0.
out$Rpointwise
#>            X1         X2        X3         X4
#> X1  1.0000000 -0.1477240 0.9990000  0.8548518
#> X2 -0.1477240  1.0000000 0.3523666 -0.5030324
#> X3  0.9990000  0.3523666 1.0000000  0.9114307
#> X4  0.8548518 -0.5030324 0.9114307  1.0000000
out$R
#>            X1         X2        X3         X4
#> X1  1.0000000 -0.1053533 0.9232992  0.9048072
#> X2 -0.1053533  1.0000000 0.2372115 -0.4244433
#> X3  0.9232992  0.2372115 1.0000000  0.7723678
#> X4  0.9048072 -0.4244433 0.7723678  1.0000000

As a result, R and Rpointwise could be quite different when sample size \(n\) is small. When \(n\) is large and \(p\) is moderate, the difference is typically driven by parameter nu.

set.seed("234820")
X = GenData(n = 100, types = c("con", "bin", "ter", "tru"))$X
out = estR(X, types = c("con", "bin", "ter", "tru"), nu = 0.001)
out$Rpointwise
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5528213 0.4476819 0.5823120
#> X2 0.5528213 1.0000000 0.4058967 0.5808889
#> X3 0.4476819 0.4058967 1.0000000 0.4567771
#> X4 0.5823120 0.5808889 0.4567771 1.0000000
out$R
#>           X1        X2        X3        X4
#> X1 1.0000000 0.5522684 0.4472342 0.5817297
#> X2 0.5522684 1.0000000 0.4054908 0.5803080
#> X3 0.4472342 0.4054908 1.0000000 0.4563203
#> X4 0.5817297 0.5803080 0.4563203 1.0000000

Appendix

Derivation of bridge function \(F\) for ternary/truncated case

Without loss of generality, let \(j=1\) and \(k=2\). By the definition of Kendall’s \(\tau\), \[ \tau_{12}=E(\hat{\tau}_{12})=E[\frac{2}{n(n-1)}\sum_{1\leq i\leq i' \leq n} sign\{(X_{i1}-X_{i'1})(X_{i2}-X_{i'2})\}]. \] Since \(X_{1}\) is ternary, \[\begin{align} &sign(X_{1}-X_{1}') \nonumber\\ =&[I(U_{1}>C_{11},U_{1}'\leq C_{11})+I(U_{1}>C_{12},U_{1}'\leq C_{12})-I(U_{1}>C_{12},U_{1}'\leq C_{11})] \nonumber\\ &-[I(U_{1}\leq C_{11}, U_{1}'>C_{11})+I(U_{1}\leq C_{12}, U_{1}'>C_{12})-I(U_{1}\leq C_{11}, U_{1}'>C_{12})] \nonumber\\ =&[I(U_{1}>C_{11})-I(U_{1}>C_{11},U_{1}'>C_{11})+I(U_{1}>C_{12})-I(U_{1}>C_{12},U_{1}'>C_{12}) \nonumber\\ &-I(U_{1}>C_{12})+I(U_{1}>C_{12},U_{1}'>C_{11})] \nonumber\\ &-[I(U_{1}'>C_{11})-I(U_{1}>C_{11},U_{1}'>C_{11})+I(U_{1}'>C_{12})-I(U_{1}>C_{12},U_{1}'>C_{12}) \nonumber\\ &-I(U_{1}'>C_{12})+I(U_{1}>C_{11},U_{1}'>C_{12})] \nonumber\\ =&I(U_{1}>C_{11})+I(U_{1}>C_{12},U_{1}'>C_{11})-I(U_{1}'>C_{11})-I(U_{1}>C_{11},U_{1}'>C_{12}) \nonumber\\ =&I(U_{1}>C_{11},U_{1}'\leq C_{12})-I(U_{1}'>C_{11},U_{1}\leq C_{12}). \end{align}\] Since \(X_{2}\) is truncated, \(C_{1}>0\) and \[\begin{align} sign(X_{2}-X_{2}')=&-I(X_{2}=0,X_{2}'>0)+I(X_{2}>0,X_{2}'=0) \nonumber\\ &+I(X_{2}>0,X_{2}'>0)sign(X_{2}-X_{2}') \nonumber\\ =&-I(X_{2}=0)+I(X_{2}'=0)+I(X_{2}>0,X_{2}'>0)sign(X_{2}-X_{2}'). \end{align}\] Since \(f\) is monotonically increasing, \(sign(X_{2}-X_{2}')=sign(Z_{2}-Z_{2}')\), \[\begin{align} \tau_{12}=&E[I(U_{1}>C_{11},U_{1}'\leq C_{12}) sign(X_{2}-X_{2}')] \nonumber\\ &-E[I(U_{1}'>C_{11},U_{1}\leq C_{12}) sign(X_{2}-X_{2}')] \nonumber\\ =&-E[I(U_{1}>C_{11},U_{1}'\leq C_{12}) I(X_{2}=0)] \nonumber\\ &+E[I(U_{1}>C_{11},U_{1}'\leq C_{12}) I(X_{2}'=0)] \nonumber\\ &+E[I(U_{1}>C_{11},U_{1}'\leq C_{12})I(X_{2}>0,X_{2}'>0)sign(Z_{2}-Z_{2}')] \nonumber\\ &+E[I(U_{1}'>C_{11},U_{1}\leq C_{12}) I(X_{2}=0)] \nonumber\\ &-E[I(U_{1}'>C_{11},U_{1}\leq C_{12}) I(X_{2}'=0)] \nonumber\\ &-E[I(U_{1}'>C_{11},U_{1}\leq C_{12})I(X_{2}>0,X_{2}'>0)sign(Z_{2}-Z_{2}')] \nonumber\\ =&-2E[I(U_{1}>C_{11},U_{1}'\leq C_{12}) I(X_{2}=0)] \nonumber\\ &+2E[I(U_{1}>C_{11},U_{1}'\leq C_{12}) I(X_{2}'=0)] \nonumber\\ &+E[I(U_{1}>C_{11},U_{1}'\leq C_{12})I(X_{2}>0,X_{2}'>0)sign(Z_{2}-Z_{2}')] \nonumber\\ &-E[I(U_{1}'>C_{11},U_{1}\leq C_{12})I(X_{2}>0,X_{2}'>0)sign(Z_{2}-Z_{2}')]. \end{align}\] From the definition of \(U\), let \(Z_{j}=f_{j}(U_{j})\) and \(\Delta_{j}=f_{j}(C_{j})\) for \(j=1,2\). Using \(sign(x)=2I(x>0)-1\), we obtain \[\begin{align} \tau_{12}=&-2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq \Delta_{12},Z_{2}\leq \Delta_{2})]+2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq \Delta_{12},Z_{2}'\leq \Delta_{2})] \nonumber\\ &+2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq \Delta_{12})I(Z_{2}>\Delta_{2},Z_{2}'>\Delta_{2},Z_{2}-Z_{2}'>0)] \nonumber\\ &-2E[I(Z_{1}'>\Delta_{11},Z_{1}\leq \Delta_{12})I(Z_{2}>\Delta_{2},Z_{2}'>\Delta_{2},Z_{2}-Z_{2}'>0)] \nonumber\\ =&-2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq \Delta_{12}, Z_{2}\leq \Delta_{2})]+2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq \Delta_{12}, Z_{2}'\leq \Delta_{2})] \nonumber\\ &+2E[I(Z_{1}>\Delta_{11},Z_{1}'\leq\Delta_{12},Z_{2}'>\Delta_{2},Z_{2}>Z_{2}')] \nonumber\\ &-2E[I(Z_{1}'>\Delta_{11},Z_{1}\leq\Delta_{12},Z_{2}'>\Delta_{2},Z_{2}>Z_{2}')]. \end{align}\] Since \(\{\frac{Z_{2}'-Z_{2}}{\sqrt{2}}, -Z{1}\}\), \(\{\frac{Z_{2}'-Z_{2}}{\sqrt{2}}, Z{1}'\}\) and \(\{\frac{Z_{2}'-Z_{2}}{\sqrt{2}}, -Z{2}'\}\) are standard bivariate normally distributed variables with correlation \(-\frac{1}{\sqrt{2}}\), \(r/\sqrt{2}\) and \(-\frac{r}{\sqrt{2}}\), respectively, by the definition of \(\Phi_3(\cdot,\cdot, \cdot;\cdot)\) and \(\Phi_4(\cdot,\cdot, \cdot,\cdot;\cdot)\) we have \[\begin{align} F_{NT}(r;\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k})= & -2\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k};\begin{pmatrix} 1 & 0 & -r \\ 0 & 1 & 0 \\ -r & 0 & 1 \end{pmatrix} \right\} \nonumber\\ &+2\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k};\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & r & 1 \end{pmatrix}\right\}\nonumber \\ & +2\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & 0 & \frac{r}{\sqrt{2}} \\ 0 & 1 & -r & \frac{r}{\sqrt{2}} \\ 0 & -r & 1 & -\frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\} \nonumber\\ &-2\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & r & -\frac{r}{\sqrt{2}} \\ 0 & 1 & 0 & -\frac{r}{\sqrt{2}} \\ r & 0 & 1 & -\frac{1}{\sqrt{2}} \\ -\frac{r}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\}. \end{align}\] Using the facts that \[\begin{align} &\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & r & -\frac{r}{\sqrt{2}} \\ 0 & 1 & 0 & -\frac{r}{\sqrt{2}} \\ r & 0 & 1 & -\frac{1}{\sqrt{2}} \\ -\frac{r}{\sqrt{2}} & -\frac{r}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\} \nonumber\\ &+\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & r & \frac{r}{\sqrt{2}} \\ 0 & 1 & 0 & \frac{r}{\sqrt{2}} \\ r & 0 & 1 & \frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\} \nonumber\\ =&\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k};\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & r & 1 \end{pmatrix}\right\} \end{align}\] and \[\begin{align} &\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k};\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & r & 1 \end{pmatrix}\right\}+\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k};\begin{pmatrix} 1 & 0 & -r \\ 0 & 1 & 0 \\ -r & 0 & 1 \end{pmatrix} \right\} \nonumber\\ =&\Phi_{2}(-\Delta_{j}^{1},\Delta_{j}^{2};0) =\Phi(-\Delta_{j}^{1})\Phi(\Delta_{j}^{2}). \end{align}\] So that, \[\begin{align} F_{NT}(r;\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k})= & -2\Phi(-\Delta_{j}^{1})\Phi(\Delta_{j}^{2}) \nonumber\\ &+2\Phi_{3}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},\Delta_{k};\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & r \\ 0 & r & 1 \end{pmatrix}\right\}\nonumber \\ & +2\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & 0 & \frac{r}{\sqrt{2}} \\ 0 & 1 & -r & \frac{r}{\sqrt{2}} \\ 0 & -r & 1 & -\frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\} \nonumber\\ &+2\Phi_{4}\left\{-\Delta_{j}^{1},\Delta_{j}^{2},-\Delta_{k},0;\begin{pmatrix} 1 & 0 & r & \frac{r}{\sqrt{2}} \\ 0 & 1 & 0 & \frac{r}{\sqrt{2}} \\ r & 0 & 1 & \frac{1}{\sqrt{2}} \\ \frac{r}{\sqrt{2}} & \frac{r}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 1 \end{pmatrix}\right\}. \end{align}\]

It is easy to get the bridge function for truncated/ternary case by switching \(j\) and \(k\).

Derivation of approximate bound for the ternary/continuous case

Let \(n_{0x}=\sum_{i=1}^{n_x}I(x_{i}=0)\), \(n_{2x}=\sum_{i=1}^{n_x}I(x_{i}=2)\), \(\pi_{0x}=\frac{n_{0x}}{n_{x}}\) and \(\pi_{2x}=\frac{n_{2x}}{n_{x}}\), then \[\begin{align} |\tau(\mathbf{x})|\leq & \frac{n_{0x}(n-n_{0x})+n_{2x}(n-n_{0x}-n_{2x})}{\begin{pmatrix} n \\ 2 \end{pmatrix}} \nonumber\\ = & 2\{\frac{n_{0x}}{n-1}-(\frac{n_{0x}}{n})(\frac{n_{0x}}{n-1})+\frac{n_{2x}}{n-1}-(\frac{n_{2x}}{n})(\frac{n_{0x}}{n-1})-(\frac{n_{2x}}{n})(\frac{n_{2x}}{n-1})\} \nonumber\\ \approx & 2\{\frac{n_{0x}}{n}-(\frac{n_{0x}}{n})^2+\frac{n_{2x}}{n}-(\frac{n_{2x}}{n})(\frac{n_{0x}}{n})-(\frac{n_{2x}}{n})^2\} \nonumber\\ = & 2\{\pi_{0x}(1-\pi_{0x})+\pi_{2x}(1-\pi_{0x}-\pi_{2x})\} \end{align}\]

For ternary/binary and ternary/ternary cases, we combine the two individual bounds.

Derivation of approximate bound for the ternary/truncated case

Let \(\mathbf{x}\in\mathcal{R}^{n}\) and \(\mathbf{y}\in\mathcal{R}^{n}\) be the observed \(n\) realizations of ternary and truncated variables, respectively. Let \(n_{0x}=\sum_{i=0}^{n}I(x_{i}=0)\), \(\pi_{0x}=\frac{n_{0x}}{n}\), \(n_{1x}=\sum_{i=0}^{n}I(x_{i}=1)\), \(\pi_{1x}=\frac{n_{1x}}{n}\), \(n_{2x}=\sum_{i=0}^{n}I(x_{i}=2)\), \(\pi_{2x}=\frac{n_{2x}}{n}\), \(n_{0y}=\sum_{i=0}^{n}I(y_{i}=0)\), \(\pi_{0y}=\frac{n_{0y}}{n}\), \(n_{0x0y}=\sum_{i=0}^{n}I(x_{i}=0 \;\& \; y_{i}=0)\), \(n_{1x0y}=\sum_{i=0}^{n}I(x_{i}=1 \;\& \; y_{i}=0)\) and \(n_{2x0y}=\sum_{i=0}^{n}I(x_{i}=2 \;\& \; y_{i}=0)\) then \[\begin{align} |\tau(\mathbf{x}, \mathbf{y})|\leq & \frac{\begin{pmatrix}n \\ 2\end{pmatrix}-\begin{pmatrix}n_{0x} \\ 2\end{pmatrix}-\begin{pmatrix}n_{1x} \\ 2\end{pmatrix}-\begin{pmatrix} n_{2x} \\ 2 \end{pmatrix}-\begin{pmatrix}n_{0y} \\ 2\end{pmatrix}+\begin{pmatrix}n_{0x0y} \\ 2 \end{pmatrix}+\begin{pmatrix}n_{1x0y} \\ 2\end{pmatrix}+\begin{pmatrix}n_{2x0y} \\ 2\end{pmatrix}}{\begin{pmatrix}n \\ 2\end{pmatrix}} \nonumber \end{align}\] Since \(n_{0x0y}\leq\min(n_{0x},n_{0y})\), \(n_{1x0y}\leq\min(n_{1x},n_{0y})\) and \(n_{2x0y}\leq\min(n_{2x},n_{0y})\) we obtain \[\begin{align} |\tau(\mathbf{x}, \mathbf{y})|\leq & \frac{\begin{pmatrix}n \\ 2\end{pmatrix}-\begin{pmatrix}n_{0x} \\ 2\end{pmatrix}-\begin{pmatrix}n_{1x} \\ 2\end{pmatrix}-\begin{pmatrix} n_{2x} \\ 2 \end{pmatrix}-\begin{pmatrix}n_{0y} \\ 2\end{pmatrix}}{\begin{pmatrix}n \\ 2\end{pmatrix}} \nonumber\\ & + \frac{\begin{pmatrix}\min(n_{0x},n_{0y}) \\ 2 \end{pmatrix}+\begin{pmatrix}\min(n_{1x},n_{0y}) \\ 2\end{pmatrix}+\begin{pmatrix}\min(n_{2x},n_{0y}) \\ 2\end{pmatrix}}{\begin{pmatrix}n \\ 2\end{pmatrix}} \nonumber\\ \leq & \frac{\begin{pmatrix}n \\ 2\end{pmatrix}-\begin{pmatrix}\max(n_{0x},n_{1x},n_{2x},n_{0y}) \\ 2\end{pmatrix}}{\begin{pmatrix}n \\ 2\end{pmatrix}} \nonumber\\ \leq & 1-\frac{\max(n_{0x},n_{1x},n_{2x},n_{0y})(\max(n_{0x},n_{1x},n_{2x},n_{0y})-1)}{n(n-1)} \nonumber\\ \approx & 1-(\frac{\max(n_{0x},n_{1x},n_{2x},n_{0y})}{n})^{2} \nonumber\\ =& 1-\{\max(\pi_{0x},\pi_{1x},\pi_{2x},\pi_{0y})\}^{2} \nonumber\\ =& 1-\{\max(\pi_{0x},(1-\pi_{0x}-\pi_{2x}),\pi_{2x},\pi_{0y})\}^{2} \end{align}\]

It is easy to get the approximate bound for truncated/ternary case by switching \(\mathbf{x}\) and \(\mathbf{y}\).

References

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