# Lower and Upper Semi-Correlations of a Copula Compute the **lower semi-correlations** (bottom-left) [REMOVE_ME]$$ \rho^{N{-}{-}}_\mathbf{C}(u,v; a) = \rho_N^{{-}{-}}(a) \mbox{ and} [REMOVE_ME_2]$$ compute the **upper semi-correlations** (top-right) [REMOVE_ME]$$ \rho^{N{+}{+}}_\mathbf{C}(u,v; a) = \rho_N^{{+}{+}}(a) [REMOVE_ME_2]$$ of a copula $\mathbf{C}(u,v)$ (Joe, 2014, p. 73) using numerical simulation. The semi-correlations are defined as [REMOVE_ME]$$ \rho_N^{{-}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 < -a]\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \rho_N^{{+}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 > +a]\mbox{,\ and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \rho_N(a > -\infty) = \mathrm{cor}[Z_1, Z_2]\mbox{,} [REMOVE_ME_2]$$ where $\mathrm{cor[z_1, z_2]}$ is the familiar **Pearson correlation function**, which is in the syntax `cor(...,` `method="pearson")`, parameter $a \ge 0$ is a truncation point that identifies **truncated tail regions** (Joe, 2014, p. 73), and lastly $(Z_1, Z_2) \sim \mathbf{C}(\Phi, \Phi)$ and thus from the standard normal distribution $(Z_1, Z_2) = (\Phi^{-1}(u), \Phi^{-1}(v))$ where the random variables $(U,V) \sim \mathbf{C}$. The semi-correlations are extended for the `copBasic` package into bottom right and top left versions as well by [REMOVE_ME]$$ \rho_N^{{+}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 < -a]\mbox{,\ and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \rho_N^{{-}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 > +a]\mbox{.} [REMOVE_ME_2]$$ As a result, the notations ${-}{-}$, ${+}{+}$, ${+}{-}$, and ${-}{+}$ can be used to represent each of the respective corners bottom-left, top-right, bottom-right, and top-left of the $(u,v)$ domain with the respective truncation. These words are used in the variable names of the returned list from the function. utf8 ## Description Compute the **lower semi-correlations** (bottom-left) $$ \rho^{N{-}{-}}_\mathbf{C}(u,v; a) = \rho_N^{{-}{-}}(a) \mbox{ and} $$ compute the **upper semi-correlations** (top-right) $$ \rho^{N{+}{+}}_\mathbf{C}(u,v; a) = \rho_N^{{+}{+}}(a) $$ of a copula $\mathbf{C}(u,v)$ (Joe, 2014, p. 73) using numerical simulation. The semi-correlations are defined as $$ \rho_N^{{-}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 < -a]\mbox{,} $$ $$ \rho_N^{{+}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 > +a]\mbox{,\ and} $$ $$ \rho_N(a > -\infty) = \mathrm{cor}[Z_1, Z_2]\mbox{,} $$ where $\mathrm{cor[z_1, z_2]}$ is the familiar **Pearson correlation function**, which is in the syntax `cor(...,` `method="pearson")`, parameter $a \ge 0$ is a truncation point that identifies **truncated tail regions** (Joe, 2014, p. 73), and lastly $(Z_1, Z_2) \sim \mathbf{C}(\Phi, \Phi)$ and thus from the standard normal distribution $(Z_1, Z_2) = (\Phi^{-1}(u), \Phi^{-1}(v))$ where the random variables $(U,V) \sim \mathbf{C}$. The semi-correlations are extended for the `copBasic` package into bottom right and top left versions as well by $$ \rho_N^{{+}{-}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 > +a, Z_2 < -a]\mbox{,\ and} $$ $$ \rho_N^{{-}{+}}(a) = \mathrm{cor}[Z_1, Z_2 \mid Z_1 < -a, Z_2 > +a]\mbox{.} $$ As a result, the notations ${-}{-}$, ${+}{+}$, ${+}{-}$, and ${-}{+}$ can be used to represent each of the respective corners bottom-left, top-right, bottom-right, and top-left of the $(u,v)$ domain with the respective truncation. These words are used in the variable names of the returned list from the function. ```r semicorCOP(cop=NULL, para=NULL, truncation=0, n=0, as.sample=FALSE, ...) ``` ## Arguments - `cop`: A copula function; - `para`: Vector of parameters or other data structure, if needed, to pass to the copula; - `truncation`: The truncation value for $a$, which is in standard normal variates, and the default of zero is the origin (medians); - `n`: The sample size $n$ for simulation estimates of the $\rho_N$; - `as.sample`: A logical controlling whether an optional `data.frame` in `para` is used to compute the $\hat\rho_N$ (see Note ); and - `...`: Additional arguments to pass to the copula. ## Returns The value(s) for $\rho_N$, $\rho_N^{{-}{-}}$, $\rho_N^{{+}{+}}$, $\rho_N^{{+}{-}}$, and $\rho_N^{{-}{+}}$ are returned. ## Note The sample semi-correlations can be computed from a two-column table that is passed into the function using the `para` argument. Although the truncation point $a \ge 0$, as $a$ increases and focus is increasingly made into one or the other truncated tail regions, the sample version with data becomes decreasing well estimated because the available sample size diminishes. The `para` argument can contain probabilities or raw data because internally the function computes the **Hazen plotting positions** (**e.g.** $u_i = (i-0.5) / n$ for rank $i$ and sample size $n$) because Joe (2014, pp. 9, 17, 245, 247--248) repeatedly emphasizes this form of plotting position when normal scores are involved. ## References Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p. ## Author(s) W.H. Asquith ## See Also `giniCOP`, `rhoCOP`, `tauCOP`, `COP` ## Examples ```r ## Not run: # Gumbel-Hougaard copula with Pearson rhoN = 0.4 (by definition) run <- sapply(1:50, function(i) semicorCOP(cop=GHcop, para=1.350, n=600)) mean(unlist(run[1,])) # cor.normal.scores mean(unlist(run[2,])) # botleft.semicor mean(unlist(run[3,])) # topright.semicor sd( unlist(run[1,])) # cor.normal.scores (These are our sampling variations sd( unlist(run[2,])) # botleft.semicor for the n=600 used as a Monte sd( unlist(run[3,])) # topright.semicor Carlo simulation.) # The function returns: rhoN = 0.392112, rhoN--= 0.117674, rhoN++= 0.404733 # standard deviations (0.038883) (0.073392) (0.073942) # Joe (2014, p. 72) shows: rhoN = 0.4_____, rhoN--= 0.132___, rhoN++= 0.415___ # standard deviations (not avail) (0.08____) (0.07____) # We see alignment with the results of Joe with his n=600. # ## End(Not run) ## Not run: p <- 0.5 # Reasonable strong positive association for the Raftery copula, # but then we are going to be reflecting and rotating the copula. # See similar Example under COP() function. "RFcop1" <- function(u,v, para) COP(u,v, cop=RFcop, para=para, reflect="1") "RFcop2" <- function(u,v, para) COP(u,v, cop=RFcop, para=para, reflect="2") "RFcop3" <- function(u,v, para) COP(u,v, cop=RFcop, para=para, reflect="3") "RFcop4" <- function(u,v, para) COP(u,v, cop=RFcop, para=para, reflect="4") RF <- NULL; n <- 10000 RF <- rbind(RF, as.data.frame(semicorCOP(cop=RFcop1, para=p, n=n))[,1:5]) RF <- rbind(RF, as.data.frame(semicorCOP(cop=RFcop2, para=p, n=n))[,1:5]) RF <- rbind(RF, as.data.frame(semicorCOP(cop=RFcop3, para=p, n=n))[,1:5]) RF <- rbind(RF, as.data.frame(semicorCOP(cop=RFcop4, para=p, n=n))[,1:5]) print(RF[,1:3]) # total sample and lower and upper semi-correlations # cor.normal.scores botleft.semicor topright.semicor # 1 0.5587837 0.74124686 0.10027641 # 2 0.5567889 0.10302772 0.73729702 # 3 -0.5807201 -0.04683536 -0.01714573 # see near zeros --,++ # 4 -0.5698139 0.03040520 0.05125916 # see near zeros --,++ print(RF[,2:5]) # now look at all four corners # botleft.semicor topright.semicor topleft.semicor botright.semicor # 1 0.74124686 0.10027641 0.01529508 0.02046530 # 2 0.10302772 0.73729702 -0.05195628 0.01747874 # 3 -0.04683536 -0.01714573 -0.11106842 -0.74077321 # 4 0.03040520 0.05125916 -0.74516061 -0.07636233 # Notice how the tight tail of the copula, being reflected and rotated # into each of the four corners, shows semi-correlation magnitude of 0.74. # See the copula density plots of COP() Examples section. ## End(Not run) ```

semicorCOP function