# make_deadata_stoch This function creates, from a `deadata` object, a `deadata_stoch` object by adding the corresponding covariance matrices. These objects are prepared to be passed to a modelstoch_* function. We consider $\mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} $ a set of $n$ DMUs with $m$ stochastic inputs and $s$ stochastic outputs. Matrices $\tilde{X}=(\tilde{x}_{ij})$ and $\tilde{Y}=(\tilde{y}_{rj})$ are the input and output data matrices, respectively, where $\tilde{x}_{ij}$ and $\tilde{y}_{rj}$ represent the $i$-th input and $r$-th output of the $j$-th DMU. Moreover, we denote by $X=(x_{ij})$ and $Y=(y_{rj})$ their expected values. We suppose that $\tilde{x}_{ij}$ and $\tilde{y}_{rj}$ follow a multivariate probability distribution with means $E(\tilde{x}_{ij})=x_{ij}$, $E(\tilde{y}_{rj})=y_{rj}$ and covariance matrix [REMOVE_ME]$$ \Delta = \begin{pmatrix}\Delta ^{II}_{11} & \cdots & \Delta ^{II}_{1m} & \Delta ^{IO}_{11} & \cdots & \Delta ^{IO}_{1s} \\\vdots & & \vdots & \vdots & & \vdots \\\Delta ^{II}_{m1} & \cdots & \Delta ^{II}_{mm} & \Delta ^{IO}_{m1} & \cdots & \Delta ^{IO}_{ms} \\ \\\Delta ^{OI}_{11} & \cdots & \Delta ^{OI}_{1m} & \Delta ^{OO}_{11} & \cdots & \Delta ^{OO}_{1s} \\\vdots & & \vdots & \vdots & & \vdots \\\Delta ^{OI}_{s1} & \cdots & \Delta ^{OI}_{sm} & \Delta ^{OO}_{s1} & \cdots & \Delta ^{OO}_{ss} \\\end{pmatrix}_{n(m+s)\times n(m+s)} [REMOVE_ME_2]$$ where $\Delta ^{II}_{ik}$, $\Delta ^{OO}_{rp}$, $\Delta ^{IO}_{ir}$ and $\Delta ^{OI}_{ri}$ are $n\times n$ matrices, for $1\leq i,k\leq m$ and $1\leq r,p\leq s$, such that [REMOVE_ME]$$ \left( \Delta ^{II}_{ik}\right) _{jq}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{kq}), [REMOVE_ME_2]$$ [REMOVE_ME]$$ \left( \Delta ^{OO}_{rp}\right) _{jq}=\mathrm{Cov}(\tilde{y}_{rj}, \tilde{y}_{pq}), [REMOVE_ME_2]$$ [REMOVE_ME]$$ \left( \Delta ^{IO}_{ir}\right) _{jq}=\left( \Delta ^{OI}_{ri}\right) _{qj}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{y}_{rq}), [REMOVE_ME_2]$$ with $1\leq j,q\leq n$. - If we have the covariances matrix in the general form above, it can be introduced directly by parameter `cov_matrix`. Since this matrix is supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. - Alternatively, we can introduce the covariances matrix using parameters `cov_II`, `cov_OO` and `cov_IO`, that are 4-dimensional arrays of size c("$m\\times\n$", "$m\\times n\\times n$"), $s\times s\times n\times n$ and $m\times s\times n\times n$, respectively, such that [REMOVE_ME]$$ \texttt{cov\_II[i, k, , ]}=\Delta ^{II}_{ik}, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \texttt{cov\_OO[r, p, , ]}=\Delta ^{OO}_{rp}, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \texttt{cov\_IO[i, r, , ]}=\Delta ^{IO}_{ir}, [REMOVE_ME_2]$$ for $1\leq i,k\leq m$ and $1\leq r,p\leq s$. Since matrices $\Delta ^{II}_{ii}$ and c("$\\Delta\n$", "$^{OO}_{rr}$") are supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. Moreover, since $\Delta ^{II}_{ki}$ is the transpose of $\Delta ^{II}_{ik}$, and $\Delta ^{OO}_{pr}$ is the transpose of $\Delta ^{OO}_{rp}$, only matrices $\Delta ^{II}_{ik}$ and $\Delta ^{OO}_{rp}$ with $i\leq k$ and $r\leq p$ are necessary, ignoring those with $i>k$ and $r>p$. - If covariances between different inputs/outputs are zero, we can make use of parameters `cov_input` and `cov_output`, that are arrays of size $m\times n\times n$ and $s\times n\times n$, respectively, such that [REMOVE_ME]$$ \texttt{cov\_input[i, , ]}=\Delta ^{II}_{ii}, [REMOVE_ME_2]$$ [REMOVE_ME]$$ \texttt{cov\_output[r, , ]}=\Delta ^{OO}_{rr}, [REMOVE_ME_2]$$ for $1\leq i\leq m$ and $1\leq r\leq s$. By symmetry of c("$\\Delta\n$", "$^{II}_{ii}$") and $\Delta ^{OO}_{rr}$, only values above the diagonal are read, ignoring values below the diagonal. - Finally, if all the variables are independent then the covariances matrix is diagonal. Hence, we might use parameters `var_input` and `var_output`, that are matrices of size $m\times n$ and $s\times n$, respectively, such that [REMOVE_ME]$$ \texttt{var\_input[i, j]}=\mathrm{Var}\left( \tilde{x}_{ij}\right) , [REMOVE_ME_2]$$ [REMOVE_ME]$$ \texttt{var\_output[r, j]}=\mathrm{Var}\left( \tilde{y}_{rj}\right) , [REMOVE_ME_2]$$ for $1\leq i\leq m$, $1\leq r\leq s$ and $1\leq j\leq n$. ```r make_deadata_stoch(datadea = NULL, var_input = NULL, var_output = NULL, cov_input = NULL, cov_output = NULL, cov_II = NULL, cov_OO = NULL, cov_IO = NULL, cov_matrix = NULL) ``` ## Arguments - `datadea`: The `deadata` object. - `var_input`: A matrix of size `m` x `n`, where `m` is the number of inputs and `n` is the number of DMUs. It contains the variances of each input of each DMU, such that `var_input[i, j]` is the variance of the input `i` of DMU `j`. Use this parameter if all covariances are 0. - `var_output`: A matrix of size `s` x `n`, where `s` is the number of outputs and `n` is the number of DMUs. It contains the variances of each output of each DMU, such that `var_output[r, j]` is the variance of the output `r` of DMU `j`. Use this parameter if all covariances are 0. - `cov_input`: An array of size `m` x `n` x `n` containing the covariances of each input between DMUs, such that `cov_input[i, j, k]` is the covariance between the input `i` of DMU `j` and the input `i` of DMU `k`. The corresponding variances are in the diagonal of each `n` x `n` submatrix. Since these submatrices are supposed to be symmetric, only values above the diagonal are read in order to reconstruct the symmetric submatrices, ignoring values below the diagonal. Use this parameter if covariances between different inputs are 0. - `cov_output`: An array of size `s` x `n` x `n` containing the covariances of each output between DMUs, such that `cov_output[r, j, k]` is the covariance between the output `r` of DMU `j` and the output `r` of DMU `k`. The corresponding variances are in the diagonal of each `n` x `n` submatrix. Since these submatrices are supposed to be symmetric, only values above the diagonal are read in order to reconstruct the symmetric submatrices, ignoring values below the diagonal. Use this parameter if covariances between different outputs are 0. - `cov_II`: An array of size `m` x `m` x `n` x `n` containing the covariances between inputs and between DMUs, such that `cov_II[i1, i2, j, k]` is the covariance between the input `i1` of DMU `j` and the input `i2` of DMU `k`. For the input `i`, the corresponding variances are in the diagonal of the `n` x `n` submatrices of the form `cov_II[i, i, , ]`. Since these submatrices are supposed to be symmetric, only values above the diagonal are read in order to reconstruct the symmetric submatrices, ignoring values below the diagonal. Moreover, since `cov_II[i2, i1, , ]` is the transpose of `cov_II[i1, i2, , ]`, only submatrices `cov_II[i1, i2, , ]` with `i1 <= i2` are necessary, ignoring those with `i1 > i2`. Use this parameter if covariances between inputs are nonzero. - `cov_OO`: An array of size `s` x `s` x `n` x `n` containing the covariances between outputs and between DMUs, such that `cov_OO[r1, r2, j, k]` is the covariance between the output `r1` of DMU `j` and the output `r2` of DMU `k`. For the output `r`, the corresponding variances are in the diagonal of the `n` x `n` submatrices of the form `cov_OO[r, r, , ]`. Since these submatrices are supposed to be symmetric, only values above the diagonal are read in order to reconstruct the symmetric submatrices, ignoring values below the diagonal. Moreover, since `cov_OO[r2, r1, , ]` is the transpose of `cov_OO[r1, r2, , ]`, only submatrices `cov_OO[r1, r2, , ]` with `r1 <= r2` are necessary, ignoring those with `r1 > r2`. Use this parameter if covariances between outputs are nonzero. - `cov_IO`: An array of size `m` x `s` x `n` x `n` containing the covariances between inputs and outputs, and between DMUs, such that `cov_IO[i, r, j, k]` is the covariance between the input `i` of DMU `j` and the output `r` of DMU `k`. Use this parameter if covariances between inputs and outputs are nonzero. - `cov_matrix`: A matrix of size `n(m+s)` x `n(m+s)` following the notation of Cooper et al. (1998). Since this matrix is supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. ## Returns An object of class `deadata_stoch`. ## Description This function creates, from a `deadata` object, a `deadata_stoch` object by adding the corresponding covariance matrices. These objects are prepared to be passed to a modelstoch_* function. We consider $\mathcal{D}=\left\{ \textrm{DMU}_1, \ldots ,\textrm{DMU}_n \right\} $ a set of $n$ DMUs with $m$ stochastic inputs and $s$ stochastic outputs. Matrices $\tilde{X}=(\tilde{x}_{ij})$ and $\tilde{Y}=(\tilde{y}_{rj})$ are the input and output data matrices, respectively, where $\tilde{x}_{ij}$ and $\tilde{y}_{rj}$ represent the $i$-th input and $r$-th output of the $j$-th DMU. Moreover, we denote by $X=(x_{ij})$ and $Y=(y_{rj})$ their expected values. We suppose that $\tilde{x}_{ij}$ and $\tilde{y}_{rj}$ follow a multivariate probability distribution with means $E(\tilde{x}_{ij})=x_{ij}$, $E(\tilde{y}_{rj})=y_{rj}$ and covariance matrix $$ \Delta = \begin{pmatrix}\Delta ^{II}_{11} & \cdots & \Delta ^{II}_{1m} & \Delta ^{IO}_{11} & \cdots & \Delta ^{IO}_{1s} \\\vdots & & \vdots & \vdots & & \vdots \\\Delta ^{II}_{m1} & \cdots & \Delta ^{II}_{mm} & \Delta ^{IO}_{m1} & \cdots & \Delta ^{IO}_{ms} \\ \\\Delta ^{OI}_{11} & \cdots & \Delta ^{OI}_{1m} & \Delta ^{OO}_{11} & \cdots & \Delta ^{OO}_{1s} \\\vdots & & \vdots & \vdots & & \vdots \\\Delta ^{OI}_{s1} & \cdots & \Delta ^{OI}_{sm} & \Delta ^{OO}_{s1} & \cdots & \Delta ^{OO}_{ss} \\\end{pmatrix}_{n(m+s)\times n(m+s)} $$ where $\Delta ^{II}_{ik}$, $\Delta ^{OO}_{rp}$, $\Delta ^{IO}_{ir}$ and $\Delta ^{OI}_{ri}$ are $n\times n$ matrices, for $1\leq i,k\leq m$ and $1\leq r,p\leq s$, such that $$ \left( \Delta ^{II}_{ik}\right) _{jq}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{x}_{kq}), $$ $$ \left( \Delta ^{OO}_{rp}\right) _{jq}=\mathrm{Cov}(\tilde{y}_{rj}, \tilde{y}_{pq}), $$ $$ \left( \Delta ^{IO}_{ir}\right) _{jq}=\left( \Delta ^{OI}_{ri}\right) _{qj}=\mathrm{Cov}(\tilde{x}_{ij}, \tilde{y}_{rq}), $$ with $1\leq j,q\leq n$. - If we have the covariances matrix in the general form above, it can be introduced directly by parameter `cov_matrix`. Since this matrix is supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. - Alternatively, we can introduce the covariances matrix using parameters `cov_II`, `cov_OO` and `cov_IO`, that are 4-dimensional arrays of size c("$m\\times\n$", "$m\\times n\\times n$"), $s\times s\times n\times n$ and $m\times s\times n\times n$, respectively, such that $$ \texttt{cov\_II[i, k, , ]}=\Delta ^{II}_{ik}, $$ $$ \texttt{cov\_OO[r, p, , ]}=\Delta ^{OO}_{rp}, $$ $$ \texttt{cov\_IO[i, r, , ]}=\Delta ^{IO}_{ir}, $$ for $1\leq i,k\leq m$ and $1\leq r,p\leq s$. Since matrices $\Delta ^{II}_{ii}$ and c("$\\Delta\n$", "$^{OO}_{rr}$") are supposed to be symmetric, only values above the diagonal are read, ignoring values below the diagonal. Moreover, since $\Delta ^{II}_{ki}$ is the transpose of $\Delta ^{II}_{ik}$, and $\Delta ^{OO}_{pr}$ is the transpose of $\Delta ^{OO}_{rp}$, only matrices $\Delta ^{II}_{ik}$ and $\Delta ^{OO}_{rp}$ with $i\leq k$ and $r\leq p$ are necessary, ignoring those with $i>k$ and $r>p$. - If covariances between different inputs/outputs are zero, we can make use of parameters `cov_input` and `cov_output`, that are arrays of size $m\times n\times n$ and $s\times n\times n$, respectively, such that $$ \texttt{cov\_input[i, , ]}=\Delta ^{II}_{ii}, $$ $$ \texttt{cov\_output[r, , ]}=\Delta ^{OO}_{rr}, $$ for $1\leq i\leq m$ and $1\leq r\leq s$. By symmetry of c("$\\Delta\n$", "$^{II}_{ii}$") and $\Delta ^{OO}_{rr}$, only values above the diagonal are read, ignoring values below the diagonal. - Finally, if all the variables are independent then the covariances matrix is diagonal. Hence, we might use parameters `var_input` and `var_output`, that are matrices of size $m\times n$ and $s\times n$, respectively, such that $$ \texttt{var\_input[i, j]}=\mathrm{Var}\left( \tilde{x}_{ij}\right) , $$ $$ \texttt{var\_output[r, j]}=\mathrm{Var}\left( \tilde{y}_{rj}\right) , $$ for $1\leq i\leq m$, $1\leq r\leq s$ and $1\leq j\leq n$. ## Examples ```r # Example 1 library(deaR) data("Coll_Blasco_2006") ni <- 2 # number of inputs no <- 2 # number of outputs data_example <- make_deadata(datadea = Coll_Blasco_2006, ni = ni, no = no) nd <- length(data_example$dmunames) # number of DMUs # All variances are 1. var_input <- matrix(1, nrow = ni, ncol = nd) var_output <- matrix(1, nrow = no, ncol = nd) data_stoch <- make_deadata_stoch(datadea = data_example, var_input = var_input, var_output = var_output) # All covariances are 1. cov_input <- array(1, dim = c(ni, nd, nd)) cov_output <- array(1, dim = c(no, nd, nd)) data_stoch2 <- make_deadata_stoch(datadea = data_example, cov_input = cov_input, cov_output = cov_output) # Example 2. Deterministic data with one stochastic input. library(deaR) dmunames <- c("A", "B", "C") nd <- length(dmunames) # Number of DMUs inputnames <- c("Input_1", "Input_2") ni <- length(inputnames) # Number of Inputs outputnames <- c("Output_1", "Output_2", "Output_3") no <- length(outputnames) # Number of Outputs X <- matrix(c(5, 14, 8, 15, 7, 12), nrow = ni, ncol = nd, dimnames = list(inputnames, dmunames)) Y <- matrix(c(9, 4, 16, 5, 7, 10, 4, 9, 13), nrow = no, ncol = nd, dimnames = list(outputnames, dmunames)) datadea <- make_deadata(inputs = X, outputs = Y) covX <- array(0, dim = c(2, 3, 3)) # The 2nd input is stochastic. # Since the corresponding 3x3 covariances matrix is symmetric, only values # above the diagonal are necessary. covX[2, 1, ] <- c(1.4, 0.9, 0.6) covX[2, 2, 2:3] <- c(1.5, 0.7) covX[2, 3, 3] <- 1.2 # Alternatively (note that values below the diagonal are ignored). covX[2, , ] <- matrix(c(1.4, 0.9, 0.6, 0, 1.5, 0.7, 0, 0, 1.2), nrow = 3, byrow = TRUE) datadea_stoch <- make_deadata_stoch(datadea, cov_input = covX) # Example 3. Replication of Program Follow Through data in Land et al. (1993) library(deaR) data("PFT1981") # Selecting DMUs in Program Follow Through (PFT) PFT <- PFT1981[1:49, ] PFT <- make_deadata(PFT, inputs = 2:6, outputs = 7:9) c <- 0.5 var_output <- matrix(c^2, nrow = 3, ncol = 49) PFT_stoch <- make_deadata_stoch(datadea = PFT, var_output = var_output) # Example 4. 5 random observations of 3 DMUs with 2 inputs and 2 outputs. library(deaR) # Generate random observations. input1 <- data.frame(I1D1 = rnorm(5, mean = sample(5:10, 1)), I1D2 = rnorm(5, mean = sample(5:10, 1)), I1D3 = rnorm(5, mean = sample(5:10, 1))) input2 <- data.frame(I2D1 = rnorm(5, mean = sample(5:10, 1)), I2D2 = rnorm(5, mean = sample(5:10, 1)), I2D3 = rnorm(5, mean = sample(5:10, 1))) output1 <- data.frame(O1D1 = rnorm(5, mean = sample(5:10, 1)), O1D2 = rnorm(5, mean = sample(5:10, 1)), O1D3 = rnorm(5, mean = sample(5:10, 1))) output2 <- data.frame(O2D1 = rnorm(5, mean = sample(5:10, 1)), O2D2 = rnorm(5, mean = sample(5:10, 1)), O2D3 = rnorm(5, mean = sample(5:10, 1))) # Generate deadata with means of observations. inputs <- matrix(mapply(mean, cbind(input1, input2)), nrow = 2, ncol = 3, byrow = TRUE, dimnames = list(c("I1", "I2"), c("D1", "D2", "D3"))) outputs <- matrix(mapply(mean, cbind(output1, output2)), nrow = 2, ncol = 3, byrow = TRUE, dimnames = list(c("O1", "O2"), c("D1", "D2", "D3"))) datadea <- make_deadata(inputs = inputs, outputs = outputs) # Generate covariances matrix cov_matrix. cov_matrix <- cov(cbind(input1, input2, output1, output2)) # Generate deadata_stoch datadea_stoch <- make_deadata_stoch(datadea, cov_matrix = cov_matrix) ``` ## References Cooper, W.W.; Huang, Z.; Lelas, V.; Li, S.X.; Olesen, O.B. (1998). “Chance Constrained Programming Formulations for Stochastic Characterizations of Efficiency and Dominance in DEA", Journal of Productivity Analysis, 9, 53-79. Land, K.C; Lovell, C.A.K.; Thore, S. (1993). "Chance-constrained data envelopment analysis", Managerial and Decision Economics, Vol. 14, No. 6, 541-554. ## Author(s) Vicente Bolós (vicente.bolos@uv.es ). **Department of Business Mathematics** Rafael Benítez (rafael.suarez@uv.es ). **Department of Business Mathematics** Vicente Coll-Serrano (vicente.coll@uv.es ). **Quantitative Methods for Measuring Culture (MC2). Applied Economics.** University of Valencia (Spain)

make_deadata_stoch function