Stochastic Models to Generate Geospatial-, Temporal-, and Cross-Correlated Daily Maximum and Minimum Temperatures

Table 2

Variable description.

Symbol	Definition	Units
𝐂𝐗, 𝐂𝐈	Correlation matrices of maximum and minimum temperatures	Unitless
𝐂𝐗𝐈	Cross-correlation matrix between maximum and minimum temperatures	Unitless
E	Eigen decomposition matrix	Unitless
F	Toeplitz-Cholesky factorization matrix	Unitless
𝑖,𝑗	Locations	Unitless
𝐼,𝐼,̂𝐼	Observed, expected and mean daily minimum temperature	°C
𝑛	Total number of locations	Unitless
𝜂	Total number of pairwise daily observations	Unitless
𝜑𝑖𝑗,𝜑𝑖𝑗	Cross-correlation between locations 𝑖,𝑗 based on observed and generated data, respectively	Unitless
𝑟	Random number from a Gaussian 𝑁[0,1] distribution	Unitless
𝑟^𝜓	Correlated random number from a Gaussian 𝑁[0,1] distribution	Unitless
R	Vector of random numbers following a Gaussian 𝑁[0,1] distribution	Unitless
𝐑^𝚿	Vector of spatially correlated random numbers following a Gaussian 𝑁[0,1] distribution after matrix-multiplying a vector of random numbers by F or E	Unitless
𝜌𝑖𝑗,𝜌𝑖𝑗	Pearson’s correlation between locations 𝑖,𝑗 based on observed and generated data, respectively	Unitless
𝜎𝑖,𝜎𝑗	Standard deviation of daily temperature observations	°C
𝜎𝑋𝑖,𝜎𝐼𝑗	Standard deviation of daily observations of maximum and minimum temperatures	°C
𝑡	Notation of time for a specific day	Days
𝑇𝑖𝑡,𝑇𝑗𝑡	Pairwise temperature observations on day 𝑡 for locations 𝑖,𝑗	°C
𝑇,𝑇,𝑇	Observed, expected and mean daily temperatures	°C
𝑋,𝑋,𝑋	Observed, expected and mean daily maximum temperatures	°C