／var／logmarcus chiu

Distance Covariance／Covariation - Properties

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  • 𝑑𝐶𝑜𝑣(𝑋,𝑌) = 𝑑𝐶𝑜𝑣(𝑌,𝑋)
  • 𝑑𝐶𝑜𝑣(𝑎1 + 𝑏1𝐶1𝑋, 𝑎2 + 𝑏2𝐶2𝑌) = |𝑏1𝑏2| 𝑑𝐶𝑜𝑣(𝑋,𝑌) for all:
    • constant vectors 𝑎1 and 𝑎2
    • scalars 𝑏1 and 𝑏2
    • orthonormal matrices 𝐶1 and 𝐶2
  • 𝑑𝐶𝑜𝑣(𝑋1 + 𝑋2, 𝑌1 + 𝑌2) ≤ 𝑑𝐶𝑜𝑣(𝑋1, 𝑌1) + 𝑑𝐶𝑜𝑣(𝑋2, 𝑌2)
    • if the random vectors (𝑋1, 𝑌1) and (𝑋2, 𝑌2) are independent then:
  • 𝑑𝐶𝑜𝑣(𝑋1 + 𝑋2, 𝑌1 + 𝑌2) = 𝑑𝐶𝑜𝑣(𝑋1, 𝑌1) + 𝑑𝐶𝑜𝑣(𝑋2, 𝑌2)
    • If either:
      • 𝑋1and 𝑌1 are both constants
      • 𝑋2 and 𝑌2 are both constants
      • 𝑋1, 𝑌1, 𝑋2, 𝑌2 are mutually independent
  • 𝑑𝐶𝑜𝑣(𝑌,𝑋) = 0, iff 𝑋 and 𝑌 are mutually independent