Analisis Multivariado 1 (Apunte basado en notas de clases del ...

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Distancia de Mahalanobis
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14.1 Descomposicion de la Distancia de Mahalanobis
Let T 2 be the Mahalanobis distance, the MYT decomposition is
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T 2 = T 2 1 +T 2 2/1 + T 2 3/1,2 + T 2 4/1,2,3 + . . . + T 2 p/1,2,3,4,...,p−1
where the first and last term are
T 2
1 = (x1−x2 1 )2
S2 2
Tp/1,2,3,4,...,p−1 1 and =
(xp−x 2 p/1,2,3,4,...,p−1 )2
S2 p/1,2,3,4,··· ,p−1
We define the contribution of variable p to the distance as
Cp =
T 2
p/1,2,3,4,...,p−1
T 2
because all terms are positive, the contribution satisfies
0 ≤ Cp ≤ 1
From now on we will call
If in a certain day happens that T 2 is big, and T 2 p/1,2,3,4,...,p−1 is
close to 1, then the station/variable p is wrong, because
T 2 j/Aj
is small
for all j∈ {1, 2, 3, 4, . . . , p − 1} and Aj ⊆ {1, 2, 3, 4, . . . , j − 1, j + 1, . . . , p − 1}
The p relevant decompositions are
T 2 = T 2 2 +T 2 3/2 + T 2 4/2,3 + T 2 5/2,3,4 + . . . + T 2 1/2,3,4,...,p
T 2 = T 2 1 +T 2 3/1 + T 2 4/1,3 + T 2 5/1,3,4 + . . . + T 2 2/1,3,4,...,p
T 2 = T 2 1 +T 2 2/1 + T 2 4/1,2 + T 2 5/1,2,4 + . . . + T 2 3/1,2,4,...,p
.
T 2 = T 2 1 +T 2 2/1 + T 2 3/1,2 + T 2 4/1,2,3 + . . . + T 2 p−1/1,2,3,4,...,p−2,p
T 2 = T 2 1 +T 2 2/1 + T 2 3/1,2 + T 2 4/1,2,3 + . . . + T 2 p/1,2,3,4,...,p−1
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