The
answer given by mathematicians and other "autistic quants" leads to a
"correlation matrix" and thus
a formula used by econometrists:
R2 = c' * Rxx−1 * c where: c' is the transpose vector of
correlations involving the dependent variable and the independent variables, Rxx−1 is the inverse matrix of the
correlations among the independent variables only and c is the vector of
correlations involving the dependent variable and the independent variables.
So far,
so good. The main pitfall and flaw behind this formula lays on the fact that at
least one of the "series" or "variables" to be analyzed has
to be dependent. Furthermore, the formula is not conmutative.
But what
if we want to deploy a conmutative formula assuming total independence about
the birth of the n series targeted to study... The solution is not far away.
The Pearson product has the answer...
The
original formula was thought for 2 series, it reads:
Covariance
(ab)/(σa * σb) = ρ Even if we want a Pearson product for a lonesome
series, the formula holds on tight by respecting its attribute...
Covariance (aa)/(σa * σa) = ρ Both the upper factor and the lower
factor are always SECOND DEGREE.
if we have 3 series:
Covariance (abc) = Σ{ (ai – A)*(bi – B)*(ci – C)}^(2/3)
--------------------------------------------------
n -1
if we have k series:
Covariance (abc...k) = Σ {(ai – A)*(bi – B)*(ci – C)*...*(ki – K)}^(2/k)
-------------------------------------------------------------
n -1
where ai, bi ci and ki are data at the same time, A, B, C and K are the averages of each series, k is the
number of series and n is the number
of summands.
The trick behind this algorithm is to keep the accurate “+” or “-” after
the conversion of every subproduct. Noteworthy, the conversion has to be
deployed on the absolute values...And the coupled correlations have to fall on
the same grounds (i.e. Strong direct relation, Strong inverse relation or
strong independent relation)
So therefore, the correlation for both examples would remain:
ρ(a,b,c) = Covariance (abc)
-------------------------
(σa*σb*σc)^2/3
ρ(a,b,c...k) =
Covariance (abc...k)
---------------------------
(σa*σb*σc*...σk)^2/k
Part 2 of this essay will render nummerical examples based upon all of
the above. An alternate approach would be a fictional sample or time series
built by 1 unit out of the independent variables.
Sources: RiskCenter, Investopedia, Wikipedia,
EdX.org, Coursera.org
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