# How to find the correlation coefficient between two technologies when those are sub correlated?

Suppose I have two power generating technologies, Coal and Oil, of which the generating cost components(total generating cost = Capital cost+Fuel cost+Variable O&M cost+Fixed O&M cost) are correlated in the following manner:

                                         Oil
Fuel           Variable O&M             Fixed O&M
Fuel       0.48              0                       0
Coal    Var O&M       0                 0.7                     0.1
Fixed O&M       0                 0.1                     0.7


Suppose Technology A is COAL and Technology B is OIL. How do I find the correlation between the total costs of the two technologies, that is COAL and OIL (one correlation coefficient) when three categories of costs are already correlated as given above? I have the weights of cost components of each total technology cost and the standard deviation of each cost component of the two technologies as well.

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How do you obtain diagonal values that are not identically $1$? Is this perhaps a covariance matrix rather than a correlation matrix? – whuber Dec 9 '12 at 17:48
It says cross-correlation matrix, but I'm not too sure. Can you kindly refer the page 37 of this file --> awerbuch.com/shimonpages/shimondocs/iea-portfolio.pdf ,maybe you'll get a clear idea than me explaining since my English is not good. Thanks. – Kanch Dec 9 '12 at 17:51
Thank you: I had misunderstood this table. It shows "assumed cross-correlations for the cost streams of existing generation assets." In order for your question to be answerable, you will also need to assume correlations among the different cost streams (within each technology). – whuber Dec 9 '12 at 18:04
Lets assume that there is no correlations among different cost streams within the same technology. – Kanch Dec 9 '12 at 18:15
Mods, can you please forward this to someone who can answer this.. – Kanch Dec 20 '12 at 5:19

Let me update your table to reduce confusion by making different variable names unique:

                                         Oil
Fuel.O          VarO&M.O               FixedO&M.O
Fuel.C     0.48              0                       0
Coal     VarO&M.C     0                 0.7                     0.1
FixedO&M.C     0                 0.1                     0.7


You have given 9 entries of a correlation matrix, but there should be a further 6 subdiagonal elements, being the correlations within Coal and within Oil.

For further simplicity:

$C_f$ Fuel.C
$C_v$ VarO&M.C
$C_x$ FixedO&M.C
$O_f$ Fuel.O
$O_v$ VarO&M.O
$O_x$ FixedO&M.O

$C = C_f + C_v + C_x$

$O = O_f + O_v + O_x$

$$\operatorname{Cov}(C,O) = \operatorname{Cov}(C_f + C_v + C_x,O_f + O_v + O_x)$$

$$= σ_{C_f}⋅σ_{O_f}⋅0.48+ σ_{C_v} ⋅σ_{O_v}⋅0.7+ σ_{C_x}⋅σ_{O_x}⋅0.7\\ + σ_{C_v}⋅σ_{O_x}⋅0.1+ σ_{C_x}⋅σ_{O_v}⋅0.1$$

Using the "no correlation within" specified in comments:

$σ_C= \sqrt{σ_{C_f}^2+ σ_{C_v}^2+ σ_{C_x}^2}$
$σ_O = \sqrt{σ_{O_f}^2+ σ_{O_v}^2+ σ_{O_x}^2}$

$\text{Corr}(C,O)= \text{Cov}(C,O)/ (σ_C⋅σ_O)$

From there, it's impossible to simplify further, but the question states that the remaining quantities $σ_{C_f}, ..., σ_{O_v}$ are all known so it's a simply matter of substitution from here.

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I think those sub-diagonal elements are zero. "Lets assume that there is no correlations among different cost streams within the same technology. – Kanch Dec 9 '12 at 18:15" and can you kindly explain how to calculate σC and σO if those elements are zero. Thank you. – Ryan Apr 20 '13 at 17:21
@Ryan You're right; I didn't see that important information buried in comments. (This is why it's important for posters who update important information in comments to edit it back into the question.). I have edited my answer (and corrected an error in what I had) - given the additional information that's unspecified but stated by the original poster to be known, this should now be a solved problem. (Are you the original poster under a new account?) – Glen_b Apr 20 '13 at 23:44
@Ryan I just had one final read through and noticed an error - where I had $σ_O^2 = \sqrt{}...$ that should have been $σ_O = \sqrt{}...$ (this was caused by initially giving the formulas in terms of the variances and then editing to give in terms of s.d.s). I hope that change doesn't cause any problems. – Glen_b Apr 25 '13 at 22:26
@Glen_b , When we find the Corr(C,O) as stated above, can't it give an answer greater than 1 which is not acceptable for a Correlation coefficient? (In case all S.d.s values are smaller than one say 0.1) – Kanch Aug 7 '13 at 19:57
The only way it's possible to get it to give a correlation greater than one is if there's an error. Can you give some numbers where you think it leads to a correlation greater than 1? – Glen_b Aug 7 '13 at 22:52