In order to estimate color difference, it is possible that we calculate CIELAB ∆E using its three components: ∆L, ∆a and ∆b by using the following formula: ∆E = sqrt(∆L^2 + ∆a^2 + ∆b^2)
However, I would like to also estimate the standard error (SE) or standard deviation (SD) of ∆E based on the SE or SD of its components (∆L, ∆a and ∆b). How can I do so?
I believe I cannot simply apply the same formulae as used in the ∆E calculation, right?
Let DE = y, DL = x1, Da = x2, Db = x3, D stands for the delta, then the function can be denoted as y= f(x1,x2,x3)
y= sqrt(x1^2 + x2^2 + x3^2)
se(y) denote std. deviation of y, s1, s2 and s3 are std. dev of the input variables x1, x2 and x3 respectively.
se(y) = sqrt(gf(X)'*S*gf(X)), here gf(X) stands for the gradient of the function f evaluated at X = (x1, x2, x3), i.e the vector ( 3 by 1) of the partial derievatives of f evaluated at X, the symbol gf(X) ' stands for transpose of vector gf(X) and S is the variance covariance matrix of the input variables x1, x2, x3.
In case the situation is simplified such that there are no co variances among the three inputs then S = diag( s1^2, s2^2, s3^2 ), the se(y) in presence of this simplification has the following expression:
se(y) = sqrt( ( (x1^2 × s1^2) + (x2^2 × s2^2) + (x3^2 × s3^2) ) / (x1^2 + x2^2 + x3^2 ) )
You can look into the topic of propagation of errors for more details about derievation of the general expression, it involves linearization of the function f , in neighborhood of the experimental point of interest X, using the taylor series expansion formula and then applying the statistical formulas for expectation and variance on the linearized form to obtain the quadratic form and from it its squared root as the expression for std. Deviation of function f.
This topic is explained in great details in the ACOL Series book : Measurement , Statistics and Computation by Mc Cormick and Roach.
What Debopam Ghosh says is quite true. But you also need a distribution to do much with the se(y) plus I believe that you need the se's of the delta(xi) which begs the question. I would be tempted to look this link:
https://www.amazon.com/s?k=dental+statistics&i=stripbooks&ref=nb_sb_noss_2 and this link before embarking on the above computations:
https://www.google.com/search?q=Statistics+of+dental+whitening&rlz=1C1CHBF_enUS847US847&oq=Statistics+of+dental+whitening&aqs=chrome..69i57j33.23386j1j4&sourceid=chrome&ie=UTF-8
Best wishes, D. Booth
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics