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Linear Regression Models
Published in Norman Matloff, Statistical Regression and Classification, 2017
We can also use (2.124) to derive the Law of Total Variance, (1.62). For convenience in present notation, rewrite that equation as Var(Y)=E[Var(Y|X)]+Var[E(Y|X)] $$ Var(Y) = E[Var (Y \vert X)]+ Var [E(Y \vert X)] $$
Interference and Its Impact on System Capacity
Published in Jerry D. Gibson, Mobile Communications Handbook, 2017
where E{Y1}=1−∑i=1M(E{Ri,u2−α}−E{Ri,l2−α}). For variance, the law of total cumulance degenerates to the law of total variance and we have
Global sensitivity analysis in the design of rockfill dams
Published in Jean-Pierre Tournier, Tony Bennett, Johanne Bibeau, Sustainable and Safe Dams Around the World, 2019
Equation 3 determines the contribution of Xm to the variance of Y; a smaller value of (3) implies that Xm plays a greater role in Y. From the law of total variance (Lilburne et al 2009; Sobol 1993), the following relation is obtained DY=DXmEX−mY|Xm+EXmDX−mY|Xm
Global sensitivity analysis based on random variables with interval parameters by metamodel-based optimisation
Published in International Journal of Systems Science: Operations & Logistics, 2018
Sinan Xiao, Zhenzhou Lu, Liyang Xu
The main effect index Si proposed by Sobol' (1993) for an individual input Xi is defined as where E( · ) and V( · ) are the expectation and variance operators, respectively. E(Y|Xi) is the conditional expectation of Y when Xi is fixed. According to the law of total variance, it can be obtained that V(E(Y|Xi)) = V(Y) − E(V(Y|Xi)). V(Y|Xi) is the conditional variance of Ywhen Xi is fixed. E(V(Y|Xi)) denotes the average residual variance of Y when Xi is fixed in its full support. Thus, the partial variance V(E(Y|Xi)) represents the average reduction of output variance when Xi is fixed in its full support. Due to the fact that 0 < V(E(Y|Xi)) < V(Y), Si is the normalisation of V(E(Y|Xi)) in the bound [0,1]. Thus, Si can reflect the effect of the uncertainty of Xi on the output variance V(Y). The main effect index is just a first order index, which cannot reflect the interaction effects among input variables. Thus, the total effect index for Xi is proposed by Homma and Saltelli (1996), and it is defined as where denotes the vector containing all the input variables except Xi. represents the average variance of output when all the input variables except Xi are fixed over their supports, thus, it measures both the individual effect of Xi and the interaction effects between Xi and the other input variables . STi is still the normalisation of in the bound [0,1] since , and it leads to 0 < Si < STi < 1. STi can reflect the effects of both the individual uncertainty of Xi and the uncertainty of the interaction terms between Xi and other input variables on the output variance V(Y). Thus STi can reflect the importance of the input variable more sufficiently, and the total effect index will be mainly considered in the following sections.