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Simultaneous Equation Models
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
To understand the issues involved in estimating simultaneous equations model parameters, a natural starting point is to consider a reduced form solution. In Equations (5.1) and (5.2), the problem becomes a simple one of solving two equations and two unknowns to arrive at reduced forms. Substituting Equation (5.2) for u2 in Equation (5.1) gives () u1=β1Z1+α1X+λ1[β2Z2+α2X+λ2u1+ε2]+ε1
Systems of Linear Equations
Published in James R. Kirkwood, Bessie H. Kirkwood, Elementary Linear Algebra, 2017
James R. Kirkwood, Bessie H. Kirkwood
In Exercises 1 through 4, we give the row reduced form of the augmented matrix for a system of linear equations. Tell whether the system of equations is consistent, and if it is consistent give the solution in standard form.
Systems of Linear Equations
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
In Exercises 1–4, we give the row reduced form of the augmented matrix for a system of linear equations. Tell whether the system of equations is consistent, and if it is consistent give the solution in standard form.(12000−112340−2)(1060101203000100000000000)(100010001)(14000010630000010020)
Newsgroup topic extraction using term-cluster weighting and Pillar K-Means clustering
Published in International Journal of Computers and Applications, 2022
Sigit Adinugroho, Randy C. Wihandika, Putra P. Adikara
In a rank-reduced form, where , the A matrix is approximated by the reduced form of U, S, and V as follow: where is (d is the number of documents) matrix of left singular vectors, is diagonal matrix of k largest singular values in non-decreasing order, and is (w is the number of terms) matrix of right singular vectors. The rank-reduced form of A is expressed as [22]:
Enhancing cyber-physical security in manufacturing through game-theoretic analysis
Published in Cyber-Physical Systems, 2018
Zach DeSmit, Aditya U. Kulkarni, Christian Wernz
Upon inspecting the payoffs for the reduced game, we now see that for the defender, the mixed strategy or tossing an unbiased coin to decide between securing the computer and securing the lathe, strictly dominates the defender’s pure strategy of not securing either system, that is the strategy Similar to the analysis of the strategy space of the attacker, rationality requires both players to ignore the defender’s pure strategy of not securing either system from further consideration, since it is not rationalisable due to the existence of at least one mixed strategy that strictly dominates it. Thus, we can reduce the game further to just considering those mixed strategies for each player that either affects the computer or the lathe. The payoff matrix for the most reduced form of the game is shown in Table 4.
Incorporating the extended theory of planned behavior in a school travel mode choice model: a case study of Shaoxing, China
Published in Transportation Planning and Technology, 2018
Peng Jing, Jing Wang, Long Chen, Qi-fen Zha
In order to examine the complicated interrelationships among the TPB’s latent variables of adults and children’s school travel mode choice with the socioeconomic status variables, an MIMIC model is estimated. In terms of the multivariate regression of indicators on causes, the model implies restrictions of two types: (i) the regression coefficient matrix has a rank of one; and (ii) the residual variance–covariance matrix satisfies a factor analysis model with one common factor. The MIMIC model is in fact a special form of structural equation modeling. The specification of the model is as follows:where equation (1) is the structural equation and equation (2) is the measurement equation. The latent variable vector η is linearly determined, subject to disturbances ζ, by vector of observable exogenous causes x. The latent variable determines, linearly, subject to disturbance ε, a vector of observable endogenous indicators y. Γ and Λ are matrices of unknown parameters to be estimated. The operational implications of the model appear when we solve for the reduced-form relation connecting the observables:where the reduced-form coefficient matrix isand the reduced-form disturbance vector is