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Product Matrix

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Journal Article
Demography (1985) 22 (3): 455–468.
Published: 01 August 1985
... 1985 Product Matrix Vital Rate Eventual Population Size Theoretical Population Biology Relative Population Size References Chrystal G. ( 1959 ). Algebra. An Elementary Textbook, Vol. II . 6th Edition New York : Chelsea Publishing Company . Cohen J. E. ( 1977...
Journal Article
Demography (2020) 57 (2): 779–797.
Published: 24 March 2020
...Robert Schoen Abstract Cross-product ratios (αs), which are structurally analogous to odds ratios, are statistically sound and demographically meaningful measures. Assuming constant cross-product ratios in the elements of a matrix of multistate transition probabilities provides a new basis both...
Includes: Supplementary data
Journal Article
Demography (2003) 40 (4): 621–635.
Published: 01 November 2003
... transform in Eq. (13) and avoids the need to assume constant mortality. 2. A Rank 1 matrix is a matrix whose rows and columns are proportional and hence can be written as a constant times the product of a column vector and a row vector. For human populations, Leslie matrices, when raised to a sufficiently...
Journal Article
Demography (1966) 3 (1): 19–34.
Published: 01 March 1966
... = ~ aijbj k, i=1 The hypothesis which we propose to study herein is whether or not the matrix product AB = C. In case this hypothesis is true, then we can see from the rules of matrix multiplication that ones which would be found if the associa- tion between a person's own occupation and his grandfather's...
Journal Article
Demography (1986) 23 (1): 91–104.
Published: 01 February 1986
..., causative matrices always have rows summing to one. It is easily shown that the rows of the inverse of a stochastic matrix add to one, and it follows that the product matrix of two matrices with unitary row sums also possesses unit sums. C, however, is not necessarily stochastic in that individual elements...
Journal Article
Demography (2006) 43 (3): 553–568.
Published: 01 August 2006
...-state-speci c rates of transfer can be arrayed in a matrix, but it is usually not possible to nd a solution to the matrix version of Eq. (1). The reason is that matrix multiplication is generally not commutative, so a product of exponentiated matrices is not the exponential of a simple sum...
Journal Article
Demography (1978) 15 (4): 559–569.
Published: 01 November 1978
... . For readers without matrix algebra, this equation tells us to interpret the product matrix M t = (mt,l1 m t,12) mt,21 mt,22 as follows: the upper left element mt,l1 gives the number of persons in the first age group at time t per person in the first age group initially [i.e., for an initial popu- lation Xu...
Journal Article
Demography (1969) 6 (2): 185–221.
Published: 01 May 1969
... . Sankhya , 6 , 93 – 96 . Pollard A. H. ( 1948 ). The measurement of reproductivity . Journal of the Institute of Actuaries , 74 , 288 – 318 . Pollard J. H. ( 1966 ). On the use of the direct matrix product in analyzing certain stochastic population models . Biometrika , 53...
Journal Article
Demography (2013) 50 (5): 1615–1640.
Published: 17 September 2013
... the transpose of x . The symbol diag( x ) denotes the matrix with the vector x on the diagonal and zeros elsewhere. The vector e is a vector of ones, and the vector e i is the i th unit vector—that is, the vector with a 1 in the i th location and zeros elsewhere. The Hadamard product (or element...
FIGURES | View All (9)
Includes: Supplementary data
Journal Article
Demography (1968) 5 (1): 382–409.
Published: 01 March 1968
... Matrix Product in Analyzing Certain Stochastic Population Models . Biometrika , LIII , 397 – 416 . 30 Keyfitz , N. ( 1964 ). The Population Projection as a Matrix Operator . Demography , I , 62 – 62 . 32 Leslie , P. H. ( 1945 ). On the Use of Matrices in Population...
Journal Article
Demography (1986) 23 (2): 247–259.
Published: 01 May 1986
... males. When r = - .2, the corresponding r-equilibrium population is given by F = M = [I, 1.25/, 1.56/, 1.95 The BMMR model is r-productive at r = -.2 if the mating rule and the birth matrix imply that this population produces at least .8/ female and .8/ male newborns in the next petiod. The r...
Journal Article
Demography (1964) 1 (1): 56–73.
Published: 01 March 1964
... = 900 400 64 DEMOGRAPHY = ID - XII; .32167 .68154 .12110 (since the determinant of a matrix prod- uct is the product of the determinants of its factors) o o o o .97203 D = 0 o M = .98610 o where Xl, X2, X3 are the roots of the de- terminantal equation 1M - XII = O. For since M and D are similar...
Journal Article
Demography (2023) 60 (6): 1675–1688.
Published: 01 December 2023
... on the transition probabilities as l x   = l x   −   1 P x   −   1   = l α ∏ k   =   α x   −   1 P k , (1) where the product operator ∏ k   =   α x   −   1 P k invokes matrix products. This survivorship function...
FIGURES
Includes: Supplementary data
Journal Article
Demography (1971) 8 (4): 441–450.
Published: 01 November 1971
... in the appendix leads to the approximation 'Yj ~ !Fj(1 - P j ) ; j = 0, 1, , m. The recurrence relations (1) to (5) are easily represented in the framework of direct matrix products. Following Pollard (1966), we write Uj = var Y/ P, = EX/, P, = EY/, 'Yj = cov (Y/, X Tj = var X/ = P j(1 - Pj), (6) whenever Z...
Journal Article
Demography (1964) 1 (1): 194–211.
Published: 01 March 1964
... of variables. RESEARCH DESIGN A matrix of product-moment correla- tion coefficientswas constructed for all 42 variables as a basic, preliminary, analytic step. The matrix was then subjected to a multiple-stage factor analysis. 4 Throughout the study, the following factor analytic techniques were utilized...
Journal Article
Demography (1969) 6 (3): 287–299.
Published: 01 August 1969
..., t to t + w, where w is the width of the age intervals used to define the age structure and the fertility and mor- tality rates, then the projected age struc- ture of the female population at t + w is given by the product M X F(O), where M is an appropriate matrix constructed using f and 8...
Journal Article
Demography (1966) 3 (1): 259–275.
Published: 01 March 1966
.... Latent Root Real Root Quadratic Formula Matrix Algebra Characteristic Matrix References 1. Leslie P. H. ( 1945 ). On the Use of Matrices in Certain Population Mathematics . Biometriks , XXXIII , 183 – 212 . 10.1093/biomet/33.3.183 2 For example, see N. Keyfitz, “Mathematical...
Journal Article
Demography (1968) 5 (1): 449–459.
Published: 01 March 1968
...-the proportion of stores in an enumeration district which stock the product. There are five independent vari- ables. Income.-In the last section we hypoth- esized an income effect although we saw that for peculiar statistical reasons that are related to difficulties with matrix inversion income turned out...
Journal Article
Demography (1979) 16 (3): 481–484.
Published: 01 August 1979
...) For all s groups, the parity progression ratios can be simultaneously calculated by F(p) = exp [Kjlog, A p]} where A = A* (2)1., K = K*@ I., I. is an s X s identity matrix and @ is the outer or Kronecker product. A consistent estimate of the covariance matrix of F(p) is given by VF = O,K D, -lA V A' D...
Journal Article
Demography (2008) 45 (1): 157–171.
Published: 01 February 2008
... for this estimation namely, the matrix product integral is dif cult to implement. For general cases, explicit expressions of variance of the estimators are cumbersome. Because of these dif culties in implementation, only limited applications of MSLT models have appeared in the literature. To overcome these dif...