| Title: | Analysis of Age-Structured Demographic Models |
|---|---|
| Description: | Construction and analysis of matrix population models in R. |
| Authors: | James Holland Jones [aut] <[email protected]>, Jim Oeppen [ctb] |
| Maintainer: | Hana Sevcikova <[email protected]> |
| License: | GPL (>=2) |
| Version: | 0.6.0 |
| Built: | 2025-03-04 04:32:10 UTC |
| Source: | https://github.com/hanase/demogr |
demogR is a collection of functions to construct and analyze
age-structured population models. For a list of functions, type:
help(package="demogR").
| Package: | demogR |
| Type: | Package |
| Version: | 0.6.0 |
| Date: | 2018-09-13 |
| License: | GPL version 2.0 or newer |
James Holland Jones
Department of Anthropological Sciences
Stanford University
Contributor: Jim Oeppen, Southern Denmark University, Odense, Denmark
Maintainer: Hana Sevcikova <[email protected]>
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
Coale, A.J. 1972. The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press.
Jone, J.H. (2007). demogR: A Package for the Construction and Analysis of Age-structured Demographic Models in R. Journal of Statistical Software, 22(10), 1-28.
Keyfitz, N. 1977. Introduction to the mathematics of populations. 2nd ed. Menlo Park: Addison-Wesley.
Keyfitz, N., and H. Caswell. 2005. Applied mathematical demography. 3rd ed. New York: Springer.
Leslie, P.H. 1945. On the use of matrices in certain population mathematics. Biometrics 33:213-245.
Preston, S.H., P. Heuveline, and F. Guillot. 2001. Demography: Measuring and modeling population processes. Oxford: Blackwell.
Tuljapurkar, S. 1990. Population dynamics in variable environments. Vol. 85, Lecture notes in biomathematics. Berlin: Springer-Veralg.
Calculate the net reproduction number () from an age or
stage-classified demographic projection matrix.
calc.ro(A, N.out = FALSE)calc.ro(A, N.out = FALSE)
A |
A demographic projection matrix |
N.out |
Return the fundamental matrix (N) of the Markov chain? |
Calculates the net reproduction number () from an age or
stage-classified demographic projection matrix by first decomposing
the k x k projection matrix A into two component matrices, T and F. T
collects the transitions between life-cycle stages while F collects
the fertility transitions. For an age-classified Leslie matrix, T
will contain only the sub-diagonal of A and F will contain only the
first row of A. The fundamental matrix is given by ,
where I is a k x k identity matrix. is the leading eigenvalue
of the matrix FN.
If the (default) option N.out=FALSE is used, the net reproduction number is returned as a single value. If N.out=TRUE, the returned value is a list of two items:
ro |
Net reproduction number |
N |
Fundamental matrix |
...
James Holland Jones
Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation, Second edition. Sinauer, Sunderland, Massachusetts, USA.
see leslie.matrix
## Create a Leslie matrix Px <- c(0.77, 0.95, 0.98, 0.97) Fx <- c(0,0,1,1.2,1) L <- odiag(Px,-1) L[1,] <- Fx ## Calculate R_0 calc.ro(L) ## look at the fundamental matrix calc.ro(L, N.out=TRUE)## Create a Leslie matrix Px <- c(0.77, 0.95, 0.98, 0.97) Fx <- c(0,0,1,1.2,1) L <- odiag(Px,-1) L[1,] <- Fx ## Calculate R_0 calc.ro(L) ## look at the fundamental matrix calc.ro(L, N.out=TRUE)
Coale-Demeny regional model life tables
cdmltw(sex = "F") cdmlte(sex = "F") cdmlts(sex = "F") cdmltn(sex = "F")cdmltw(sex = "F") cdmlte(sex = "F") cdmlts(sex = "F") cdmltn(sex = "F")
sex |
female or male life tables. Defaults to 'F' (female). |
Provides region-specific model life tables for 25 different levels of mortality, indexed by life expectancy at age 10 (with 1 being the lowest and 25 being the highest life expectancy at age 10).
The regions are West (cdmltw), East (cdmlte), South (cdmlts), and North (cdmltn).
A list containing two 21 element vectors, one 25 element vector and eight 25 x 21 matrices: 25 mortality levels by 21 age classes. The eight matrices contain the standard columns of a life table for each mortality level:
age |
the 21 age classes |
width |
widths of the 21 age classes |
e10 |
the 25 mortality levels as life expectancy at age 10 |
lx |
survival probabilities to exact age x |
nqx |
probabilities of death between ages x and x+n |
nax |
person-years lived by those dying between ages x and x+n |
ndx |
proportion of all deaths occurring between ages x and x+n |
nLx |
person-years lived in the interval x to x+n |
nmx |
central death rate in the interval x to x+n |
Tx |
person-years of life remaining to the cohort at age x |
ex |
life expectancy at age x |
Kenneth Wachter, Department of Demography, University of California, Berkeley.
Modified by James Holland Jones, Department of Anthropological Sciences, Stanford University
Corrected by Jim Oeppen, Southern Denmark University, Odense, Denmark
Coale, A., P. Demeny, and B. Vaughn. 1983. Regional model life tables and stable populations. 2nd ed. New York: Academic Press.
## Generate West model life table family for females Wf <- cdmltw() ## Plot survivorship for the 25 levels plot(Wf$age,Wf$lx[1,], type="l", xlab="Age", ylab="Survivorship") for(i in 2:25) lines(Wf$age, Wf$lx[i,])## Generate West model life table family for females Wf <- cdmltw() ## Plot survivorship for the 25 levels plot(Wf$age,Wf$lx[1,], type="l", xlab="Age", ylab="Survivorship") for(i in 2:25) lines(Wf$age, Wf$lx[i,])
Utility used by life.table to graduate the person-years lived
by those dying in the interval by the method of Coale and Demeny.
coale(b1, b4, nMx)coale(b1, b4, nMx)
b1 |
two element vector of regression coefficients for graduating 1a0 provided in Coale et al. (1983) |
b4 |
two element vector of regression coefficients for graduating 4a1 provided in Coale et al. (1983) |
nMx |
period central death rates: |
Utility function called by life.table.
The first two values (age classes 0-1 and 1-5) of the nax column of a period life table.
Coale, A., P. Demeny, and B. Vaughn. 1983. Regional model life tables and stable populations. 2nd ed. New York: Academic Press.
data(goodman) vlt1 <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx, type="cd"))data(goodman) vlt1 <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx, type="cd"))
Dummy function called in life.table.
cohort(width12)cohort(width12)
width12 |
width of the first two age classes |
This is essentially a place-holder function used when passing the
option type="cohort" to life.table.
A two element vector representing the first two values of the nax column of the life table.
data(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.ltdata(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.lt
Calculates the asymptotic growth rate and related quantities from a demographic projection matrix.
eigen.analysis(A)eigen.analysis(A)
A |
demographic projection matrix |
Calculates the asymptotic growth rate (lambda) of a population described by demographic projection matrix A. The asymptotic growth rate of the population is given by the dominant eigenvalue of the projection matrix. By the Perron-Frobenius Theorem, this eigenvalue is guaranteed to be real, positive and strictly greater than all the other eigenvalues if the matrix A is non-negative, irreducible, and primitive (for details see Caswell (2001)).
Also calculates the damping ratio (rho), eigenvalue sensitivities, eigenvalue elasticities, the stable age distribution (for the communicating parts of the life cycle), and scaled reproductive values.
The damping ratio is the ratio of the dominant eigenvalue and the absolute value of the second eigenvalue. rho is a measure of the rate of convergence to the stable age-distribution. A population characterized by damping ratio rho will converge asymptotically to the stable age distribution exponentially with rate at least as fast as log(rho). Clearly, a population already at or very near the stable age distribution will converge faster, but rho provides an upper bound.
The eigenvalue sensitivities are the partial derivatives of lambda
with respect to a perturbation in matrix element . The
sensitivities measure the selection gradient on the life-cycle
(Lande 1982). The eigenvalue elasticities are scaled to be
proportional sensitivities of lambda to a perturbation in .
Elasticities have a number of desirable properties including, their
sum across all life-cycle transitions is unity and the sum of the
elasticities of all incoming arcs to a life-cycle stage must equal
the sum of all outgoing arcs (van Groenendael et al 1994).
The stable age distribution is normalized to represent the proportion in each of the communicating age classes. If the population is characterized by post-reproductive survival (and hence age classes that do not communicate with the rest of the life cycle graph), then other methods should be used to calculate to stable distribution. For example, from classic stable population theory, we know that the stable age distribution of the population c(x) is given by the relationship:
c(x) = b l(x) exp(-r*x)
where b is the gross birth rate, l(x) is survivorship to age x and r is the rate of increase of the population (=log(lambda)). See Coale (1972) or Preston et al. (2001) for details.
The age-specific reproductive values are normalized so that the reproductive value of the first age class is unity. Problems associated with post-reproductive survival are irrelevant for reproductive value since the reproductive value of post-reproductive individuals is, by definition, zero.
A list with six components:
lambda1 |
the asymptotic growth rate (dominant eigenvalue) of A |
rho |
damping ratio of A |
sensitivities |
eigenvalue sensitivities of A |
elasticities |
eigenvalue elasticities of A |
stable.age |
stable age distribution of A |
repro.value |
reproductive values of A |
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
Coale, A.J. 1972. The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press.
Lande, R. A. 1982. A quantitative genetic theory of life history evolution. Ecology 63:607-615.
van Groenendael, J., H. De Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: Evaluating life history pathways in population projection matrices. Ecology 75 (8):2410-2415.
data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) eav <- eigen.analysis(A) eau <- eigen.analysis(B) ## compare rates of increase eav$lambda1 eau$lambda1 ## compare stable age distributions for Venezuela and the USA op <- par(no.readonly = TRUE) age <- seq(0,50,by=5) par(mfrow=c(1,2)) barplot(eav$stable.age, names.arg=age, horiz=TRUE, col="grey", ylab="Age") barplot(eau$stable.age, names.arg=age[-11], horiz=TRUE, col="blue", ylab="Age", xlim=c(0,0.2)) par(op) ## Compare reproductive values for Venezuela and the USA plot(age, eav$repro.value, type="l", xlab="Age", ylab="Reproductive Value") lines(age[-11], eau$repro.value, lty=2) legend("topright", c("Venezuela (1965)", "USA (1967)"), lty=c(1,2))data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) eav <- eigen.analysis(A) eau <- eigen.analysis(B) ## compare rates of increase eav$lambda1 eau$lambda1 ## compare stable age distributions for Venezuela and the USA op <- par(no.readonly = TRUE) age <- seq(0,50,by=5) par(mfrow=c(1,2)) barplot(eav$stable.age, names.arg=age, horiz=TRUE, col="grey", ylab="Age") barplot(eau$stable.age, names.arg=age[-11], horiz=TRUE, col="blue", ylab="Age", xlim=c(0,0.2)) par(op) ## Compare reproductive values for Venezuela and the USA plot(age, eav$repro.value, type="l", xlab="Age", ylab="Reproductive Value") lines(age[-11], eau$repro.value, lty=2) legend("topright", c("Venezuela (1965)", "USA (1967)"), lty=c(1,2))
Calculates the partial derivatives of the eigenvalue elasticities of a demographic projection matrix.
elassens(A, k, l)elassens(A, k, l)
A |
a demographic projection matrix |
k |
row index of the transition that the partial is calculated with respect to |
l |
column index of the transition that the partial is calculated with respect to |
See Caswell (2001) section 9.7.1.
A matrix of class "leslie.matrix".
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
secder, leslie.matrix,
eigen.analysis
data(goodman) mlt <- with(goodman, life.table(x=age, nDx=mad.nDx, nKx=mad.nKx)) mx <- goodman$mad.bx/goodman$mad.nKx ## make the Leslie matrix mad <- leslie.matrix(lx=mlt$lx, mx=mx) ## now calculate the sensitivities of the elasticitities with respect ##to infant survival es21 <- elassens(mad,2,1)data(goodman) mlt <- with(goodman, life.table(x=age, nDx=mad.nDx, nKx=mad.nKx)) mx <- goodman$mad.bx/goodman$mad.nKx ## make the Leslie matrix mad <- leslie.matrix(lx=mlt$lx, mx=mx) ## now calculate the sensitivities of the elasticitities with respect ##to infant survival es21 <- elassens(mad,2,1)
Calculates the second derivatives of the dominant eigenvalue of a square matrix, A, with respect to all non-zero elements of A.
fullsecder(A)fullsecder(A)
A |
a demographic projection matrix |
Produces a matrix the columns of which are re-shaped matrices of second derivatives of the dominant eigenvalue of the projection matrix A with respect to each non-zero element in A.
See Caswell (1996, 2001) for details on second derivatives of the dominant eigenvalue.
A square matrix. If A is a Leslie matrix of rank k, then the maximum rank of the resulting matrix is 2k-1 (since there are at most k non-zero fertilities and k-1 survival probabilities in a k x k Leslie matrix).
Caswell, H. 1996. Second derivatives of population growth rate: Calculation and applications. Ecology 77 (3):870-879.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
secder, elassens,
eigen.analysis, stoch.sens
## eigenvalue second derivatives of the US projection matrix from 1967 data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) fs <- fullsecder(usa) ## plot the survival cross-second derivatives of lambda with respect ## to infant survival plot( seq(0,40,by=5), fs["21",10:18], type="l", xlab="Age (j)", ylab=expression(paste(partialdiff^2 , lambda, "/", partialdiff, P[1] , partialdiff, P[j]))) abline(h=0,lty=3)## eigenvalue second derivatives of the US projection matrix from 1967 data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) fs <- fullsecder(usa) ## plot the survival cross-second derivatives of lambda with respect ## to infant survival plot( seq(0,40,by=5), fs["21",10:18], type="l", xlab="Age (j)", ylab=expression(paste(partialdiff^2 , lambda, "/", partialdiff, P[1] , partialdiff, P[j]))) abline(h=0,lty=3)
Calculates the generation time for an age or stage-classified demographic projection matrix.
gen.time(A, peryear = 5)gen.time(A, peryear = 5)
A |
demographic projection matrix |
peryear |
width of the age classes |
Calculates the generation time (T) for an age or stage-classified demographic projection matrix using the identity
where is the net reproduction number and r is the intrinsic rate
of increase (=log(lambda)).
Generation time is the amount of time that it takes a typical female
to produce offspring or, equivalently, the amount of time it
takes a population growing with instantaneous rate r to increase by
a factor of .
The generation time implied by the demographic projection matrix.
Calls function calc.ro, which calculates from the
fundamental matrix of the Markov transition matrix (Caswell 2001).
Keyfitz, N., and H. Caswell. 2005. Applied mathematical demography. 3rd ed. New York: Springer.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
Preston, S.H., P. Heuveline, and F. Guillot. 2001. Demography: Measuring and modeling population processes. Oxford: Blackwell.
## compare generation times from two populations with very different ## R_0 values data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) gen.time(A) gen.time(B)## compare generation times from two populations with very different ## R_0 values data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) gen.time(A) gen.time(B)
Aggregate vital registration data for female from Venezuela (1965), Madagascar (1966), and the United States (1967) used in Goodman, Keyfitz, and Pullam (1974).
data(goodman)data(goodman)
A data frame with 19 observations on the following 10 variables.
ageage classes
ven.nKxmid-year population structure for Venezuela
ven.nDxenumerated deaths for Venezuela
ven.bxenumerated births for Venezuela
mad.nKxmid-year population structure for Madagascar
mad.nDxenumerated deaths for Madagascar
mad.bxenumerated births for Madagascar
usa.nKxmid-year population structure for the United States
usa.nDxenumerated deaths for the United States
usa.bxenumerated births for the United States
Vital event data tabulated by Keyfitz and Flieger (1990) and used by Goodman et al. (1974) to illustrate their kinship frequency calculations. The three female populations represent tremendous diversity in age-specific demographic schedules. Venezuela (1965) is characterized by low mortality and high fertility; Madagascar (1966) is characterized by very high mortality and fertility, while the United States (1967) is characterized by low mortality and low fertility.
Keyfitz and Flieger (1990), Goodman et al. (1974)
Keyfitz, N., and W. Flieger. 1990. World population growth and aging: Demographic trends in the late twentieth century. Chicago: University of Chicago Press.
Goodman, L. A., N. Keyfitz, and T. W. Pullum. 1974. Family formation and the frequency of various kinship relationships. Theoretical Population Biology 5:1-27.
data(goodman) ## Plot age-specific mortality rates for the three populations with(goodman, plot(age, usa.nDx/usa.nKx, type="l", log="y", xlab="Age", ylab="nMx")) with(goodman, lines(age, mad.nDx/mad.nKx, lty=2)) with(goodman, lines(age, ven.nDx/ven.nKx, lty=3)) legend("bottomright", c("USA","Madagascar","Venezuela"), lty=c(1,2,3))data(goodman) ## Plot age-specific mortality rates for the three populations with(goodman, plot(age, usa.nDx/usa.nKx, type="l", log="y", xlab="Age", ylab="nMx")) with(goodman, lines(age, mad.nDx/mad.nKx, lty=2)) with(goodman, lines(age, ven.nDx/ven.nKx, lty=3)) legend("bottomright", c("USA","Madagascar","Venezuela"), lty=c(1,2,3))
Utility used by life.table to graduate the person-years lived
by those dying in the interval by the method of Keyfitz and Flieger
(1990).
keyfitz(b0, nMx)keyfitz(b0, nMx)
b0 |
two element vector of regression coefficients provided in
Keyfitz and Flieger (1990). Default value: |
nMx |
period central death rates: |
Utility function called by life.table.
The first two values (age classes 0-1 and 1-5) of the nax column of a period life table.
Keyfitz, N., and W. Flieger. 1990. World population growth and aging: Demographic trends in the late twentieth century. Chicago: University of Chicago Press.
data(goodman) ## default type="kf" vlt <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx))data(goodman) ## default type="kf" vlt <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx))
Calculates the logarithm of the stochastic growth rate using Tuljapurkar's second order approximation for independent and identically distributed environments.
lams(aseq, n = 5)lams(aseq, n = 5)
aseq |
sequence of matrices with each matrix given as a re-shaped column of aseq |
n |
width of the projection interval/age-class |
Uses Tuljapurkar's second order approximation for independent and identically distributed (i.i.d.) environments.
The long-run growth rate for the population with projection matrices given by aseq.
Tuljapurkar, S. 1990. Population dynamics in variable environments. Edited by S. A. Levin. Vol. 85, Lecture notes in biomathematics. Berlin: Springer-Veralg.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
## simulate two matrices: good year/bad year ## beta distributed survival, gamma fertility px1 <- rbeta(5,9,1) px2 <- rbeta(5,7,3) mx1 <- c(0,rgamma(5,10,10)) mx2 <- c(0,rgamma(5,7,10)) ## good year matrix A1 <- odiag(px1,-1) A1[1,] <- mx1 ## bad year matrix A2 <- odiag(px2,-1) A2[1,] <- mx2 aseq <- cbind(matrix(A1,nr=36,nc=1), matrix(A2,nr=36,nc=1)) lams(aseq)## simulate two matrices: good year/bad year ## beta distributed survival, gamma fertility px1 <- rbeta(5,9,1) px2 <- rbeta(5,7,3) mx1 <- c(0,rgamma(5,10,10)) mx2 <- c(0,rgamma(5,7,10)) ## good year matrix A1 <- odiag(px1,-1) A1[1,] <- mx1 ## bad year matrix A2 <- odiag(px2,-1) A2[1,] <- mx2 aseq <- cbind(matrix(A1,nr=36,nc=1), matrix(A2,nr=36,nc=1)) lams(aseq)
Generates a Leslie matrix for demographic projection from vectors of age-specific cumulative survival and fertility.
leslie.matrix(lx, mx, L = TRUE, peryear = 5, one.sex = TRUE, SRB = 1.05, infant.class = TRUE)leslie.matrix(lx, mx, L = TRUE, peryear = 5, one.sex = TRUE, SRB = 1.05, infant.class = TRUE)
lx |
vector of either age-specific cumulative survival or person-years lived in the interval |
mx |
age-specific fertility rates |
L |
logical; if 'TRUE', |
peryear |
Multiplier for fertility. Defaults to |
one.sex |
logical; If 'TRUE', fertility rates will be divided by (1+SRB). |
SRB |
sex ratio at birth. Defaults to |
infant.class |
logical; 'TRUE' if |
Constructs a k x k age-classified demographic projection matrix with age-specific survival probabilities along the sub-diagonal and age-specific fertilities along the first row of the matrix.
lx and mx are assumed to be of the same
length. The resulting matrix is truncated to insure that there are
no post-reproductive classes. This is important for ensuring
irreducibility of the resulting matrix.
If mx is longer than lx, mx is trucated to be
the same length as lx. If lx is longer than
mx, a warning is issed and lx is truncated to be the
same length as mx.
Fertility is assumed to be birth-flow (Caswell 2001). That is,
breeding is assumed to be continuous and the individual elements of
the first row of the Leslie matrix are averaged over successive
age-classes. Fertility rates are typically given in annualized
form. If this is the case and the age-classes are wider than one
year, then peryear can be used to appropriately scale up the
annual values.
The default behavior is to use person-years lived in the interval as
the survival measure. If infant.class=TRUE, lx is taken to
have a value for the infant age class (i.e., a shorter class width
than the other elements of lx. What is done when there is an
infant class depends on what the values in lx represent. If
L=TRUE, then the first two values of lx are combined to form
the total person-years for the first age-class in the Leslie
matrix. Human demographic data from abridged life tables typically
come with age classes x = 0, 1, 5, 10, ... Thus, combining the
person-years for the first two age classes gives an initial age
class of the correct width. If infant.class=TRUE and
L=FALSE, the second element of lx is deleted.
Creating a Leslie matrix from other forms of non-standard early
age-classes can be accomplished by pre-processing lx and
using the option infant.class=FALSE.
The human sex ratio at birth (male births/female births) is
remarkably close to SRB=1.05 across a wide range of populations and
this is the default value for SRB.
The resulting matrix has class "leslie.matrix". This class is not used extensively but will be in future development.
A k x k age-classified demographic projection matrix with class "leslie.matrix".
Keyfitz, N. 1977. Introduction to the mathematics of populations. 2nd ed. Menlo Park: Addison-Wesley.
Preston, S.H., P. Heuveline, and F. Guillot. 2001. Demography: Measuring and modeling population processes. Oxford: Blackwell.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
## Construct lx and mx values data(goodman) mlt <- with(goodman, life.table(x=age, nDx=mad.nDx, nKx=mad.nKx)) mx <- goodman$mad.bx/goodman$mad.nKx ## Now make the Leslie matrix mad <- leslie.matrix(lx=mlt$nLx, mx=mx) ## Might as well do something with it too! eigen.analysis(mad)## Construct lx and mx values data(goodman) mlt <- with(goodman, life.table(x=age, nDx=mad.nDx, nKx=mad.nKx)) mx <- goodman$mad.bx/goodman$mad.nKx ## Now make the Leslie matrix mad <- leslie.matrix(lx=mlt$nLx, mx=mx) ## Might as well do something with it too! eigen.analysis(mad)
A utility for calculating the first row of a Leslie matrix from vectors of interval survival probabilities and age-specific fertility rates.
leslie.row1(mx, px, L = NULL, SRB = 1.05, peryear = 5, one.sex = TRUE)leslie.row1(mx, px, L = NULL, SRB = 1.05, peryear = 5, one.sex = TRUE)
mx |
vector of age-specific fertilities (length=k) |
px |
vector of interval survival probabilities (length=k-1) |
L |
person-years lived in the first interval; default
|
SRB |
sex ratio at birth; default |
peryear |
width of the age-class/projection interval |
one.sex |
should each element be reduced by a factor
1/(1+SRB)? default |
Calculates the first-row entries for a Leslie matrix. This utility is particularly useful for constructing simulated Leslie matrices.
A vector of length k, where k is the rank of the projection matrix A.
## simulate survival values from a beta(9,1) distribution px <- rbeta(10,9,1) ## simulate fertility values from a gamma(2,10) distribution with age ## at first reproduction = 10 mx <- c(0,0,0,rgamma(8,2,10)) A <- odiag(px,-1) Fx <- leslie.row1(mx,px) A[1,] <- Fx## simulate survival values from a beta(9,1) distribution px <- rbeta(10,9,1) ## simulate fertility values from a gamma(2,10) distribution with age ## at first reproduction = 10 mx <- c(0,0,0,rgamma(8,2,10)) A <- odiag(px,-1) Fx <- leslie.row1(mx,px) A[1,] <- Fx
Construct either a period or cohort life table from enumerated deaths and mid-interval population estimates.
life.table(x, nDx, nKx, b0 = c(0.07, 1.7), b1 = c(0.053, 2.8), b4 = c(1.522, 1.518), type = "kf", nxx = 0, iwidth = 5, width12 = c(1, 4))life.table(x, nDx, nKx, b0 = c(0.07, 1.7), b1 = c(0.053, 2.8), b4 = c(1.522, 1.518), type = "kf", nxx = 0, iwidth = 5, width12 = c(1, 4))
x |
age at the beginning of the age classes of the life table |
nDx |
deaths |
nKx |
population size |
b0 |
coefficients used in Keyfitz-Flieger graduation |
b1 |
first set of coefficients used in Coale-Demeny graduation |
b4 |
second set of coefficients used in Coale-Demeny graduation |
type |
type of life table calculation: "kf", "cd", or "cohort". Default is "kf". |
nxx |
person-years lived by those dying in the last (possibly
open) age-class. If |
iwidth |
width of the age intervals. Default |
width12 |
width of the first two age classes. Default
|
Constructs a period or cohort life tables from enumerated deaths and
mid-interval population sizes (period) or enumerated deaths and
person-years at risk (cohort). x, nDx, and nKx must all
the be same length.
There are currently three options for life table construction. The
first two are for the construction of period life tables. They differ
only in the way that person-years lived by those dying in the first
two intervals are handled. For type="kf", the default, the
first two values of nax estimated using Keyfitz and Fleiger's (1990)
regression method. For type="cd", Coale and Demeny's method
(1983) is used. The Coale-Demeny method uses different coefficients
depending on the level of early mortality. As a result, this method
may work better for high-mortality populations.
The third type of life table is a cohort life table, for which the conversion from mortality rates to probabilities of death is unnecessary, so the nax column of the life table is of limited interest.
A dataframe with nine columns:
x |
age at the beginning of the interval |
nax |
person-years lived by those dying in the interval x to x+n |
nMx |
period central death rate |
nqx |
probability of death between ages x and x+n |
lx |
probability of survival to exact age x |
ndx |
proportion of deaths occurring between ages x and x+n |
nLx |
person-years lived in the interval x to x+n |
Tx |
person-years of life left in the cohort at age x |
ex |
life expectancy at age x |
Keyfitz, N. 1977. Introduction to the mathematics of populations. 2nd ed. Menlo Park: Addison-Wesley.
Coale, A., P. Demeny, and B. Vaughn. 1983. Regional model life tables and stable populations. 2nd ed. New York: Academic Press.
Keyfitz, N., and W. Flieger. 1990. World population growth and aging: Demographic trends in the late twentieth century. Chicago: University of Chicago Press.
Preston, S.H., P. Heuveline, and F. Guillot. 2001. Demography: Measuring and modeling population processes. Oxford: Blackwell.
data(goodman) ## default type="kf" vlt <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ## compare nax values for cd vs kf life tables vlt1 <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx, type="cd")) vlt$nax[1:2] vlt1$nax[1:2] ## now construct a cohort life table for Himalayan thar, ## (Hemitargus jemlahicus) data(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.ltdata(goodman) ## default type="kf" vlt <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ## compare nax values for cd vs kf life tables vlt1 <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx, type="cd")) vlt$nax[1:2] vlt1$nax[1:2] ## now construct a cohort life table for Himalayan thar, ## (Hemitargus jemlahicus) data(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.lt
Calculate and plot loop elasticities for a demographic projection matrix .
loop.elas(A, draw.plot = TRUE, peryear = 5, xlab = "Loop Elasticity", ylab = "Age", xlim = c(0, (maxe + 0.02)), ...)loop.elas(A, draw.plot = TRUE, peryear = 5, xlab = "Loop Elasticity", ylab = "Age", xlim = c(0, (maxe + 0.02)), ...)
A |
demographic projection matrix |
draw.plot |
should a plot be drawn? default:
|
peryear |
width of the age-class/projection interval |
xlab |
x-axis label; default |
ylab |
y-axis label; default |
xlim |
plotting limits for x-axis |
... |
arguments to be passed to/from other methods |
Calculates and plots the life-cycle decomposition of van Groenendael et al (1994).
A vector of length k (the rank of the projection matrix A) giving the loop elasticities for each of the loops in the life-cycle graph.
Van Groenendael, J., H. De Kroon, S. Kalisz, and S. Tuljapurkar. 1994. Loop analysis: Evaluating life history pathways in population projection matrices. Ecology 75 (8):2410-2415.
data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) eav <- eigen.analysis(A) eau <- eigen.analysis(B) ## Compare the loop elasticities of Venezuela (1965) and the USA (1967) le.usa <- loop.elas(A) le.usa <- loop.elas(B)data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) eav <- eigen.analysis(A) eau <- eigen.analysis(B) ## Compare the loop elasticities of Venezuela (1965) and the USA (1967) le.usa <- loop.elas(A) le.usa <- loop.elas(B)
Utility that takes a matrix n x m and stacks its columns to form a column vector of length mn.
m2v(A)m2v(A)
A |
an m x n matrix |
For an m x n input matrix, returns a m*n x 1 matrix (i.e., a column vector).
Meant to capture the MATLAB functionality of the colon operator, A(:)
A <- matrix(rnorm(9), nr=3, nc=3) m2v(A)A <- matrix(rnorm(9), nr=3, nc=3) m2v(A)
Utility function that creates a square matrix with a vector on the off-diagonal or extracts the specified off-diagonal vector.
odiag(A, at = 0)odiag(A, at = 0)
A |
either a vector or a matrix |
at |
which diagonal? |
If the first argument is a vector of length k, the result will be a
square matrix of rank k+at with the provided vector along the 'at'
diagonal. Positive values for 'at' place the vector above the
diagonal, negative values below the diagonal, and at=0 (the
default) places the vector on the diagonal.
A vector if argument A is a matrix and a matrix if A is
a vector.
odiag is meant to capture some of the functionality of the
MATLAB function diag().
## Construct a matrix from a vector ## random survival probabilities with mean 0.9 and variance 0.0082 y <- rbeta(4,9,1) A <- odiag(y,-1) ## add fertilities F <- c(0,rep(1,4)) A[1,] <- F ## Extract a vector from a matrix A <- matrix(rnorm(25), nr=5, nc=5) odiag(A,2)## Construct a matrix from a vector ## random survival probabilities with mean 0.9 and variance 0.0082 y <- rbeta(4,9,1) A <- odiag(y,-1) ## add fertilities F <- c(0,rep(1,4)) A[1,] <- F ## Extract a vector from a matrix A <- matrix(rnorm(25), nr=5, nc=5) odiag(A,2)
Plots Leslie matrices and other objects of class "leslie.matrix".
## S3 method for class 'leslie.matrix' plot(x, y=NULL, ..., main="", sub="", ann=TRUE, xlim=NULL, ylim=NULL, axes=TRUE, col=c("black","grey"), lwd=2, xlab="Age", ylab="Sensitivity", peryear=5 )## S3 method for class 'leslie.matrix' plot(x, y=NULL, ..., main="", sub="", ann=TRUE, xlim=NULL, ylim=NULL, axes=TRUE, col=c("black","grey"), lwd=2, xlab="Age", ylab="Sensitivity", peryear=5 )
x |
demographic projection matrix |
y |
y axis values; NULL |
... |
arguments to be passed to/from other methods |
main |
an overall title for the plot |
sub |
a subtitle for the plot |
ann |
logical; if TRUE, annotate plot |
xlim |
plotting limits for x-axis; if |
ylim |
plotting limits for y-axis; if |
axes |
logical; if TRUE, draw axes and box |
col |
vector (length 2) of line colors for the plot; default
|
lwd |
thickness of the plotted lines; default |
peryear |
width of the age-class/projection interval; default |
xlab |
x-axis label; default |
ylab |
y-axis label; default |
Plots the subdiagonal and the first row of an object of class leslie.matrix on common axes. Probably most useful for plotting either eigenvalue sensitivities or elasticities of a projection matrix.
NULL
data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) ea.ven <- eigen.analysis(A) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) ea.usa <- eigen.analysis(B) ## Compare sensitivities for Venezuela and the USA plot(ea.ven$sens, main="Venezuela") plot(ea.usa$sens, main="USA")data(goodman) ven <- with(goodman, life.table(x=age, nKx=ven.nKx, nDx=ven.nDx)) ven.mx <- with(goodman, ven.bx/ven.nKx) A <- leslie.matrix(lx=ven$nLx,mx=ven.mx) ea.ven <- eigen.analysis(A) usa <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) usa.mx <- with(goodman, usa.bx/usa.nKx) B <- leslie.matrix(lx=usa$nLx,mx=usa.mx) ea.usa <- eigen.analysis(B) ## Compare sensitivities for Venezuela and the USA plot(ea.ven$sens, main="Venezuela") plot(ea.usa$sens, main="USA")
Projects an population vector tmax intervals by pre-multiplication with a Leslie matrix.
project.leslie(A, no, tmax, pop.sum = FALSE)project.leslie(A, no, tmax, pop.sum = FALSE)
A |
a k x k projection matrix |
no |
a k x 1 population vector |
tmax |
number of time steps to project the vector |
pop.sum |
logical; If 'TRUE', the age-classes of the projected population are summed, yielding a single total population vector |
Takes an initial population vector, no, and pre-multiplies by the demographic projection matrix, A, tmax times. This projection will be tmax*n years into the future, where n is the width of the age-classes in the Leslie matrix, A.
If pop.sum=FALSE (the default), the value will be a k x
tmax+1 matrix. The first column of the matrix is no and each
subsequent column represents the population structure at time
step 1, 2, ..., tmax.
If pop.sum=TRUE, the value will be a vector of length
tmax+1, where each element of the vector is the total
population at time t=0, 1, ..., tmax.
data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) ## initial population is the age structure in 1967 ## this includes the number of 0-1 year-olds and the number of 1-4 ## year-olds, which need to be combined into a single 5-year wide ## element no <- goodman$usa.nKx[3:11] no <- c(sum(goodman$usa.nKx[1:2]),no)/1e6 N <- project.leslie(usa,no,tmax=20) plot( seq(1967,2067,by=5), apply(N,2,sum), type="l", xlab="Years", ylab="Projected Population (millions)") title("Projected Total Population Size (Ages 0-45) of the United States") ## Note that this is a lousy projection since it assumes constant ## demographic rates and a closed population and only accounts for ## ages 0-45data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) ## initial population is the age structure in 1967 ## this includes the number of 0-1 year-olds and the number of 1-4 ## year-olds, which need to be combined into a single 5-year wide ## element no <- goodman$usa.nKx[3:11] no <- c(sum(goodman$usa.nKx[1:2]),no)/1e6 N <- project.leslie(usa,no,tmax=20) plot( seq(1967,2067,by=5), apply(N,2,sum), type="l", xlab="Years", ylab="Projected Population (millions)") title("Projected Total Population Size (Ages 0-45) of the United States") ## Note that this is a lousy projection since it assumes constant ## demographic rates and a closed population and only accounts for ## ages 0-45
Calculates the second derivatives of the dominant eigenvalue of the demographic projection matrix for all non-zero transitions with respect to one specified transition.
secder(A, k, l)secder(A, k, l)
A |
demographic projection matrix |
k |
row index for the specified transition |
l |
column index for the specified transition |
See Caswell (1996, 2001) for details on second derivatives of the dominant eigenvalue.
A square matrix of the same rank as A where each element is the
second derivative of the dominant eigenvalue of A, .
The eigenvalue second derivatives are essential for calculating both
perturbation analyses of the eigenvalue elasticities and stochastic
sensitivities. secder is used in functions to calculate both
these quantities.
Caswell, H. 1996. Second derivatives of population growth rate: Calculation and applications. Ecology 77 (3):870-879.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
fullsecder, elassens,
eigen.analysis, stoch.sens
## eigenvalue second derivatives of the US projection matrix from 1967 ## with respect to infant survival data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) sd21 <- secder(usa,2,1)## eigenvalue second derivatives of the US projection matrix from 1967 ## with respect to infant survival data(goodman) ult <- with(goodman, life.table(x=age, nKx=usa.nKx, nDx=usa.nDx)) mx <- goodman$usa.bx/goodman$usa.nKx usa <- leslie.matrix(lx=ult$nLx,mx=mx) sd21 <- secder(usa,2,1)
Calculates the sensitivity of the stochastic growth rate to perturbations in the mean demographic projection matrix.
stoch.sens(env, amat, k)stoch.sens(env, amat, k)
env |
environmental sequence |
amat |
matrices describing the population dynamics in the different environmental states, organized as columns of a matrix |
k |
rank of the projection matrices |
See Caswell (2001, section 14.4) for details.
A list with two elements:
sensitivities |
sensitivities of the stochastic growth rate |
elasticities |
elasticities of the stochastic growth rate |
This is essentially a translation of Caswell's (2001, section 14.4.1.1) MATLAB code.
Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.
Haridas, C. V., and S. Tuljapurkar. 2005. Elasticities in variable environments: Properties and implications. American Naturalist 166 (4):481-495.
Tuljapurkar, S. 1990. Population dynamics in variable environments. Vol. 85, Lecture notes in biomathematics. Berlin: Springer-Veralg.
Tuljapurkar, S., and C. V. Haridas. 2006. Temporal autocorrelation and stochastic population growth. Ecology Letters 9 (3):324-334.
## Simulate an i.i.d. sequence of 3 environmental states env <- floor(runif(100,0,3))+1 px1 <- rbeta(4,9.5,0.5) px2 <- rbeta(4,7.5,2.5) px3 <- rbeta(4,5,5) mx <- c(0,rgamma(4,5,10)) A1 <- odiag(px1,-1) A2 <- odiag(px2,-1) A3 <- odiag(px3,-1) A1[1,] <- leslie.row1(mx,px1) A2[1,] <- leslie.row1(mx,px2) A3[1,] <- leslie.row1(mx,px3) amat <- cbind(matrix(A1,nr=25), matrix(A2,nr=25), matrix(A3,nr=25)) stoch.sens(env,amat,k=5)## Simulate an i.i.d. sequence of 3 environmental states env <- floor(runif(100,0,3))+1 px1 <- rbeta(4,9.5,0.5) px2 <- rbeta(4,7.5,2.5) px3 <- rbeta(4,5,5) mx <- c(0,rgamma(4,5,10)) A1 <- odiag(px1,-1) A2 <- odiag(px2,-1) A3 <- odiag(px3,-1) A1[1,] <- leslie.row1(mx,px1) A2[1,] <- leslie.row1(mx,px2) A3[1,] <- leslie.row1(mx,px3) amat <- cbind(matrix(A1,nr=25), matrix(A2,nr=25), matrix(A3,nr=25)) stoch.sens(env,amat,k=5)
Caughley's (1966) survival data on Himalayan thar (Hemitargus jemlahicus).
data(thar)data(thar)
A data frame with 13 observations on the following 3 variables.
ageage class
countthe number of individuals at the start of the age class
deathsobserved deaths in the age class
Caughley (1966) smoothed the counts within age classes and this data set used those reported smoothed counts.
Caughley, G. 1966. Mortality patterns in mammals. Ecology 47:906-918.
data(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.ltdata(thar) thar.lt <- with(thar, life.table(x=age, nDx=deaths, nKx=count, type="cohort", iwidth=1, width12=c(1,1))) thar.lt