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Leslie Matrices are tables used for population projection.A transition matrix is a table with rows and columns showing the expected number of people who END up in the state with the label on the ROW per person at the START in the state with the label on the COLUMN.ALeslie matrix is a special case of a transition matrix in which the states correspond to age groups.We begin by defining a transition matrix.Although the choice of assumptions about the future course of fertility, mortal ity, marriage, divorce, and immigration is fraught with problems, the methods for using those assumptions to calculate future popu lation sizes and age distributions are very well-developed and sat isfactory.Much of the mathematics of projection was developed treating time and age as continuous variables, but projections on comput ers generally are done with populations split into age groups and with discrete steps of time.The choice of credible assump tions about vital rates is as much a part of the art and science of projection as are the formulas we use to implement the calcula tions.Sophisticated projections treat a population classified by many characteristics including race, ethnicity, marital status, income, education, and locality as well as sex and age.Current thinking about plausible assumptions for predicting populations around the world is presented in a recent book called Beyond Six Billion edited by Bongaarts and Bulatao (2000).Of f icial presentations of projections are often filled with disclaimers cautioning the reader that projections are not predictions.It largely failed to predict the continuing trend toward lower mortality in industrialized countries.But the basic ideas are well-illustrated by simpler projections which focus on a single sex and on all races and ethnicities together and which subdivide the population only by age.When demographers project a distribution of marital status, parity, educational level, or other variables into the future, they use general transition ma trices.TRANSITION MATRICES 111 The kind of calculations we shall be doing are closely related to the subject of Markov chains in probability theory.Generally speaking, however, we shall be projecting expected numbers of people rather than probabilities, and that makes the demographic theory slightly different from the theory of Markov chains.They do not tell us what the world will be like but only what the world would be like if a particular set of stated assumptions about future vital rates turned out to be true.Here we study the discrete versions of projection, which use matrices and project populations one step at a time.are called "Leslie matrices". It was P. H. Leslie who published a description of their use in 1945. The same theory was developed a few years earlier by H. Bernardelli and by E. G. Lewis, but their work did not become widely known until Leslie published his. Leslie matrices are a special case of "transition matrices".Such disclaimers, however, are disingenuous.5.1.

Leslie Matrices are tables used for population projection. Of
f
icial presentations of projections are often filled with disclaimers
cautioning the reader that projections are not predictions. They
do not tell us what the world will be like but only what the world
would be like if a particular set of stated assumptions about future
vital rates turned out to be true. The assumptions may or may
not bear any relation to what actually happens.
Such disclaimers, however, are disingenuous. Projection is not
just a game with computers and pieces of paper. We do projec
tions for a purpose, and that purpose is to foresee, as best we can,
the future state of the population. The choice of credible assump
tions about vital rates is as much a part of the art and science of
projection as are the formulas we use to implement the calcula
tions. Current thinking about plausible assumptions for predicting
populations around the world is presented in a recent book called
Beyond Six Billion edited by Bongaarts and Bulatao (2000).
Over recent history, the record of demographers at guessing
future vital rates for use in projection has not been good. The de
mographic community as a whole failed to predict the baby boom
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CHAPTER 5. POPULATION PROJECTION
of the 1950s and 1960s, and then it failed to predict the baby lull
of the 1970s and 1980s. It largely failed to predict the continuing
trend toward lower mortality in industrialized countries. These
failures show that we do not yet understand the mechanisms that
drive demographic change. We need deeper theories with better
predictive power.
Although demographers have a poor record of prediction, their
record is probably better than the records of economists, weather
forecasters, and those who bet on football games. Although the
choice of assumptions about the future course of fertility, mortal
ity, marriage, divorce, and immigration is fraught with problems,
the methods for using those assumptions to calculate future popu
lation sizes and age distributions are very well-developed and sat
isfactory. In this chapter we study such methods of calculation.
Much of the mathematics of projection was developed treating
time and age as continuous variables, but projections on comput
ers generally are done with populations split into age groups and
with discrete steps of time. Here we study the discrete versions of
projection, which use matrices and project populations one step
at a time.
Sophisticated projections treat a population classified by many
characteristics including race, ethnicity, marital status, income,
education, and locality as well as sex and age. But the basic ideas
are well-illustrated by simpler projections which focus on a single
sex and on all races and ethnicities together and which subdivide
the population only by age. We shall concentrate here on this
simple cases. Matrices used to project the size and age distri
bution of a population forward through time. are called “Leslie
matrices”. It was P. H. Leslie who published a description of their
use in 1945. The same theory was developed a few years earlier
by H. Bernardelli and by E. G. Lewis, but their work did not
become widely known until Leslie published his. Leslie matrices
are a special case of “transition matrices”. When demographers
project a distribution of marital status, parity, educational level,
or other variables into the future, they use general transition ma
trices. When they project age structure, they use the special kind
of transition matrices that Leslie defined.
5.1. TRANSITION MATRICES
111
The kind of calculations we shall be doing are closely related
to the subject of Markov chains in probability theory. Generally
speaking, however, we shall be projecting expected numbers of
people rather than probabilities, and that makes the demographic
theory slightly different from the theory of Markov chains.
We begin by defining a transition matrix.
A transition matrix is a table with rows and columns
showing the expected number of people
who
END up in the state with the label on the ROW
per person
at the START in the state with the label on the COLUMN.
ALeslie matrix is a special case of a transition matrix in which
the states correspond to age groups. With Leslie matrices, the
processes of transition are surviving and giving birth. The Leslie
matrix describes a “one-step transition”. We project the popula
tion forward one step at a time. The time between START and
END, the projection step, should be equal to the width n of all
the age groups.
The fact that the step size has to equal the age group width is
an important point. For projection, we generally pick one sex, usu
ally females, and we divide the female population into age groups
of width n. This width may be one year, five years, fifteen years,
or some other convenient number. If we are using age groups that
are one year wide, then we have to project forward one year at a
time, so to project ten years into the future requires ten projection
steps. If we are using five-year age groups, then we have to project
forward five years at a time, so to project ten years into the future
would require only two projection steps.
For our application, we shall assume a closed population. This
assumption can be relaxed in more advanced applications. People
enter the population only by being born to members already in
the population, and people leave it only by dying. Our projection
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CHAPTER 5. POPULATION PROJECTION
treats birth as a possible transition along with survival. That
means that the people who END up in some state may not be
the same people who START in any one of the states. They may,
instead, be the babies of the people who start in the various states.
We are concerned with the expected numbers in the state for the
row per person in the state for the column, however the people are
channeled there. وملخصصص ترجمه مع شرحححح