From: pfreed@rp.CSIRO.AU ()
Subject: Re: Population Modelling (Amateur attempt)
Keywords: AR modelling, Indian Population
Organization: CSIRO Division of Radiophysics/Australia Telescope National Facility
Date: Tue, 28 Jul 1992 07:05:55 GMT

It seems you only have data for 8 years. To obtain your
AR coefficients I presume you have either used (1) a
auto-correlation technique, and used Levinson's algorithm
to get the AR coeff's. or (2) used a covariance technique
and used a Cholesky type inverter to obtain the coefficients.
Either way, the variance in the correlation estimates will
be very high due to the small size of your data set, and the
resulting AR model will itself display large error.
I guess what I'm saying, is if you want to fit an AR(3) model,
you will need a bit more raw data to do your least squares fit.In article <1992Jul14.012645.15216@talon.ucs.orst.edu>, rajkumar@ece.orst.edu (Rajkumar Vedam) writes:
> 
> The following is a beginner's attempt at modelling
> the Indian population, and a prelude to suggesting
> control policies. I have not read any literature
> on population modelling, and so this attempt can be
> described at best as "pseudo-scientific." 
> 
> 
> Data Used:
> ----------
> 
> Growth of Population (1911-1981) (From "India 1988-89",
> published by the director publications division, Ministry
> of I & B, Govt. of India, New Delhi-110 001)
> ------------------------------------------
> Year		Count		  Label
> ------------------------------------------	
> 1911		25,20,93,390      y(1)
> 1921		25,13,21,213      y(2)
> 1931		27,89,77,238      y(3)
> 1941		31,86,60,580      y(4)
> 1951		36,10,88,090      y(5)
> 1961		43,92,34,771      y(6)
> 1971		54,81,59,652      y(7)
> 1981		68,51,84,692      y(8)
> ------------------------------------------
> 
> 
> Method:
> -------
> 
> We can attempt to fit Auto-Regressive (AR) models
> of an appropriate order to the above data. i.e., we 
> prescribe a model as:
> 
> y(k)=a(1)y(k-1) + a(2)y(k-2) +...+a(n)y(k-n)
> 
> where a(i) = parameters to be identified, i=1,2,..,n
> 	n = specified model order
>       y(.) = measured data at discrete time steps k
> 	k = discrete time step = [1,2,3,4,5,6,7,8,9,10,11,..]
> 
> Results:
> --------
> 
> (A) 	As a first attempt, a 3rd order Least Squares (LS) fit
> 	was attempted. However, this model was found to be biased
> 	and made poor predictions even within the data set.
> 
> (B)	Next, a 4th order LS fit was attempted. The following model
> 	was obtained:
> 
> 	y(k)=1.4075y(k-1) - 0.1444y(k-2) + 0.8449y(k-3) - 1.0293y(k-4)
> 
> 	This model has discrete roots at the following locations:
> 
> 	-1.8788
> 	-0.0859 +/- 0.9189i (complex conjugates)
> 	0.6432
> 
> Discussion
> ----------
> 
> 	People working in Systems and Control will note that 
> 	this model exhibits one unstable mode, corresponding 
> 	to the root at -1.8788
> 
> 	In layman terms, this root corresponds to an exponential
> 	growth of the population.
> 
> Predictions obtained with this model:
> -------------------------------------
> 
> 	It can be verified that for k=5, the model
> 	uses the data y(4),y(3),y(2) and y(1) and
> 	correctly predicts y(5). Similarly, we can
> 	check that it predicts y(6),y(7) and y(8)
> 	reasonably well. 
> 
> 	We now use this model to predict beyond the
> 	data set:
> 
> 	---------------------------------
> 	Label	Year		Estimate
> 	---------------------------------
> 	y(9)	1991		88,46,70,697
> 	y(10)	2001	      1,15,72,50,025
> 	y(11)	2011	      1,51,57,48,535
>         y(12)	2021	      2,00,84,71,640
> 	--------------------------------	
> 
> Notes:
> ------
> 
> 	(1) The official Indian census figure released for 1991
> 	indicates a population of about 85,40,00,000
> 
> 	(2) The reason for the discrepancy could be either
> 	the model order is not correct, or more
> 	plausibly, the model structure is inaccurate.
> 
> 	(3) If the latter hypothesis is true, then we can
> 	conjecture an ARMA model, where the MA corresponds
> 	to the Moving Average effect caused by an exogenous
> 	variable such as birth control policies. (Does anybody
> 	know which agency in India can supply figures such as
> 	# of condoms sold in a particular region, etc. ?)
> 
> 	(4) Thus, without reckoning the effect of the control
> 	variable, these figures are at best, a projection
> 	of CURRENT GROWTH rates. Control can very definitely
> 	alter this profile (O.K. If it is found	controllable).
> 
> 
> Can a 5th order model do a better job at prediction ?
> -----------------------------------------------------
> 
> 	It may be verified that one obtains the following
> 	LS model:
> 
>  y(k)=1.2269y(k-1) + 0 y(k-2) + 0.3216 y(k-3) + 0 y(k-4) -0.3709 y(k-5)
> 
> 	The parameter estimates 0 must be a warning indicator
> 	that the model is overparameterized.
> 
> 	However, proceeding as above, we get the following
> 	discrete time roots:
> 
> 	-1.2857
> 	0.1201 +/- 0.7652i (complex conjugates)
> 	-0.79
> 	0.6089
> 
> 	Very clearly, this model also indicates instability.
> 
> 	The model predicts well within the data set. The following
> 	future predictions were obtained:
> 	----------------------------
> 	y(9)	1991	86,37,11,930   -> closer to 850 million than 884 mil.
> 	y(10)	2001  1,10,20,37,827
> 	y(11)	2011  1,40,95,18,908
> 	y(12)	2021  1,80,37,77,115
> 	y(13)   2031  2,31,33,09,814
> 	----------------------------	
> 
> 	The same comments as before hold regarding CURRENT GROWTH
> 	rates prediction.
> 
> 
> 	In my future work, I shall attempt predictive
> 	control methods using short-term birth control
> 	policies - such as # of contraceptive devices
> 	sold in the country - to stabilize the model.
> 	Such studies might indicate that certain #s of
> 	contraceptive devices must be sold if short-term
> 	stabilization is desired.
> 
> 	Your comments, criticism and input (sic) is welcome.
> 
> --------------------------------
> Rajkumar Vedam
> 
> "A friend of mine at Purdue
>  said - here's the ultimate clue
>  my studies shows
>  that the universe grows
>  as Xdot = Ax + Bu"
> 
> Aftermath:
> 
> "He linearized to the left,
>  he linearized to the right,
>  till nothing was right,
>  and nothing was left !"
> 
> Moral: GO NONLINEAR !!


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