Lithuanian Journal of Statistics
Lietuvos statistikos darbai
2011, vol. 50, No 1, pp. xx-yy
2011, 50 t., Nr. 1, xx-yy p.
www.statisticsjournal.lt
ESTIMATES OF PARAMETERS FOR STOCHASTIC LOGISTIC GROWTH LAWS
THROUGH THE MAXIMUM LIKELIHOOD AND THE L1 DISTANCE PROCEDURES
Petras Rupšys
Dept. of Mathematics, Institute of Information Technologies, Lithuanian University of Agriculture.
Address: Universiteto 10, Akademija, Kaunas distr., Lithuania. E-mail: [email protected]
Received:
Revised:
Published:
Abstract. Stochastic logistic type growth models of a single species population have been considered. Six alternative stochastic logistic growth models, the exponential, the Verhulst, the Gompertz, the Mitcherlich, the Bertalanffy, the Richards
were used for modeling of the growth process. The objective was working out a procedure on the estimation of parameters
for all these stochastic logistic growth models. To estimate parameters the maximum likelihood procedure with the local linearization method was applied. As the second alternative approven for the estimate of parameters the L1 distance procedure
was proposed. As an illustrative experience, the actual data to model the height of an individual tree and the EAT in a mouse
were used. Numerical experiments use a Monte Carlo approach. Numerical approximations of the trajectories for the stochastic logistic growth laws were simulated by the Milshtein method. In addition, to test the performance of models: the Akaike’s
Information Criterion, the L1 norm and the efficiency (R2) were used. The results have been implemented in the symbolic
computational language MAPLE.
Keywords: stochastic differential equation, maximum likelihood method, L1 norm, estimate.
1. Introduction
The study of growth is a main problem in ecology, economy, biology and ect. For describing a growth dynamics
we use ordinary stochastic differential equations. This paper discusses the application of dynamic stochastic growth
laws when we model the growth of the height of an individual tree and the growth of the Ehrlich ascities tumour (EAT)
in a mouse. Although the growth of the height and the EAT are complicated processes, the most of current growth models remains deterministic. For most applications the prediction of mean trends is usually associated with logistic growth
models (e.g. the exponential, the Verhulst, the Gompertz, the Bertalanffy, the Richards [1]-[11]).
……………………………………………………………………
In the present paper we discuss the performance of the applied models too. Among these six stochastic logistic
growth models, we expect to select the best. In order to rank the performance of stochastic growth models when apply
the maximum likelihood procedure we’ll use Akaike’s Information Criterion (AIC), which has the form [22]
AIC  2  LOGL  2  NOP , (2)
where LOGL is a maximum value of the maximum likelihood function, NOP is the number of parameters in the
model. In order to rank the performance of stochastic growth models when apply the L1 procedure we’ll use the L1
norm.
As an alternative comparison of the stochastic logistic growth models fitted by the maximum likelihood and L1
procedures we’ll use the coefficient of determination.
Lithuanian Statistical Association, Statistics Lithuania
Lietuvos statistikų sąjunga, Lietuvos statistikos departamentas
ISSN 1392-642X print
ISSN 2029-7262 online
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Estimates of parameters for stochastic logistic growth laws
2. Stochastic logistic growth models
We now consider the stochastic logistic growth model as the Ito type ordinary stochastic differential equation.
Random perturbations consist of both residual variability (measurements error) and individual variability (heterogeneity
between subjects). The individual variability is an intrinsic part of the growth process.
Let us return to the deterministic growth model (1). It is clear that the parameters r and K vary with time. In the
sequel we suppose that the right-hand side of Equation (1) is as a random variable in the form

  xt    
r x(t )  1  
   x(t )W t ,
  K  


(3)
where r and K are constants, W (t ) is a scalar Brownian motion (white noise) and satisfies the following three
conditions:
1) W t 0   0 (with probability 1),
2) for t0  s  t  T the random variable W t   W s  is normally distributed with mean zero and variance t  s ,
3) for t0  s  t  u  v  T the increments W t   W s  and W v   W u  are independent,  is the intensity of
noise.
By using the random perturbations of the form (3), the generalized deterministic logistic growth model (1) can be
rewritten in the stochastic form

  xt    
dx(t )  r x(t )  1  
  dt  x(t )dW t ,
  K  


(4)
xt0   x0 , t0  t  T .
We have simply added a general diffusion term x(t ) to the deterministic growth model (1) with the drift term
r xt 
 xt  
1 

  K 



 
 . A list of the used stochastic logistic growth models is presented in Table 2.


Table 2. List of stochastic logistic growth laws
Model

Parameter

Drift term

Diffusion term
Exponential
1
1
0
rx(t )
x(t )
Verhulst
1
1
1
x(t ) 

rx (t )1 

K 

x(t )
Gompertz
1
 0
1
  

rx(t ) ln 
 x(t ) 
x(t )
Mitscherlich or
monomolecular
0
1
1
x(t ) 

r 1 

K 

x(t )
Bertalanffy
Richards
2
3
1

1
3
1
  1
1
1
xt   3 
r xt 
 
  K  


  xt    
rx (t )1  
 
  K  


2

3 1  
3. Maximum likelihood procedure
……………………………………..
x(t )
x(t )
Petras Rupšys
3
5. Illustrative numerical examples
Let us discuss two numerical examples to illustrate theory established in the previous sections. The height of an individual tree is important component in long-term forest planning systems. In the second example we analyze the growth
of the Ehrlich ascities tumour (EAT) in a mouse [25]. So, two sets of data will be discussed in this section. The first set
consists of 5 sample plots with 890 measurements of the height in pine forests of Lithuania (Fig. 1). The second set
consists of 13 measurements of the EAT in a mouse (Fig. 2). The estimates of parameters, the ranking of performance
of the stochastic logistic growth models (4) when apply the maximum likelihood procedure for the estimate of parameters are summarized in Table 4. Table 5 shows the estimates of parameters and the ranking when we apply the L1 procedure.
Fig. 1. Plot of the height including data from pine forests in
Lithuania
Fig. 2. Plot of the Ehrlich ascities tumour including data from
[25]
………………………………….
6. Conclusions
1. As the summary of the ranking of all the used stochastic growth models we can say that parameter estimations by
the maximum likelihood procedure are generally slower, but provide considerably better precision for the fit. The
best one is the Richards stochastic growth model for the modeling of the growth of the height of an individual tree and
the growth of the Ehrlich ascities tumour (EAT) in a mouse.
2. The L1 distance procedure can be recommended when the number of observations is large and observations are uniformly distributed at all over the age (time) interval.
3. To cover more stochastic models we can change the diffusion term.
4. In order to propose more precise stochastic logistic growth laws, we can use stochastic delay differential equations.
References
1. Verhulst P. F. Notice sur la loi que la population suit dans son accroissement, Curr. Math.Phys., 10, p. 13, 1838.
2. Gompertz B. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value
of life contingencies, Phil. Trans. Roy. Soc., 115, p. 513–585, 1825.
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Estimates of parameters for stochastic logistic growth laws
3. Von Bertalanffy L. A Quantitative theory of organic growth, Human Biol., 10(2), p. 181–213, 1938.
…………………………………….
16. Rupšys P. Stochastic analysis of logistic growth models. In: Proceedings of the Fourth Nordic–Baltic Agrometrics Conference
held in Swedish University of Agricultural Sciences, Department of Biometry and Informatics. Olsson U., Sikk J. (Eds), Report
81, Uppsala, Sweden, p. 127–139, 2003.
……………………………………...
23. Oksendal B. Stochastic Differential Equations. An Introduction with Applications. Second Edition, Springer-Verlag. Berlin 1989.
24. Borodin A., Salminen P. Handbook of Brownian Motion – Facts and formulae. Birkhouser, Basel, 1996.
25. Foryš U., Marciniak-Czochra A. Logistic equations in tumor growth modeling, Int. J. Appl. Math.13, p. 317–325, 2003.
STOCHASTINIO AUGIMO MECHANIZMO SKIRSTINIŲ ĮVERČIAI TAIKANT DIDŽIAUSIO TIKĖTINUMO IR L1
ATSTUMO PROCEDŪRAS
Petras Rupšys
Santrauka. Nagrinėjamas tam tikros veislės populiacijos stochastinis augimo mechanizmo modelis. Augimo proceso
modeliavimui taikomi 6 rūšių modeliai: eksponentinis, Verhulsto, Gompertzo, Mitcherlicho, Bertalanffy ir Richardso. Darbo tikslas –
sukurti procedūrą stochastinio augimo mechanizmo modelio parametrams vertinti. Parametrų vertinimui naudojama didžiausio
tikėtinumo procedūra ir lokalaus ištiesinimo metodas. Kaip alternatyvus parametrų vertinimo metodas buvo pasiūlyta L1 atstumo
procedūra. Iliustracijai naudojami realūs duomenys medžio aukščiui ir pelių pilvo naviko dydžiui modeliuoti. Skaitiniams
bandymams naudojamas Monte-Karlo metodas. Stochastinio augimo mechanizmo skirstinio trajektorijų skaitinės aproksimacijos
buvo modeliuojamos Milshteino metodu. Be to, modelių tinkamumui vertinti naudojamas Akaike informacinis kriterijus, L1 norma ir
modelio efektyvumo rodiklis R2. Skaičiavimai atlikti naudojant simbolinę programavimo kalbą MAPLE.
Reikšminiai žodžiai: stochastinė diferencialinė lygtis, didžiausio tikėtinumo metodas, L1 norma, įvertis.