Growth Models with Oblique Asymptote

. A class of smooth functions which can be used as regression models for modelling phenomena requiring an oblique asymptote is analyzed. These types of models were deﬁned as a product of a linear function and some well known growth models. In addition to their increasing character with an oblique asymptote, the resulting models provide curves with a single inﬂection point


Introduction
Growth curve modelling using functions with an oblique asymptote and one or several inflection points are of interest in practice.Indeed, in economy smooth convex increasing curves with an oblique asymptote were considered in [30] to establish a relationship between household income and expenditure.In hydrology, for fitting the rating curves, it could be useful to have growth functions with one or more than one inflection point (see the U.S. Geological Survey's National Streamflow Information Program web page for more details).Also modelling with a varying carrying capacity leads to growth curve with more than one inflection point (see [16,17,29]).
In this paper we extend our previous study about the already mentioned models [4,5,6].We modify them to obtain models with an oblique asymptote while keeping the single inflection point property.The structure of the present paper is as follows.In Section 2 we start with a general study of our modified model and obtain its main properties.In Section 3, using the analysis done in the preceding section, we modify some growth functions having a horizontal asymptote to obtain models having an increasing linear asymptote.Finally in Section 4 we present some numerical examples on synthetic data.

Main Results
Let us start with a general growth model f (t), defined and twice continuously differentiable on an interval (T 0 , +∞), and having the following basic properties: • f (t) ≥ 0 is an increasing function on (T 0 , +∞), • f has one and only one inflection point on (T 0 , +∞) at (t * , f * = f (t * )), Let us assume that the second derivative of this growth functions can be decomposed as follows where Remember that the Michaelis-Menten, Richards (generalized Logistic), Gompertz, Logistic, and Bridge models given in Table 1 have these properties, see [4].
In order to realize our goal of obtaining a growth function with an oblique asymptote, we multiply the growth function f (t) by a linear increasing function given by m(t) = pt + q with p > 0 and q ≥ 0 to obtain Moreover the zero of m(t), given by −q/p is required to be much less than t * , the abscissa of the inflection point of f (t), then −q/p < t * .We are going to consider F (t) for t > max {−q/p, T 0 } = T 0 .Then we have Hence F (t) is a positive increasing function for t > T 0 .Also, from (2.1), and it follows that Math.Model.Anal., 18(2):204-218, 2013.
Richards (1) f Moreover, suppose we can write where g(t) > 0 for t ∈ (T 0 , +∞).Then Q(t) > 0 for t ∈ (T 0 , t * ], and Since we have to show that F (t) has one and only one inflection point, it remains to prove that the sign of Q(t) changes only once on (t * , +∞).Some conditions on the expression of Q(t) which ensure this fact are given in the next two theorems.
Theorem 1.Let Q(t) be a function defined by where B > 0 and D > 0 are two positive real numbers and γ ≥ 1.If there exists a positive real number t * ≥ 0 such that Q(t * ) ≥ 0, then there exists a unique T * > t * such that (2.4) Proof.Let us suppose that there exists a t * ≥ 0 such that Q(t * ) > 0. Under the assumptions on D and γ, we also have For the first derivative of Q(t), we have In the above equation the linear factor in brackets is strictly increasing since B and D are strictly positive.So to determine the sign of the term between braces we have to study two cases.
Remark.We can allow D = 0 in Theorem 1 and the conclusion holds under the following conditions : In this case, to ensure that the condition (2.4) holds we must have B < C , because otherwise for all t ≥ t * we have Q(t) ≥ Q(t * ) > 0.
Theorem 2. Let Q(t) be a function defined by where B > 0 and D ≥ 0 are two positive real numbers and γ > 0. Suppose there exists a real number t * such that Q(t * ) > 0, then there exists a unique (2.5) Proof.Let us assume that there exists t * ∈ R such that Q(t * ) > 0. Under the assumptions on B and γ we also have For the first derivative of Q(t), we have In the above equation the linear factor in brackets is strictly decreasing since B, D, and γ are strictly positive (or a constant if D = 0).Moreover, the exponential function t → e γt is strictly increasing with lim t→−∞ e γt = 0 and lim t→+∞ e γt = +∞.Thus, there exists one and only one τ ∈ R such that Q (1) (τ ) = 0 and then Hence the function Q(t) is strictly increasing for t ≤ τ and strictly decreasing for t ≥ τ .Since Q(t * ) > 0, there exists a unique T * > t * such that the condition (2.5) holds.
From the two theorems, we have (2.6) The function F (t) is a positive strictly increasing function, convex on (T 0 , T * ), concave on (T * , +∞), with a single inflection point at (T * , F * = F (T * )).Moreover the oblique asymptote is given by y = [pt + q]f +∞ .
In the next section, we apply these previous ideas on the six models mentioned in Table 1.This study does not cover all the existing models used in practice (see [11,12,14,23,34,35]), but any other model with a sigmoidal behavior can be modified in a similar way with the use of Theorem 1 or Theorem 2.

Models
The basic models are given in Table 1.We use them and consider the product to obtain the modified models listed in Table 2. Let us observe that the multiplicative constants, ω f or a, in the basic models are included in the parameters p and q of the modified models, and f +∞ = 1.Using the decomposition (2.3), we look for a unique T * > t * such that to obtain the desired shape for the graph of F (t) for the values of the parameters (p, q, θ) in the feasible set Θ specified in the last two columns of the Table 2.The graphs of f and F for all these models look like the graphs given in Figure 1.
Table 2. Growth models with an oblique asymptote y = pt + q.
Graphic representation of a growth model f (t) and its modified model F (t) with their respective asymptotes y = 1 and y = pt + q and inflection points (t * , f * ) and (T * , F * ).

Michaelis
To ensure the increasing character of this function with the existence of an inflection point we assume that k > 0, c ≥ 1 and 0 ≤ ω 0 < 1.The Michaelis-Menten function is well defined when t ≥ T 0 = 0. Moreover we have and f (2) .
The function is defined as long as 1 − be −ct ≥ 0. In this case we have From (2.2) and (3.1), we can write

It follows that
Two sets of conditions on the parameters are allowed here to ensure the increasing character of Richards function with the existence of an inflection point.
Then, Theorem 2 with B = cp, D = cmbp > 0 and γ = c > 0 ensures in both cases the existence of a unique T * > t * such that the condition (3.2) holds.

Gompertz model
It is well known that the Gompertz model is a limiting case of Richards (1) This function is well defined on R and T 0 = −∞.It is a strictly increasing function with one inflection point for b > 0 and c > 0. We have Hence, from (2.2) and (3.1), we get and g(t) = 1.Thus, Theorem 2 with B = cp, D = bcp and α = c > 0 provides that there exists a unique T * > t * such that the condition (3.2) holds.

Logistic model f
Obviously, the Logistic model is a particular case of the Richards (2) model with the parameter m equal to −1 and b is replaced by −b.It is a well defined function on R and T 0 = −∞.We must have b > 0 and c > 0 to have one inflection point at t * = ln(b) c .Since Hence, from (2.2) and (3.1), it follows that

Hence we set
From Theorem 2 with B = cp, D = bcp, and α = c > 0 we have the existence of a unique T * > t * such that the condition (3.2) holds.

Bridge model f
The Bridge function is well defined when t ≥ T 0 = 0.Moreover, we assume that b ≥ 1 and m > 0 to have an increasing function with one inflection point.We have f B (t).

Numerical Examples
In this section we use the six models presented in the last section to fit some synthetic data sets {(t k , y k )} K k=1 .Because the numerical behavior of the original models of Table 1 has been studied elsewhere, in this section we focus on the asymptote of the modified models of Table 2. Firstly, we compare the values of the two parameters of the asymptote, the slope p and the y-intercept q, computed from the fitting of the six models on the same data sets.Because a comparison of the models based on their best fits depends widely on the data set, we briefly mention the quality of the fitting, we compare and rank the fitting of the models using the value of the optimal criteria.Secondly, for a given model, the Richards (2) model, we illustrate the effect of increasing the density of data to recover the values of the parameters p and q of the asymptote.

Numerical computation
The parameter vector (p, q, θ) of each model F (t) is estimated by minimizing the least squares criteria over the feasible set of parameters Θ as defined for each model in the last two columns of Table 2.We used the MATLAB least squares function "lsqnonlin" to solve these problems.The MATLAB function "fsolve" was used to solve the equation Q(t) = 0 to obtain an approximation T * * of the abscissa T * of the inflection point (T * , F * ) for each modified model, and the approximate ordinate F * * is given by F * * = F (T * * ).Numerical results of our simulations include the optimal solution (p * , q * , θ * ), the optimal value of the criteria Z * , its mean optimal value Z * /K = Z(p * , q * , θ * )/K, and an approximation (T * * , F * * ) of the inflection point (T * , F * ).
The six models of Table 2 were then fitted to the 20 data on (0, 4000].With each of these optimal models we have generated 20 other data points, ten on each subintervals (0, 2000] and (2000, 4000], as we did before with the exponential model.We have added together these data points to get two sets of 70 data points on each intervals (0, 2000] and (2000,4000].In this way we can say that the data represent all the models.
We have considered the best fitting for the following situations: • on the interval (0, 2000] with 70 data points; • on the interval (0, 4000] with 140 data points; • on the interval (0, 4000] with 70 data points (chosen randomly in the preceding data set with 35 data points on each subinterval (0, 2000] and (2000, 4000]); The results are reported in Table 3.For each data set, and on each model F , we have tested 10 different starting points (p 0 , q 0 , θ 0 ) randomly chosen in the feasible set Θ using continuous uniform distributions.For each model and each data set, when the starting values are chosen around the values of the parameters used to generate the data, the algorithm has always converged to the same optimal solution from any one of these starting points.

Numerical results and discussion
From Table 3 we observe that the asymptotes recovered by the Gompertz, Logistic, Richards (1) , Richards (2) , and Bridge models have the slope p * and y-intercept q * relatively close together.These models recover almost the same asymptote.For the Michaelis-Menten model the slope p * is also close to the values obtained by the preceding 5 models, but the y-intercept q * is slightly bigger.This differences in q * decreases if we increase the length T of the interval, from 2000 to 4000, maintaining the same density K/T of data as shown when we compare the parameter values of Table 3.
Based on the optimal value of the criteria Z * /K, the numerical results summarized in Table 4 indicates that the best fit is obtained for the Richards (2)  model followed by the Bridge model, but the differences between the Z * /K for the 6 models are rather small.We also observe the same results for the Richards (1) and the Gompertz models because m R b R ≈ b G .Obviously, the rank for the models might change depending on the data set under consideration.We have included, in Figures 2 and 3, the graphs of the best model, the Richards (2)  model, and the graphs of the worst one, the Michaelis-Menten model, on the intervals (0, 2000] and (0, 4000] to illustrate that globally the fitting is good for all models.We can see slight differences on the graphs of the asymptotes.Let us observe that when we increase the length T with the same data density, the mean value Z * /K decreases, which means we get more information for the fitting and obtain a more reliable asymptote when T is large.

Numerical results and discussion
The results of the Table 5 indicate a monotonic convergence of the estimated values of the parameters through the true values of the parameters as the density K of the data increases in the interval.

Conclusion
In this paper we have extended a class of standard growth curves with horizontal asymptote in order to obtain growth curves with linear asymptote.The Table 5.
Richards (2) model ( T = 4000): Optimal parameters, optimal value and an approximation of the inflection point (T * * , F * * ) for different data sets with an increasing degree of freedom.
Richards (2) model: F R (t; p, q, b, c, m) objective of the present study was to present such models and show their performances on synthetic data sets.In addition to their increasing character with an oblique asymptote, the resulting models provide curves with a single inflection point.Other extensions of these models to obtain curvilinear asymptote and growth curves with several inflection points could be of interest.These extensions and their applications will be presented in a future paper.

Table 1 .
Parameter values and inflection point of the growth models.

Table 3 .
Optimal parameters, optimal value and an approximation of the inflection point (T * * , F * * ) for each data set (K, T ).

Table 4 .
Ranking criterion values for the models.