IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ CITATION: Ricker, M., & D. von Rosen. 2018. A generalization of the exponential function to model growth. IAENG International Journal of Applied Mathematics 48(2): 152-167.
A Generalization of the Exponential Function to Model Growth Martin Ricker and Dietrich von Rosen Abstract—We generalize the exponential function to model instantaneous relative growth. The modified function is defined by a linear relationship between a continuous quantity (rather than time) and logarithmic relative growth. The
generalized exponential function, to derive the corresponding function of q t . Furthermore, we derive formulas for the conversion of a segmented curve of logarithmic relative growth as a function of time, into an equivalent growth curve of q t .
corresponding formula is ln q t q t a b q t , where
Finally, the generalized exponential function is compared with a 2nd-degree polynomial and the nonlinear Schnute function. In conclusion, the generalized exponential function is useful for modeling a path of changing relative growth continuously, and to translate it into a growth curve of quantity as a function of time.
q t q t is instantaneous relative growth of a quantity q , t refers to time, a denotes initial logarithmic relative growth, and b is a shape parameter in terms of its sign, as well as a scaling parameter in terms of its magnitude. For calculating
q t , the exponential integral Ei b q exp b q q dq is needed. The problem of taking the inverse of Ei x zEi is
Index terms - Exponential growth, Exponential function, Exponential integral Ei x , Growth curves, Inverse of the
addressed. In order to distinguish two possible solutions for given z Ei , we define the two inverse functions Ei(x1)0 z Ei and
exponential integral function, Explosive growth, Schnute function, Sigmoid growth
Ei(x1)0 z Ei . An indirect method for their numerical evaluation is developed. With the generalized exponential function, one can model sigmoid growth (b 0) , exponential
I. INTRODUCTION
growth (b 0) , and explosive growth (b 0) , where the term
Growth curves are monotonically increasing functions over time. In the case of negative growth, they can also be monotonically decreasing functions. Growth curves are of interest in a variety of fields, such as biology, medicine, engineering, or economics [1]. In forest science, there is a long history of applying statistical growth curve models to predict tree size and yield (e.g., [2-3]). To model various types of growth processes, the main aim of our article is to study a generalization of the exponential function [4]. This generalization is constructive from a mathematical point of view, although we are also interested in subsequent statistical applications of tree growth data or other empirical growth phenomena. Here, the non-statistical, mathematical aspects are treated.
“explosive growth” refers to a relative growth rate that increases with time. The resulting formula of generalized exponential growth is
1 ( 1) b Ei x 0 t tC exp a +Ei b qC for b 0, q t 1 Ei ( 1) t t exp a +Ei b q for b 0, C C x0 b where qC is a calibrating quantity at time tC . For b 0 , the two functions equal the (standard) exponential function
q t qC exp t tC exp a . In the case of sigmoid growth, the inflection point quantity is 1 / b , which depends only on one parameter ( b ). Negative growth can be modeled by substituting t tC with tC t . Any two points of logarithmic
Let q t represent growth as a function of time, and let
relative growth can be connected unambiguously with the
q t denote dq dt . Nota that the term “relative growth” will always refer to instantaneous relative growth q t q t . The basic characteristic of the exponential function is that relative growth satisfies
Manuscript received on September 22, 2017; revised on April 18, 2018. The Dirección General de Asuntos del Personal Académico (DGAPA) of the Universidad Nacional Autónoma de México (UNAM) provided a stipend for the first author (MR), to spend a sabbatical year at the Swedish University of Agricultural Sciences in 2014-15.
q t q t = exp c t ,
(1)
where c is a constant. Now we modify this model:
M. Ricker is with the Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), Mexico City (e-mail:
[email protected],
[email protected]).
q t q t exp a b q t ,
D. von Rosen is with the Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden, as well as with the Department of Mathematics, Linköping University, Linköping, Sweden (e-mail:
[email protected]).
(2)
where the term c t has been substituted with a b q , with
a and b being constants. The original motivation for this substitution was to establish a linear relationship with
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ where | b v | refers to the absolute value of b v , and i !
relative growth of the form logarithmic ln q t q t a b q t , when time t is unknown [5].
denotes the factorial of i . Furthermore, the Euler-
However, it turns out that the resulting model is interesting also when time is known. It will be shown that the growth function q t can be sigmoid, exponential, or explosive,
Mascheroni
(exp[0 v] / v)dv (1/ v)dv ln[v] . The integral from qC to q can be calculated numerically as Ei b q Ei b qC . For example, in the program Mathematica (www.wolfram.com/mathematica/), one can use the function “ExpIntegralEi”. Ricker and del Río (Appendix 3 in [5]) also derived a relatively easy-toimplement procedure to calculate the integral up to a desired accuracy, with a remainder R x, i :
Ei x ln x
and
transform
it
into
R x, i
dt dq
.
(5b)
6 in [8].
(3)
Thus the time to grow from qC 0 (at tC ) to q 0
Integrating both sides of this equation with respect to q , and
q qC
letting qC at tC be a calibrating point on the growth curve, yields
For q qC the resulting time period t tC is negative. From (6), the initial logarithmic relative growth rate is derived:
1 Note that dv equals ln v . The exponential integral v exp b v v dv , however, represents an infinite series, and is denoted Ei b v (page 176 in [6]):
b v 2! 2
can be calculated as
Ei b q Ei b qC for b 0, tC exp a t q (6) ln q ln qC for b 0. tC exp a
q exp b v tC exp a dv for b 0, v qC t q q 1 exp dv for b 0. t a C qC v
1!1
2 x i ! i 1 1 i2
For an interpretation of Ei b q Ei b qC , see pages 5-
exp a b q . tq q
Ei b v
xi 1
we can write
1
(5a)
where i refers to a chosen integer number. For i x 2 :
dt q
1 q exp a b q . Let dt q dq t q : given q 0 ,
b v ln b v
x1 x2 xi ... 1!1 2! 2 i ! i
R x, i ,
Since q t is a monotone function, we can first consider
exp a b q
mathematically in detail in [7]. Note that Ei[0] , but
II. DERIVATION OF THE GROWTH CURVE FUNCTION
dq
1 k 1 k
theoretical basis for taking the integral over x 0. The Ei x has been discussed exponential integral
final formulas of the resulting growth function are presented. The inflection point in sigmoid growth, and the relationship among the parameters are discussed. Negative growth is also treated. Subsequently, the new function is compared with that of standard exponential growth. Furthermore, a procedure is derived for the conversion of a segmented curve of logarithmic relative growth as a function of time, into a growth curve of quantity as a function of time. Finally, the new function is compared with a polynomial function and the nonlinear Schnute function.
q t . Write (2) as
n
has a singularity at x 0 (Figure 1 top). Whereas in a definite integral cancels out, adding provides a
The article starts with the derivation of the growth curve function. Then, the numerical calculation of the inverse of the exponential integral function Ei x is treated, and the
instead of
nlim ln n
0.577215... has been added in (4), so that (4) represents the Cauchy principal value of Ei x . The reason is that Ei x
where the term “explosive” refers to a relative growth rate that increases with time. Furthermore, the resulting growth function has some interesting mathematical properties.
t q
constant
Ei[b q] Ei[b qC ] ln for b 0, t tC a ln[q] ln[qC ] for b 0. ln t tC
(7)
Since x in Ei x or ln x has to be positive (for non-
2
... , (4)
complex solutions), one gets the following restrictions for (7):
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ q qC for t tC , and q qC for t tC .
x zEi Ei(x1)0 zEi . It is clear which case applies in (6):
(8)
q in Ei b q is always positive, and therefore
The restriction of (8) is derived in the following way: With t tC in the case of b 0 in (7), one needs
with b 0 b q Ei ( 1) z , x 0 Ei zEi Ei b q ( 1) with b 0 b q Ei x 0 z Ei .
Ei b q Ei b qC . Using the notation shown in Figure 1, for
b0
and positive
q , this translates into:
Ei x 0 b q Ei x 0 b qC b q b qC q qC . On the other hand, for b 0 and positive q , one gets:
Ei x0 b q Ei x0 b qC b q b qC q qC .
10
The reason for b q b qC in the latter case (rather than zEi
b q b qC ) is that Ei x 0 x decreases with increasing
Ei x
x. Therefore, for both b 0 and b 0 one gets the condition q qC . For t tC , the inverse logic applies. For b 0 , the parameter a refers only to the initial logarithmic relative growth, i.e., infinitesimally close to zero: a lim ln q t q (see Figure 3 bottom). For the q 0
Eix
5
Eix
0
0
0
x
x
5
logarithmic relative growth at any quantity q 0 , we
2
1
0
1
introduce the variable y : y q ln q t q t for q 0.
2
3
x
In correspondence with (2), the relationship between a and y is
3
y a bq .
2
Eix
(9)
If b 0 (no slope in Figure 3 bottom), we get y a for any
1 0
zEi
1
x
q , i.e., y does not change with q ; this is the case for the (standard) exponential function.
0 1 Eix
III. THE INVERSE OF THE EXPONENTIAL INTEGRAL FUNCTION Ei x
1 0
zEi
2 5
0
5
10
zEi
With (7), one can calculate t q , i.e., time as a function of Figure
quantity. To calculate q t for given t , the inverse of the
1.
Top:
The 1
Ei x ln x x 1!1 x
exponential
integral
function
2
2! 2 ... with a singularity at x 0 , where both branches Ei x0 x and Ei x0 x approach . Upwards, Ei x0 x goes to , and Ei x0 x to zero. Bottom: For zEi 0 , the inverse of Ei x zEi presents two ( 1) solutions of x . Note that Ei x 0 0 , corresponding to Ei 0 ; Ei(x1)0 z Ei 0.372507 (when given with six ( 1) ( 1) decimals); Ei x0 0 ; and Ei x 0 z Ei approaches quickly
exponential integral function Ei x is needed. The inverse
Ei(1) zEi x , however, is more problematic to calculate than Ei x zEi . There are two reasons for this: First, for (1) zEi has always two solutions (see negative z Ei , Ei ( 1) zEi does not represent a Figure 1 bottom), so that Ei
true function. One solution corresponds to the case where x is positive in Figure 1 top, and the other where x is negative. To address the problem of calculating the inverse, one has to distinguish between the two branches of the original function (Figure 1 top):
zero for more negative z Ei , so that one has to consider sufficient accuracy of the numbers involved in the calculation of z Ei .
Ei x 0 x for x 0, z Ei Ei x0 x for x 0.
The second reason, why the inverse function of Ei x represents a problem, is that there is no formula for calculating it. Only some functions for numerical approximations within certain intervals of x exist (see [9]). One (not very elegant) way to calculate the inverse
We define the solutions for the inverse in Figure 1 bottom ( 1) as two distinct (true) functions: x zEi Ei x 0 zEi and
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________
For example, Ei 1.5 is 3.301, when given with three
always in the domain of the complex logarithm, but the imaginary part of Ei x 0 x at the end becomes zero in the
decimals, so that the inverse of the exponential integral function at 3.301 is 1.5 (see Figure 1). To solve, for example, Ei x 3 , one has to search iteratively below 1.5 to find x.
Thus, the problem has been converted in the following way: We wish to find x for given z Ei , in order to find the
indirectly is to repeatedly solve numerically Ei x zEi :
solution.
( 1) value of the inverse functions x zEi Ei x 0 zEi or
In the following, we present a new way of solving this problem. We can express (4) as
x zEi Ei(x1)0 zEi :
k
x . k 1 k ! k
Ei x ln x
(10)
ln 1 0, x for x 0,
zEi
0, x
The natural logarithm can be represented as an infinite series (page 137 in [6]):
2 x 1 2 1 k x 1 k 1
ln x
Whereas this still does not provide an explicit formula for
2k 1
x zEi , the advantage of (13) is that there is no ambiguity
for x 0. (11)
anymore of two function values for Ei
2k 1 x 1 2 xk Ei x . k ! k k 1 2 k 1 x 1
x 1 0, x ln x , (12) x 1
Table 1. The formulas from (13) are given in Table 1, the core code is shown, as well as recommended starting values for the indirect search are provided (see also the section “Supporting Information” at the end of the article).
where atanh refers to the inverse hyperbolic tangent function (page 142 in [6]), and to the incomplete gamma function (page 236 in [6]):
1 1 z , and atanh z ln 2 1 z
With z x 1
x 1
v
exp v dv.
Substituting
ln x 2
x z Ei
=Ei(x1)0 zEi
in atanh z :
x 1 1 x 1 atanh ln 1 x 1 x 1 2
TABLE 1. SUMMARY FOR CALCULATING NUMERICALLY THE INVERSE OF THE EXPONENTIAL INTEGRAL FUNCTION
x
x 1 1 x 1
Formula
.
atanh x 1
x 1
since the
Concerning computation, the numerical evaluation of the incomplete gamma function has been treated in [10], pages 221-224 of [11], and pages 560-567 of [12]. When using Mathematica, the computation of x z Ei is summarized in
Ei x 2 atanh
1
x ,
finding algorithm (see chapter 9 of [11]).
It turns out that this is equivalent to
0, x
( 1)
upper and lower branches have been separated (Figure 1 bottom). To find x for given z Ei , one has to use a root-
Inserting (11) into (10) yields
(13)
for x 0.
in
zEi Ei x
=Ei(x1)0 zEi
Given z Ei , solve zEi
Given zEi , with zEi 0 , solve
ln 1 0, x
zEi 0, x
numerically for x in the domain of real numbers
numerically for x in the domain of real numbers ( zEi 0 does not exist in this branch)
(12),
distinguishing between positive and negative x , and simplifying, the following two equations are obtained:
Mathematica code
Ei x 0 x ln 1 0, x , Ei x 0 x 0, x . These equations provide a mathematical filter to distinguish between the two branches in Ei x , for x 0 and x 0 ,
Re[ FindRoot[ zEi = = Log[1] Gamma[0, x], {x, zStart}, WorkingPrecision >30, MaxIterations >1000]
Re[ FindRoot[ zEi = = Gamma[0, x], {x, zStart}, WorkingPrecision >30, MaxIterations >1000]
for z Ei 0.5: Recommended values for zStart
by keeping the branch of interest for given z Ei and either positive or negative x , in the domain of real numbers, while moving the other branch into the domain of complex numbers. Not that in the case of x 0 , ln 1 i is
for zEi 1: 10
zEi
, and
for zEi 1: ln zEi
exp z Ei , and for 0.5 z Ei 0: ln z Ei
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ There is one closely related, as well as another, completely different approach to find numerically the inverse of the exponential integral function. First, one can use (5a) with ln x for Ei x 0 x and ln x for
Ei 0 q Ei 0 qC
Ei x 0 x , instead of ln x . The accuracy is controlled with the remainder of (5b). The second approach, given in the supporting information, consists of a segment-wise numerical simulation of the two branches of the function in Figure 1 bottom, with high-degree polynomials (e.g., 31 coefficients) and a high number of calculated data points (e.g., 10,000) in each of several segments.
ln q ln qC .
1
ln q ln qC
q t ln q ln qC t tC , suggested already as the “correct” formula by Sir Ronald Fisher in 1921 [15]. It reflects exponential growth. Whereas this formula is relatively well-known by field biologists, (14) with b 0 for sigmoid growth and for explosive growth, as the more general cases, are new. For calculating the logarithmic relative growth at any time, q in (9) is substituted by q t from (14):
t tC exp a 1 Ei( 1) for b 0, x 0 b +Ei b qC q t qC exp t tC exp a for b 0, (14) t tC exp a 1 for b 0, Ei(x1) 0 b +Ei b qC
t tC exp a a Ei(1) x 0 +Ei b qC for b 0 and a y t 0, y t a for b 0, ( 1) t tC exp a a Ei x0 +Ei b qC for b 0 and a y t 0,
where b 0 represents sigmoid growth, b 0 exponential growth, and b 0 explosive growth, and applies.
(16)
where for b 0 the restriction of (15) applies. With the lefthand side y t involved in the condition to choose among
Rearranging this restriction leads to
options for b 0 , this formula is only practical if one knows that a y t will be positive or negative. Otherwise, if one
(15)
does not want to calculate t y instead of
The restriction in (15) represents an upper bound for t , and reflects the fact that for explosive growth with given coefficients, there is a maximum time t that can be reached for given parameters a , b , qC , and tC , no matter how large
alternative is to employ (9), where
y t , the
q t has to be
calculated first with (14). Furthermore, the instantaneous increment as a function of time is
q is. We call (14) “generalized exponential function”. The term has rarely been used before for other functions. In unrelated form, the term “generalized ( q -)exponential function”
...
Equation (14) with b 0 represents the formula for calculating relative growth from a smaller qC to a larger
we define the generalized exponential function as:
Ei b qC for b 0 . exp a
2! 2
2
q t 0 remains always positive.
to calculate q t , rather than t q . With t tC and qC 0,
t tC
1!1
C
will not converge to a maximum quantity. Using (2), one can see that q t q t exp[ a b q t ] , which for increasing
Employing the inverse of Ei x , (6) can now be re-arranged
t tC exp a +Ei b qC 0
C
For sigmoid growth ( b 0 ) in (14), and t , q t
IV. THE FINAL FORMULAS OF THE GENERALIZED EXPONENTIAL FUNCTION
where for b 0 the restriction
0 q q 0 q q
q t exp a b q t for b 0, (17) qC exp a t tC exp a for b 0,
q t
is used in [13] (page 2923) to refer to 1 q '
eq x lim 1 q ' x . In [14] (page 574), the term
where in the case of b 0 , q t has to be calculated first
“generalized exponential growth model” is employed for
with (14). Introducing (14) for q t in (17) would not lead
q ' q
1
f t a b exp t
to simplification, but would involve twice the inverse of Ei x , causing in turn complicated conditions if
.
Note that in computations with (14) one can also use the formulas for b 0 arbitrarily close to b 0 , instead of using the one for b 0 . One can show this with (6), the formulas that form the basis for (14). Using (4) and b 0 , the formulas convert into each other:
Ei(x1)0 zEi or Ei(x1)0 zEi applies. For b 0 , the formula is derived from (2). For b 0 , the formula is derived from (14) by taking the derivative. 156
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ formula for the inflection point is of interest for several reasons:
V. DETERMINATION OF THE FUNCTION PARAMETERS FROM THREE DATA POINTS
(i) The inflection point characterizes the shape and scale of a growth curve. In a nonlinear growth curve model, there are few points on the curve that can be used to compare different curves of a given function: In the case of the generalized exponential function, there is one calibrating point ( qC at tC ), as well as for b 0
The generalized exponential function of (14) with b 0 is a three-parameter model, whose parameters a , b , and qC (at tC ) or tC (at qC ) are thus determined exactly by three points q1 at t1 , q2 at t2 , and q3 at t3 , with t1 t2 t3 and q1 q2 q3 or q1 q2 q3 . Given such three data
the inflection point ( qIP at t IP ).
points, any one can represent qC at tC . To determine b ,
(ii) The formula for calculating the location of the inflection point reveals important information about the dependence of the growth curve’s shape, location, and scale on the different parameters: How does changing the function’s parameters affect the inflection point, and consequently the whole growth curve?
one has to solve the following relationship numerically and indirectly (in Mathematica with the FindRoot function, with 1 being generally a recommended starting value):
Ei b q3 Ei b q1 Ei b q2 Ei b q1
t3 t1 . t2 t1
(18)
The formula is derived by transforming the equation of (7) for b 0 with q1 at t1 being the calibrating point qC at tC ,
resulting in exp a Ei b q Ei b q1
(iii) It can also be of interest to develop a growth curve around a known inflection point, i.e., to derive the growth function’s parameters with a known (fixed) inflection point.
t q t1 .
By substituting q at t q with q2 at t2 on the one hand,
The inflection point quantity ( qIP ) is derived by taking
and with q3 at t3 on the other, one obtains two equations
the second derivative of (6) for b 0 with respect to q :
for exp a , which are set equal. Having determined b , one
d 2t dq 2 1 b q exp a b q q 2 . (19)
calculates
Setting this expression equal to zero, and solving for q ,
Ei b q Ei b q 3 1 . t3 t1
a ln
yields:
qIP 1 b ,
Since the generalized exponential function represents a continuous spectrum of growth functions from sigmoid to explosive, one expects to always find a solution for b in (18), and subsequently for a. Note that the three points may lie on a straight line, but that the resulting function will represent nevertheless in that case a sigmoid curve among the points.
(20)
which is positive only for b 0 . This relation reveals a surprisingly simple interpretation of b as the negative inverse of the inflection point quantity. For b 0 , i.e., explosive growth, qIP would be negative, which however has no real-world interpretation, and thus will not be considered further.
An alternative method to determine a and b for b 0 is to use nonlinear regression with (6), which (at least in Mathematica) works well. The regression equation is
Inserting (20) into (6) for b 0 corresponding inflection point time ( t IP ):
t t1 Ei b q Ei b q1 exp a . With three data
tIP tC
points of t q , the resulting coefficient of determination has
results in the
Ei 1 Ei b qC for b 0, exp a
(21)
where Ei 1 1.89512 , when given with five decimals.
to be 1. The regression can also be used for more than three data points, in which case it becomes a true regression analysis, with (generally) variance of the residuals and
Furthermore, as a necessary condition, one has to show that there is a change of sign of the second-derivative at the inflection point (page 231 in [16]). With q 0 , the second
R 2 1 . In that case, one can also let the regression determine either qC or tC . Note, however, that the residuals
factor exp a b q q 2 in (19) will always be positive.
present time, rather than quantity.
Therefore, if the second derivative will be positive or negative depends only on the first factor 1 b q in (19). For b 0 , 1 b q 0 implies q 1 b , whereas
VI. INFLECTION POINT IN NON-EXPONENTIAL GROWTH
1 b q 0
An inflection point refers to the point on a growth curve, where the curve switches from being left-winged (locally convex) to being right-winged (locally concave), or vice versa (circles in Figure 2). Knowing and analyzing a
implies
q 1 b .
For
b0,
1 b q 0 implies q 1 b , whereas 1 b q 0 implies q 1 b . 157
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25
Quantity q
15
0.2
a a
a
0.5 0
Quantity q
b
10
a
0.5
0
20
b
0.1
15
b
0.2
b
0.3
10
5 5 0
0 2
4
6
8
10
2
4
Time t
8
10
Time t
15
20
a qC qC qC
10
0, b
a
0.2 15
1.9 1 0.1
Quantity q
Quantity q
6
5
10
0
b qC qC qC
1 1.5 1 1 0.5
b
5
0
0.2 1.5
b
0.2
0.2 0.5
0 2
4
6
8
10
2
4
Time t
6
8
10
Time t
Figure 2. Variations of sigmoid growth curves ( b 0 ), when changing symmetrically the growth rate parameter a , the shape and scaling parameter b , or the quantity qC of the calibrating point at tC 0.5 in (14) with b 0 . Top left: Change of a 0 (initial relative growth is 100%) by 0.5 . Top right: Change of b 0.2 ( qIP 5 ) by 0.1 . The new inflection point quantity is qIP 1 b b , and the new inflection point time has to be calculated with (21). Bottom left: Change of qC 1 by 0.9 without changing b , i.e., the inflection point quantity (circles) moves horizontally. Bottom right: Changing qC and b according to (23) changes the growth curve’s scaling, and moves the inflection point quantity vertically. Note that relative growth remains the same for all three curves at any time in this graph, since y a b f q f a bq .
winged (concave) part of the growth curve appears for positive time.
Consequently, there is always a change of sign at qIP . Since the second derivative switches from negative to positive at the inflection point with increasing q , the formula also proves that the function switches always from right-winged to left-winged for b 0 . Recall, however, that (6) is the formula for t q . In the case of q t in (14), the
VII. THE RELATIONSHIP AMONG THE PARAMETERS a , b , AND qC AT tC
curve’s shape changes to its mirror image, resulting for b 0 always in a growth curve that switches inversely from being left-winged to being right-winged.
In the generalized exponential function, the three parameters a, b, and qC at tC have straight-forward interpretations:
Depending on tC , b , and qC , the inflection point time
Whereas qC at tC calibrates the position of the growth
tIP can be positive or negative; a negative inflection point time for b 0 means that tIP tC , i.e., only the right-
curve segment, a is the initial logarithmic relative growth, i.e., the intercept at q 0 (see Figure 3 bottom). It serves as a standardized growth rate parameter (see Figure 1 in [8]). 158
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ On the other hand, b is a shape parameter in terms of its sign, as well as a scaling parameter in terms of its magnitude.
VIII. NEGATIVE GROWTH WITH THE GENERALIZED EXPONENTIAL FUNCTION
Different combinations of parameter values between a and b cause different curvatures. Note that the second derivatives of (6) for b 0 , as measures of the curvature, depend on both a and b :
According to the standard exponential function, growth can be positive or negative. To describe negative growth with the generalized exponential function, presented in (14), one has to replace t tC with tC t . We add the subindex NG for negative growth, so that (14) changes to
d 2t Ei b q Ei b qC , exp a da 2
(22a)
t t exp a 1 Ei( 1) C for b 0, x 0 b +Ei b qC qNG t qC exp tC t exp a for b 0, (24) tC t exp a 1 for b 0, Ei(x1) 0 +Ei b qC b
d 2t 1 2 exp a b qC q 2 db b (22b) . 1 exp b q b q 1 b q exp b q C C
For example, the more negative a in (22a), the larger is the curvature. Extreme curvatures are associated with extreme parameter values of a and b .
where b 0 represents negative sigmoid growth, b 0 negative exponential growth, and b 0 negative explosive growth. The restrictions in (8), when b 0 , convert to
Figure 2 shows different combinations among the three parameters a , b , and qC in the case of sigmoid curves
(b 0) . Changing tC is not shown in Figure 2, because it only moves the whole curve to the left or right, as can be seen from (6). The inflection points are indicated with circles in Figure 2. In Figure 2 top left, the parameter a is varied around a 0 (i.e., initial relative growth is 100%). One sees that a symmetrical change of a (here 0.5 ) leads to symmetrical growth curves around the original curve, with constant qIP but different inflection point times tIP .
qNG qC for t tC , and qNG qC for t tC . Furthermore, (15) changes for negative growth:
t tC
Ei b qC for b 0 and negative growth. exp a
Finally, (21) converts to
tIP tC
This is not the case for a symmetrical change of b (here 0.1 ), which results in asymmetrical curves around the original growth curve (Figure 2 top right). In Figure 2 bottom left, qC is moved up and down by the same amount
Ei 1 Ei b qC for b 0 & negative growth. exp a
For negative growth, one also substitutes t tC with tC t in (7) for calculating a , in (16) for calculating y t , and
(here 0.9 ). The qIP , however, are kept constant, which causes asymmetrical shapes of the resulting growth curves around the original one. Changing both the quantities of qC
in (17) for calculating q t .
and qIP (via b ) together by the same proportion results in a change of scale, shown in Figure 2 bottom right and explained next.
IX. GENERALIZED VERSUS STANDARD EXPONENTIAL GROWTH
by some factor, and divide b by the same factor (say f ),
To show how the formulas work, and how the generalized exponential function compares with the standard exponential function, some sample growth curves are shown in Figures 3 and 4. To get a sigmoid growth curve with inflection point t IP 7 , qIP 5 , and calibrating point
to model a growth curve of q f t f q t :
tC 2 , qC 1 , according to (20) one uses b 0.2 . To
In addition to b determining if the growth curve is sigmoid, exponential, or explosive, the variables b and qC are scaling q t . In (14) with b 0 , one can multiply qC
calculate the corresponding a , one has to transform (21):
t t exp a C f ( 1) q f t Ei b . b +Ei qC f f
Ei 1 Ei b qC for b 0. t IP tC
(23)
a ln
The restrictions from (8) become qIP qC for t tC , and
The shape of the curve remains the same, with the new inflection point quantity f qIP , but unchanged inflection
qIP qC for t tC . For negative growth, one has to
substitute tC tIP for tIP tC , with qIP qC for t tC ,
point time. The factor f does neither affect the initial logarithmic growth rate a , nor y in (9) or (16).
and qIP qC for t tC . Here, one gets a 0.610.
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10
explosive t 6.54
15
sigmoid
Quantity q
Quantity q
8 10
inflection point 7, 5
5
calibrating point 2, 8
negative exponential
6 inflection point 7, 5
4
exponential
negative sigmoid
2 calibrating point 2, 1
0 0
5
negative explosive 0.054 t
0
10
15
20
25
0
5
10
Time t
15
20
25
Time t
a
1
explosive b 0.05
0.6100 sigmoid b
a
2
0.2 exponential b
2.111
Logarithmic relative growth y q
Logarithmic relative growth y q
0.5 a 0.6100
0
0
3
4
a 1.073
1.0
a
explosive b 0.05
1.073 exponential b
1.5
a
0
1.465
2.0
sigmoid b
0.2
2.5 3.0
0
5
10
15
0
2
Quantity
4 6 Quantity q
8
Figure 3. Sigmoid growth ( b 0 ), exponential growth ( b 0 ), and explosive growth ( b 0 ), positive (left) and negative (right). Top: Quantity q as a function of time, as calculated with (14) for positive growth (left), and with t tC substituted by tC t in (24) for negative growth (right). Whereas the parameter b determines the shape and scale of the function, a determines the relative growth rate, and the calibrating point qC tC the position of the growth curve in the upper graphs. Bottom: The underlying linear relationship of logarithmic relative growth y as a function of q, as calculated with (9) for positive or negative growth, which cannot be distinguished in these two graphs. Whereas b is the slope in this functional relationship, a is the intercept at a quantity of zero.
Using (14) with b 0 , Figure 3 top left shows the resulting sigmoid growth curve. Going through qC 1 at
Figure 3 bottom shows the corresponding linear relationships between q t and logarithmic relative growth,
tC 2 , the curve reaches q 16.2 at t 25 . Employing
as given in (2). In those graphs, a is the intercept and b is the slope of the lines ( qC at tC does not appear in this
(7) for b 0 , one can calculate the logarithmic relative growth rate for an exponential growth curve that also goes through q 1 at t 2 and q 16.2 at t 25 , which
mathematical relationship). Figure 4 shows two more functional relationships for the growth curves from Figure 3 top. For sigmoid growth, there appears a maximum of the instantaneous increment with time. This maximum is at the inflection point time. First, the relationship between t IP and qIP is given by substituting
results in a 2.11. The corresponding growth curve, calculated with (14), is shown in Figure 3 top left. Negative growth is shown in Figure 3 top right, employing (24) for negative sigmoid growth, negative exponential growth, and negative explosive growth, with the same parameters of b as for positive growth, but choosing qC qIP (and calculating the corresponding parameters for
b in (21) with b 1 qIP from (20), and rearranging:
tIP tC
a ).
Ei 1 q exp a Ei C . exp a qIP
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(25)
IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________
2
1
Logarithmic relative growth y t
Logarithmic relative growth y t
2 explosive
0 1 exponential
2 3
sigmoid
4 0
5
10
15
20
1
negative explosive
0 1 negative exponential
2 negative sigmoid
3
25
0
5
10
Time t
20
25
Time t
3.0
3.0 explosive
negative explosive
2.5
2.5
Increment q' t
Increment q' t
15
2.0 1.5
exponential maximum 7, 0.999
1.0
2.0 1.5 1.0
negative exponential maximum 7, 0.629
0.5
0.5
negative sigmoid
sigmoid
0.0
0.0 0
5
10
15
20
25
0
5
Time t
10
15
20
25
Time t
Figure 4. Two additional functional relationships for the positive (left) and negative (right) growth curves of Figure 3 top. Top: Logarithmic relative growth y as a function of time, calculated with (16) for positive growth, and with t tC substituted by tC t for negative growth. Bottom: Instantaneous increment q t as a function of time, calculated with (17) for positive growth, and with t tC substituted by tC t for negative growth. For sigmoid growth ( b 0 ), there is a maximum at the inflection point time, calculated with (26).
Next, to calculate the maximum instantaneous increment, one substitutes t in (17) for b 0 , with tIP from (25),
larger i refers to a later time point). Then one can find the corresponding ai and bi for the generalized exponential
which yields
function that goes from the first to the second point. The procedure will be useful, for example, for statistical applications. There may be more than two subsequent points yi at ti , and one wants to use (16) to get a continuous
1 b
q t max exp a 1 for b 0.
(26)
The formula applies for both positive and negative growth.
function through all points, employing splines (Figure 5 top left). This segmented function can then be converted into a growth curve of quantity as a function of time (Figure 5 bottom left). The method works only for segments with b 0, but b can be arbitrarily close to 0.
X. CONVERSION OF A SEGMENTED CURVE OF y t INTO A GROWTH CURVE OF q t
Transforming (16) for b 0 , one gets two equations for a single segment of the generalized exponential function, where the subindex i refers to the function segment (spline) in between time points ti and ti 1 :
Assume that there are two known points of logarithmic relative growth, yi at ti and yi 1 at ti 1 , where the integer subindex i serves to index and distinguish the points (a
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i
i
ri bi qC and ri 1 i
Ei ai yi 1 ti 1 tC exp ai +Ei bi qC . tC ti ,
With
i
the
i
first
equation
i
simplifies
exp ai
to
to the left-hand side: If ci ri 1 0 , then Ei (x1)0 z Ei applies, and if ci ri 1 0 , then Ei (x1)0 z Ei applies. Given
The variables bi and qC occur always together as bi qC i i
that ci ri 1 bi qC
i1
qIP within a
ri 0 bi 0 ci ri 1 0 Ei(x1)0 z Ei .
ti 1 ti 0,
(28)
Consequently, the formula for calculating ci equals
Ei( 1) t t exp a Ei r i i x 0 i 1 i for ri 0, ri 1 ci (30) ( 1) Ei x0 ti 1 ti exp ai Ei ri for ri 0. ri 1
with yi yi 1 (which is equivalent to b 0 ). With (28), one can indirectly find the ri , given yi , yi 1 , and ti 1 ti , with a root-finding method (in Mathematica the function “FindRoot”; a recommended starting value is 1). In accordance with (9), ai is calculated for the segment
The case ri qC
between the two points subsequently as
i
ai yi ri ,
If one qC
i
i
are needed separately,
variables of bi and nt 2 variables of qC . All bi and qC i i are found simultaneously with a method for solving a system of linear equations (in Mathematica one can use the function “NSolve”). For negative growth, one has to adapt (27), (28), and (30a-b), by substituting ti 1 ti with ti ti 1.
rather than combined in ri . Assuming that there are nt points yi at ti , one has to determine nt 1 segments as continuous functions that connect the points non-linearly, according to (16). With the nt points of logarithmic relative growth yi ti , the nt 1 parameters of ri and ai , and a single known calibrating point qC , one can find bi and qC i
A small example will illustrate the procedure with data from growth of tree trunk radiuses. We will use four points of logarithmic relative growth y t , shown in Figure 5 top
i
of all segments simultaneously: Using (7) for b 0 , one obtains
1 ti 1 ti
is known, then one gets 2 nt 1 1 =
variables: nt 2 equations of bi ci bi 1 0 , and nt 1 equations of ri bi qC 0 , corresponding to nt 1
Next, for calculating the growth curve of q t , the
1 Ei bi qC Ei bi qC i1 i ti 1 ti
i
2 nt 3 equations with the same number of unknown
data of each ai and ri , but nt data of yi ).
exp ai
q IP 0 does not exist for positive, finite
quantities.
(29)
where the last yi ( yn ) is not being used (there are nt 1 t
i
0 , the sign of bi
ri 0 bi 0 ci ri 1 0 Ei(x1)0 zEi ,
root-finding, (27) becomes
numerical values of bi and qC
i1
i
i
exp yi ri
, and qC
determines which inverse function has to be used. Since ri bi qC :
i
given segment. With the definition of the ratio ri bi qC qC qIP , and rearranging for more robust
Ei yi yi 1 ri Ei ri
The inverse exponential integral function has to be applied
(27)
i
ti 1 ti exp yi bi qC 0 for bi 0. i
i
Ei ci ri 1 ti 1 ti exp ai Ei ri .
Ei yi yi 1 bi qC Ei bi qC i i
i
1 Ei ci ri 1 Ei ri . ti 1 ti
yields
in the second equation with yi bi qC yields i
in (27). With (20), this term converts to qC
i 1
Now only the ci are still unknown. Rearranging further
ai yi bi qC , which corresponds to (9). Substituting ai i
ci bi bi 1 , and as before bi 1 qC , one gets
Defining the coefficient
Ei ai yi ti tC exp ai +Ei bi qC ,
left; thus there are three segments. With (28), the parameters of ri are determined as r1 0.1538 , r2 1.458 , and
r3 6.893 . With (29), the parameters of ai are calculated (given in Figure 5 top right). With three segments, there are two ci , calculated with (30) for positive ri : c1 0.1124
b Ei i bi 1 qC Ei bi qC . b i1 i i 1
and c2 0.2245 . Finally, qC is taken as one known time2
quantity point (1.41 cm at 10 years, in Figure 5 bottom left).
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2.7 2.8 2.9
y1
2.76
y2
2.77
y3
tC1
Logarithmic relative growth y q
Logarithmic relative growth y t
IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________
2.86
tC2
9 yrs
10 yrs
tC3 11 yrs
3.0
y4 3.20
3.1 tC4
3.2 9.0
9.5
12 yrs
10.0 10.5 11.0 Time t years
11.5
2.8 2.9 3.0 3.1
12.0
1.45 1.40 1.35 1.30
12 yrs
1.50 cm
tC3 qC2
tC2
1.32 cm
tC1
9.0
11 yrs
1.41 cm
qC1
1.50 cm
4.03 1.31 2.61
piecewise linear relationship qC4
1.35
10 yrs
1.40 1.45 1.50 Quantity q radius in cm
0.80
11.5
1.55
0.838
0.75 0.70 0.65
10.0 10.5 11.0 Time t years
1.57 cm
0.884
0.85
9 yrs
9.5
b3 4.60 cm
qC3
0.857
Increment q' t in mm year
Quantity q radius in cm
qC3
a2 a1
1.41 cm
0.90 tC4
1.50
a3
1.03 cm
qC2
1.32 cm
1.30
1.57 cm
1.55
b2 qC1
3.2
1.60 qC4
b1 0.116 cm
2.7
12.0
0.640
9.0
9.5
10.0 10.5 11.0 Time t years
11.5
12.0
Figure 5. Conversion of four points of logarithmic relative growth as a function of time into a continuous function of quantity as a function of time, with the generalized exponential function. The data represent tree trunk radiuses, measured at 9, 10, 11, and 12 years. Top left:
From four points of yi ti , the parameters ai , bi , and qC are determined for each of three segments ( i = 1 to 3), as explained in the text, i and the continuous function determined with (16). Top right: Only the relationship between quantity and logarithmic relative growth is piecewise linear, as given with (9), and represents the fundamental assumption of a linear relationship between instantaneous logarithmic relative growth and quantity in the generalized exponential function. All other graphs present slightly curved lines. The variable ai represents the y-intercept at a quantity of zero, and bi the slope of the line in segment i . Bottom left: Given the determined parameters, the time-quantity curve can now be calculated in three segments with (14). Bottom right: The instantaneous annual increment is calculated with (17).
Given these parameters, the system of equations for nt 4
are shown. Given the parameters ai , bi , and qC for each
data points contains 2 4 3 5 equations, as well as five variables:
segment the continuous generalized exponential function of quantity with time can now be calculated with (14) for each segment (Figure 5 bottom left). Finally, the continuous function of instantaneous increment with time is calculated with (17), and shown in Figure 5 bottom right.
i
b1 c1 b2 0 b1 0.1124 b2 0, b2 c2 b3 0 b2 0.2245 b3 0, r1 b1 qC 0 0.1538 b1 qC 0, 1
Two segments do connect in Figure 5 top (with dependent variable y ) only under the condition that
1
r2 b2 qC 0 1.458 b2 1.41 0, 2
qC ri yi yi 1 ri 0 . i
r3 b3 qC 0 6.893 b3 qC 0. 3
3
The
condition
arises
by
assuming that quantities are always positive. It is derived by solving the following system of five equations for ai , ai 1 ,
The numerical results are given in Figure 5 top right, where the piecewise linear relationships of y q in three segments
bi , bi 1 , and qC
i1
: yi ai bi qC , yi 1 ai bi qC
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i
i1
,
IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________
when assumed to be positive, results in the above condition. Technically, qC 0 is implemented in (30), where it was
i.e., the Schnute function can only model certain shapes of a growth curve through four data points, even if the quantity q increases. If there is no solution, one can use nonlinear regression for an approximation to the four points. For example, for qC 0.25 at tC 0 , q1 0.64 at t1 1 ,
qC 0 to decide between
q2 0.83 at t2 2 , and qC 1.04 at tC 3 , there is an
Thus, when a negative qC
exact solution with aS 1.198 and bS 5.609 . Increasing
yi 1 ai 1 bi 1 qC , ri bi qC , and ri 1 bi 1 qC . i
i1
The solution includes qC
i1
i1
qC ri yi yi 1 ri , which i
1
i1
assumed that ci ri 1 qC
i1
Ei (x1)0
z Ei
and
Ei (x1)0
z Ei .
1
2
i
i
2
q1 from 0.64 to 0.74, however, there is no exact solution
would be necessary to connect yi and yi 1 (which at this
anymore. Using nonlinear regression, the closest leastsquares solution is with aS 35.106 and bS 187.928 .
stage is not known yet), the “wrong” option for the inverse is chosen, resulting in a “jump”, but complying with qC 0.
For this reason, we will use here the Schnute function as a growth curve for interpolating three (rather than four) data points. Using bS 1 results in the formulas given in Table
i 1
2, which will always go through three quantities that increase with time. There is, however, no inflection point anymore with bS 1 , only a curved function. Two data
XI. COMPARISON OF THE GENERALIZED EXPONENTIAL FUNCTION WITH THE NONLINEAR SCHNUTE FUNCTION AND A POLYNOMIAL
points are used as calibrating points, and the parameter a S has to be determined with a root-finding algorithm, such that q at t represents the third data point (see supporting
The generalized exponential function is determined with only two known data points of relative growth, whereas other functions usually need more than two points. It is informative to compare the generalized exponential function with a polynomial, which is linear in its parameters, and with the Schnute function, a flexible nonlinear growth function [17, 18]. A polynomial function is the typical function that underlies the classical growth curve model in statistics, the Generalized Multivariate Analysis of Variance (GMANOVA) model ([19], chapter 4 in [20]). The seconddegree polynomial q t b0 b1 t b2 t 2 (with b0 , b1 ,
information). Figure 6 shows that the shapes of the generalized exponential function, the 2nd-degree polynomial, and the Schnute function are different: The 2nd-degree polynomial function and the Schnute function with bS 1 are either strictly concave or convex, whereas the generalized exponential function is sigmoid. Furthermore, the curvature of the generalized exponential function is more pronounced. Despite being more flexible, the polynomial function has the undesirable property for a growth function that it has a minimum or maximum in between the data points.
and b2 being coefficients) connects any three data points, no matter their constellation, and thus in this regard is more flexible than the generalized exponential function, for which the quantity q must always increase with time t . On the
Table 2 provides the different characteristics and formulas of the three functions in comparison. The derivations of the formulas are given as supporting information. Some aspects of interest are the following:
other hand, the Schnute model has the following functional form:
1 bS
1 exp aS t tC b b b 1 q t qCS qCS qCS 1 2 1 1 exp aS tC tC 2 1
1) Interpolation of three q t points: Only the generalized exponential function can be sigmoid with an inflection point.
,
2) Connecting two points of logarithmic relative growth as a function of quantity or time: Only the generalized exponential function is uniquely determined between the two points, because it is based on the assumption of a linear change of logarithmic relative growth with quantity. For the other two functions, one can see in Figure 6 right that there is a limited range of q for which
with four parameters a S , bS , qC at tC , and qC at tC . 1
1
2
2
The function has two calibrating data points. For a 0 and b 0, the Schnute function represents a sigmoid curve with an inflection point quantity at 1
qIP
1 b exp a t qbS exp a t qbS S S C2 C2 S C1 C1 exp aS tC exp aS tC 2 1
bS .
y q can be calculated. In the case of the second-degree polynomial, there is a limiting minimum or maximum quantity, calculated as b0 b12 4 b2 , where growth
Note that this inflection point quantity depends on all parameters, in contrast to the generalized exponential function, where qIP depends on a single parameter, as given in (20).
switches from positive to negative or vice versa. The Schnute function with bS 1 has a limiting, lower or
Even though there are four distinct parameters, the Schnute function cannot go through any four data points, 164
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Quantity q
0.8
Logarithmic relative growth y q
6 Generalized exponential function
2, 0.83
1, 0.74
0.6
2nd degree polynomial
0.4 0, 0.25
0.2
Schnute function with bS
1
0.0 0.0
0.5
1.0 Time t
1.5
2.0
4
Generalized exponential function
2
2nd degree polynomial
0
qL 0.845
2 Schnute function with bS
4
2.5
0.0
1
qL 0.850
0.2
0.4 0.6 Quantity q
0.8
2, 0.83
0.8
Quantity q
Logarithmic relative growth y q
1.0
0.6
Schnute function with bS nd
2
1
degree polynomial
0.4
1, 0.34
Generalized exponential function
0, 0.25
0.2 0.0
0.5
1.0 Time t
1.5
2.0
Generalized exponential function
2 1 0
2nd degree polynomial
yL 0.527
y
1 Schnute function with bS
2 3 0.0
qL 0.235
0.2
1
qL 0.230
0.4 0.6 Quantity q
0.8
1.0
Figure 6. The generalized exponential function in comparison with a second-degree polynomial and the Schnute function with bS 1 . Left: All three models are able to pass exactly through three points, which in the case of the generalized exponential function and the Schnute function have to represent increasing quantities with time (hence, these are growth curve functions). In contrast, the polynomial connects any three data points, but may have a maximum (above at t 1.725 ) or minimum (below at t 0.275 ) in between points. Only the generalized exponential function can be sigmoid with three points to go through (above). The functions can be calculated for negative time. Right: The underlying relationship of instantaneous logarithmic relative growth as a function of quantity is linear for the generalized exponential function, but highly nonlinear for the other two functions. The second-degree polynomial presents limits qL , where it reaches an extreme and growth switches between negative and positive. The Schnute function with bS 1 has an asymptote of the quantity as a lower or upper limit.
asymptotic
upper
qC qC qC 1
2
1
quantity,
exp a t S
C
2
calculated
as
relative growth with increasing quantity in an unlimited way.
tC 1 . 1
4) Moving the three data points together by a time period t : The generalized exponential function develops i, relative to tC , so that moving tC by t moves the
3) Explosive growth: In the lower right graph of Figure 6, one can see that only the generalized exponential function (with b 0 ) goes from y to . The Schnute function with
bS 1 and
aS 0
whole curve accordingly to tC t . In the 2nd-degree polynomial, the three data points can be moved in time, resulting in changed coefficients: b0,new b0
goes
asymptotically towards ln aS , and the 2nd-degree polynomial reaches the
maximum (limit)
yL
t b1 b2 t , b1, new b1 2 b2 t and b2, new b2 . The Schnute model develops relative to tC and tC ,
ln 2 b2 4 b0 b2 b at q 2 b0 b 2 b2 . Both functions contain phases of explosive growth, but the generalized exponential function can increase 2 1
2 1
1
which then become tC t and tC t . 1
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2
2
IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ TABLE 2. MAIN FORMULAS OF THE GENERALIZED EXPONENTIAL FUNCTION IN COMPARISON WITH A POLYNOMIAL AND THE NONLINEAR SCHNUTE FUNCTION
Variable with interpretation
Generalized exponential function
2rd-degree polynomial
Schnute function with bS = 1
Model parameters
a , b , qC tC
b0 , b1 , b2
aS , qC at tC , qC at tC
Interpretation of model parameters
Instantaneaous initial logarithmic relative growth, shape and scaling parameter, calibrating quantity
Positioning coefficient, growth rate-related coefficient, shape-related coefficient
Growth rate-related coefficient, two positioning and scaling coefficients
1
1
2
2
qC qC qC 1
q t see (14)
quantity as a function of time
2
1
1 exp aS
qIP and tIP inflection point quantity and time
tC
Ei 1 Ei b qC exp[a ]
No inflection point
C2
C1
No inflection point
q
b1 2 b2 t
see (17)
C2
exp a t
qC 1
b2 4 b q b 1 2 0 ln q
a bq
as a function of quantity
ln aS
,
where the argument of the square root and q must be positive for non-complex solutions
aS exp aS tC t 2 S
y q instantaneous logarithmic relative growth ln dq dt q
1
for b 0 : 1 b ,
q t instantaneous increment dq dt as a function of time
t t
1 exp aS t tC
b0 b1 t b2 t 2
C2
tC 1 1
q q q q C1 C2 exp a t t S C2 C1
ln q exp aS tC tC 1 ,
2
1
where the argument of each logarithm must be positive for non-complex solutions
ln qC qC aS exp aS tC tC
y t instantaneous logarithmic relative growth as a function of time
b1 2 b2 t 2 , b0 b1 t b2 t
ln see (16)
where the argument of the logarithm must be positive for non-complex solutions
1
2
2
1
qC1 exp aS t tC1 ln qC exp aS t 2 tC tC , 2 1 2 q q exp a t t C1 S C2 C1 C2
where the argument of each logarithm must be positive for non-complex solutions
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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______________________________________________________________________________________ [3] A. R. Weiskittel, D. W. Hann, J. A. Kershaw, and J. K. Vanclay, Forest Growth and Yield Modeling. Chichester, West Sussex: Wiley-Blackwell, 2011. [4] E. E. Maor, The Story of a Number. Princeton, New Jersey: Princeton University Press, 1994. [5] M. Ricker and R. del Río, “Projecting diameter growth in tropical trees: A new modeling approach,” Forest Science, vol. 50, no. 2, pp. 213-224, 2004. [6] A. Jeffrey and H. H. Dai, Handbook of Mathematical Formulas and Integrals, 4th edition. Amsterdam: Elsevier, 2008. [7] C. G. van der Laan and N. M. Temme, Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-like Functions. Amsterdam: Centrum voor Wiskunde en Informatica (CWI), 1984. [8] M. Ricker, V. M. Peña Ramírez, and D. von Rosen, “A new method to compare statistical tree growth curves: the PLGMANOVA model and its application with dendrochronological data,” PLOS ONE, vol. 9, no. 14, e112396, 2014. [9] P. Pecina, “On the function inverse to the exponential integral function,” Bulletin of the Astronomical Institutetutes of Czechoslovakia, vol. 37, no. 1, pp. 8-12, 1986. [10] S. Winitzki, "Computing the incomplete gamma function to arbitrary precision," Lecture Notes in Computer Science, vol. 2667, pp. 790-798, 2003. [11] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, 2nd edition. Cambridge: Cambridge University Press, 2005. [12] N. H. F. Beebe, Programming Using the MathCW Portable Software Library. Springer International Publishing, 2017. [13] A. Souto Martinez, R. Silva González, and A. Lauri Espíndola, “Generalized exponential function and discrete growth models,” Physica A: Statistical Mechanics and its Applications, vol. 388, no. 14, pp. 2922-2930, 2009. [14] H. S. Migon and D. Gamerman, “Generalized exponential growth models: a Bayesian approach,” Journal of Forecasting, vol. 12, pp. 573-584. [15] R. A. Fisher, “Some remarks on the methods formulated in a recent article on ‘The quantitative analysis of plant growth’,’’ Annals of Applied Biology, vol. 7, pp. 367-372, 1921. [16] I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics, 5th edition. Heidelberg: Springer, 2007. [17] J. Schnute, “A versatile growth model with statistically stable parameters,” Canadian Journal of Fisheries and Aquatic Sciences, vol. 38, pp. 1128-1140, 1981. [18] Y. Lei and S. Y. Zhang, “Comparison and selection of growth models using the Schnute model,” Journal of Forest Science, vol. 52, no. 4, pp. 188-196, 2006. [19] J. X. Pan and K. T. Fang, Growth Curve Models and Statistical Diagnostics. New York: Springer, 2002. [20] T. Kollo and D. von Rosen, Advanced Multivariate Statistics with Matrices. Springer, 2005.
5) Multiplying q t by a factor: In all three functions, the relative growth rate remains unchanged for any given time after multiplication by a factor. 6) Shifting the logarithmic relative growth rate y q ln q t q up or down: In the generalized exponential function, the shift can be combined with a single parameter as anew a shift , for example to calculate confidence curves that correspond only to the relative growth rate. This is possible because the shift does not affect the inflection point quantity. In the 2nddegree polynomial, the shift can be incorporated into the parameters as b1, new exp shift b1 and b2, new
exp shift
2
b2 , whereas b0 remains unchanged. In
the Schnute model with b 1 , the shift of y q is equivalent to changing the and aS , new aS exp shift
t
C2
tC
1
coefficients to tC ,new tC 2
1
exp shift . SUPPORTING INFORMATION
Three files are available from the first author (
[email protected],
[email protected]), and will also be made available on the Internet: 1) The file “S1 Derivations for the polynomial and Schnute model.pdf” provides the derivations of the two models that are compared with the generalized exponential function in the last section. 2) The file “S2 Inverse of the exponential integral function Ei[x].pdf” consists of a Mathematica notebook converted to a PDF file, where the methods to calculate Ei (x1)0 z Ei and Ei (x1)0 z Ei were programmed.
3) The file “S3 Additional information about functions shown in figures.pdf” contains detailed explanations about the computational methods and numerical results for generating the figures.
ACKNOWLEDGMENT We thank Christina Siebe and Víctor M. Peña Ramírez (Geology Institute, UNAM) for their support in developing this article.
Article modified on 25 July 2018: Equations (16) and (17) were corrected, the legend of Figure 1 was complemented, and some information in sections V and X was added. Article modified on 9 August 2018: Some formatting matters were corrected, and the file names of the supporting information modified.
REFERENCES [1] M. J. Panik, Growth Curve Modeling: Theory and Applications. Hoboken, New Jersey: John Wiley & Sons, 2014. [2] H. E. Burkhart and M. Tomé, Modeling Forest Trees and Stands. Dordrecht: Springer, 2012.
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(Revised online publication: 9 August 2018)
26 July 2018
Supporting Information for Ricker, M., & D. von Rosen. 2018. A generalization of the exponential function to model growth. IAENG International Journal of Applied Mathematics 48(2): 152-167.
S1. Derivations for the polynomial and Schnute model Martin Ricker and Dietrich von Rosen (
[email protected],
[email protected])
Derivation of the formulas for the 2nd-degree polynomial: The 2nd-degree polynomial is expressed as
q t b0 b1 t b2 t 2 ,
(1)
where q t is a quantity as a function of time t , and b0 , b1 , and b2 are coefficients. Taking the second derivative of (1) with respect to t yields q t 2 b2 , which cannot be zero for any t . Therefore, the function has no inflection point. It does have an extreme value. The first derivative of (1) with respect to t is the instantaneous increment:
q t b1 2 b2 t. Setting equal to 0, and solving for t results in tExtreme
b1 . 2 b2
Substituting t in (1) with the formula for tExtreme yields
qExtreme b0
1
b12 . 4 b2
(2)
The logarithmic instantaneous relative growth as a function of time is the natural logarithm of (2) divided by (1): q t b1 2 b2 t , y t ln t ln 2 b0 b1 t b2 t q t
(3)
where the argument of the logarithm must be positive for non-complex solutions. Solving (1) for time results in two solutions: t q
b1 b12 4 b2 q b0
(4)
2 b2
Replacing time in (3) with (4), and simplifying, results in b2 4 b q b q t 1 2 0 , y q ln q ln q q t where the argument of the square root and q must be positive for non-complex solutions. With q 0 , furthermore only the positive argument of the logarithm results in noncomplex logarithmic relative growth y q . Next, we consider a time shift t of the polynomial function: q t b0 b1 t t b2 t t . 2
Expanding the products and re-arranging slightly leads to
q t b0 b1 t b2 t 2 b1 2 b2 t t b2 t 2 , so that b0,new b0 b1 t b2 t 2 , b1, new b1 2 b2 t , and b2,new b2 . Finally, shifting the whole curve of y q results in
2
b2 4 q b b 1 0 2 y q shift ln q
ln exp shift
b2 4 q b b 1 0 2 ln exp shift q b 2 4 q b b exp shift 2 1 0 2 ln q ln
exp shift b
2 4 q b0 exp shift b2 . q
2
1
Therefore b0, new b0 , b1,new exp shift b1 , and b2,new exp shift b2 . 2
Derivation of the formulas for the Schnute model with b = 1: The function of the Schnute model with b 1 is
q (t ) qC1 qC2 qC1
1 exp aS t tC1 , 1 exp aS tC2 tC1
(5)
where a quantity qC1 at time point tC1 grows to qC1 at time point tC2 . The coefficient a S represents a growth rate parameter of the growth curve. To be realistic, all q should positive. With aS 0 and t , one calculates the asymptotic quantity (5) as
q ( ) qC1 qC2 qC1 qC1
q
C2
1 exp aS tC1 1 exp aS tC2 tC1 qC1
1 exp aS tC2 tC1 3
.
Alternatively, aS 0 and t gives the same result. Taking the first derivative of (5) with respect to t yields the instantaneous increment:
aS exp aS tC2 t dq . q t qC2 qC1 dt exp aS tC2 tC1 1
(6)
With aS 0 and qC2 qC1 0 , there is no solution for t that would result in q t 0 . Consequently, there is no extreme point for (5). The second derivative of (5) with respect to time is
aS2 exp aS tC2 t dq . q t qC1 qC2 ddt exp aS tC2 tC1 1
Setting q t equal to zero has no solution for t , and consequently there is no inflection point. Next, dividing (6) by (5) and taking the natural logarithm, results in the formula of logarithmic instantaneous relative growth as a function of time:
q t y t ln t q t
ln qC1 qC2 aS exp a tC2 tC1 q exp a t t q exp a t 2 t t C1 C1 C2 C1 C2 S S , ln qC2 qC1 exp aS tC2 tC1
(7)
where the argument of each logarithm must be positive for non-complex solutions. Solving (5) for t :
qC1 qC2 1 . t (q ) tC1 ln q q q q exp a t t aS C2 C1 S C1 C2
4
(8)
We substitute t in (7) with (8), in order to get the logarithmic relative growth as a function of q , and simplify:
q t y q ln q q t
ln aS q qC1 qC2 q exp aS tC2 tC1 ln q exp aS tC2 tC1 1 ,
(9)
where again the argument of each logarithm must be positive for non-complex solutions. A vertical shift of y q in (9) leads to the following changes in the parameters: aS , new aS exp shift ,
tC2 ,new tC1
tC2 tC1
exp shift
.
Derivation: Adding on both sides of (9) the parameter shift :
a q q q q exp a t t S C1 C2 S C2 C1 exp shift . (10) y q shift ln q exp aS tC2 tC1 1
On the other hand, introducing aS ,new and tC2 ,new , instead of adding shift on the right-hand side:
a S , new qC1 q q qC2 exp aS , new tC2 , new tC1 . y q shift ln q exp aS ,new tC2 ,new tC1 1
(11)
When the whole curve of (9) is shifted vertically, there is no change of its shape. Therefore, the derivatives of (10) and (11) with respect to q can be set equal:
5
d y q shift dq d y q shift dq
qC2 exp aS tC2 tC1 qC1 q qC1 q exp aS tC2 tC1 q qC2
qC2 exp aS ,new tC2 ,new tC1 q qC1 q exp aS ,new tC2 ,new tC1
q q q
,
C1
C2 , new
.
Setting the right-hand sides of the two derivatives equal and simplifying:
exp a
q q q C1
C2
tC2 tC1 exp aS ,new tC2 ,new tC1 qC1 qC2 0. exp aS tC2 tC1 qC1 q qC2 exp aS ,new tC2 ,new tC1 S , new
The left-hand side will be zero, when the numerator is zero:
exp a
aS tC1 tC2
.
S , new
aS ,new
tC2 tC1 exp aS ,new tC2 ,new tC1 qC1 qC2 0
tC1 tC2 ,new
(12)
Next, we set the right-hand sides of (10) and (11) equal, substitute aS ,new with (12), and simplify:
a q q q q exp a t t C1 C2 S C2 C1 S tC2 tC1 exp shift tC1 tC2 ,new q exp aS tC2 tC1 1 tC1 tC2 ,new
0.
Solving this equation for tC2 ,new results in:
tC2 ,new t1
t2 t1 . exp shift
Substituting tC2 ,new from (13) in (12) results in anew a exp shift .
6
(13)
Finally, one can show that for aS 0 the limit yL ln aS for q in (9). Starting with (9):
a q q q q exp a t t S C1 C2 S C2 C1 y q ln q exp aS tC2 tC1 1
q q exp a t t
q qC1 C2 S a S q ln exp aS tC2 tC1 1
C2
C1
q exp aS tC2 tC1 q exp aS tC2 tC1 a q qC1 C2 S q q q q ln exp aS tC2 tC1 1
exp a
q exp aS tC2 tC1 a 1 qC1 C2 S q q ln exp aS tC2 tC1
S
tC2 tC1
1
For q :
a 1 0 0 exp a t t S S C2 C1 yL ln exp aS tC2 tC1 1 1 exp aS tC2 tC1 ln aS exp aS tC2 tC1 1
ln aS .
7
.
26 July 2018
Supporting Information for Ricker, M., & D. von Rosen. 2018. A generalization of the exponential function to model growth. IAENG International Journal of Applied Mathematics 48(2): 152-167.
S2. Inverse of the exponential integral function Ei[x] Martin Ricker and Dietrich von Rosen (
[email protected],
[email protected])
Introduction Mathematica includes the function “ExpIntegralEi” to handle and evaluate numerically the exponential integral function Ei[x]. Let us call any value of the function zEi , i.e., zEi = Ei[x]. A graph from x = -2 to x = 3 shows a negative branch, denominated here
Eix0[x] (in blue): EiPoints = Table[ y = ExpIntegralEi[x]; {x, y}, {x, - 2.5, 3.5, 0.001}];
negEiPoints = Cases[EiPoints, {x_ /; x < 0, _}];
posEiPoints = Cases[EiPoints, {x_ /; x > 0, _}];
Ricker S2 Inverse of the exponential integral function Ei[x].nb
Show[ ListLinePlot[negEiPoints, PlotStyle {Red, AbsoluteThickness[2]}, PlotRange Full], ListLinePlot[posEiPoints, PlotStyle {ColorData[59, 1], AbsoluteThickness[2]}, PlotRange Full], PlotRange {{- 2, 3}, {- 4, 11}}, Frame {{True, True}, {True, True}}, FrameStyle 14, FrameLabel {Style["x", Black, FontSize 14], Style["Ei[x] = zEi ", Black, FontSize 14]}, ImageSize 500, AspectRatio 0.75, ImageMargins 8, Epilog { {Text[Style["Eix>0 [x]", ColorData[59, 1], FontSize 14], {1.5, 6}]}, {Text[Style["Eix0 [zEi ]", ColorData[59, 1], FontSize 14, {3.4, 2.6}, (-1)
TextStyle"Eix
