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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

A Generalization of the Exponential to Model Growth

Martin Ricker and Dietrich von Rosen

Abstract—We generalize the to model generalized exponential function, to derive the corresponding instantaneous relative growth. The modified function is function of q t . Furthermore, we derive formulas for the defined by a linear relationship between a continuous quantity conversion of a segmented curve of logarithmic relative growth (rather than time) and logarithmic relative growth. The 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 q t q t is instantaneous relative growth of a quantity q , a 2nd-degree and the nonlinear Schnute function. In conclusion, the generalized exponential function is useful for t refers to time, a denotes initial logarithmic relative growth, modeling a path of changing relative growth continuously, and and b is a shape parameter in terms of its sign, as well as a to translate it into a growth curve of quantity as a function of scaling parameter in terms of its magnitude. For calculating time. 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, addressed. In order to distinguish two possible solutions for Exponential integral Eix , Growth curves, Inverse of the ( 1) given zEi , we define the two inverse functions Eix0 zEi  and exponential integral function, Explosive growth, Schnute Ei( 1) z  . An indirect method for their numerical function, Sigmoid growth x0 Ei  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 “explosive growth” refers to a relative growth rate that Growth curves are monotonically increasing functions over increases with time. The resulting formula of generalized time. In the case of negative growth, they can also be exponential growth is monotonically decreasing functions. Growth curves are of  1 interest in a variety of fields, such as , medicine,   Ei( 1) t t  expa +Ei  b  q  for b  0,  b x0  C C  engineering, or economics [1]. In forest science, there is a q t    1 long history of applying statistical growth curve models to   Ei( 1) t t  exp a +Ei  b  q  for b  0,  b x0  C C  predict tree size and yield (e.g., [2-3]). To model various types of growth processes, the main aim of our article is to where qC is a calibrating quantity at time tC . For b  0 , the study a generalization of the exponential function [4]. This two functions equal the (standard) exponential function generalization is constructive from a mathematical point of view, although we are also interested in subsequent q t qC  exp t tC  exp a . In the case of sigmoid growth, statistical applications of tree growth data or other empirical the inflection point quantity is 1/ b , which depends only on growth phenomena. Here, the non-statistical, mathematical one parameter ( b ). Negative growth can be modeled by aspects are treated. 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 Manuscript received on September 22, 2017; revised on April 18, 2018. function is that relative growth satisfies The Dirección General de Asuntos del Personal Académico (DGAPA) of the Universidad Nacional Autónoma de México (UNAM) provided a q t q t =exp c t  , (1) stipend for the first author (MR), to spend a sabbatical year at the Swedish University of Agricultural Sciences in 2014-15. 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: q t q texp  a  b q t  , (2) [email protected], [email protected]).        D. von Rosen is with the Department of Energy and Technology, where the term c t has been substituted with a b  q , with Swedish University of Agricultural Sciences, Uppsala, Sweden, as well as with the Department of Mathematics, Linköping University, Linköping, a and b being constants. The original motivation for this Sweden (e-mail: [email protected]). substitution was to establish a linear relationship with

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

logarithmic relative growth of the form where |b  v | refers to the absolute value of b  v , and i! denotes the of i . Furthermore, the Euler- ln q t q t   a  b q t , when time t is unknown [5].  n 1  However, it turns out that the resulting model is interesting Mascheroni constant  lim  ln n   n      also when time is known. It will be shown that the growth  k1 k  function q t can be sigmoid, exponential, or explosive, 0.577215... has been added in (4), so that (4) represents the where the term “explosive” refers to a relative growth rate Cauchy principal value of Eix . The reason is that Eix that increases with time. Furthermore, the resulting growth has a singularity at x  0 (Figure 1 top). Whereas in a function has some interesting mathematical properties. 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 theoretical basis for taking the integral over x  0. The the exponential integral function Eix is treated, and the exponential integral Eix has been discussed final formulas of the resulting growth function are mathematically in detail in [7]. Note that Ei[0]   , but presented. The inflection point in sigmoid growth, and the (exp[0v ] / v )dv   (1/ v )d v  ln[]v . relationship among the parameters are discussed. Negative

growth is also treated. Subsequently, the new function is The integral from qC to q can be calculated numerically compared with that of standard exponential growth. as Eib  q Ei b  q  . For example, in the program Furthermore, a procedure is derived for the conversion of a C segmented curve of logarithmic relative growth as a function Mathematica (www.wolfram.com/mathematica/), one can of time, into a growth curve of quantity as a function of time. use the function “ExpIntegralEi”. Ricker and del Río Finally, the new function is compared with a polynomial (Appendix 3 in [5]) also derived a relatively easy-to- function and the nonlinear Schnute function. implement procedure to calculate the integral up to a desired accuracy, with a remainder R x,i : x1 x 2 xi Eix    ln  x    ...  II. DERIVATION OF THE GROWTH CURVE FUNCTION   1!1 2!  2i ! i (5a)

R x,,i  Since q t is a monotone function, we can first consider where i refers to a chosen integer . For i x  2 : t q instead of q t . Write (2) as dq d t q  xi1 expa b q and transform it into dt d q  R x,i   . (5b) 2 x  1q exp a  b q . Let dt q d q  t q : given q  0 , i! i  1  1          i  2  we can write For an interpretation of Eib  q Ei b  qC  , see pages 5- exp a  bq  t q     . (3) 6 in [8]. q 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  can be calculated as letting qC at tC be a calibrating point on the growth curve,  yields Eib  q   Ei  b  qC  tC  for b  0,  q  exp a  exp b  v  t q    (6) tC exp  a   dv for b  0,     v  lnq   ln  qC   qC t  for b  0. t q   C    q  exp a   1 texp  a   d v for b  0. C    v For q qC the resulting time period t tC is negative. From  qC (6), the initial logarithmic relative growth rate is derived: 1 Note that dv equals lnv . The exponential integral  Ei[b  q] Ei[ b  q ]  v ln  C  for b  0, t t expb  v   C   dv , however, represents an infinite series, and a   (7) v  ln[q] ln[ qC ] ln   for b  0. is denoted Eib  v (page 176 in [6]):   t tC 

1 2 Since x in Ei x or ln x has to be positive (for non- b  v   b  v      Ei b  v   ln  b  v   ... , (4) complex solutions), one gets the following restrictions for   1!1 2!  2 (7):

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

q q for t  t , and q q for t  t . (8) ( 1) C C C C x zEi   Eix0 zEi  . It is clear which case applies in (6): The restriction of (8) is derived in the following way: With q in Eib  q is always positive, and therefore

t tC in the case of b  0 in (7), one needs  ( 1) with b0   b q  Eix0  zEi   , Eib  q Ei b  qC . Using the notation shown in Figure zEi  b q Ei    ( 1) 1, for b  0 and positive q , this translates into: with b0   b  q  Eix0  zEi  . Ei b  q Ei  b q  b q b q  q q . x0 x0 C C C On the other hand, for b  0 and positive q , one gets:

Eix0 b  q Eix0  b qC   b q b qC  q qC . 10

The reason for b  q  b  qC in the latter case (rather than

b  q  b  q ) is that Ei x decreases with increasing Eix 0 x C x0 5 Ei x. Therefore, for both b  0 and b  0 one gets the z

condition q qC . For t tC , the inverse logic applies. x Ei x

Ei x 0 For b  0 , the parameter a refers only to the initial 0 logarithmic relative growth, i.e., infinitesimally close to   zero: a lim ln  qt q   (see Figure 3 bottom). For the q0 5 logarithmic relative growth at any quantity q  0 , we 2 1 0 1 2 3 introduce the variable y : y q  ln q t q t  for q  0. x

In correspondence with (2), the relationship between a and 3 y is 1 Eix 0 zEi y a  b q . (9) 2

If b  0 (no slope in Figure 3 bottom), we get y a for any 1 q , i.e., y does not change with q ; this is the case for the x (standard) exponential function. 0

1 1 III. THE INVERSE OF THE EXPONENTIAL INTEGRAL Eix 0 zEi 2 FUNCTION Eix 5 0 5 10

zEi With (7), one can calculate t q , i.e., time as a function of quantity. To calculate q t for given t , the inverse of the Figure 1. Top: The exponential integral function Eix   ln x x1  1!1   x2  2!  2  ... with a singularity at exponential integral function Eix is needed. The inverse ( 1) x  0 , where both branches Eix0x and Eix0x approach Ei zEi   x , however, is more problematic to calculate  . Upwards, Eix0x goes to  , and Eix0 x to zero. than Eix  zEi . There are two reasons for this: First, for Bottom: For zEi  0 , the inverse of Eix  zEi presents two Ei( 1) z  negative zEi , Ei  has always two solutions (see ( 1) solutions of x . Note that Eix0  0 , corresponding to ( 1) Figure 1 bottom), so that Ei zEi  does not represent a ( 1) Ei0   ; Eix0 z Ei   0.372507 (when given with six true function. One solution corresponds to the case where x decimals); Ei( 1) 0   ; and Ei( 1) z  approaches quickly is positive in Figure 1 top, and the other where x is x0 x0 Ei zero for more negative z , so that one has to consider sufficient negative. To address the problem of calculating the inverse, Ei one has to distinguish between the two branches of the accuracy of the involved in the calculation of z Ei . original function (Figure 1 top):

Eix0 x for x  0, The second reason, why the inverse function of Eix zEi   Eix0 x  for x  0. represents a problem, is that there is no formula for calculating it. Only some functions for numerical We define the solutions for the inverse in Figure 1 bottom approximations within certain intervals of x exist (see [9]). ( 1) as two distinct (true) functions: x zEi   Eix0 zEi  and One (not very elegant) way to calculate the inverse

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

always in the domain of the complex , but the indirectly is to repeatedly solve numerically Eix  zEi : imaginary part of Ei  x at the end becomes zero in the For example, Ei1.5 is 3.301, when given with three x0 solution. decimals, so that the inverse of the exponential integral function at 3.301 is 1.5 (see Figure 1). To solve, for example, Thus, the problem has been converted in the following Eix  3, one has to search iteratively below 1.5 to find x. way: We wish to find x for given zEi , in order to find the ( 1)   In the following, we present a new way of solving this value of the inverse functions x zEi   Eix0 zEi  or problem. We can express (4) as ( 1) x zEi   Eix0 zEi  :  xk Eix  ln x  . (10)     k1 k! k ln  1    0, x  for x  0, zEi   (13) The can be represented as an infinite series  0,x  for x  0. (page 137 in [6]): Whereas this still does not provide an explicit formula for   2k  1   2 x  1  x zEi  , the advantage of (13) is that there is no ambiguity ln x     for x  0. (11)  2k  1 x 1   ( 1) k1  anymore of two function values for Ei x  , since the Inserting (11) into (10) yields upper and lower branches have been separated (Figure 1 bottom). To find x for given z , one has to use a root- 2k  1 Ei   k   2 x 1  x  finding algorithm (see chapter 9 of [11]). Eix        . k1 2k  1 x 1k !  k      Concerning computation, the numerical evaluation of the incomplete gamma function has been treated in [10], pages It turns out that this is equivalent to 221-224 of [11], and pages 560-567 of [12]. When using

x 1  Mathematica, the computation of x zEi  is summarized in Eix  2  atanh    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 where atanh refers to the inverse hyperbolic tangent for the indirect search are provided (see also the section function (page 142 in [6]), and  to the incomplete gamma “Supporting Information” at the end of the article). function (page 236 in [6]): 1 1 z  atanhz    ln  , and 2 1 z  TABLE 1. SUMMARY FOR CALCULATING NUMERICALLY THE  INVERSE OF THE EXPONENTIAL INTEGRAL FUNCTION zEi  Ei x 1 0, x   v exp v d v. x x z  ( 1)   ( 1)    Ei  =Eix0 zEi  =Eix0 zEi  With z x 1  x  1 in atanhz : Given zEi , with x 1  1 x1   x  1  Given z , solve z z  0 , solve atanh    ln 1   1   Ei Ei Ei x  21 x 1  x  1         ln1   0, x zEi  0,  x Formula ln x  numerically for x in numerically for x in    . the domain of real the domain of real 2 numbers numbers ( zEi  0 does not exist in this branch) Substituting atanh  x1  x  1 in (12), distinguishing between positive and negative x , and Re[ FindRoot[ zEi = = Re[ FindRoot[ zEi = = simplifying, the following two equations are obtained: Log[1]  Gamma[0,  Gamma[0, x], {x, Mathematica x], {x, zStart}, zStart}, code WorkingPrecision WorkingPrecision Eix0  x  ln 1   0, x  , >30, MaxIterations >30, MaxIterations Eix0 x    0,. x  >1000] >1000] These equations provide a mathematical filter to distinguish for zEi   0.5: between the two branches in Ei x , for x  0 and x  0 , z   Recommended Ei  expz  , and for zEi  1: 10 , and  Ei  by keeping the branch of interest for given zEi and either values for zStart for z 1: ln z  for  0.5  zEi  0: positive or negative x , in the domain of real numbers, while Ei Ei  ln z  moving the other branch into the domain of complex  Ei  numbers. Not that in the case of x  0 , ln1  i  is

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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, Ei 0 q Ei  0  qC  completely different approach to find numerically the 1 2 inverse of the exponential integral function. First, one can 0 q qC    0  q  qC   lnq   ln  qC    ... use (5a) with lnx for Eix0 x and lnx for 1!1 2! 2

 lnq  ln  qC  . Eix0 x, instead of ln x  . The accuracy is controlled with the remainder of (5b). The second approach, given in For sigmoid growth ( b  0 ) in (14), and t   , q t the supporting information, consists of a segment-wise will not converge to a maximum quantity. Using (2), one can numerical simulation of the two branches of the function in see that q t q t exp[] a b  q t , which for increasing Figure 1 bottom, with high-degree (e.g., 31       coefficients) and a high number of calculated data points q t  0 remains always positive. (e.g., 10,000) in each of several segments. Equation (14) with b  0 represents the formula for

calculating relative growth from a smaller qC to a larger

q t ln q  ln q t t IV. THE FINAL FORMULAS OF THE GENERALIZED       C   C  , suggested already as the EXPONENTIAL FUNCTION “correct” formula by Sir Ronald Fisher in 1921 [15]. It reflects exponential growth. Whereas this formula is Employing the inverse of Eix , (6) can now be re-arranged relatively well-known by field biologists, (14) with b  0 to calculate q t , rather than t q . With t t and q  0, for sigmoid growth and for explosive growth, as the more C C general cases, are new. we define the generalized exponential function as: For calculating the logarithmic relative growth at any    1 t tC  exp a  time, q in (9) is substituted by q t from (14):  Ei( 1)   for b  0,  b x0     +Ei b  qC     t t exp a    ( 1) C     a  Ei   q t q exp t t  exp a for b  0,  x0      C C    +Ei b qC         for b 0 and a  y t   0,  1 t tC  exp  a  ( 1)      Eix0 for b  0, y t a for b  0, (16)  b +Ei b  q        C     t t exp a  (14)  ( 1) C    a  Ei   x0 +Ei b  q   where b  0 represents sigmoid growth, b  0 exponential    C    growth, and b  0 explosive growth, and where for b  0  for b 0 and a  y t   0, the restriction t t exp a+ Ei b  q  0 applies.  C     C  where for b  0 the restriction of (15) applies. With the left- Rearranging this restriction leads to hand side y t involved in the condition to choose among Ei b  qC   options for b  0 , this formula is only practical if one knows t tC  for b  0 . (15) exp a  that a y t will be positive or negative. Otherwise, if one The restriction in (15) represents an upper bound for t , and does not want to calculate t y instead of y t , the reflects the fact that for explosive growth with given alternative is to employ (9), where q t has to be coefficients, there is a maximum time t that can be reached calculated first with (14). for given parameters a , b , qC , and tC , no matter how large q is. Furthermore, the instantaneous increment as a function of time is We call (14) “generalized exponential function”. The    term has rarely been used before for other functions. In qt exp  a  b qt  for b  0, unrelated form, the term “generalized ( q -)exponential q t    (17)   qC exp a   t tC  exp a  for b  0, function” is used in [13] (page 2923) to refer to    1q ' e x  lim 1  q ' x  . In [14] (page 574), the term q q' q   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 1  f t   a  bexp  t  . to simplification, but would involve twice the inverse of Note that in computations with (14) one can also use the Ei x, causing in turn complicated conditions if formulas for b  0 arbitrarily close to b  0 , instead of ( 1) ( 1) Eix0 zEi  or Eix0 zEi  applies. For b  0 , the using the one for b  0 . One can show this with (6), the formula is derived from (2). For b  0 , the formula is formulas that form the basis for (14). Using (4) and b  0 , derived from (14) by taking the . the formulas convert into each other:

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

V. DETERMINATION OF THE FUNCTION PARAMETERS FROM formula for the inflection point is of interest for several THREE DATA POINTS reasons:

(i) The inflection point characterizes the shape and scale The generalized exponential function of (14) with b  0 is of a growth curve. In a nonlinear growth curve model, a three-parameter model, whose parameters a , b , and qC there are few points on the curve that can be used to

(at tC ) or tC (at qC ) are thus determined exactly by three compare different curves of a given function: In the case of the generalized exponential function, there is points q1 at t1 , q2 at t2 , and q3 at t3 , with t1 t 2  t3 one calibrating point ( qC at tC ), as well as for b  0 and q1 q 2  q3 or q1 q 2  q3 . Given such three data the inflection point ( qIP at tIP ). points, any one can represent qC at tC . To determine b , (ii) The formula for calculating the location of the one has to solve the following relationship numerically and inflection point reveals important information about indirectly (in Mathematica with the FindRoot function, with the dependence of the growth curve’s shape, location, 1 being generally a recommended starting value): and scale on the different parameters: How does     changing the function’s parameters affect the Ei b  q3  Ei   b  q1  t t  3 1 . (18) inflection point, and consequently the whole growth     t t Eib  q2  Ei  b  q1  2 1 curve? The formula is derived by transforming the equation of (7) (iii) It can also be of interest to develop a growth curve around a known inflection point, i.e., to derive the for b  0 with q1 at t1 being the calibrating point qC at tC , growth function’s parameters with a known (fixed) resulting in expa   Ei  b q   Ei   b  q1  t q   t1 . inflection point. By substituting q at t q with q at t on the one hand, 2 2 The inflection point quantity ( q ) is derived by taking and with q at t on the other, one obtains two equations IP 3 3 the second derivative of (6) for b  0 with respect to q : for expa , which are set equal. Having determined b , one 2 2 2 calculates dt dq1 b q  exp   a  b  q  q . (19)

      Setting this expression equal to zero, and solving for q , Ei b  q3  Ei  b  q1  a  ln  . yields:  t t   3 1  qIP  1 b, (20) Since the generalized exponential function represents a continuous spectrum of growth functions from sigmoid to which is positive only for b  0 . This relation reveals a explosive, one expects to always find a solution for b in surprisingly simple interpretation of b as the negative (18), and subsequently for a. Note that the three points may inverse of the inflection point quantity. For b  0 , i.e.,

lie on a straight line, but that the resulting function will explosive growth, qIP would be negative, which however represent nevertheless in that case a sigmoid curve among has no real-world interpretation, and thus will not be the points. considered further. An alternative method to determine a and b for b  0 is Inserting (20) into (6) for b  0 results in the to use nonlinear regression with (6), which (at least in corresponding inflection point time ( t ): Mathematica) works well. The regression equation is IP

t t Ei b  q  Ei  b  q  exp a . With three data Ei 1  Ei  b qC  1    1    t t      for b  0, (21) IP C exp a  points of t q , the resulting coefficient of determination has   to be 1. The regression can also be used for more than three where Ei1  1.89512 , when given with five decimals. data points, in which case it becomes a true regression Furthermore, as a necessary condition, one has to show analysis, with (generally) variance of the residuals and 2 that there is a change of sign of the second-derivative at the R  1. In that case, one can also let the regression inflection point (page 231 in [16]). With q  0 , the second determine either q or t . Note, however, that the residuals C C factor expa  b  q q2 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).

VI. INFLECTION POINT IN NON-EXPONENTIAL GROWTH For b  0 , 1  b q  0 implies q 1 b , whereas

1  b q  0 implies q 1 b . For b  0 , An inflection point refers to the point on a growth curve,   where the curve switches from being left-winged (locally 1  b q  0 implies q 1 b , whereas 1  b q  0 convex) to being right-winged (locally concave), or vice implies q 1 b . versa (circles in Figure 2). Knowing and analyzing a

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

25 b 0.2 a 0 15 a 0.5 20 b 0.1 q q a 0 10 15 b 0.2 antity antity a 0.5 10 Qu Qu 5 b 0.3 5

0 0 2 4 6 8 10 2 4 6 8 10 Time t Time t

15 20 a 0, b 0.2 a 0

15 b 0.2 1.5 10 q 1.9 q 1 1.5

q C q C qC 1 qC 1 10 q 1 0.5 b 0.2 antity qC 0.1 antity C

Qu 5 Qu 5 b 0.2 0.5

0 0 2 4 6 8 10 2 4 6 8 10 Time t 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  b  q .

Consequently, there is always a change of sign at qIP . winged (concave) part of the growth curve appears for Since the second derivative switches from negative to positive time. 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 VII. THE RELATIONSHIP AMONG THE PARAMETERS a , b ,

(6) is the formula for t q . In the case of qt in (14), the 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 In the generalized exponential function, the three parameters being left-winged to being right-winged. 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 curve segment, a is the initial logarithmic relative growth, i.e., the intercept at q  0 (see Figure 3 bottom). It serves as time for b  0 means that tIP  tC , i.e., only the right- a standardized growth rate parameter (see Figure 1 in [8]).

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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, VIII. NEGATIVE GROWTH WITH THE GENERALIZED as well as a scaling parameter in terms of its magnitude. EXPONENTIAL FUNCTION

Different combinations of parameter values between a According to the standard exponential function, growth can and b cause different curvatures. Note that the second be positive or negative. To describe negative growth with of (6) for b  0 , as measures of the the generalized exponential function, presented in (14), one curvature, depend on both a and b : has to replace t tC with tC  t . We add the subindex NG 2 for negative growth, so that (14) changes to d t Eib  q   Ei  b  qC     , (22a) da2 exp a       1 tC  t  exp a   ( 1)    Eix0 for b  0, 2  b   dt 1 +Ei b  qC   exp a  b q  q      2 2  C   db b (22b)   (24) qNG  t   qC exp tC  t   exp  a  for b  0, 1b  q exp b  q   1  b  q  exp  b  q  .  C      C     1 tC  t  exp  a   Ei( 1)   for b  0, For example, the more negative a in (22a), the larger is the  b x0 +Ei b  q     C   curvature. Extreme curvatures are associated with extreme parameter values of a and b . where b  0 represents negative sigmoid growth, b  0 Figure 2 shows different combinations among the three negative exponential growth, and b  0 negative explosive

parameters a , b , and qC in the case of sigmoid curves growth. The restrictions in (8), when b  0 , convert to (b  0) . Changing t is not shown in Figure 2, because it C qNG  qC for t  tC , and qNG q C for t tC . only moves the whole curve to the left or right, as can be Furthermore, (15) changes for negative growth: seen from (6). The inflection points are indicated with circles in Figure 2. In Figure 2 top left, the parameter a is Ei b  q  t  t   C  for b  0 and negative growth. varied around a  0 (i.e., initial relative growth is 100%). C exp a  One sees that a symmetrical change of a (here 0.5 ) leads to symmetrical growth curves around the original curve, Finally, (21) converts to

with constant qIP but different inflection point times tIP . Ei 1   Ei  b qC  This is not the case for a symmetrical change of b (here tIP tC  for b  0 & negative growth. exp a  0.1), which results in asymmetrical curves around the For negative growth, one also substitutes with original growth curve (Figure 2 top right). In Figure 2 t tC tC  t

bottom left, qC is moved up and down by the same amount in (7) for calculating a , in (16) for calculating y t , and

(here 0.9 ). The qIP , however, are kept constant, which in (17) for calculating q t . causes asymmetrical shapes of the resulting growth curves

around the original one. Changing both the quantities of qC

and q (via b ) together by the same proportion results in IP IX. GENERALIZED VERSUS STANDARD EXPONENTIAL a change of scale, shown in Figure 2 bottom right and GROWTH explained next. In to b determining if the growth curve is To show how the formulas work, and how the generalized sigmoid, exponential, or explosive, the variables b and q exponential function compares with the standard C exponential function, some sample growth curves are shown are scaling q t . In (14) with b  0 , one can multiply qC in Figures 3 and 4. To get a sigmoid growth curve with

by some factor, and divide b by the same factor (say f ), inflection point tIP  7 , qIP  5 , and calibrating point  2 , q  1, according to (20) one uses b  0.2 . To to model a growth curve of qf t  f  q t : tC C calculate the corresponding a , one has to transform (21):   t tC  exp a      f ( 1) Ei 1  Ei  b qC  qf  t    Ei   b . (23) a  ln   for b  0. b +Ei  q  f   t t   C   IP C    f  The restrictions from (8) become q q for t t , and The shape of the curve remains the same, with the new IP C C q q for t t . For negative growth, one has to inflection point quantity f qIP , but unchanged inflection IP C C point time. The factor f does neither affect the initial substitute tC  tIP for tIP  tC , with qIP  qC for t tC ,

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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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

10 15 explosive t 6.54 8 calibrating point 2, 8 sigmoid

q q negative exponential 10 6

inflection point 7, 5 antity antity

Qu Qu 4 inflection 5 exponential negative sigmoid point 7, 5 2 negative explosive 0 calibrating point 2, 1 0 t 0.054 0 5 10 15 20 25 0 5 10 15 20 25 Time t Time t

0.5 0 a a q explosive q explosive y 0.6100 y 1.073 b 0.05 1.0 b 0.05 a 0.6100 a 1.073 owth 1 owth sigmoid b 0.2 exponential b 0 gr gr 1.5 a 1.465 a 2.111 exponential b 0 lative 2 lative re re 2.0 sigmoid b 0.2

3 2.5 garithmic garithmic Lo 4 Lo 3.0 0 5 10 15 0 2 4 6 8 Quantity Quantity q

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 qCCt  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 Figure 3 bottom shows the corresponding linear relationships between q t and logarithmic relative growth, resulting sigmoid growth curve. Going through qC  1 at  

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

(7) for b  0 , one can calculate the logarithmic relative the slope of the lines ( qC at tC does not appear in this growth rate for an exponential growth curve that also goes mathematical relationship). through q  1 at t  2 and q  16.2 at t  25 , which Figure 4 shows two more functional relationships for the results in a  2.11. The corresponding growth curve, growth curves from Figure 3 top. For sigmoid growth, there calculated with (14), is shown in Figure 3 top left. appears a maximum of the instantaneous increment with time. This maximum is at the inflection point time. First, the Negative growth is shown in Figure 3 top right, relationship between and is given by substituting employing (24) for negative sigmoid growth, negative tIP qIP exponential growth, and negative explosive growth, with the b in (21) with b 1 qIP from (20), and rearranging: same parameters of b as for positive growth, but choosing Ei 1   qC  q qIP (and calculating the corresponding parameters for C tIP  tC  exp a   Ei   . (25) a ). exp a  qIP 

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

2 2 t t y y 1 explosive 1 negative explosive owth

gr 0 0

lative 1

re 1 exponential 2 negative exponential 2 3 garithmic

sigmoid garithmic relative growth

Lo negative sigmoid 4 Lo 3 0 5 10 15 20 25 0 5 10 15 20 25 Time t Time t

3.0 3.0 explosive negative explosive 2.5 2.5 t t ' 2.0 ' 2.0 q q

1.5 exponential 1.5 maximum 7, 0.999 crement crement

1.0 In negative exponential

In 1.0 maximum 7, 0.629 0.5 0.5 negative sigmoid sigmoid 0.0 0.0 0 5 10 15 20 25 0 5 10 15 20 25 Time t 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, larger i refers to a later time point). Then one can find the corresponding a and b for the generalized exponential one substitutes t in (17) for b  0 , with tIP from (25), i i which yields function that goes from the first to the second point. The 1 q t    exp a 1  for b  0. (26) procedure will be useful, for example, for statistical max b   applications. There may be more than two subsequent points y at t , and one wants to use (16) to get a continuous The formula applies for both positive and negative growth. i i 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 X. CONVERSION OF A SEGMENTED CURVE OF y t INTO A   b  0, but b can be arbitrarily close to 0. GROWTH CURVE OF q t  Transforming (16) for b  0 , one gets two equations for

a single segment of the generalized exponential function, Assume that there are two known points of logarithmic where the subindex i refers to the function segment (spline) relative growth, y at t and y at t , where the integer i i i1 i1 in between time points t and t : subindex i serves to index and distinguish the points (a i i1

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Defining the coefficient c b b , and as before Ei a y   t t exp  a  +Ei b  q  , i i i1 i i   i Ci  i   i Ci  r  b  q and r  b  q , one gets i i Ci i1 i  1 Ci1 Ei a y   t  t exp  a +Ei b  q  . i i1  i  1 C i   i C  i  i  1 exp ai   Eici ri1  Ei  r i  . With t t , the first equation simplifies to ti1  t i Ci i a y  b q , which corresponds to (9). Substituting a Now only the c are still unknown. Rearranging further i i i Ci i i in the second equation with y b  q yields yields i i Ci       Ei ci  ri1   t i  1  ti  exp  ai   Ei r i  . Ei  y y b q Ei   b  q   ii1 i Ci  i Ci   The inverse exponential integral function has to be applied (27)   ( 1) t  texp y  b  q 0 for b  0. to the left-hand side: If ci r i1  0 , then Eix0 zEi  i1 i  i i Ci  i   ( 1) applies, and if ci r i1  0 , then Eix0 zEi  applies. Given The variables b and q occur always together as b  q i Ci i Ci that c r  b  q , and q  0 , the sign of b i i1 i Ci1 Ci1 i in (27). With (20), this term converts to q q within a Ci IPi determines which inverse function has to be used. Since given segment. With the definition of the ratio r  b  q : i i Ci r  b q  q q , and rearranging for more robust i i CCi i IPi ( 1) ri 0  bi  0  ci  ri1  0  EizEi  , root-finding, (27) becomes x0 ( 1) ri 0  bi  0  ci  r i1  0  Eix0 zEi  .     Ei  yi yi1  r i   Ei  ri  t  t  0, (28) Consequently, the formula for calculating c equals   i1 i  i exp yi r i   ( 1)      Eix0 ti1  t i  expai   Ei  r i  with yi y i1 (which is equivalent to b  0 ). With (28),  for r  0,  r i one can indirectly find the r , given y , y , and t t  i1 i i i1 i1 i ci   (30) ( 1)   , with a root-finding method (in Mathematica the function Eix0 ti1  t i  exp a i   Ei  r i    for r  0. “FindRoot”; a recommended starting value is 1). In  i  ri1 accordance with (9), ai is calculated for the segment The case r q q  0 does not exist for positive, finite between the two points subsequently as i Ci IPi quantities. ai y i  ri, (29) If one q is known, then one gets = where the last y ( y ) is not being used (there are n 1 C 2nt  1 1 i nt t i 2n  3 equations with the same number of unknown data of each ai and ri , but nt data of yi ). t variables: nt  2 equations of bi c i bi1  0 , and nt 1 Next, for calculating the growth curve of q t , the equations of ri b i qC  0 , corresponding to nt 1 numerical values of b and q are needed separately, i i Ci variables of bi and nt  2 variables of qC . All bi and qC rather than combined in r . Assuming that there are n i i i t are found simultaneously with a method for solving a system points yi at ti , one has to determine nt 1 segments as of linear equations (in Mathematica one can use the function continuous functions that connect the points non-linearly, “NSolve”). For negative growth, one has to adapt (27), (28), according to (16). With the nt points of logarithmic relative and (30a-b), by substituting ti1  t i with ti t i1.

growth yi t i  , the nt 1 parameters of ri and ai , and a A small example will illustrate the procedure with data from growth of tree trunk radiuses. We will use four points single known calibrating point qC , one can find bi and qC i i of logarithmic relative growth y t , shown in Figure 5 top of all segments simultaneously: Using (7) for b  0 , one left; thus there are three segments. With (28), the parameters obtains of ri are determined as r1  0.1538 , r2  1.458 , and 1 exp a   Ei  b q  Ei  b  q  r3  6.893 . With (29), the parameters of ai are calculated i  t t  i Ci1  i Ci  i1 i (given in Figure 5 top right). With three segments, there are

1   b   two ci , calculated with (30) for positive ri : c1  0.1124  Ei i b  q Ei   b  q  .  i1 C  i C  t t  b i1  i   and c  0.2245 . Finally, q is taken as one known time- i1 i  i1   2 C2 quantity point (1.41 cm at 10 years, in Figure 5 bottom left).

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q b

t 1 2.7 y1 2.76 2.7

y 2.77 y y 2 0.116 cm b2 1.03 cm 2.8 y3 2.86 2.8

owth q owth t C1 t C1 qC C2 2 gr 9 yrs gr 1.32 cm 1.41 cm b3 2.9 10 yrs 2.9 q C3 4.60 tC3 lative lative a3 4.03 1.50 cm cm

re 11 yrs 3.0 re 3.0 a2 1.31 y4 a 2.61 piecewise linear 3.1 1 3.1 3.20 relationship garithmic t 12 yrs garithmic 3.2 q 1.57 cm Lo 3.2 C4 C4 Lo 9.0 9.5 10.0 10.5 11.0 11.5 12.0 1.30 1.35 1.40 1.45 1.50 1.55 Time t years Quantity q radius in cm

1.60 0.90 qC4 1.57 cm 0.857

1.55 ar t 0.85 0.884 C ye

cm 4

in 12 yrs 1.50 qC 1.50 cm 0.838 3 mm 0.80

t 11 yrs in dius C3

1.45 t ra qC 1.41 cm ' 2 q 0.75 q 1.40 q C1 t 10 yrs ntity C 1.32 cm 2 0.70

1.35 crement Qua In 0.65 0.640 1.30 tC1 9 yrs 9.0 9.5 10.0 10.5 11.0 11.5 12.0 9.0 9.5 10.0 10.5 11.0 11.5 12.0 Time t years Time t years

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 y t , the parameters a , b , and q are determined for each of three segments ( i = 1 to 3), as explained in the text, i i  i i Ci 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 n  4 are shown. Given the parameters a , b , and q for each t i i Ci data points contains 2 4  3 5 equations, as well as five segment the continuous generalized exponential function of variables: quantity with time can now be calculated with (14) for each segment (Figure 5 bottom left). Finally, the continuous b c b 0  b  0.1124  b  0, 1 1 2 1 2 function of instantaneous increment with time is calculated b2 c 2  b 3 0  b2  0.2245  b3  0, with (17), and shown in Figure 5 bottom right. r b  q 0  0.1538 b  q  0, Two segments do connect in Figure 5 top (with dependent 1 1 C1 1 C1 variable y ) only under the condition that r2 b 2  qC 0  1.458 b2  1.41  0, 2 q r  y y r  0 . The condition arises by Ci  i i i1  i r3 b 3  qC 0  6.893 b3  qC  0. 3 3 assuming that quantities are always positive. It is derived by

The numerical results are given in Figure 5 top right, where solving the following system of five equations for ai , ai1 , the piecewise linear relationships of y q in three segments b , b , and q : y a  b q , y a  b q , i i1 Ci1 i i i Ci i1 i i Ci1

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yi1 a i  1  bi1  q C , ri  b i  qC , and ri1  b i  1  qC . i.e., the Schnute function can only model certain shapes of a i1 i i1 growth curve through four data points, even if the quantity The solution includes q q r y  y r , which CCi1 i i i i1 i q increases. If there is no solution, one can use nonlinear when assumed to be positive, results in the above condition. regression for an approximation to the four points. For Technically, q  0 is implemented in (30), where it was example, for qC  0.25 at tC  0 , q1  0.64 at t1 1 , Ci1 1 1

assumed that c r  q q  0 to decide between q  0.83 at t  2 , and qC 1.04 at tC  3, there is an i i1 Ci1 Ci 2 2 2 2 ( 1) ( 1) Ei z and Ei z . Thus, when a negative q exact solution with aS  1.198 and bS  5.609 . Increasing x0 Ei  x0 Ei  Ci q from 0.64 to 0.74, however, there is no exact solution would be necessary to connect y and y (which at this 1 i i1 anymore. Using nonlinear regression, the closest least- stage is not known yet), the “wrong” option for the inverse squares solution is with a  35.106 and b  187.928 . is chosen, resulting in a “jump”, but complying with S S

qC  0. For this reason, we will use here the Schnute function as i1 a growth curve for interpolating three (rather than four) data

points. Using bS  1 results in the formulas given in Table 2, which will always go through three quantities that XI. COMPARISON OF THE GENERALIZED EXPONENTIAL increase with time. There is, however, no inflection point FUNCTION WITH THE NONLINEAR SCHNUTE FUNCTION AND anymore with b  1, only a curved function. Two data A POLYNOMIAL S points are used as calibrating points, and the parameter aS The generalized exponential function is determined with has to be determined with a root-finding algorithm, such that only two known data points of relative growth, whereas q at t represents the third data point (see supporting other functions usually need more than two points. It is information). informative to compare the generalized exponential function Figure 6 shows that the shapes of the generalized with a polynomial, which is linear in its parameters, and exponential function, the 2nd-degree polynomial, and the with the Schnute function, a flexible nonlinear growth Schnute function are different: The 2nd-degree polynomial function [17, 18]. A polynomial function is the typical function and the Schnute function with b  1 are either function that underlies the classical growth curve model in S strictly concave or convex, whereas the generalized statistics, the Generalized Multivariate Analysis of Variance exponential function is sigmoid. Furthermore, the curvature (GMANOVA) model ([19], chapter 4 in [20]). The second- 2 of the generalized exponential function is more pronounced. degree polynomial qt  b0  b 1  t  b2 t (with b0 , b1 , Despite being more flexible, the polynomial function has the

and b2 being coefficients) connects any three data points, undesirable property for a growth function that it has a no matter their constellation, and thus in this regard is more minimum or maximum in between the data points. flexible than the generalized exponential function, for which Table 2 provides the different characteristics and the quantity q must always increase with time t . On the formulas of the three functions in comparison. The other hand, the Schnute model has the following functional derivations of the formulas are given as supporting form: information. Some aspects of interest are the following: 1 b S 1) Interpolation of three q t points: Only the generalized  1 exp a t  t    b b b  SC   q t  qSS  q  q S   1  , exponential function can be sigmoid with an inflection  C1 CC 2 1    1 exp  a t  t   point.   SCC2 1   2) Connecting two points of logarithmic relative growth as with four parameters a , b , q at t , and q at t . a function of quantity or time: Only the generalized S S C1 C1 C2 C2 exponential function is uniquely determined between the The function has two calibrating data points. For a  0 and two points, because it is based on the assumption of a b  0, the Schnute function represents a sigmoid curve with linear change of logarithmic relative growth with an inflection point quantity at quantity. For the other two functions, one can see in 1 Figure 6 right that there is a limited range of q for which b b b  1b   exp  a t  qS exp  a t   q S  S y q can be calculated. In the case of the second-degree  S  S CC2  2 SCC1  1     q  . IP   polynomial, there is a limiting minimum or maximum  exp a t  exp a t    SC2  SC1   2 quantity, calculated as b0 b 1 4 b2  , where growth Note that this inflection point quantity depends on all switches from positive to negative or vice versa. The parameters, in contrast to the generalized exponential Schnute function with bS  1 has a limiting, lower or function, where qIP depends on a single parameter, as given in (20). Even though there are four distinct parameters, the Schnute function cannot go through any four data points,

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IAENG International Journal of Applied Mathematics, 48:2, IJAM_48_2_08 ______

6 Generalized exponential 0.8 q Generalized function 2, 0.83 y 4 exponential function 1, 0.74

0.6 owth q nd gr 2 2 degree polynomial 2nd degree 0.4 lative antity 0 polynomial re qL Qu 0, 0.25 0.845 0.2 2 Schnutefunction with bS 1

garithmic Schnutefunction with b 1 0.0 S

Lo 4 qL 0.850 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 Time t Quantity q

1.0

q Generalized 2 y exponential function 2, 0.83 0.8 1 owth nd q gr 2 degree yL 0.527 0 polynomial 0.6 lative antity Schnutefunction with bS 1 y re

Qu 1 2nd degree polynomial 0.4 1, 0.34 Schnutefunction with b 1 Generalized 2 S 0, 0.25 garithmic exponential function qL qL Lo 0.2 3 0.235 0.230 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 Time t Quantity q 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.

upper asymptotic quantity, calculated as relative growth with increasing quantity in an unlimited way. q q  q exp a t  t  1 . CC1 2 C1    S  CC2 1   4) Moving the three data points together by a time period 3) Explosive growth: In the lower right graph of Figure 6, t : The generalized exponential function develops i, one can see that only the generalized exponential relative to tC , so that moving tC by t moves the function (with b  0 ) goes from y   to  . The nd whole curve accordingly to tC   t . In the 2 -degree Schnute function with b  1 and a  0 goes S S polynomial, the three data points can be moved in time, asymptotically towards lna  , and the 2nd-degree S  resulting in changed coefficients: b0,new  b 0 

polynomial reaches the maximum (limit) yL  t b1 b 2   t  , b1,new  b 1  2b2  t and b2,new  b 2.  2  2 ln 2b2 4  b 0 b2  b 1 at q2  b0b 1  2  b2 . The Schnute model develops relative to t and t ,   C1 C2 Both functions contain phases of explosive growth, but which then become tC   t and tC   t . the generalized exponential function can increase 1 2

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TABLE 2. MAIN FORMULAS OF THE GENERALIZED EXPONENTIAL FUNCTION IN COMPARISON WITH A POLYNOMIAL AND THE NONLINEAR SCHNUTE FUNCTION

Variable with Generalized 2rd-degree polynomial Schnute function with b = 1 interpretation exponential function S

b b , b , b a , q at t , q at t Model parameters a , , qCCt  0 1 2 S CC1 1 CC2 2

Instantaneaous initial Positioning coefficient, logarithmic relative Interpretation of growth rate-related Growth rate-related coefficient, two growth, shape and scaling model parameters coefficient, shape-related positioning and scaling coefficients parameter, calibrating coefficient quantity

q q  q  CC1 2 C1  q t     see (14) b b tb  t2 1 exp a t  t quantity as a 0 1 2  SC 1  function of time 1exp  a t  t   SCC 2 1 

for b  0 : 1 b , qIP and tIP  inflection point Ei 1   Ei  b q  No inflection point No inflection point t    C  quantity and time C exp[a ]

qt   aexp  a  t t  instantaneous SS  C2  b2  b t q q    increment dq d t see (17) 1 2  CC  2 1 exp a t  t  1 as a function of  SCC 2 1  time

    2  q qCC  q q  y q  b4  b q  b   1 2    1 2 0   ln , ln aS    instantaneous  q   exp a       S tCC t   logarithmic      2 1   relative growth a b q where the argument of the   ln dqdt q  ln qexp  a   1 ,       SCt tC   square root and q must be    2 1   as a function of positive for non-complex quantity solutions where the argument of each logarithm must be positive for non-complex solutions

   ln  qCC q aSSCCexp a t  t    1 2    2 1   b 2 b t  qexp  a  t t    y t   ln  1 2  ,  C1 S  C1    b b t b  t2    instantaneous  0 1 2      logarithmic see (16) lnqC  expaS  t  2 tCC  t  ,  2   1 2   relative growth as where the argument of the   a function of time logarithm must be positive     qCC q exp aSCCt  t for non-complex solutions  2 1    2 1   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 ______

5) Multiplying q t  by a factor: In all three functions, the [3] A. R. Weiskittel, D. W. Hann, J. A. Kershaw, and J. K. Vanclay, Forest Growth and Yield Modeling. Chichester, West Sussex: relative growth rate remains unchanged for any given Wiley-Blackwell, 2011. time after by a factor. [4] E. E. Maor, The Story of a Number. Princeton, New Jersey: Princeton University Press, 1994. 6) Shifting the logarithmic relative growth rate [5] M. Ricker and R. del Río, “Projecting diameter growth in tropical y q  ln qt q  up or down: In the generalized trees: A new modeling approach,” Forest Science, vol. 50, no. 2, exponential function, the shift can be combined with a pp. 213-224, 2004. single parameter as a a  shift , for example to [6] A. Jeffrey and H. H. Dai, Handbook of Mathematical Formulas new and Integrals, 4th edition. Amsterdam: Elsevier, 2008. calculate confidence curves that correspond only to the [7] C. G. van der Laan and N. M. Temme, Calculation of Special relative growth rate. This is possible because the shift Functions: The Gamma Function, the Exponential Integrals and does not affect the inflection point quantity. In the 2nd- Error-like Functions. Amsterdam: Centrum voor Wiskunde en degree polynomial, the shift can be incorporated into the Informatica (CWI), 1984. parameters as b   exp shift b and b  [8] M. Ricker, V. M. Peña Ramírez, and D. von Rosen, “A new 1,new   1 2,new method to compare statistical tree growth curves: the PL- 2 GMANOVA model and its application with dendrochronological exp shift   b2 , whereas b0 remains unchanged. In data,” PLOS ONE, vol. 9, no. 14, e112396, 2014. the Schnute model with b 1, the shift of y q is [9] P. Pecina, “On the function inverse to the exponential integral function,” Bulletin of the Astronomical Institutetutes of equivalent to changing the coefficients to Czechoslovakia, vol. 37, no. 1, pp. 8-12, 1986.

aSS,new  a exp shift and tC, new  tC  [10] S. Winitzki, "Computing the incomplete gamma function to 2 1 arbitrary precision," Lecture Notes in Computer Science, vol. 2667, pp. 790-798, 2003. tCC t expshift  .  2 1  [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. SUPPORTING INFORMATION [12] N. H. F. Beebe, Programming Using the MathCW Portable Software Library. Springer International Publishing, 2017. Three files are available from the first author [13] A. Souto Martinez, R. Silva González, and A. Lauri Espíndola, ([email protected], [email protected]), “Generalized exponential function and discrete growth models,” and will also be made available on the Internet: Physica A: Statistical Mechanics and its Applications, vol. 388, no. 14, pp. 2922-2930, 2009. 1) The file “S1 Derivations for the polynomial and Schnute [14] H. S. Migon and D. Gamerman, “Generalized exponential growth model.pdf” provides the derivations of the two models models: a Bayesian approach,” Journal of Forecasting, vol. 12, that are compared with the generalized exponential pp. 573-584. function in the last section. [15] R. A. Fisher, “Some remarks on the methods formulated in a recent article on ‘The quantitative analysis of plant growth’,’’ 2) The file “S2 Inverse of the exponential integral function Annals of Applied Biology, vol. 7, pp. 367-372, 1921. Ei[x].pdf” consists of a Mathematica notebook [16] I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, converted to a PDF file, where the methods to calculate Handbook of Mathematics, 5th edition. Heidelberg: Springer, 2007. Ei( 1) z and Ei( 1) z were programmed. x0 Ei  x0 Ei  [17] J. Schnute, “A versatile growth model with statistically stable parameters,” Canadian Journal of Fisheries and Aquatic 3) The file “S3 Additional information about functions Sciences, vol. 38, pp. 1128-1140, 1981. shown in figures.pdf” contains detailed explanations [18] Y. Lei and S. Y. Zhang, “Comparison and selection of growth about the computational methods and numerical results models using the Schnute model,” Journal of Forest Science, vol. for generating the figures. 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 ACKNOWLEDGMENT Matrices. Springer, 2005.

We thank Christina Siebe and Víctor M. Peña Ramírez (Geology Institute, UNAM) for their support in developing Article modified on 25 July 2018: Equations (16) and (17) this article. 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 REFERENCES matters were corrected, and the file names of the supporting information modified. [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.

(Revised online publication: 9 August 2018)