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ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
Subject ECONOMICS
Paper No and Title 2: Quantitative Methods-II (Statistical Methods)
Module No and Title 35: Gompertz and Logistic Regression
Module Tag ECO_P2_M35
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
TABLE OF CONTENTS
1. Learning Outcomes
2. Introduction
3. Sigmoid Function
4. Logistic Function
5. Generalized logistic function
6. Gompertz Function
7. Problems of Non-Linear Models
8. Summary
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
1. Learning Outcomes
After studying this module, you shall be able to
Know about a class of non-linear functions under the mould of Sigmoid
Functions.
Know about the nature of a Logistic Function.
Extend the Logistic Function to a Generalized Logistic Function.
Understand the properties of a Gompertz function.
Evaluate the problems of non-linear functions
2. Introduction
The study of non-linear functions helps in understanding and estimating certain real
world phenomena. Mostly these functions are used for biological and natural
phenomenon. We are aware, however, that some of the non-linear functions find
applications in economics as well.
The U-shaped cost function like Average Variable Cost or Average Total Cost are very
well explained with the help of Quadratic Functions. Similarly, Total Cost Curve is
explained with the help of a Cubic Function. There is a class of non-linear functions
which is generally based on a function called the ‘sigmoid’ function.
The four functions that we will be considering are:
Sigmoid Function
Logistic Function
Generalized Logistic function.
Gompertz Function
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
3. Sigmoid Function
A sigmoid function is a mathematical relationship that yields an "S" shape and hence, is
called a sigmoid curve. While sometimes a sigmoid function is referred to as a special
case of the logistic function it could better be treated as the general form of all such S-
shaped curves. There are other S-shaped curves like the Gompertz curve, the Ogee curve
and the Logistic Curve. Sigmoid curves are most appropriate for describing cumulative
distribution functions, such as the integrals of the logistic distribution, the normal
distribution, and Student's t probability density functions. The integral of any smooth,
positive, "bump-shaped" function will be sigmoidal.
Definition
“A sigmoid function is a bounded differentiable real function (y) that is defined for all
real input values (of x or ‘t’, that is, time) and has a positive derivative at each point. This
implies that while ‘t’ may change from -∞ to +∞ the value of its function (y) will always
remain positive”.
The Sigmoid Function is given by the formula:
Properties
1. There are also a pair of horizontal asymptotes as .
2. The differential equation
,
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
with the inclusion of a boundary condition providing a third degree of
freedom, , provides a class of functions of this type.
Figure 1: Sigmoid Function
Many biological processes like the growth of a tumor or demographic processes like
population growth can be captured by this class of functions. Management concepts like
learning curves which show a initial growth rate followed by an accelerated pace and a
tapering off as the climax approaches over time, are described and can be measured by
one or the other type of a sigmoid function. When specific parameters are not known a
sigmoid function is useful in measurement.
Besides the logistic function, sigmoid functions include the ordinary arctangent,
the hyperbolic tangent, the Gudermannian function, and the error function, but also
the generalised logistic function
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
4. Logistic Function
In the case of a logistic function the initial stage of growth is approximately exponential;
then, as saturation begins, the growth slows, and at maturity, growth stops.
A logistic function or logistic curve is a common "S" shape (sigmoid curve), with
equation:
where
e = the natural logarithm base (also known as Euler's number),
x0 = the x-value of the sigmoid's midpoint,
L = the curve's maximum value, and
k = the steepness of the curve.
For values of x in the range of real numbers from −∞ to +∞, the S-curve is obtained (with
the graph of f approaching L as x approaches +∞ and approaching zero as x approaches
−∞).
The function was named in 1844–1845 by Pierre François Verhulst, who applied it to the
study of population growth. The logistic function is used in other fields. It is used in
neural networks, ecology, psychology, etc.
The logistic function combines two growth patterns: one of exponential growth and the
other is of exponential decay. This gives rise to the peculiar s-shape. Given below is a set
of figures (Figure 2) which show this phenomenon. The first curve is a curve that
captures exponential growth. The second shows a slowing down of the growth process.
The third shows how if we combine the first two we would get a logistic curve.
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
Figure 2: Deriving a Logistic Curve
Generalized logistic function
The generalised logistic function or curve is also known as Richards' curve. It is the
generalized form of the logistic function. The function allows for a more flexible S-
shaped curves:
The formula is:
Where
Y = population, and t = time.
These are the parameters:
A: the lower asymptote;
K: the upper asymptote. If A = 0 then K is called the carrying capacity;
B : the growth rate;
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
υ > 0 : affects near which asymptote maximum growth occurs;
: is related to the value Y (0); and
C: usually takes a value of 1.
Figure 3: Generalized Logistic Function
The equation can also be written:
where can be thought of a starting time, (at which), including both and can
be convenient:
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
this representation simplifies the setting of both a starting time and the value of Y at that
time.
The logistic, with maximum growth rate at time , is the case where = 1. A
further extension of the logistic function yields another curve called the Gompertz
Function.
5. Gompertz Function
A Gompertz curve or Gompertz function was first established by Benjamin Gompertz.
It is a sigmoid function used in modeling any growth curve that reaches a plateau after a
long time period (t). Typically, this model is used for mathematical models for time
series. Here, growth is very slow both in the beginning and end. The right-hand or future
value asymptote of the function is approached much more gradually by the curve than the
left-hand or lower valued asymptote, in contrast to the simple logistic function in which
both asymptotes are approached by the curve symmetrically. It is a special case of the
generalized logistic function.
Features
b, c are positive numbers
b sets the displacement along the x-axis (translates the graph to the left or right)
c sets the growth rate (y scaling)
e is Euler's Number (e = 2.71828...).
These are some of the examples of the use of Gompertz curve:
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
Diffusion of technology: Initially, mobile were very costly. Hence, the adoption
of this technology was very slow. Then a period of rapid growth followed. Finally,
when even rural India will adopt mobile phones there would be a slowing down of
adoption as the saturation level is reached.
Theory of demographic transition clearly stipulates such a pattern exists. Initially
birth rates are low and death rates are high. The net population growth is low. Later the
death rate falls drastically and birth rate continues to be almost the same rate such that the
growth rate of population is high. Slowly the birth rate also falls. But the death rate falls
slower. Hence, the population growth rate beings to decline. Finally, birth and death
rates fall to the minimum. Marginally births exceed deaths. Initially addition to total
population growth is very low. But slowly the population starts rising and hence total
population rises fast. Finally, population growth tapers off till it reaches a plateau.
Modeling market development: Initially it is difficult to push a new product in a
market. The growth in sales is very low. Then there is a period of accelerated growth in
sales revenue. As the base keeps growing then fewer and fewer people are left who do
not know of the new product or have not bought it.
The difference between Gompertz and Logistic function is somewhat subtle.
Figure 4: Logistic Function vs. Gompertz.
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
This could be a function that describes total sales revenue. In this sense it is nearly a
Product Life Cycle. Here ‘n’ stands for total sales. These two curves are generated with
the help of certain parameters, that is , certain values of α K and r.
Both appear to be following the same sigmoid form but the logistic curve is squarer. This
makes it useable for binary classes. It is often approximated by to discrete categories.
This is known as a logistic approximation. The growth in the case of Gompertz is gradual
compared to the logistic function. The Logistic function suddenly starts to rise and
equally suddenly falls up to a level of saturation. Finally, of course, they both coincide.
Problems of Non-Linear Models
Some of these problems for fitting nonlinear regression models common, but some are
unique. The basic problems are the following.
1. The choice of nonlinear model needs to be made separately for each
problem. There is no such thing as a generic nonlinear function, only a lot
of special cases. Genuinely, the choice of nonlinear models should be
based on theory or by precedent. But, in fact, in practice the choice is
made through ad hoc choice. This problem does not arise when dealing
with linear models.
2. Nonlinear models require a clear understanding about how the different
parameters affect the function's graph.
3. To look at many of the graphs may look similar. So they belong to the
same functional form – Sigmoid. But each curve derived from the Sigmoid
Curve has different parameters which perform differently. Once the
parameters change they have different interpretations. Yet, they yield
different graphs. As a result there could be many curves, all of which are
referred to as Gompertz models. For choosing a particular nonlinear model
we also need to be clear about what we are demanding of the curve or
function. We need to know which non-linear curve best suits our needs.
____________________________________________________________________________________________________
ECONOMICS
Paper 2: Quantitative Methods-II (Statistical Methods)
Module 35: Gompertz and Logistic Regression
4. In a nonlinear model it is difficult to envisage the ultimate effect of a
change in any parameter.
5. Parameter estimation in nonlinear regression is more complicated and
delicate than in linear regression. Sometimes we may have to get initial
estimates of the growth parameters. This is another problem that does not
arise in linear regression (OLS).
6. Summary
The four functions that we will be considering are:
Sigmoid Function
Logistic Function
Generalized Logistic function.
Gompertz Function
A sigmoid function is a mathematical relationship that yields an "S" shape and hence, is
called a sigmoid curve. While sometimes a sigmoid function is referred to as a special
case of the logistic function it could better be treated as the general form of all such S-
shaped curves.
In the case of a logistic function the initial stage of growth is approximately exponential;
then, as saturation begins, the growth slows, and at maturity, growth stops.
The generalized logistic function or curve is also known as Richards' curve. It is the
generalized form of the logistic function. The function allows for a more flexible S-
shaped curves.
A Gompertz curve or Gompertz function was first established by Benjamin Gompertz. It
is a sigmoid function used in modeling any growth curve that reaches a plateau after a
long time period (t). Typically, this model is used for mathematical models for time
series. Here, growth is very slow both in the beginning and end.
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