Subject ECONOMICS Paper No and Title 2: Quantitative

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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

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ECONOMICS



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

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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

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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

,

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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

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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.

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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;

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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:

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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:

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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.

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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.

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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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Subject ECONOMICS Paper No and Title 2: Quantitative