“parametric analysis of the bass model”

“Parametric analysis of the Bass model”

AUTHORS Yair Orbach

ARTICLE INFOYair Orbach (2016). Parametric analysis of the Bass model. Innovative Marketing

, 12(1), 29-40. doi:10.21511/im.12(1).2016.03

DOI http://dx.doi.org/10.21511/im.12(1).2016.03

RELEASED ON Wednesday, 27 April 2016

JOURNAL "Innovative Marketing "

FOUNDER LLC “Consulting Publishing Company “Business Perspectives”

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© The author(s) 2022. This publication is an open access article.

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Innovative Marketing, Volume 12, Issue 1, 2016

29

Yair Orbach (Israel)

Parametric analysis of the Bass model

Abstract

In this research, the authors explore the influence of the Bass model p, q parameters values on diffusion patterns and

map p, q Euclidean space regions accordingly. The boundaries of four different sub-regions are classified and defined,

in the region where both p, q are positive, according to the number of inflection point and peak of the non-cumulative

sales curve. The researchers extend the p, q range beyond the common positive value restriction to regions where either

p or q is negative. The case of negative p, which represents barriers to initial adoption, leads us to redefine the motiva-

tion for seeding, where seeding is essential to start the market rather than just for accelerating the diffusion. The case of

negative q, caused by a declining motivation to adopt as the number of adopters increases, leads us to cases where the

saturation of the market is at partial coverage rather than the usual full coverage at the long run. The authors develop a

solution to the special case of p + q = 0, where the Bass solution cannot be used. Some differences are highlighted be-

tween the discrete time and continuous time flavors of the Bass model and the implication on the mapping. The distor-

tion is presented, caused by the transition between continuous and discrete time forms, as a function of p, q values in

the various regions.

Keywords: Bass model, mapping, diffusion patterns, discrete time, continuous time, seeding.

JEL Classification: M3.

Introduction

The theory of diffusion of innovation, proposed by

Bass (1969), has been explored, implemented and

extended by numerous researches. Some researchers

extend the model to explicitly consider market or

industry characteristics or behavior. Others examine

implementation of the model on various cases and

show that adding more factors improves the capabil-

ity of the model to capture complex behavior in de-

tails. Another direction taken by some researchers

was to explore the factors that influence parameters

values estimation and forecasts accuracy. Some re-

searches explore empirically how the Bass model

parameters vary across products and markets, usual-

ly at the ranges when p << q and 0.1< q < 0.7. Bass

(1969) notes that there are two different categories

of diffusion curve patterns. When q > p, the period-

ic sales grow until they reach a peak and, then, de-

cline asymptotically to zero. When q < p, periodic

sales decline asymptotically to zero starting from

launch. Little attention has been directed since then

to a further comprehensive exploration of the con-

straints and classification of the Bass model parame-

ters and how they affect the adoption curve patterns.

Another issue that has attracted little attention in

diffusion theory is the transition between the conti-

nuous time form and the discrete time form of the

Bass model. While many researchers switch be-

tween these two flavors without further notice, at-

tention needs to be paid to verifying, for each im-

plementation, that this transition has little impact on

parameters’ values and on forecasts’ accuracy.

Yair Orbach, 2016.

Yair Orbach, Ph.D., Dr., Assistant Professor of Marketing, School of

Business Administration, Bar-Ilan University, Israel.

In this paper, we perform a comprehensive explora-

tion and classify and map the different diffusion pat-

terns over the p, q space. The exploration is theoretic

and is not limited to specific empiric cases. We refer

to the different flavors (continuous vs. discrete) and

highlight the differences in mapping, constraints and

classification. We also define the conditions that

need to be checked when switching between them.

1. Literature review

The diffusion of innovation model, known as the

Bass model, has several flavors and numerous ex-

tensions. In this paper we focus on the basic model

of Bass (1969), which has a simple analytic solution

and not to extensions that are usually solved numer-

ically. We brief the different basic models and high-

light the differences between them. We use these

models for mapping the patterns and constraints cat-

egories over the p, q space and show that each fla-

vor has a slightly different map.

The Bass (1969) presents the diffusion of innovation

dynamic equation:

( )( ) ( ) 1 ( ) .

dF tf t p q F t F t

dt (1)

This equation represents the innovation and imita-

tion influence on the remaining potential market.

The periodic sales, or the rate of cumulative sales

change, which is the derivative of the cumulative

adoption, are proportional to the multiplication of

the remaining market by the sum of innovation and

imitation influence. The analytic continuous time

solution that Bass presents to his equation is:

1( ) .

1

p q t

p q t

eF t

qe

p

(2)


30

The cumulative sales function, as well as its deriva-

tive or periodic sales function, can be outlined as a

graph where time is the X axis and sales, or sales

rate, are the Y axis. The shapes of both curves, the

cumulative and periodic sales, are determined by the

values of the model parameters p and q. Srinivasan

and Mason (1986) note that both p and q must be

non-negative. Acemoglu and Ozdaglar (2009) note

that the levels of p and q scale time, while the ratio

q/p determines the overall shape of the curve.

Bass (1969) refers to the effect of the q/p ratio or q,

p phase and differentiates between two categories.

He notes that when q/p > 1, i.e., p, q phase = 45°,

the product is successful and sales experience

growth and then decline due to saturation. When q/p

< 1, which represents an unsuccessful product, sales

will start at a certain level and keep declining. Bass

(1969) also presents the influence of (p + q) values,

between 0.3 and 0.9, on the growth rate. Sultan et al.

(1990) performed a meta-analysis of 213 applica-

tions of diffusion models from 15 articles published

between 1950 and 1980. They compare several pa-

rameters estimation methods (OLS, MLE, Bayesian

and non-linear least square) and also how the num-

ber of sampling points influences accuracy. They

found that the average p value is 0.03, while the av-

erage q value is 0.38. Van den Bulte (2002) ex-

plored how p and q vary across products and coun-

tries, based on a database containing 1586 sets of p

and q parameters, from 113 papers published be-

tween January 1969 and May 2000. He explains that

the parameters p and q provide information about

the speed of diffusion. A high value for p indicates

that the diffusion has a quick start, but also tapers

off quickly. A high value of q indicates that the dif-

fusion is slow at first, but accelerates after a while.

He also notes that, when q is larger than p, the cu-

mulative number of adopters follows the type of S-

curve often observed for radically innovative prod-

uct categories. When q is smaller than p, the cumu-

lative number of adopters follows an inverse J-curve

often observed for less risky innovations such as

new grocery items, movies, and music CDs.

Lilien et al. (2000) formulate a discrete difference equ-

ation to model diffusion of innovation, used by many

previous and later of the Bass model extensions.

( ) ( ) ( 1) ( )

( ) ( ) 1 ( ) ,

d d d d

d d d d

f n F n F n F n

X n p q F n F n (3)

where Fd(n) is the cumulative adoption and fd(n) is

the periodic sales at period n.

We use the notation pd and qd for the discrete time

model parameters to distinguish them from the p, q

parameters of the continuous time form. The X(n)

function can capture many factors the original Bass

model ignores such as advertisement of price

changes. When X(n) = 1, it converges back to the

original Bass model. For example, Bass et al. (1994)

propose including the influence of advertisement of

price changes by using:

Pr( ) Pr( 1) ( ) ( 1)( ) 1 0, .

Pr( 1) ( 1)

n n A n A nx n Max

n A n (4)

When coefficient captures the percentage increase

in diffusion speed resulting from a 1% decrease in

price, Pr(n) is the price in period n, coefficient

captures the percentage increase in diffusion speed

resulting from a 1% increase in advertising and A(n)

is advertising in period n. The main advantage of the

discrete model is that it can be solved numerically,

in cases where there is no analytic solution. While

Bass (1969) presents an analytic general solution for

the continuous time differential equation (1), neither

he nor others propose an analytic solution for the

discrete time difference equation (3). Bass (1969)

does refer implicitly to the discrete model by pro-

viding an insight that the likelihood of a purchase at

time t, P(n) is calculated by:

)()(1

)()( nFqp

nF

nfnP ddd

d

d. (5)

While many researchers switch between the discrete

model, based on a difference equation (3), and the

continuous time model, based on the differential

equation (1), the transition is not trivial. Van den

Bulte and Lilien (1997) show that OLS or NLS es-

timations of the continuous Bass model parameters

using discrete time data are biased and that they

change systematically as one extends the number of

observations. An analytic discrete time solution for

the Bass continuous time differential equation (1),

developed by Satoh (2001), is:

2

2

1 ( )1

1 ( )( ) .

1 ( )1

1 ( )

n

s s

s s

n

s s s

s s s

p q

p qF n

q p q

p p q

(6)

However, this solution does not solve the difference

equation (3). Satoh (2001) notes that the relation

between his discrete time Bass model parameters ps

and qs and the corresponding continuous time Bass

model parameters p and q is:

)(2

)(2

1

11qp

qp

e

e

qpk ;

qkq

pkp

s

s. (7)

Table 1 compares the Bass solution (2) for typical

values p = 0.01, q = 0.3, Satoh solution (6) for the

corresponding parameters values pS = 0.0097, qS =


31

0.2907 and a numeric solution for the Lilien equation

(3) that has minimal RMSE with them with pd =

0.0138, qd = 0.2865. While the parameters values and

forecasts of all three flavors are very close, we still

notice that, while Satoh solutions perfectly match

Bass, Lilien solution is very close but not identical.

Table 1. A comparison between Bass, Satoh and Lilien solution

t Bass F(t) Satoh F(n) Lilien Fd(n)

0 0 0 0

1 0.011588 0.011588 0.013808

2 0.026960 0.026960 0.031327

3 0.047166 0.047166 0.053396

4 0.073399 0.073399 0.080948

5 0.106923 0.106923 0.114953

6 0.148907 0.148907 0.156323

7 0.200171 0.200171 0.205758

8 0.260871 0.260871 0.263546

9 0.330179 0.330179 0.329323

10 0.406107 0.406107 0.401864

11 0.485607 0.485607 0.478990

12 0.565000 0.565000 0.557684

13 0.640625 0.640625 0.634465

14 0.709486 0.709486 0.705958

15 0.769662 0.769662 0.769491

16 0.820388 0.820388 0.823493

17 0.861864 0.861864 0.867574

18 0.894941 0.894941 0.902319

19 0.920798 0.920798 0.928920

20 0.940698 0.940698 0.948819

The cumulative sales of Satoh (2001) from (6) per-

fectly match the continuous time Bass sales solution

from (2) for t = 0, 1, 2…n. Satoh (2001) compares se-

veral methods for estimating the discrete time Bass

model parameters for nine data sets, and concludes

that, for most cases, when the time interval is small

enough, the exact solution of the discrete Bass model

provides a very good approximation of the solution of

the conventional Bass model. He also proves that,

when using the transformation (7) to p and q, a solu-

tion of the discrete Bass model (6) provides identical

values to the solution of the continuous model (2). For

two cases (out of nine) where the estimated value of

the p parameter is negative, he notes that the wrong

sign indicates that the data are not appropriate for the

Bass model.

2. Mapping the Bass model parameters

We perform the mapping separately for each fla-

vor of the Bass model. First, we develop a formu-

la of the periodic sales inflection points’ times,

for the continuous time form, and classify the dif-

fusion patterns according to the number of inflec-

tion point and whether there is peak. Second, we

map each category classified to a sub-region of

the positive p and q values of the Euclidean space.

Then, we extend the map to include regions where

either p or q is negative, which add some insight

about seeding and about market saturation. For

the discrete time model, equation (3), which is

more restricted, we remap the p, q space. The map

is similar but has additional constraints about the

absolute values of p, q and their sum. We redo the

mapping for the Satoh model which is a discrete

time solution (6, 7) for the continuous time mode

of equation (1, 3).

2.1. Mapping the Bass model parameters ratio

space. Bass (1969) and Van den Bulte (2002) distin-

guish between cases where q > p, and periodic sales

have a peak after launch, to cases where q < p and pe-

riodic sales keep declining since launch. The periodic

sales peak time, as calculated by Bass (1969), is:

1* ln .

qt

p q p (8)

From (8), we see that when q > p, the peak time is positive. When q < p, peak time is negative, thus, sales keep declining since launch time (t = 0). There are no sales before product launch.

The inflection points’ times (see Appendix A) are calculated as:

qpt

qp

p

q

t)32ln(

*

)32ln(ln

** (9)

Result 1: the time between the inflection points

and between them to peak depends only on (p + q) and not on (p/q).

Innovative Ma

32

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nd Shapiro (

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ns (A to D) proximation

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ribution, (b)

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1985) men-

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mith (1981)

oducer inte-

he ability to

cal mass is

vercome and

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ass, or seed

r method is

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oner (1986)

ted this me-

ses, seeding

n, as in Jain

starting the

) do not re-

fer to such

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can overco

the negativ

condition i

cannot be o

interest rat

justify the

negative p

justification

low and th

will keep t

( qpF0

higher, the

larger.

Cumulative

according

when q va

equal:

tF

1

1

)(

1*t

p q

Without se

duces back

seed size is

we always

since (9) a

for having

0

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(1 )q F

p qF

Condition

close to its

Figure 2 pr

curves of t

tive p. Wh

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has two in

when p va

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and peak tim

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With negative

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for negative

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eleration doe

of seeding.

with seeding

en interest ra

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its minimal

he ratio p

start the mark

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are applied

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equation (10

e p and whe

mit ( pF0

eak. Furtherm

e p, the cond

when seed si

r any q/p ratio

egative p and

olute value)

s may have a

rresponding c

cases. Note

g is essential

ue to the req

ess than 100

onding positi

egion A or B

k.

itive p can b

at least p

ly for

eding

ed by

n this

option

en the

es not

With

g, the

ate is

n, we

limit

q is

ket is

eding,

(only

p and

(10)

0) re-

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qp ),

more,

dition

(11)

ize is

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d two

posi-

a sin-

curve

that,

l and

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B and

be re-

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12, Issue 1, 2016

33

represents a

required to

p values

meters rangeand Golden-

ord-of-mouthowever, theyive WOM ofoverall posi-

progress evene q does notppointed anda case wheremore people

and level forsold to thosee who do notmmute whileold, the over-uced, and thereserved traine to buy a re-ng a trainingobile applica-e translation.her rare skill,ecomes moreose who haveternalities ef-less valuable,explains thatat occurs dues curve with the cumula-

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Innovative Ma

34

tive and nontive and neg

ratio is presing effect tdamping ef

increases an

Result 3: T

leased whe

sales will e

lative ado

equilibrium

the potenti

when p

a portion o

When p + q

though the

this case, th

(( )

dF tf t

dt

The solution

( ) 1F tp

A summary

is presented

It presents t

p – q space

region rathe

Region

A

B

C

D

arketing, Volume

n-cumulative gative q valu

ented too. Ththat contradiffect strength

nd, when p

Non-cum

Fig. 3. Co

The assumpt

en p is posi

xperience st

ption asym

m. The equ

ial market, a

q . The equ

of the potent

q = 0, Bass

discrete time

he Bass equat

)1 (

tp F

n for this cas

1.

1p t

y map of the

d in Figure 4.

the classic (p

e regions, b

er than two)

Table 2. The

Lower bound oq/p

32

1

32

0

e 12, Issue 1, 201

sales curve oues. The effe

he negative qicts the drivhens as cum

qtF )( , t

mulative adop

omparing cum

tion of posit

tive. At bot

teady declin

mptotically

ilibrium ma

as in a “reg

uilibrium is

tial market,

solution (2)

e (3) can stil

tion is:

2( ) .t

se is:

e combinatio

.

positive p an

but in a grea

). The map a

e seven regio

of Upper bounq/p

32

1

32

16

of cases with pfect of the p

q creates a dave to adopt. mulative adop

the drive to a

ption

mulative and no

tive q can be

th, the peri

ne, while, cu

approaches

ay be 100%

gular” diffus

qp / , whic

when qp

is undefined

ll be applied.

ons we discu

nd q) Bass m

ater details (

also presents

ons of q/p, bo

nd of Lower phof q/p

75º

45º

15º

0º

posi-qp /

amp-The

ption

adopt

is b

level

adop

decli

qp /

in Fi

on-cumulative

e re-

iodic

umu-

s its

% of

sion,

ch is

q .

d, al-

. For

(12)

(13)

ussed

model

(four

s re-

gion

dere

We

mine

cumu

Tabl

at ea

flect

satur

oundaries, pe

hasep

Upper phaof q/p

90º

75º

45º

15º

alanced by t

l is a stable

ption starts at

ine (i.e., neg

q . This is de

igure 3.

Cu

adoption with

ns with negat

d to be out-o

Fig. 4

summarize a

e the existe

ulative sales

le 3 presents

ach region a

tion points,

ration level.

eak, inflectio

ase Number inflection po

2

1

1

0

the q dampin

e equilibrium

t a higher lev

gative non-c

emonstrated b

umulative ad

positive and n

tive p or q

of-scope in p

4. Extended p –

all seven reg

ence of thes

in Table 2.

examples fo

and whether

as well as a

on points and

of oints

Peak e

Yes

Yes

No

No

ng effect. Th

m and if the

vel, or even 1

cumulative a

by the broken

doption

negative q

values that w

revious resea

– q space map

gions of q/p

se points on

or the values

there are pe

a need for s

d equilibrium

exists E(

he saturation

cumulative

00%, it will

adoption) to

n line curves

were consi-

arches.

p that deter-

n the non-

s of p and q

eak and in-

seeding and

m

Equilibriumsaturation)

100%

100%

100%

100%

Tab

Region

E

F

G

Table 3. Ex

Re

p – q phase

Peak

Inflection points

Minimal seed

Saturation

p

q

q/p

Note that m

Lilien et al

found that

side in reg

tions, prod

Products in

be adopted

3. Mappin

parameter

When refer

tion (3), fo

the likeliho

(1969), it a

ratio, as pr

solute valu

When refer

non-negativ

sume that a

purchase a

pd. Since P

ly that the

ter a lon

)( nd nF

imply that

range [0:1]

mention th

the range o

the continu

thus, can v

ters of the

bounded in

For region

negative, th

pd and with

ood , see (5

(1) dP p q

ble 2 (cont.).

Lower boundq/p

-1

xamples of di

egion

s

most empiric

., 2000; Cha

most succes

gion A. Still

ducts at reg

n region G c

d only by a po

ng the discre

rs space

rring to the

ormulated by

ood of purch

adds some b

resented abov

ues.

rring to regio

ve, and seed

at launch F(0

at time n = 0

P(n) represen

value of pd i

ng time t

1, thus, P

the value of

]. Indeed, No

hat the values

of [0:1]. Whi

uous Bass m

vary to any p

discrete tim

n the range

s F and G,

he likelihood

h no seeding

5), during the

(1)d d dq F p

. The seven r

of Upper bouq/p

-1

0

-1

ifferent param

A

75°-90°

yes

2

0

100%

c studies (Su

ndrasekaran

ssful product

l, with prop

ion E can

can succeed

ortion of the

ete time Bas

discrete tim

y Lilien et a

hased (5), m

oundaries no

ve, but also t

ons A to D, w

ding in opti

0) = 0, thus,

0, according

nts a probabil

is bounded b

the whole

n pnP )(

f pd+qd is als

oratikah and

s of pd and q

ile the param

model repre

positive num

me Bass mode

of [0:1] and

where p is p

d probability

g Fd(1) = pd,

e following p

(1d d dq p p

regions of q/p

und of Lower pof q/

90º

-45º

-90º

meters sets th

B

5°- °

yes

1

0

100%

ultan et al., 1

and Tellis, 2

t p – q value

per marketing

succeed as

as well, but

potential ma

ss model

e diffusion e

al. (2000), an

entioned by

ot only to th

to parameter

where p and

onal, we ca

the probabili

to (5), is P(

lity, one can

between [0:1]

market ad

dd qp . W

so bounded i

Ismail (201

qd are bound

meters p and

esent a rate

mber, the par

el (pd and qd

d also their

positive, but

y at launch P

purchase lik

period is:

1 ).dq

/p, boundarie

hase/p

Upper phof q/p

º 135º

º 0º

º -45º

hat belong to d

C

5°- °

no

1

0

100%

1990;

2008)

es re-

g ac-

well.

t will

arket.

equa-

nd to

Bass

he p/q

rs ab-

q are

n as-

ity of

(0) =

imp-

]. Af-

dopts

e can

in the

3) do

ded in

q, of

and,

rame-

d) are

sum.

t q is

(0) =

kelih-

(14)

It m

prob

be h

tive

as p

not

valu

abso

thei

Figu

disc

of th

Res

spac

spac

resi

In

es, peak, infle

hasep

Numberinflection p

2

0

0

different regi

D

°- °

no

0

0

100%

means that,

bability, lim

higher than –

, seeding is

dd qp , mu

have further

ues of dp and

olute values

r sum dp

ure 5 presen

crete model

he continuou

Fig. 5.

sult 4. Discr

ce is more c

ce of the co

de in a sub-a

nnovative Mark

ection points

r of points

Peak

Yes

No

No

ons of p – q s

E

°- °

yes

2

|p|/q

100%

-0.1

0.2

-2

since dp is

ited to [0:1]

–1. For regio

essential. Th

ust be positiv

r limitations

d dq . There

of dp and q

dq is betwe

nts the pd – q

(3). Note tha

us model (1)

Discrete time p

rete time mo

onstrained c

ontinuous tim

area of the co

keting, Volume 1

s and equilibr

exists

space and the

F

-45°-0°

no

0

0

100%

-

-

positive an

], the value

on E, where

he values of

ve. For regio

regarding th

e may be case

dq are highe

een 0 and 1.

qd space map

at it include

map (see Fig

pd - qd space m

odel parame

compared w

me map. Its

ontinuous tim

12, Issue 1, 2016

35

rium

Equilibrium(Saturation)

100%

100%

p/|q|

eir properties

G

-90°-45°

no

0

0

p/|q|

0.1

-0.2

-2

nd P(1) is a

of dq must

dp is nega-

dq , as well

on E, we do

he individual

es where the

er than 1, but

pping of the

s a sub-area

gure 4).

map

eters pd – qd

with the p – q

valid areas

me map.

6

5

a

t

-

l

o

l

e

t

e

a

d

q

s

Innovative Ma

36

4. Mapping

solution pa

The Satoh (

continuous

matches per

an integer bu

tions. When

are non-neg

both sp an

Note that t

which appl

0, and also

absolute va

higher than

The p – q sp

arketing, Volume

g the Satoh

arameters sp

(2001) model

diffusion m

rfectly the B

ut it still follo

n referring to r

gative, we can

nd sq are wit

Fig.

there is sing

ies also for

that, for reg

alues of bot

n 1, while th

Table 4

pace map of

e 12, Issue 1, 201

discrete tim

pace

l is a discrete

model (1).

ass solution

ows the prob

regions A to

n directly im

thin the range

. 6. Relation of

gularity wh

p + q = 0 a

gions E to G

th ps and q

heir sum is

. Examples o

Regio

Bass p

Bass q

Satoh k

Satoh ps

Satoh qs

Lillien pd

Lillien qd

the Satoh (2

Fi

16

me analytic

e solution fo

Its solution

(2) when tim

ability (5) res

D, where p a

mply from (7)

e of [0:1]. Fo

f Satoh discrete

en ps + qs

and for pd +

G, the indivi

s may be m

bounded by

of ps, pd and q

on E

-1

20

0

-1

2

-

3

001) is prese

ig. 7. The p – q

or the

(6)

me is

stric-

and q

) that

or re-

gion

may

are h

the

relati

ters t

p –

qs /

e parameters a

= 0,

qd =

idual

much

y –1

and

wher

whil

the p

indiv

qs, qd that ha

E

90 1

00 -2

0.1 0

19

20 -

-2 0

3 -0

ented in Figu

space map of

s E to G, whe

be cases wh

higher than 1

1 ss qp

ion between

to the Bass (1

q phase

qpps / so

and continuous

1. Table 4

re both ps a

le their sum

parameters o

vidual absol

ave absolute v

G

190

200

0.1

19

-20

0.95

0.99 -

ure 7.

the Satoh (200

ere p and q h

here the absol

1. Still, Satoh

1s constra

Satoh (2001)

1969) qp .

of Bass an

o it maintains

s Bass parame

4 presents

and qs have

is 1. Note t

of the discre

lute value hi

values highe

F

200

-190

0.1

20

-19

0.99

-0.95

01)

have opposite

lute values of

h parameters

aint. Figure 6

) k and sp

From (7) we

nd Satoh ar

s the same reg

eters

such three

e high abso

that, for reg

ete model (3

igher than 1.

er than 1

e signs, there

f sp and sq

s still follow

presents the

sq parame-

e see that the

re identical

gion.

examples,

lute values

gion E also,

3) can have

.

Result 5. T

discrete tim

is less cons

5. Switchinand discre

Many rese

between th

time form

many case

cast seem t

time, in som

casts is sig

the properti

tinuous tim

to have eno

how many

Shannon (1

the samplin

quist (1928

in maintain

signal, and

nuous time

little distort

tween suc

In almost a

in the low

between th

seamless a

are some o

in the yello

cases, whe

areas, a tr

crete time

quires care

Result 6: T

time and

most existi

ues reside

where p,

The Satoh pme model p

strained com

ng betweente time form

earchers, inc

he continuou

(3) without

s, the discre

to be very cl

me cases, the

gnificant. The

ies of the cur

me to discrete

ough data po

data points

1948) who p

ng rate boun

8) determines

ning the info

d being able

signal from

tion, we nee

ccessive cur

all empiric ca

distortion (g

he continuou

and causes in

ther cases, w

ow areas wi

ere the value

ransition betw

causes a s

ful adjustme

The transiti

discrete tim

ing empiric

in regions w

q values re

pS, qS space

parameters p

mpared in re

n continuousms

cluding Bass

us form (1)

an explicit

ete time para

ose to those

e difference

e major issu

rve, in the tra

e time form, i

oints per tim

are required

provided a th

ndaries found

s that, when

rmation of th

to restore th

its discrete t

d at least tw

rve peaks.

ases, the val

green) areas w

us and discre

nsignificant

where the val

ith minor im

es of p, q r

ween the con

significant d

ents.

ion between

me form is

c researches,

with low dis

eside in hig

is similar to

pd – qd space

egions F and

s time

s (1969), sw

and the dis

notice. Whi

ameters and

of the contin

between the

e for mainta

ansition from

is the require

me. The theo

d was outline

heoretic basi

d by Nyquist

we are inter

he harmonies

he original c

time samples

wo data point

In the co

Fig. 8. Distor

ues of p, q r

where a trans

ete time form

distortion. T

lues of p, q r

mpact. In the

reside in the

ntinuous and

distortion an

n the contin

justified, a

, where p, q

stortion. In c

gh distortion

o the

e but

d G.

witch

screte

le, in

fore-

nuous

fore-

aining

m con-

ement

ory of

ed by

is for

t. Ny-

rested

s as a

conti-

s with

ts be-

ontext

of t

man

infle

infle

min

(Ro

disc

each

RM

Whe

satu

the

corr

are u

Figu

acro

area

is b

area

0.15

and

tion (RMSE) p

reside

sition

ms is

There

reside

e rare

e red

d dis-

d re-

nuous

as in

q val-

cases

n re-

gion

imp

Con

In thcurvramguisas ireginumanalalsogionprovning

In

he Bass mod

ny data point

ection point

ection point

ned by p, q v

ot Mean Sq

crete time fo

h p, q coordi

0

1 n

i

MSEn

ere n is the

uration and F

continuous

respondingly

used ( pp

ure 8 presen

oss the p, q

as is between

between 0.0

as is more th

54. The map

q values.

p – q space map

ns, such a tr

pact on the f

nclusion

his paper, weve patterns a

meters valuesshing betweein previous ions {A, B, C

mber of the inlytic formula

o extend the ns {E, F, G}vide marketig of the nega

nnovative Mark

del curve, w

ts are betwe

(when we h

s and peak.

values. The d

quare Error)

orecasts diffe

nate as:

1

0

( ) (dF i F

number of

F, Fd are the

(2, 10, 13)

y, when the

dd qqp ; ).

nts how the d

space. The

n 0 and 0.015

15 and 0.07

han 0.07 an

p was genera

p

ransition wo

forecasts.

e explored thand how theys and p/q raen two regionresearch, we

C, D} that arenflection poia for the inflcommon p

} with negatiing intuitionative values

keting, Volume 1

we need to co

en start (t =

have one), a

These time

distortion, o

of the con

erence, is ca

2) .i

periods from

e forecasts a

and discret

same param

distortion (RM

e distortion a

5. At the yel

7. The disto

nd up to a m

ated with 0.0

ould have a

he propertiesy depend on atio. Rather ns, with or we refer to foe categorizedints. We alsoection points– q space toive values o

n or insight and to the m

12, Issue 1, 2016

37

onsider how

0) and first

and between

es are deter-

r the RMSE

ntinuous and

alculated for

(15)

m launch to

according to

te (3) forms

meters values

MSE) varies

at the green

llow areas, it

ortion at red

maximum of

01 steps of p

a significant

of diffusionthe p, q pa-than distin-

without peak,our differentd also by theo develop ans’ times. Weo include re-f p or q andto the mea-

market beha-

6

7

w

t

n

-

E

d

r

o

o

s

s

s

n

t

d

f

p

t

n--,tene-d--


38

vior. In region E, we define a new motivation for seeding which, unlike previous research, is not used only for accelerating the diffusion, but, in certain conditions is essential for starting the market. Another contribution of this paper is defining the conditions to saturation below 100% (unlike pre-vious concept that any product will finally cover the entire market). We also highlight some differences

between discrete time and continuous time flavors of diffusion models and the map of the regions, where a switch between them is appropriate. Future research may propose an intuition for the regions that are still white in the p – q space maps and ex-plore their properties. Another direction may be de-veloping an analytic solution for the discrete time diffusion difference equation.

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39

Appendix A: Inflection points calculations

Bass solution for the non-cumulative adoption rate is: 2

2

1

)()(

tqp

tqp

ep

q

e

p

qptf

For finding the inflection points, we need to find where the derivative equals 0.

The derivative is calculated using the formula:

)(

)()(

)()(

)(

)(

)()(

2tg

thdt

tdgtg

dt

tdh

dt

tdf

tg

thtf

4

2

2

1

))((121)()(

)(

tqp

tqptqptqptqptqp

ep

q

eqpp

qe

p

qee

p

qeqp

p

qptf

3

2

1

))((21)()(

tqp

tqptqptqptqp

ep

q

eqpp

qee

p

qeqp

p

qp

3

3

3

3

1

1)(

1

21)(

tqp

tqptqp

tqp

tqptqptqptqp

ep

q

ep

qe

p

qp

ep

q

ep

qee

p

qe

p

qp

When comparing the numerator to 0, since denominator is always positive, as Bass (1969) we can find the periodic

sales peak time t*.

p

q

qpt ln

1*

For finding the inflection points, we need to calculate when the second derivative equals 0.

Changing variables for convenience

tqpetX )( -3

3

)(1

)(1)()(

)(

tXp

q

tXp

qtX

p

qptf

6

2

23

3

)(1

)()()`()(13)(1)`()(2

1)(

)(

tXp

q

tXp

qtXtX

p

qtX

p

qtX

p

qtXtX

p

q

p

qptf

Comparing the numerator to 0


40

0)()()`()(13)(1)`()(2

1 2

23

tXp

qtXtX

p

qtX

p

qtX

p

qtXtX

p

q

0)(1)(1)(3)(1)(2

1

23

tXp

q

p

qtX

p

qtXtX

p

qtX

p

q

)(1)(3)(1)(2

1 tXp

q

p

qtXtX

p

qtX

p

q

01)(4)(2

2

tXp

qtX

p

q

3232

2

324

2

4164

)(22

22

q

p

p

q

p

q

p

q

p

q

p

q

p

q

p

q

p

q

tX

qp

p

q

tp

qtqp

q

petX

tqp

)32ln(ln

ln)32ln()(32)( )(

Since )32ln()32ln(0)32ln()32ln()32ln( 2

The times of the inflection points are:

qpt

qp

p

q

t)32ln(

*

)32ln(ln

**

Note that the inflection points are at equal distance from the peak.

“parametric analysis of the bass model”

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