DETERMINANTS OF DIFFUSION MODELS’ FORECAST ACCURACY
This paper investigates factors that affect the accuracy of forecasts generated by diffusion
models. The analysis is performed using long-enough simulated series based on 926 real new
product diffusion datasets, together with a collection of diffusion models with good forecasting
performance. The magnitude of the forecast error is modelled as function of early diffusion
summary data, structural characteristics of the diffusion models, and estimation performance
measures. Information on these factors is readily available to the forecaster prior to the generation
of the forecasts. The method is capable to explain part of the variation in forecast error, and can
be used as an aid to diffusion model selection for new product forecasting.
Keywords: Diffusion models, Technology forecasting, Forecast evaluation
Track: Modelling and Forecasting
1. Introduction
Diffusion models are used to model the temporal evolution of new product first-purchases (e.g.
Mahajan, Muller, and Wind, 2000) and to forecast the development of the diffusion process (e.g.
Parker, 1994). A large number of alternative models are available to the marketing researcher.
However little is known about their forecasting performance and the factors that affect the
magnitude of their forecast errors. This lack of knowledge makes model selection difficult when
facing a diffusion-forecasting problem. Fildes and Kumar (2002) and Armstrong (2005) are some
of the authors who point to the limited research effort in assessing and improving forecast
accuracy of diffusion models. Important exceptions are the studies of Meade and Islam (1995a,
1998) and of Islam, Fiebig, and Meade (2002). These authors have analyzed the performance of
diffusion models in telecom diffusion datasets and discussed model selection and forecast
combination. However they utilized few actual diffusion series (typically 50 or less) and did not
systematically investigate the possible sources of forecast error and the magnitude of their effect.
Of major interest to the diffusion modeler is the question of how to assess the likely forecast error
from a diffusion model utilizing the information contained in a single diffusion data series of
usually limited length. One obvious way to answer this question is by providing forecast
confidence intervals to accompany the generated point forecasts. However there are problems
associated with estimating prediction intervals for non-linear models (see for example Chatfield
1993, Meade and Islam 1995b). Most diffusion models are non-linear. One serious problem is
that prediction intervals are correct only when the diffusion model is more or less correctly
identified for the particular data series. The data generating process (DGP) for a diffusion series
is not known (and even may change over time), hence there is no guarantee about the quality of
the generated intervals. Selecting an appropriate model as an approximation to the real DGP is a
very important step in generating reliable forecasts and associated prediction intervals.
To facilitate model selection, this paper analyses measurable factors that affect the long-range
forecast error from diffusion models. The analysis is performed using long-enough simulated
series based on 926 real new product diffusion datasets, together with a collection of diffusion
models with good forecasting performance. The magnitude of the expected forecast error is
modelled as function of early diffusion summary data, structural characteristics of the employed
diffusion models, and estimation performance measures. The forecaster can use the expected
forecast error and the associated prediction interval as an indicator of model appropriateness.
Thus, the method can be used as an aid to diffusion model selection for new product forecasting.
In section 2 we present the framework for the analysis and the data. Section 3 describes the
regression model for the forecast error and the explanatory variables. In section 4 we discuss
model estimation results and the predictive validity of the model in a test dataset. Finally, we
conclude with a summary of the results and directions for future research.
2. The Analysis Framework - Data and Models
The available data cover a wide range of diffusion realizations. They are 926 annual
multinational time series (for a total of about 50 countries) for the penetration Ft ∈ (0,1] of
household appliances, home electronics, and telecommunication equipment introduced after year
1950 with 7 or more observations. 496 series have length of 8 or more and only 73 with 20 or
more observations. Given that a number of observations are reserved for model estimation, it is
clear that more series are required for detailed medium to long-term forecast error computations
and comparisons. For this reason simulated data are generated.
The simulation strategy is the following: (a) For each diffusion model included in the analysis
parameters and associated error variances are estimated in all 926 series using all available
observations. Estimation is via NLS. Arguments for using NLS estimation can be found in Meade
and Islam (1998). (b) 250 pseudo-random draws per model were obtained from the empirical
multivariate distribution of estimates and the observed minimum penetration to generate (i)
model parameters, (ii) error variance, and (iii) minimum observed penetration. (c) Finally, the
draws were used to generate artificial series of 100 observations.
This process was repeated for the 20 diffusion models listed in Table 1 at the Appendix. In such a
way about 5000 artificial diffusion series were generated. For all series every model was fitted in
estimation windows including the first 7,8, … to 25 observations respectively. Then forecasts for
up to 10 periods ahead were made for the corresponding 19 forecast origins and several forecast
accuracy measures were recorded. Forecast performance comparison is performed across all
origins and data series. The details are lengthy and are presented elsewhere. Based on these
comparisons, and in order to keep the forecast error analysis task to a manageable size, only the 5
best performing models were selected for further analysis. These are the Extended Riccati, the
Extended Logistic - which gives the cumulative penetration of the Bass (1969) model as a
function of time-, Flexible Logistic, Autoregressive Gompertz, and the Mean Field model. All
models are 4-parametric with the 4th parameter being the saturation level.
3. Modeling Forecast Error
In the diffusion forecasting literature several factors related to forecast accuracy have attracted
some - but not systematic - attention (e.g. Meade 1984; Young and Ord 1989; Young 1993;
Meade and Islam 1995a, 1998). These factors pertain to the ability of a model and the estimation
procedure to properly capture the characteristics of a diffusion process and can be classified into
three broad categories: factors related to (a) model misspecification, (b) model fit, and (c) quality
of estimates. We model forecast error as a function of these factors. In addition, we explore
possible relationships between the descriptive statistical characteristics of an observed series and
the forecasting accuracy of a diffusion model. The forecast error measure we adopt is the 10steps ahead mean absolute percentage error (10-MAPE). MAPE is the most commonly used
measure in the relevant literature. It is scale-free and allows for forecast comparisons in different
series with different diffusion levels. The forecast origin is fixed to the 10th period for all
simulated series. These choices are made because: (a) in most real-life situations long-range
forecasts are required, and (b) ten observations allow for parameter estimation and measurement
of the explanatory variables that are described in detail below.
Kolmogorov-Smirnov tests show that the natural logarithmic transformation of 10-MAPE is
approximately normal for all five models. Hence, it can be modeled as a linear in the parameters
(m )
function with a model specific normally distributed residual ei
y i( m ) = ∑ ak( m ) f k ( xi(,mk ) ) + ei
( m)
,
ei
(m)
(
)
~ N 0, σ m ,
2
(1)
k
where yi(m ) is the logarithm of 10-MAPE for diffusion model m for forecast-error observation i,
(m )
and x i is a K × 1 vector of characteristics related to model m and the data series. f k is generally
a smooth non-linear transformation function that can be determined via a linearization procedure,
for example a Box-Cox transformation or the alternating conditional expectation (ACE)
transformation method of Breiman and Friedman (1985). The explanatory variables and their
measurement are discussed below.
Model misspecification and fit
Factors related to model misspecification that may influence forecast error include
heteroscedasticity, serial correlation, and temporal model stability. The presence of
heteroscedastic and serially correlated residuals is tested using the White (1980) and Durbin
(1970) tests respectively. Results are recorded using the indicator variables HET and SER; they
assume the value 1 when the corresponding test rejects the null hypothesis. The 5% significance
level is the decision boundary in all tests we perform. Model instability over time has been
observed in many diffusion modeling occasions. We adopt the approach of Meade and Islam
(1998) who employ a test suggested by Harvey and Collier (1977). The test examines one-step
ahead forecast errors in recursive model fits within the estimation window with respect to their
estimated variance. If the null hypothesis of correct model specification is rejected, the indicator
variable STAB takes the value 1. Of particular interest is the temporal consistency of the
estimated saturation level that is the upper bound for the model forecasts. We examine the
presence of a linear trend in the recursively estimated parameter and whenever the absence of a
trend cannot be rejected the dummy SAT takes value 1. Model fit, measured by R2, is an indicator
of model adequacy in describing the observed diffusion data. A poor model fit is expected to lead
to poor forecasts. We use the logarithmic transformation of R2, ln( R2 ), coded as LRSQ.
Estimation quality
Estimation quality is judged by the statistical significance of estimated parameters and the
theoretical plausibility of their estimated values. Insignificant estimates render the model a poor
descriptor of the series and the generated forecasts are unreliable and useless (associated forecast
intervals would be very wide). At each model estimation instance we record the number of
significant parameters (SIGPAR) and the values of the corresponding t-statistics (denoted as T1,
T2, T3, and T4). Theoretically implausible estimates (e.g. wrong signs) indicate that the model’s
theoretical foundation is not adequate for the process realized in a particular series (Meade and
Islam, 1995a). In our optimization approach we constrain parameters to take values within their
theoretical bounds in order to achieve meaningful forecasts and thus increase the power of
forecast comparisons. However, we count the total number of model parameters held at a bound
at an estimation instance with the variable NBOUND. A parameter estimate at a bound may
indicate redundancy, a possible reversion to a nested model, and/or inadequacy of the model to
properly describe the process.
Data series characteristics
If relationships between descriptive characteristics of a series and forecasting accuracy of a
model exist, they can be attributed to model identification: high accuracy of forecasts when a
series exhibits certain statistical properties implies that the model is more likely to be correctly
identified. Since little is known about the relationship of a series characteristics and model
identification, we adopt a “data mining” approach by trying out a number of measures such as:
the value of the 1st observation in a series (MINPEN), the average observed growth rate (i.e. the
average of the differenced series - AVDIFF), the average of the twice differenced series
(AVDIFF2), higher moments of the data such as variance, skewness and kurtosis (STDEV,
SKEW, KURT). To capture more detailed higher order characteristics of the series and to
identify natural groupings that could be related to forecast error, we classify them to three sets of
clusters using (a) the original series (the first 10 observations), (b) the differenced series, and (c)
the twice-differenced series. The clustering algorithm led to the identification of 2, 3, and 2
corresponding clusters. The indicator variables CLUS1, CLUS2, and CLUS3 respectively record
the membership of a series to the developed clusters.
4. Results
The analysis focuses on forecasts with origin the 10th observation. A total of 4304 simulated data
series for which the estimation algorithm converged and produced proper forecasts and
parameters’ standard errors for each of the five models were employed. Of those, 3292 (76%)
were used to estimate model (1), while 1012 observations (24%) were reserved for validating
predictions. Table 2 summarizes 10-MAPE from each of the 5 diffusion models. Initially we
performed least squares (LS) estimation of regression (1) for each diffusion model. The variance
minimizing alternating conditional expectation (ACE) transformation method of Breiman and
Friedman (1985) improved greatly the LS fits (achieved R2 ranging from 0.48 to 0.54) and
reduced the residual standard errors considerably. The residual standard error of the regression is
important as it determines the width of prediction intervals for the forecast error. Due to the
presence of influential observations that bias LS parameter estimates and affect the estimate of
the residual standard error, we estimated the untransformed versions of (1) with robust MMtechniques (e.g. Yohai, Stahel and Zamar 1991, Pena and Yohai 1999). The method de-biases
parameter estimates and in our case reduced further the standard error of the regressions. Thus,
predictions from (1) are less biased and more accurate. Robust estimation results, fit statistics and
prediction intervals for each of the 5 regressions are given in Table 3. Below we discuss the
estimated effects of the explanatory variables on forecast error and we present prediction results.
Model misspecification and fit
Other things being equal, the presence of heteroscedastic residuals (HET) has only a marginally
significant effect on the forecast error from the Extended Ricatti and the Extended Logistic
models. Serially correlated residuals (SER) and model stability (STAB) do not appear
informative. The presence of a linear trend in the saturation level estimate (SAT) has a significant
impact on the forecast error only for the Mean Field model; it is strongly associated with large
errors. As expected, better model fit is significantly (LRSQ) associated with smaller forecast
errors for all models.
Estimation quality
The number of significant parameters (SIGPAR) together with the value of the corresponding tstatistics (T1, T2, T3 and T4) has a significant effect on forecast error. Their net effect is
significantly negative for all models. However, due to correlations between them (not very high
though), their partial effect in the regression model may change sign and sometimes loose its
significance. A highly significant estimate of the saturation level (T4) is strongly related to higher
forecast accuracy for all five models. The number of parameters in bounds (NBOUND) is
positively associated with forecast error.
Data series characteristics
The forecast error increases significantly when the 1st observation of the penetration level
(MINPEN) in a series decreases. For 4 models (except of the Mean Field) the error is a linearly
decreasing function of the average diffusion rate (AVDIFF) in the observed series when
controlling for the effects of other variables. Combining this effect with the strong negative effect
of the standard deviation (STDEV) of the observed series we can say that the more spread-out the
observed data the better is the accuracy of the forecasts. The effect of the average rate of change
of the diffusion rate (AVDIFF2) is less important. The skewness (SKEW) has a significant and
strong effect: strong positive skew is related to high forecast errors. Kurtosis (KURT) appears
unrelated to error. Membership of a series to clusters developed using the first and the second
order differences (CLUS2 and CLUS3 respectively) has a significant impact on the forecast
accuracy of all diffusion models.
Prediction
For each diffusion model, the estimated regression (1) was applied to the test sample to provide
predictions of the forecast error. Prediction intervals were calculated using the fit statistics of
Table 3. The calculated 90% prediction intervals for the forecast errors in the test sample were
proven to be accurate, i.e. contain the actual error approximately in the 90% of the cases,
indicating the statistical validity of the prediction procedure. Figure 1 gives an example of the
predicted (a) log(10-MAPE) and (b) 10-MAPE in the test sample for the Extended Logistic
model. The average across models median width of the 90% prediction interval is quite wide
(20.32% in the MAPE scale). This is rather normal for a highly uncertain process such as
diffusion, particularly for long-range predictions made at relatively early stages (times) as in the
present analysis. Intervals are expected to have smaller width for shorter-range forecasts and
predictions made at later stages of the product life cycle.
5. Summary of the findings and concluding comments
The main findings from the robust regression fit and analysis of variance for the forecast error
from the 5 diffusion models are summarized as follows: When controlling for the effects for other
variables: (a) Model fit is the single most important factor with a negative impact on forecast
error. (b) Quality of parameter estimates is crucial for accurate forecasts. The significance of the
saturation level estimate is the second most important predictor of forecast error. The number of
parameter estimates at the optimization bounds is positively associated with forecast error. (c)
Data series characteristics are strongly related to forecast error. The penetration level at the 1st
observation has the third most important (negative) effect. Data skewness is the fourth most
important predictor, with a strong positive effect. The typical spread of the observed data has a
significant negative impact on forecast error. Higher order characteristics of the data are
informative about the accuracy of predictions. (d) Model misspecification appears to have little
impact on forecast error.
The exploratory approach adopted in this paper has identified determinants of the forecast
accuracy of diffusion models and quantified their effect. It provides an additional input to the
forecaster’s toolset for assessing the likely forecast error and for choosing among diffusion
models. The modeling effort has to be extended further to investigate more possible sources of
error in order to achieve more accurate predictions. It must also be extensively validated for more
forecast origins and lead times, not only on artificially generated data but also on a sufficient
number of long-enough real diffusion series. This requires updating the available series with the
latest released observations and collection of more datasets. Under current investigation are
complementary probabilistic modeling strategies for model selection and diffusion forecast
combination.
Appendix: Tables and Figures
Table 1: Diffusion models in forecast comparison
Model
Abbreviation
1. Bass – Skiadas
2. Cumulative log-normal
CloNo2
3. Extended Riccati
ExRic
4. Extended Logistic
ExloB
5. Floyd
Flo
6. Flexible Logistic
FLOG
7. Jeuland
Jeul
8. Gompertz
Gomp
9. Gompertz Autoregressive
GompAR
10. Harvey
Harv
11. Kumar & Kumar
KK3
12. Local Logistic
LoLog
13. Mean Field
MF
14. Mansfield
Man
15. NSRL
NSRL
16. NUI
Nui
17. Sharif – Kabir
Shk
18. Simple Logistic
Slog
19. Simple Logistic Autoregressive SlogAR
20. Weibull
Weib
(1)
S=Symmetric, NS=Non-Symmetric, F=Flexible
Reference
Skiadas (1986)
Bain (1963)
Kendall et al. (1983)
Meade (1998)
Floyd (1968), Mahajan et al (1993)
Bewley & Fiebig (1988)
Jeuland (1981)
Gompertz (1825), Stone (1980)
Meade and Islam (1995a)
Harvey (1984)
Kumar & Kumar (1992)
Meade (1988)
Emmanouilides (1997)
Mahajan et al. (1993)
Easingwood et al. (1981)
Easingwood et al. (1983)
Mahajan et al. (1993)
Verhulst (1838), Stone (1980)
Meade & Islam (1995a)
Sharif & Islam (1980)
(2)
Including the saturation level
Class(1)
F
F
F
F
NS
F
F
NS
NS
F
F
S
F
S
F
F
F
S
S
F
Number of
Parameters(2)
3
3
4
4
2
4
4
3
4
4
3
2
4
2
3
4
3
3
4
3
Table 2: 10-period lead time MAPE Statistics by Model (1) – Forecast origin is 10th observation
MAPE Statistics
Geometric
1st Quartile
Median
3rd Quartile
Mean
Model
Mean
Extended Riccati
3.65
7.69
16.67
7.78
13.55
Extended Logistic
3.44
7.12
16.13
7.21
12.58
Flexible Logistic
3.63
7.88
17.41
7.75
13.59
AR Gompertz
3.70
7.64
15.83
7.55
12.61
Mean Field
3.37
6.99
15.07
6.94
11.76
(1)
Pooled estimation and test samples of 4304 total observations per model
Standard
deviation
15.87
14.38
15.29
14.05
13.47
Table 3: Estimation results, fit statistics, and prediction intervals for the ln(10-MAPE) robust regressions(1)
Model
Extended Riccati Extended Logistic Flexible Logistic
AR Gompertz
Variable
Mean Field
(INTERCEPT)
0.46
-1.53 ****
1.99 ****
0.35
-1.85 **
Specification and fit
HET
-0.14
-0.14 **
0.13 *
-0.07
-0.05
SER
-0.04
-0.09
0.00
0.04
-0.06
STAB
-0.09
-0.09
0.62
0.27
SAT
0.08
-0.03
-0.11
-0.07
0.25 ****
LRSQ
-0.26 ****
-0.25 ****
-0.09 ****
-0.16 ****
-0.17 ****
Estimation quality
SIGPAR
-0.54 ****
-0.02
0.50 ****
-0.10
0.18 *
(T1)0.4
0.18
0.17 ***
0.43 ****
0.17 ****
0.12 ****
(T2)0.5
-0.18
0.39 ****
-0.56 ****
0.01
-0.05
T3
0.11 ***
-0.01
0.03 ***
0.00
0.01
0.4
(T4)
-0.17 ****
-0.35 ****
-0.37 ****
-0.25 ****
-0.31 ****
NBOUND
0.28 ***
0.33 ****
0.05 *
0.08 **
0.86 *
Data series characteristics
MINPEN
-3.43 ****
-4.58 ****
-2.01 ****
-5.55 ****
-6.05 ****
AVDIFF
-15.60 ***
-22.83 ****
-20.15 ***
-17.39 ****
-8.70
AVDIFF2
-3.40
-9.91 **
8.30 *
-3.70
-7.31 *
(STDEV)0.1
-3.29 ****
-1.13 **
-5.79 ****
-3.00 ****
-1.92 *
SKEW
0.46 ****
0.21 ****
0.53 ****
0.34 ****
0.29 ****
KURT
-0.01
0.01
-0.01
0.01
-0.01
CLUS1 (2)
0.03
0.10
0.04
0.09
0.04
CLUS2 (2)
0.26 ****
0.19 ***
0.14 **
0.24 ****
0.24 ****
CLUS2 (3)
0.18 ***
0.14 ***
0.21 ****
0.14 ***
0.13 **
CLUS3 (2)
0.14 **
0.10 **
0.25 ****
0.14 ***
0.14 ***
Fit statistics and prediction intervals
R2
0.38
0.41
0.41
0.44
0.39
Residual S.E.
0.70
0.62
0.67
0.70
0.71
Half width of 90% PI (log scale)
1.15
1.02
1.10
1.15
1.17
Upper 90% PI (original scale)
3.16 x fitted MAPE 2.77 x fitted MAPE 3.00 x fitted MAPE 3.16 x fitted MAPE 3.21 x fitted MAPE
Lower 90% PI (original scale)
0.32 x fitted MAPE 0.36 x fitted MAPE 0.33 x fitted MAPE 0.32 x fitted MAPE 0.31 x fitted MAPE
Median 90% PI for 10-step MAPE
(2.5, 24.2)
(2.6, 19.7)
(2.6, 23.6)
(2.5, 24.1)
(2.2, 22.4)
(1)
Estimation sample of 3292 observations per model. Significance levels for the two-sided t-test: (*) 10%, (**) 5%, (***) 1%, (****) 0.1%
(a)
(b)
Figure 1: Predicted 10 step-ahead (a) ln(MAPE) and (b) MAPE in the test sample for the Extended Logistic model. Predictions (black
line) are displayed in ascending order to facilitate exposition, together with the corresponding 90% prediction intervals (red and green
lines). Actual forecast errors are displayed as points.
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