Information Technology Reference
In-Depth Information
new technique, due to Ramsay and Silverman (1997). The basic philosophy of FDA is that we should
think of observed data as functions as opposed to a vector of observations in discrete time.
In the ITU telecommunications indicator database, we have times series of subscription data for
217 countries by various wireline and wireless technologies like landline, wireless telephony, and
internet access using several technologies like dial-up, DSL, Cable, ISDN and wireless broadband. Our
focus is in particular, understanding the dynamics of fixed broadband subscription and its impact on
growth of wireless broadband. We use functional analysis to model adoption for fixed broadband data
for different countries and to understand if the trend in these growth curves varies in longitudinal or
spatial aspects across different countries or if products exhibit different magnitudes of dynamics. We
refer to these subscription data over time as a set of functional observations and the first step in FDA
involves converting raw data into functional form. This is called “recovering, from the observed data,
the underlying functional object.
Ramsay and Silverman recommend the use of penalized smoothing splines for smoothing. Subramaniam
and Varadhan (2008) and Varadhan and Subramaniam (2009) proposed several alternative approaches for
e.g., using semi-parametric models, local polynomial smoothing with plug-in bandwidth or the Gasser-
Muller kernel global plug-in bandwidth for smoothing. These methods allow for automatic estimation
of the smoothing or bandwidth parameter. This parameter provides the tradeoff between bias and vari-
ance for the estimated models. Our analysis utilized the Gasser-Muller kernel global plug-in bandwidth
method (T. Gasser et.al., 1991) to smooth the time series of BB penetration.
R
(2005) was used for all
the statistical computations, and in particular, the “lokerns” library in
R
was used to fit these models.
The advantage of using Gasser-Muller method is that it automatically estimates the bandwidth or the
smoothing parameterusing a data driven procedure which estimates asymptotically optimal bandwidth
from the data. For this, it uses a mathematical expression that describes the asymptotic value of mean
integrated square error and that expression contain unknown functionals. We “plug-in” values for the
unknown which is estimated from the sample data. The asymptotically optimal bandwidth
b
A
: is obtained
by minimizing a large sample approximation to mean integrated square error (MISE):
1
5
c
c
2
1
s
b
A
=
⋅
1
⋅
with constants and a known function. The residual variance
σ
2
and
n
1
2
∫
′′
2
v t r
( )
( )
t dt
0
1
∫
′′
2
the functional
v t r
( )
( )
t dt
are unknown and estimators have to be plugged into the expression for to
0
make it an estimator for the optimal bandwidth (plug-in estimator). The details are given in T. Gasser
et.al. (1991) and Gasser, Th et. al (1984).
Once the data are represented by functional objects we extend the analysis to extract features from
the functional. Once such feature extraction we use here are the estimated first and second derivatives
of the functions. The derivatives give further insights on rate of change in adoption and identify the
different growth spurts associated with the curves.
Search WWH ::
Custom Search