|
ABSTRACT
We utilized a newly
developed robust forecasting technique for fertility
projection using age specific fertility rates
from 1974-2002 in Bangladesh. It combines the
ideas of functional data analysis to model age
specific fertility in time trends, non parametric
smoothing method to smooth data, decomposed
(using principal components analysis) to estimate
basic functions that represent the functional
curve and forecast the fertility trends using
an automatic modelling framework for selecting
an exponential smoothing method. Overall, the
twenty year forecast suggested that the fertility
decline of Bangladesh will continue in future.
A set of three basic functions minimized the
mean integrated squared forecasting error and
account for 98.6% of variation around the mean
fertility curve. The basis function models for
young mothers in their teens and the mothers
aged over 40 explained maximum (86.5%) of the
total variation. Forecasts show greatest decline
among these two age groups. Finally, the total
fertility rate will achieve the replacement
level of fertility by 2013. Based on the findings
of this study, it may suggest that attention
should be focused on providing educational facilities
particularly among women and encourage contraceptive
use in order to keep the declining level of
fertility in Bangladesh. All analyses were performed
using a new statistical programming language
R.
Key
Words: Robust forecasting, basic function,
functional curve, exponential smoothing, forecasting
error, fertility curve, R language, replacement
level.
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1.
Introduction
Accurate estimates
of future age-specific fertility are critical for
allocation of resources to fertility control programmes
and evaluation of family planning programmes. There
has been a surge of interest in the problem of fertility
and mortality forecasting in the last few years, driven
by the need for good forecasts to inform government
policy and planning.
Fundamental
changes in welfare policy are taking place in many
countries as a result of forecasts of an increasing
elderly population. These age-specific population
forecasts rely on age-specific forecasts of fertility
and mortality rates. Therefore, any improvements in
mortality and fertility forecasting have an immediate
impact in guiding policy decisions regarding the allocation
of current and future resources. Future fertility
forecasts are of interest to governments in planning
children's services.
Papers
by Rogers (1986) and Thompson et al. (1987) first
suggested fitting curves to annual age-specific fertility
rates, forecasting the parameters of the curves using
time series techniques, and then using the forecasted
curves to generate forecasts of future age-specific
fertility rates. Bozik and Bell ( August 1987) presented
an approach based on a principal components approximation
to the age-specific fertility rates that avoids the
problem of the error in the fitted curve that is not
negligible to a large extent, while still reducing
dimensionality. They compared this approach with direct
univariate modeling of all the age-specific rates,
and with the curve fitting approaches. The approach
appears to have potential for producing reasonable
forecasts and forecast intervals for the age-specific
rates using a small number of components. Bell et
al., (1988) projected fertility by combining short-term
forecasts from time series models with long-term demographic
judgmental projections. They had selected a particular
time series model for the series to be forecast, fits
the model to the data, and uses the fitted model to
produce point and interval forecasts.
The
autoregressive-integrated-moving average (ARIMA) models
discussed by Box and Jenkins (1970) comprise one popular
class of models. Hyndman (2003) proposed automatic
forecasts of large numbers of univariate time series.
Automatic forecasting algorithms must determine an
appropriate time series model, estimate the parameters
and compute the forecasts. The most popular automatic
forecasting algorithms are based on either exponential
smoothing or ARIMA models.
Forecasting
population has long been one of the public faces of
forecasting (along with the weather and the stock
market), and Heather Booth (2004) contributed first
in review of forecasting in demography. Her paper
contains an impressive and comprehensive critique
of the past and potential future of demographic forecasting.
In
this paper we represent a robust approach of forecasting
age-specific fertility rates that combines ideas from
functional data analysis, nonparametric smoothing
and robust statistics proposed by Hyndman and Ullah
(2005). Their approach is a generalization of the
method of Lee and Carter (1992), and has the following
advantages:
- fertility rates are
modeled as continuous functions of age so that subtle
patterns of variation between years are captured;
- data are smoothed
prior to estimating the basis functions, thus reducing
observational error;
- the approach forecasts
the entire function for future time periods with
prediction intervals;
- the method is robust
to outlying years;
- the flexibility
of the approach allows the incorporation of important
covariates such as treatment effects into the modeling.
The purpose of this
study is to demonstrate the utility of this new modeling
and forecasting method for estimating future age-specific
trends in fertility, using age-specific fertility
data of Bangladesh from 1974 to 2002. Here we will
apply functional data analysis techniques to model
age-specific fertility rates in time trends, and forecast
entire age-specific fertility functions using a state-space
approach.
2 Methodology
2.1
Sources of Data
Annual Bangladesh fertility
rates (1974-2002) for age groups 15-19, 20-24, 25-29,
30-34, 35-39, 40-44 and 45-49 were taken from the
Statistical yearbook of Bangladesh, (BBS- 2004 and
1987, Pocket Statistical Year Book- 1980). These are
defined as the number of live births during the calendar
year, according to the age of the mother, per 1,000
of the female resident population of the same age
at 30 June. Since all the age-specific population
data for the successive years are not available, we
have estimated the entire values of number of female
population using exponential growth rate. Also the
SYBs reported ASFR for different years were collected
from different sources, such as, SYBs reported ASFR
for the year 1974 based on data from the Bangladesh
Retrospective Survey of Fertility and Mortality, London;
rates for 1971-1975 from Bangladesh Population Control
and Family Planning Division, 1978; rates for 1978
from the Bangladesh Demographic Pilot Survey, BBS
1978; and rates for the years 1980 to 2002 based on
Vital Registration System, (VRS) BBS.
2.2 Functional Data Analysis
With functional data methods,
data can be smooth curves or functions. In the case
of age specific fertility, rates are treated as smooth
functions of age. The basic idea behind functional
data analysis is to express discrete observations
in the form of a function, and then draw information
from a collection of functional data by applying concepts
from multivariate data analysis. Moreover, functional
data analysis mainly uses Fourier series, spline or
B-spline smoothing techniques in transforming vector-valued
data into functions. In this study we will use constrained
and weighted penalized regression splines for fertility
data preferred by Hyndman et al. (2005).
Let  denote
the log of the observed fertility rate for age x in
year t. Let us assume there is an underlying smooth
function  that
we are observing with error.
Thus, we observe the functional time series 
t = 1 K, n, i = 1 K, p,
where
(1)
where Xi
is the center of age-group i (i = 1 ,…, p)
 is
an iid standard normal random variable and
 allows
the amount of noise to vary with x.
We are interested in forecasting
for 
and 
The error variance is  computed
as follows.
Let pt(x) denote the observed fertility
rate per thousand women for mothers of age x in year
t and Nt(x) is the female resident
population of age x in year t.
Then pt(x)
is approximately binomially distributed with estimated
variance  .
So the variance of yt(x) = log[pt(x)]
is (via a Taylor approximation)
(2)
Define weights equal to the
inverse variances  and
constrain the fitted curves to be concave.
There are various smoothing techniques available to
estimate the function from the discrete observations.
For each year t, the smooth curves  were
estimated using a weighted median smoothing B-spline,
constrained to be concave. For these data  ,
is obtained from (2) and weights set to the inverse
variances  .
The n smooth curves are our
functional observations, {ft(xi)}where
 and
For the data considered here, four of these are
shown in Figure 2.
2.3
Fitting and Decomposition of Smoothed Data
After constructing the functional
observations, we have to fit the model
(3)
where 
is the mean log fertility rate across years,
is a set of orthogonal basis functions, and
is the model error which is assumed to be serially
uncorrelated.
Now we wish to estimate the
optimal set of K orthogonal basis functions.
Specifically, for a given K, we want to find
the basis functions  which
minimize the mean integrated squared error:
(4)
This is achieved using functional
principal components decomposition (Ramsay and Silverman,
1997) applied to the smooth curves {}
which gives the least number of basis functions, enables
informative interpretations and gives coefficients
which are uncorrelated with each other.
Using functional data analysis
technique and principal component decomposition to
estimate the basis functions, a model with K = 3 basis
functions was selected. A set of K = 3 basis function
minimized the MISFE, while estimating an additional
basis function did not contribute to a further reduction
in the MISFE.
2.4
Forecasting Framework
We estimate future values
of fertility
by forecasting the entire function
for 
and .
The coefficients of the fitted  function
are forecast using time series models. The forecast
coefficients are then multiplied by the basis functions,
resulting in forecasts of mortality curves.
Let  denote
the h-step ahead forecast of  and
 let
denote the h-step ahead forecast of f n+h
(X). Then
 (5)
To forecast the coefficients in equation (5), a variety
of time series forecasting methods are available.
In this study we use state-space models for exponential
smoothing (Hyndman, et al., 2002), which underlies
the damped Holt's method. This was selected as it
extrapolates the local trends seen in the coefficient
series while damping them to avoid nonsensical long-term
forecasts. However forecasts from exponential smoothing
methods are estimated recursively where recent observations
are given more weight than historical data. Makridakis
et al. (1998) present a modeling framework based on
the taxonomy proposed by Pegels (1969) and the framework
is expanded in Hyndman et al. (2002), who show how
models can be automatically selected for a given time
series.
Hyndman and Ullah (2005) have
shown that the forecast variance can be obtained by
adding the variances from each of the terms in equations
(1) and (3). Therefore,
(6)
where,  denotes
all observed data,
 can
be obtained from time series model,
 is
the variance of the smooth estimate  ,
can be obtained from the smoothing method used,
is
computed using the approximations (2),
and v(x) is estimated by averaging  for
each x
A 100(1-?) % prediction interval
for yt(x) is then constructed as  where
 is
the 1-?/2 standard normal quantile.
The accuracy of the fertility
forecasts is computed by minimizing the mean integrated
squared forecasting error (MISFE) which is defined
as
 (7)
where  ;
h=1,…m denote the forecast error for (2) and N is
the minimum number of observations used in the fitting
model.
In our implementation the robustness parameter was
set to  =
3 and the minimum number of observations used in fitting
the models was N = 10. All analysis are performed
using the R implementation of the S language (R development
Core Team, 2006).
3 Results and Discussions
Figure 1 represents the data
of age specific fertility rates and log fertility
rates respectively, from the year 1974 to 2002 for
age groups 15-19, 20-24, 25-29, 30-34, 35-39, 40-44
and 45-49, which is shown as separate time series.
Here fertility rates are converted to functional data
by estimating a smooth curve through the observations,
taking the centre of each group as the point of interpolation.
Detailed investigation suggests
that much of the change in fertility occurred among
the younger age groups, those 15-20 years of age.
It is clear from both the figure that maximum fertility
occurs among the age group 20-24 and 25-29. There
are some up and down situations in fertility for all
age groups showing before the mid 80's. Especially
for the years 1977, 1978 and 1979, where the fertility
rates for the age group 15-19 and 20-24 are 186 and
322 in 1977, 124 and 256 in 1978 and in 1979 those
are 306 and 256. It might be the fertility rates that
are collected from different sources causing the fluctuated
situation in these particular years.
Figure 2 represents the fertility
rates of Bangladesh for the selected years 1975, 1981,
1991 and 2001. All the years showed that before age
14 and after age 50 the rates decreased to the null
as expected because we set a relatively arbitrary
value for fertility rates at age 13 and 52 being 0.001
for all the years. It is clear from this figure that
the fertility rates are declining for every age-group
and the decrease in fertility is high in the recent
years.
The first and second panel
of Figure 3 shows that the fitted basis functions
 and
associated coefficients  for
the fertility data, which are smoothed using median
smoothing B-splines. Fitting a functional regression
model with K = 3 basis functions accounts for 98.3
per cent of the variation around the mean log fertility
curve. The proportion of variation explained by each
basis function is 86.4, 8.4, 3.7 for K = 1, … , 3
(Table 1). It is apparent that the basis functions
are modeling the fertility rates in different age
ranges of mothers: "Basis function 1",  ,
models for the difference between the young mothers
in their teens and 20's, and mothers aged over 40's,
since the curve shows two picks, one at age before
20's and other at age after 40's. The basis function
1 explained 86.5% of the total variation. Therefore
this parameter controls the overall change in the
trend in age-specific fertility of Bangladesh. "Basis
function 2",  ,
is complex, and contrasts those between the young
mothers in their teens and 20s, mothers aged between
30 to 35 years, or the mothers aged over 50, with
the other ages. This basis function explained only
8.4% of the total variation. "Basis function
3",  ,
is also complex, models differences between mothers
aged between 35 to 40 years and those in their teens.
This basis function explained only 3.7% of the total
variation.
Also the second panel of Figure
represents the coefficients associated with each basis
functions. The coefficients associated with basis
function 1,  ,
shows a rapid decreasing trend. In 1974 the value
of
is 0.83 and which is decreased to -0.59 in year
2002. The maximum value of occurs
at year 1978 is 1.12. These results indicate that
the young mothers in their teens and 20's, and mothers
aged over 40's have behaved as giving less births
after 80's. The coefficient associated with basis
function 2,  ,
represents an increase around 1990 and decrease during
1997's to the recent years. A very much up and down
situation occurs before 1990's. It indicates that
the young mothers in their teens and 20, mothers aged
between 30 to 35 years and over 49 started to give
less birth in the most recent years. The coefficient
associated with basis function 3,  ,
represents, although an increase around 1980 and a
decrease during 1997's, but it is difficult to identify
its appropriate behavior. Therefore it does not indicate
that the mothers aged between 35 to 40 years are giving
less births.
We computed 20 year estimates
of future age-specific fertility rates using state
space exponential smoothing models as described by
Hyndman et al. (2002). The automatic model-selection
algorithm chose models with additive errors and a
damped trend. The model parameters were selected by
minimization of the one-step MSE. Figure 3 represents
the combination of twenty-year forecast coefficients
with the estimated basis functions yields forecasts
of the fertility curves for the years 2003-2022. The
gray shaded regions are 90% prediction intervals.
The 20 years ahead forecast
of  ,
coefficient associated with basis function 1, with
90% predicted interval is represented in figure 4.
Clearly a very much linearly decreasing line is showing
in the shaded region. Which indicates that in future
the young mothers in their teens and 20's, and mothers
aged over 40's will give less births continuously.
Figure 4 represents the forecasts of fertility for
the years 2001 and 2022, along with 90% prediction
intervals. Clearly all age of women forecasts show
a continuing decrease in fertility rates, with the
greatest declines in the age of women 30 to 40. Forecast
variance is high for age around 25. Also figure 4
indicates that young mothers and old aged mothers
are less interested to givee birth than the mothers
aged around 25. The maximum fertility contributed
by the women 20-29 years of age will continue in the
future.
Figure 5 displays the fitted
total fertility rates of Bangladesh along with twenty
years forecast of Bangladesh. The shaded region gives
90% prediction intervals. The vertical dotted line
represents the replacement level of fertility and
the horizontal line represents the year of achieving
the replacement level of fertility of the population.
It is clear from the figure that Bangladesh total
fertility rate decreasing in future will continue.
And the decreasing rate of fertility will increase
after 2010. Also this figure indicates that Bangladesh
total fertility will decrease below the replacement
level of fertility around 2013 along with 90% prediction
interval, which is a desire of population planners
and policy-makers of most developing countries.
4. Conclusion
Population futures are
central to a wide range of pressing concerns in Bangladesh
today and underlie many aspects of social, economic
and physical planning. The ability to incorporate
both greater accuracy and uncertainty in population
futures would present a major advance in decision-making
and planning. In this study we adopt a methodological
approach proposed by Hyndman and Ullah (2005) which
models age as a functional covariate rather than a
fixed variable, so that the age-shape of fertility
varies over time, thereby enabling the models and
forecasts to pick up subtle variations. To our knowledge,
no other study has modeled or forecast form a model
with age as a functional covariate of fertility over
time. We have demonstrated the utility and flexibility
of this newly developed approach to forecast age-specific
fertility rates of Bangladesh. Our estimates suggest
that fertility of Bangladesh will continue to decline
in future. The overall change (84.6%) in trend in
fertility of Bangladesh is controlled by the young
mothers in their teens and mothers aged over 40's.
Bangladesh total fertility will decrease below the
replacement level of fertility around 2013.
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Figure
1: Log fertility rates of Bangladesh
viewed as separate time trends from 1974-2002.
|
|
Figure
2: Log fertility rates viewed as functional
data and calculated using median smoothing B-splines
constrained to be concave for years 1975, 1981,
1991 and 2001.
|
|
 
Figure
3: Basis functions and associated coefficients
for the data shown in Tables 1. The gray shaded
regions are the forecasts of the coefficients
with 90% prediction intervals.
|
|
Figure
4: Forecasts of fertility rates for 2003
and 2022, along with 90% prediction intervals.
|
|
Figure
5: Forecasts of total fertility rates
for the Bangladesh population, the gray-shaded
area indicates 90% prediction intervals.
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Table 1:
Value of the fitted basis functions and the proportion
of variation explained by each basis functions.
|
Age
|
Mean
|
|
|
|
|
13 |
-5.298 |
-1.53E-08 |
-1.00E-08 |
3.29E-08 |
|
14 |
-1.342 |
0.4680 |
0.3082
|
-0.8960 |
|
15 |
1.592 |
0.7589 |
0.5002
|
-1.4112 |
|
16 |
3.503 |
0.8724 |
0.5759
|
-1.5455 |
|
17 |
4.395 |
0.8093 |
0.5356
|
-1.3004 |
|
18 |
4.741 |
0.6674 |
0.4463
|
-0.8999 |
|
19 |
5.018 |
0.5448 |
0.3748
|
-0.5692 |
|
20 |
5.228 |
0.4421 |
0.3216
|
-0.3092 |
|
21 |
5.372 |
0.3592 |
0.2866
|
-0.1200 |
|
22 |
5.448 |
0.2960 |
0.2697
|
-0.0014 |
|
23 |
5.481 |
0.2525 |
0.2632
|
0.0753 |
|
24 |
5.492 |
0.2284 |
0.2589
|
0.1389 |
|
25 |
5.483 |
0.2237 |
0.2569
|
0.1895 |
|
26 |
5.454 |
0.2383 |
0.2572
|
0.2271 |
|
27 |
5.403 |
0.2721 |
0.2597
|
0.2519 |
|
28 |
5.341 |
0.3144 |
0.2650
|
0.2753 |
|
29 |
5.277 |
0.3541 |
0.2737
|
0.3087 |
|
30 |
5.210 |
0.3913 |
0.2857
|
0.3523 |
|
31 |
5.141 |
0.4260 |
0.3009
|
0.4060 |
|
32 |
5.070 |
0.4581 |
0.3195
|
0.4699 |
|
33 |
4.995 |
0.4864 |
0.3443
|
0.5318 |
|
34 |
4.916 |
0.5098 |
0.3779
|
0.5796 |
|
35 |
4.833 |
0.5282 |
0.4205
|
0.6133 |
|
36 |
4.746 |
0.5418 |
0.4719
|
0.6329 |
|
37 |
4.654 |
0.5504 |
0.5321
|
0.6385 |
|
38 |
4.545 |
0.5619 |
0.5693
|
0.6308 |
|
39 |
4.402 |
0.5843 |
0.5514
|
0.6109 |
|
40 |
4.228 |
0.6174 |
0.4785
|
0.5787 |
|
41 |
4.021 |
0.6614 |
0.3505
|
0.5342 |
|
42 |
3.781 |
0.7163 |
0.1674
|
0.4774 |
|
43 |
3.522 |
0.7702 |
-0.0559
|
0.4092 |
|
44 |
3.257 |
0.8113 |
-0.3046
|
0.3304 |
|
45 |
2.985 |
0.8397 |
-0.5783
|
0.2410 |
|
46 |
2.706 |
0.8553 |
-0.8773
|
0.1411 |
|
47 |
2.420 |
0.8581 |
-1.2008
|
0.0308 |
|
48 |
1.879 |
0.8208 |
-1.4215
|
-0.0629 |
|
49 |
0.837 |
0.7164 |
-1.4121
|
-0.1129 |
|
50 |
-0.706 |
0.5448 |
-1.1721
|
-0.1191 |
|
51 |
-2.751 |
0.3060 |
-0.7014
|
-0.0814 |
|
52 |
-5.298 |
-5.81E-09 |
1.99E-08
|
3.78E-09 |
|
Variation (%) |
86.4 |
8.4 |
3.7 |
Table 2: Value
of the fitted coefficients using a decomposition of
order K=3.
|
Years |
Mean |
|
|
|
| 1974 |
1 |
0.8328 |
0.4507 |
0.0132 |
| 1975 |
1 |
0.9828 |
-0.2306 |
0.2882 |
| 1976 |
1 |
0.6959 |
0.2080 |
-0.1261 |
| 1977 |
1 |
0.6959 |
0.2080 |
-0.1261 |
| 1978 |
1 |
1.1189 |
-0.2069 |
0.3027 |
| 1979 |
1 |
0.8111 |
0.2178 |
-0.4192 |
| 1980 |
1 |
0.5697 |
-0.2543 |
-0.0489 |
| 1981 |
1 |
0.4468 |
-0.1003 |
-0.1488 |
| 1982 |
1 |
0.4863 |
-0.0850 |
-0.0455 |
| 1983 |
1 |
0.1546 |
0.1306 |
0.0720 |
| 1984 |
1 |
0.4264 |
-0.1347 |
0.0286 |
| 1985 |
1 |
0.1489 |
-0.1287 |
0.0218 |
| 1986 |
1 |
0.1846 |
-0.1005 |
0.0317 |
| 1987 |
1 |
-0.0219 |
0.0842 |
-0.0305 |
| 1988 |
1 |
0.1495 |
-0.2422 |
0.0083 |
| 1989 |
1 |
0.1297 |
-0.1358 |
0.0310 |
| 1990 |
1 |
0.0362 |
-0.0277 |
0.0267 |
| 1991 |
1 |
-0.0093 |
-0.0398 |
0.0181 |
| 1992 |
1 |
-0.1036 |
0.0252 |
0.0005 |
| 1993 |
1 |
-0.3287 |
0.1619 |
0.0966 |
| 1994 |
1 |
-0.4584 |
0.1946 |
0.0843 |
| 1995 |
1 |
-0.5764 |
0.2609 |
0.0544 |
| 1996 |
1 |
-0.6066 |
0.2401 |
0.0848 |
| 1997 |
1 |
-0.6912 |
0.1761 |
0.0573 |
| 1998 |
1 |
-0.7396 |
0.1404 |
0.0297 |
| 1999 |
1 |
-1.1319 |
-0.1268 |
-0.0498 |
| 2000 |
1 |
-1.2668 |
-0.0391 |
-0.0167 |
| 2001 |
1 |
-1.3448 |
-0.0932 |
0.0579 |
| 2002 |
1 |
-0.5911 |
-0.5533 |
-0.2963 |
Table 3:Twenty
years forecast values of the first coefficient, along
with 90% lower and upper prediction values.
|
Years |
Basis
1,  |
Basis
2,  |
Basis
3,  |
| Point Forecast |
90%
interval |
Point Forecast |
90% interval |
Point Forecast |
90% interval |
| 2003 |
-1.1332 |
(-1.4669,-0.7994) |
-0.1184 |
(-0.3801,0.1432) |
0.1429 |
(-0.0590,0.3448) |
| 2004 |
-1.2087 |
(-1.5424,-0.8750) |
-0.2695 |
(-0.5334,-0.0056) |
-0.1319 |
(-0.3534,0.0896) |
| 2005 |
-1.2842 |
(-1.6180,-0.9505) |
-0.2355 |
(-0.5293,0.0583) |
0 |
(-0.2489,0.2489) |
| 2006 |
-1.3598 |
(-1.6935,-1.0261) |
-0.1552 |
(-0.4813,0.1708) |
0 |
(-0.2489,0.2489) |
| 2007 |
-1.4353 |
(-1.7691,-1.1016) |
-0.0847 |
(-0.4266,0.2572) |
0 |
(-0.2489,0.2489) |
| 2008 |
-1.5109 |
(-1.8446,-1.1772) |
-0.0381 |
(-0.3854,0.3092) |
0 |
(-0.2489,0.2489) |
| 2009 |
-1.5864 |
(-1.9201,-1.2527) |
-0.0126 |
(-0.3611,0.3360) |
0 |
(-0.2489,0.2489) |
| 2010 |
-1.6620 |
(-1.9957,-1.3282) |
-0.0011 |
(-0.3498,0.3477) |
0 |
(-0.2489,0.2489) |
| 2011 |
-1.7375 |
(-2.0712,-1.4038) |
0.0028 |
(-0.3460,0.3515) |
0 |
(-0.2489,0.2489) |
| 2012 |
-1.8131 |
(-2.1468,-1.4793) |
0.0031 |
(-0.3457,0.3519) |
0 |
(-0.2489,0.2489) |
| 2013 |
-1.8886 |
(-2.2223,-1.5549) |
0.0023 |
(-0.3465,0.3511) |
0 |
(-0.2489,0.2489) |
| 2014 |
-1.9641 |
(-2.2979,-1.6304) |
0.0014 |
(-0.3474,0.3501) |
0 |
(-0.2489,0.2489) |
| 2015 |
-2.0397 |
(-2.3734,-1.7060) |
6.73E-04 |
(-0.3481,0.3495) |
0 |
(-0.2489,0.2489) |
| 2016 |
-2.1152 |
(-2.4490,-1.7815) |
2.65E-04 |
(-0.3485,0.3491) |
0 |
(-0.2489,0.2489) |
| 2017 |
-2.1908 |
(-2.5245,-1.8571) |
6.16E-05 |
(-0.3487,0.3488) |
0 |
(-0.2489,0.2489) |
| 2018 |
-2.2663 |
(-2.6000,-1.9326) |
-1.8E-05 |
(-0.3488,0.3488) |
0 |
(-0.2489,0.2489) |
| 2019 |
-2.3419 |
(-2.6756,-2.0081) |
-3.7E-05 |
(-0.3488,0.3487) |
0 |
(-0.2489,0.2489) |
| 2020 |
-2.4174 |
(-2.7511,-2.0837) |
-3.2E-05 |
(-0.3488,0.3488) |
0 |
(-0.2489,0.2489) |
| 2021 |
-2.4930 |
(-2.8267,-2.1592) |
-2.1E-05 |
(-0.3488,0.3488) |
0 |
(-0.2489,0.2489) |
| 2022 |
-2.5685 |
(-2.9022,-2.2348) |
-1.1E-05 |
(-0.3488,0.3488) |
0 |
(-0.2489,0.2489) |
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