# Hierarchical Forecasting of Dengue Incidence in Sri Lanka

L. S. Madushani
Department of Mathematics, University of Sri Jayewardenepura, Sri Lanka
and
Thiyanga S. Talagala
Department of Statistics, University of Sri Jayewardenepura, Sri Lanka
Corresponding author

###### Abstract

The recurrent thread of dengue incidence in Sri Lanka is still abundant and it
creates a huge burden to the country. Hence, the National Dengue Control Unit
of Sri Lanka propose a national action plan to prevent and control the dengue
incidence. To implement the necessary actions for short and long terms the
proposed plan operates under three levels: country-level, province-level, and
district levels. In order to optimize resource allocation, the health officers
require the forecasts for country, province and district levels, which
preserves the aggregate consistency associated with the district, province,
and country levels as well as time correlations. Hence, the objective of this
study is to forecast the dengue incidence in Sri Lanka using a hierarchical
time series forecasting approach based on the spatial and temporal
hierarchical structures. Hierarchical forecasting involves in two steps such
as generating base forecasts and reconciliation of these base forecasts. In
this study, Exponential smoothing (ETS) and Autoregressive Integrated Moving
Average (ARIMA), NAIVE, Seasonal NAIVE approaches and average method are used
to generate base forecasts. Accuracy of forecasts is evaluated using Mean
Absolute Scale Error (MASE). We compare the accuracy of hierarchical time
series forecasts with other benchmark approaches. The forecast accuracy
reveals that the best forecasting approaches for country, provinces and
districts are not limited to a single approach. Hence, we investigate reasons
for the variations of performance in different forecasting approaches based on
a time series feature-based visualization approach.

Keywords: Epidemiology, aggregate consistency, principal component analysis,
time series features

##  1 Introduction

Dengue is the most rapidly growing viral infection in the world (WHO 2014).
The causative agent of dengue infection is classified into four serotypes:
DENV-1, DENV-2, DENV-3, and DENV-4 whereas infection of one serotype does not
result in lifelong immunity against other serotypes (Sanyaolu et al. 2017).
According to the World Health Organization, 50 to 100 million new dengue cases
occur annually including 20,000 deaths in more than 100 countries of the world
(WHO 2012). A large number of dengue cases were reported in Central and South
America, South-East Asia and Western Pacific regions (Teixeira and Barreto
2009). Asia represents approximately 70$\%$ of the global burden of the
disease (Bhatt et al. 2013).

Dengue incidence of Sri Lanka shows a considerable rise during the last two
decades, causing economic and social burdens. The first serological confirmed
dengue case was reported in 1962 and the first largest dengue epidemic
outbreak occurred in 2009 with 35,008 suspected cases and 346 deaths in Sri
Lanka (Withanage et al. 2018). The largest dengue outbreak was reported in
2017, comprising 186,101 suspected cases and over 320 deaths according to the
Epidemiological Unit of the Ministry of Health, Sri Lanka (Chandrakantha
2019). This unprecedented outbreak is due to the variation of the usual
circulating DENV-2 (Ali et al. 2018; Tissera et al. 2020). More than 30,000
cases have been reported each year since 2012 and the highest number of dengue
cases is reported in the Western province of Sri Lanka (Withanage et al.
2018). At present, the spread of dengue incidence in districts outside the
Western province has indicated a dramatic change.

To date, there is no specific treatment methodology to cure the infection
(Wijegunawardana et al. 2019). However, simple fluid replacement and case
management assist to reduce the fatality rate from 20$\%$ to 1$\%$
(Chandrakantha 2019). Therefore, forecasting dengue incidence is important to
implement curative and control programs. In recent, several researchers have
made attempts to forecast dengue incidence in Sri Lanka. For example, time
series regression models were developed using different climate variables for
various districts (Withanage et al. 2018; Goto et al. 2013). Distributed lag
nonlinear modeling approach was used to identify the relationship between
climate factors and dengue incidence in Colombo district of Sri Lanka
(Talagala 2015). Talagala and Lokupitiya (2015) performed a wavelet analysis
to identify the pattern of dengue incidence in 25 districts of Sri Lanka .
More recently, Chandrakantha (2019) developed a model to predict the
likelihood of having dengue incidence in Sri Lanka based on climate factors.
Furthermore, artificial neural network approach was applied to forecast dengue
incidence in Kandy district of Sri Lanka for the period 2010 to 2012 by
considering rainfall, humidity, temperature factors, and previous dengue cases
(Nishanthi et al. 2014). Attanayake et al. (2020) used exponential smoothing
approach to forecast dengue counts in Colombo. All these studies have been
conducted to forecast dengue incidence of the whole country, particular
provinces, or districts independently. None of these approaches has focused to
cater the needs that are proposed by the national action plan to implement
dengue prevention and control programs (MOH 2019). This plan has assigned
officers for national, provincial, and district levels for achieving the
proposed objectives of dengue control and prevention activities. Hence, this
study mainly focuses on dengue incidence forecasting for districts, provinces,
and the whole country to assist the decision-making process of assigned
officers and plan and implement the programs at the district, province, and
country level.

The hierarchical time series forecasting has the ability to deal with time
series that can be aggregated at different levels based on geography. This
approach provides forecasts that are coherent across the levels of hierarchy.
That is, forecasts at the aggregate level are equal to the sum of the relevant
forecast at the desegregated level (Wickramasuriya et al. 2019). Moreover, it
allows to capture of the correlation and interaction between the series at
each level of hierarchy and it has the ability to capture different
information based on each series. In addition, the most significant aspect of
this approach is that it ensures aligned decision-making across the entire
hierarchical structure (Athanasopoulos et al. 2020). Hence, this study
considers hierarchical time series forecasting approach to obtain the
forecasts of districts, provinces, and country levels of Sri Lanka.
Furthermore, forecasts are needed for short-term and long-term periods. Hence,
this study gives attention towards the time correlation as well.

##  2 Methodology and Data

###  2.1 Forecasting hierarchical time series

Figure 1:  Two-level hierarchical tree diagram

A simple hierarchical structure is illustrated in Figure 1. In Figure 1, each
series are observed at time $t=1,2,\dots,T$ and forecasts are generated for
each series at time $T+1,T+2,\dots,T+h$. The series at the bottom level of the
hierarchy, level 2, are denoted as $AX,AY,BX,BY,CX$ and $CY$. Series at level
1, $A,B$ and $C$ can be obtained by aggregating the corresponding bottom level
series. These series are further aggregated to obtain the total series at
level 0, the ”Total”. Furthermore, $y_{i,t}$ denotes the $t^{th}$ series
corresponds to node $i$ and $m_{i}$ denotes number of series at level $i$. The
total number of series in hierarchical structure is denoted by $m$.
Hierarchical structure of Figure 1 can be denoted by matrix notation as

$\left[\begin{array}[]{r}y_{Total,t}\\\ y_{A,t}\\\ y_{B,t}\\\ y_{C,t}\\\
y_{AX,t}\\\ y_{AY,t}\\\ y_{BX,t}\\\ y_{BY,t}\\\ y_{CX,t}\\\
y_{CY,t}\end{array}\right]=\left[\begin{array}[]{rrrrrrrrr}1&1&1&1&1&1\\\
1&1&0&0&0&0\\\ 0&0&1&1&0&0\\\ 0&0&0&0&1&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\end{array}\right]\left[\begin{array}[]{r}y_{AX,t}\\\ y_{AY,t}\\\
y_{BX,t}\\\ y_{BY,t}\\\ y_{CX,t}\\\ y_{CY,t}\end{array}\right]$

In general, hierarchical time series can be denoted by the matrix notation
$y_{t}=Sb_{t}$, where $y_{t}$ denotes all series in the hierarchy and $b_{t}$
is the vector of all bottom level series in the hierarchy at time $t$. $S$ is
the summing matrix of order $m\times m_{k}$ where $m_{k}$ is the number of
series in the most disaggregate level of the hierarchical structure.

Several methods have been proposed to reconcile the time series forecasts in a
way that respects the hierarchy. However, all existing hierarchical time
series forecasting approaches can be illustrated as

$\tilde{y}_{T}(h)=SP\hat{y}_{T}(h),$

where $\tilde{y}_{T}(h)$ is $h$-step-ahead reconciled forecasts,
$\hat{y}_{T}(h)$ is $h$-step-ahead base forecasts, and $P$ is an $m_{k}\times
m$ matrix. The role of $P$ is different according to the hierarchical
forecasting approach.

Top-down and bottom-up approaches are the common approaches used in
hierarchical time series forecasting. The bottom-up approach involves
forecasting each of the bottom level series and sum these forecasts to obtain
other levels of the hierarchy (Hyndman et al. 2011). However, bottom level
data are noisy and therefore it is difficult to model bottom level series
(Zotteri et al. 2014). In the top-down approach, forecasts are generated for
the most aggregated series and then forecasts of the total series are
disaggregated based on the proportions. Here, the proportions can be obtained
based on average or historical proportions (Athanasopoulos et al. 2009;
Hyndman et al. 2011).

Another approach named as optimal combination approach was proposed by
(Hyndman et al. 2011) which performs better than top-down and bottom-up
approaches. This approach obtains the forecasts of all series of the
hierarchical structure and then optimally reconcile based on a regression
model. It requires an estimator for the covariance matrix of errors that occur
due to incoherence. However, it is impossible to estimate. Hence, minimum
trace optimal reconciliation approach (MinT) was introduced to incorporate the
information from a full covariance of forecast errors by minimizing the mean
squared error of the coherence forecasts under the assumption of unbiasedness
and the resulting reconciled forecasts are in the form

$\tilde{y}_{T}(h)=S(S^{{}^{\prime}}W_{h}^{-1}S)^{-1}S^{{}^{\prime}}W_{h}^{-1}\hat{y}_{T}(h).$

In practice, estimation of $W_{h}$ is difficult (Hyndman and Athanasopoulos
2018). Hence, (Wickramasuriya et al. 2019) proposed alternative estimators for
$W_{h}$.

The ordinary least square estimator (OLS) can be obtained with the
substitution

$W_{h}=k_{h}I\hskip 14.45377pt\forall h,$

where $k_{h}>0$ and $I$ is an identity matrix. The drawback of this
substitution is that the scale differences between the levels of the structure
are not considered. Another estimation is

$W_{h}=k_{h}diag(\hat{W}_{1})\hskip 14.45377pt\forall h,$

where $k_{h}>0$ and
$\hat{W}_{1}=\frac{1}{T}\sum_{t=1}^{T}e_{t}{e_{t}}^{{}^{\prime}}$. In this
case, $e_{t}$ is in-sample residuals of the base forecasts stacked in the same
order as the data. Here, the base forecasts are scaled using the variance of
residuals. Therefore, this approach is referred to as MinT approach based on a
weighted least square (WLS) estimator using variance scaling. Another
criterion considers that each of the bottom-level base forecasts are
uncorrelated between nodes and they have errors with equal variance $k_{h}$.
Furthermore, each element of the diagonal matrix $\Lambda$ denotes the number
of forecast error variances that contribute to that aggregation level. This
estimator focuses only on the structure of the hierarchy as

$W_{h}=k_{h}\Lambda\hskip 14.45377pt\forall h,$

where $k_{h}>0$ and $\Lambda=diag(S1)$ with 1 being a unit vector of dimension
$m_{k}$. Next,

$W_{h}=k_{h}W_{1}\hskip 14.45377pt\forall h,$

where $k_{h}>0$ is considered as the estimation. In this estimation, $W_{1}$
is full one-step covariance matrix. However, this estimator is not an
appropriate estimator when the number of bottom level series is larger than
the length of the series $T$. Hence, shrinkage estimator is provided based on
the sample covariance to a diagonal matrix as

$W_{h}=k_{h}\hat{W}_{1,D}^{*}\hskip 14.45377pt\forall h,$

where $k_{h}>0$. In this method, $\hat{W}_{1,D}^{*}$ is a shrinkage estimator
defined as

$\hat{W}_{1,D}^{*}=\lambda\hat{W}_{1,D}+(1-\lambda)\hat{W}_{1},$

where $\hat{W}_{1,D}$ is a diagonal matrix that consists of diagonal entries
of $\hat{W}_{1}$. Therefore, this estimator allows us to capture strong
interrelations between time series in the hierarchy. Schäfer and Strimmer
(2005) proposed the shrinkage intensity parameter as
$\hat{\lambda}=\frac{\sum_{i\neq j}Var(\hat{r}_{ij})}{\sum_{i\neq
j}\hat{r}_{ij}^{2}}$, where $\hat{r}_{ij}$ is the $ij^{th}$ element of
$\hat{R}_{1}$ which is the 1-step-ahead sample correlation matrix.

###  2.2 Base forecasting approaches

In hierarchical forecasting, forecasts are generated for each series at every
node of the hierarchy individually using different forecasting modelling
approaches. These individual forecasts are referred to as base forecasts. In
this study we consider Naive method, Seasonal naive method (SNAIVE), average
method, AutoRegressive Integrated Moving Average (ARIMA) approach, and
Exponential Smoothing method (ETS) to generate base forecasts.

In Naive approach, forecasts are equal to the last observed value in the
series.

$\hat{y}_{T+h|T}=y_{T}$

SNAIVE approach considers that the forecasts are equal to the last observed
value from the same season of the year. That means, forecasts for time $T+h$
is

$\hat{y}_{T+h|T}=y_{T+h-m(k+1)}$

where $m$ is the seasonal period, $k$ is the integer part of
$\frac{(h-1)}{m}$. In the average method, the forecasts of all future values
are equal to the average (or mean) of the available data. If
$y_{1},y_{2},\dots,y_{T}$ denotes the available data, then

$\hat{y}_{T+h|T}=\bar{y}=(y_{1}+y_{2}+\dots+y_{T})/{T}$

Auto-correlations in the data are described by ARIMA models. ARIMA model is a
collection of auto-regressive and moving average components which captures the
effect of past observations and error terms. ARIMA model comprises with 3
terms: $p,d,q$ where $p$ is the order of AR term, $q$ is the order of MA term,
and $d$ is the number of differences required to make the time series into
stationary. AR term considers the linear regression model which depends only
on lagged term as predictors

$y_{t}=c+\phi_{1}y_{t-1}+\phi_{2}y_{t-2}+\dots+\phi_{p}y_{t-p}+\epsilon_{t}$

MA term in ARIMA model describes the effect of lagged error terms

$y_{t}=c+\epsilon_{t}+\theta_{1}\epsilon_{t-1}+\theta_{2}\epsilon_{t-2}+\dots+\theta_{q}\epsilon_{t-q}$

Exponential Smoothing models are constructed by considering the weighted
averages of past observations in which high weights are assigned to the most
recent observation. It was proposed by Holt (2004) and Winters (1960).
Statistical frame work of exponential smoothing methods was proposed by
Hyndman et al. (2002) are illustrated by Table LABEL:table:figurechapter38.

Table 1: Exponential Smoothing methods | Seasonal Component
---|---
Trend Component | N(None) | A(Additive) | M(Multiplicative)
N (None) | (N,N) | (N,A) | (N,M)
A (Additive) | (A,N) | (A,A) | (A,M)
Ad (Additive damped) | (Ad,N) | (Ad,A) | (Ad,M)
Multiplicative (M) | (M,N) | (M,A) | (M,M)
Md (Multiplicative damped) | (Md,N) | (Md,A) | (Md,M)
Table 1: Exponential Smoothing methods (continued)

For example, cell (N, N) corresponds to simple exponential smoothing method.
Two possible state space models (additive errors or multiplicative errors) are
available for each of the methods in Table LABEL:table:figurechapter38.

###  2.3 Forecast performance evaluation

We evaluate the performance of base forecasting approaches against
hierarchical forecasting approaches. Many accuracy measures have been proposed
to check the forecast accuracy (Hyndman and Koehler 2006). However, many of
these proposed accuracy measures are not applicable due to the reasons such as
scale dependency, producing undefined or infinite value and giving misleading
results. Therefore, in this study we use Mean Absolute Scale Error (MASE) to
obtain the accuracy of forecasting methods. For $h$-step-ahead forecasts,

$MAE^{a}=\frac{1}{h}\Sigma_{j=1}^{h}|y_{j}-\hat{y}_{j}|,$

is the mean absolute error of the forecast method $a$, $y_{j}$ and
$\hat{y}_{j}$ are the actual and forecast values at period $j$ respectively.
Then,

$MASE=\frac{MAE^{a}}{Q},$

where $Q=\frac{1}{T-m}\Sigma_{t=1}^{T}|y_{t}-y_{t-m}|$ is the scaling factor
where $m$ is the sampling frequency per year (Athanasopoulos et al. 2017).

###  2.4 Data and modelling strategies

Figure 2: Sri Lanka map

Weekly notified dengue cases in 26 districts of Sri Lanka were obtained from
Epidemiological Unit, Ministry of Health, Sri Lanka for the period of
$52^{nd}$ week of 2006 to $52^{nd}$ week of 2020. This study considers the
three levels of geographic divisions in Sri Lanka. In the first level, Sri
Lanka is divided into 9 provinces such as Central (CN), Eastern (ET), North
Central (NC), Nothern (NT), North Western (NW), Sabaragamuwa (SB), Southern
(ST), UVA, and Western (WT) provinces. In the second level, these 9 provinces
are further classified into 26 districts. In addition, level 0 is the most
aggregate level or dengue counts of the whole country. The geographic
structure is illustrated as a map in Figure 2 and the corresponding spatial
hierarchical structure is illustrated by Figure 3.

Figure 3:  Spatial hierarchical structure

As shown in Figure 4, the series of each level of the spatial hierarchical
structure shown in Figure 3 reveal different time series patterns,
respectively.

Figure 4:  Time series of Dengue counts in Sri Lanka for different aggregated
levels of the spatial hierarchical structure

According to Figure 4, time series are different in scales and time series
features. In Figure 4, the panel corresponds to level 0 demonstrates dengue
counts of Sri Lanka and it indicates two significant spikes during each year
due to the monsoon season of Sri Lanka. A notable upward trend pattern is also
visible. When we focus on level 1 and level 2 of hierarchy, these pattern
variations are less prominent. Bottom level series are much noisier than other
levels. Hence, it is more challenging to model the series of bottom level.
Generating coherent accurate forecasts for the series at each levels are
extremely important to successfully plan the curative and preventive programs.

In addition, dengue curative and preventive programs require different
strategies for different time period. Hence, this study focuses on the
temporal structure of data as demonstrated by Figure 5.

Figure 5: Temporal hierarchical structure

Figure 6 represents the time series dynamic of dengue incidence for different
aggregated levels in Sri Lanka. According to Figure 6, the weekly time series
depicts the highest seasonal variation. Then, bi-weekly series have been
omitted the negligible seasonality variations. Next, monthly series have
demonstrated the important seasonality component and a small trend component.
Finally, the yearly time series depicts upward trend of the series.

Figure 6: Time series of Dengue counts in Sri Lanka according to temporal
hierarchical structure

Before fitting the model, we separate the data set into training set and test
sets. This study considers four different training sets such as 2006-2019,
2016-2019, 2006-2018, 2016-2018 and test sets are 2020, 2019 respectively.
Furthermore, weekly, monthly, quarterly, and yearly frequencies are taken into
the account in all these training and test sets. All statistical analysis
related to spatial hierarchical forecasting was performed by fabletools
package (Hyndman and Athanasopoulos 2018) whereas temporal hierarchical
forecasting was implemented by thief package in R (Hyndman and Kourentzes
2018).

In the case of average, NAIVE, SNAIVE approaches, adjusting base forecasts are
not necessary to obtain reconciled forecasts due to the fact that forecasts
already follow the aggregate constraints associated with spatial hierarchical
structure. In temporal hierarchical structure, forecasts of average and SNAIVE
approach only adhere to constraints associated with temporal generalities.
Hence, forecast reconciliation is necessary for ARIMA, ETS, and NAIVE methods
in temporal hierarchical forecasting whereas forecast reconciliation is needed
for ARIMA and ETS method in spatial hierarchical forecasting.

##  3 Results

###  3.1 Forecasting results

We compare the forecasts accuracy for all series in the spatial and temporal
hierarchical structure. We compare the accuracy of forecasts generated based
of hierarchical forecasting approach against Average method (AVG), NAIVE,
SNAIVE, ETS and ARIMA. The computed MASE values over the test sets of all
series in the spatial hierarchical structure are summarized from Table 3 to
Table 16. We then identified the best forecasting method for all series at
each level of the of the hierarchy. The corresponding results are shown in
Figure 7 \- Figure 8. According to the results it can be seen that when
generating forecasts for 2019, for most of the geographical regions
hierarchical forecasts give the best forecast. However, when generating
forecasts for the year 2020, the average method gives the best forecasts for
Colombo district, Kalutara district (level 2) and Western province (level 1).
This could be due to ongoing COVID-19 pandemic. Sometimes healthcare providers
faced challenges to distinguish COVID-19 from dengue. Hence, there were some
unusual pattern in dengue cases in 2020 specially for population density area.
For the country-level series, the lowest MASE was given by the forecasts
generated from average method based on the training sets 2006-2019 (TS1) and
2016-2019 (TS2) whereas the best forecasting approach for the training sets
2006-2018 (TS3) and 2016-2018 (TS4) is hierarchical time series forecasting
(HS) approach. Similarly, we generated weekly, monthly and quarterly forecasts
based on temporal hierarchical structure and compared the accuracy of results
with other benchmark approaches. The results are are summarized in Table 2.
Furthermore, Figure 9 and 10 show the best forecasting method for weekly,
monthly and quarterly for districts and provinces respectively. For most of
the geographical regions forecasts generated based on the hierarchical
approach gives the best forecasts.

Figure 7: The best forecasting method of level 2 according to training set Figure 8: The best forecasting method of level 1 according to training set Table 2: The best forecasting approaches for temporal granualities according to training set | Training set
---|---
| 2006-2018 | 2016-2018 | 2006-2019 | 2016-2019
Series | Method | MASE | Method | MASE | Method | MASE | Method | MASE
Annual | HS | 0.26 | HS | 0.02 | AVG | 0.20 | HS | 0.09
Semi-annual | HS | 1.28 | HS | 0.29 | AVG | 0.61 | HS | 0.14
Quarterly | HS | 1.17 | HS | 0.29 | AVG | 0.97 | HS | 0.21
Monthly | HS | 1.17 | HS | 0.31 | AVG | 0.96 | HS | 0.23
Bi-Weekly | HS | 1.18 | HS | 0.31 | AVG | 0.97 | HS | 0.24
Weekly | HS | 1.17 | HS | 0.32 | AVG | 0.96 | HS | 0.24
Figure 9: District with the best forecasting method for 2020 Figure 10:
Provinces with the best forecasting method for 2020

Based on the above results the best forecasting approaches in each series of
the spatial hierarchical structure and temporal hierarchical structure are not
limited to single approach. According to No-Free-Lunch theorem, there is no
single method that fits all situations (Kang et al. 2017). Hence, this study
attempts to identify the reasons for the best forecasting approach variations
based on time series features.

###  3.2 Visualization of model performance

In Section 3.1, we identified the best forecasting approaches corresponding to
weekly, monthly, quarterly, and yearly series for the years 2019 and 2020
under four different training sets based on the spatial or temporal
hierarchical structure. Hence, this section attempts to identify the reasons
for the forecasting model performance variations based on feature-based
visualization. In order to obtain feature-based visualization, first features
of time series are extracted from the training set corresponding to the best
forecasting method training set. The features are computed based on the feasts
package in R (O’Hara-Wild et al. 2021). We computed 48 features from each
series. They are strength of seasonality, strength of trend, number of flat
spots, statistic based on Lagrange Multiplier test, optimal $\lambda$ value of
Box-cox transformation, seasonal peak year, seasonal trough year, spikiness,
linearity, curvature, first and tenth autocorrelation coefficient of the
remainder component, first and tenth autocorrelation coefficient from the
original data, longest flat spot length, average interval between non-zero
observations, squared coefficient of variation of non-zero observations,
proportion of data which starts with zero and ends with zero, autocorrelation
function at lag 1 and lag 10 from the first differenced data, autocorrelation
function at lag 1 and lag 10 from the second differenced data, autocorrelation
function at lag 1 from first seasonal lag data, hurst coefficient, spectral
entropy, Box-Pierce statistic, p-value from Box-Pierce statistic, Ljung-Box
statistic, p-value from Ljung-Box statistic, partial autocorrelation function
at lag 5, partial autocorrelation function at lag 5 from first differenced
data, partial autocorrelation function at lag 5 from second differenced data,
partial autocorrelation at the first seasonal lag, KPSS statistic, p-value of
KPSS test, Phillips-Perron statistic, p-value from Phillips-Perron test,
required number of differences, required number of seasonal differences,
stability, lumpiness, the largest mean shift between two consecutive sliding
windows, index at which the largest mean shift occurs, the largest variance
shift between two consecutive sliding windows, index at which the largest
variance shift occurs, the largest distributional shift, index at which the
largest distributional shift occurs, and number of times time series crosses
the median. A detailed description of the features is provided in Talagala et
al. (2018). Next, the principal component analysis (PCA) was performed to
reduce the dimension of features. Furthermore, PCA was conducted separately
for district-wise or province-wise weekly, monthly, and quarterly series.
Principal component 1 versus principal component 2 is plotted to visualize
feature variation according to the modeling approach and the corresponding
visualizations are presented from Figures 11 to 16. The points in the
resulting scatter plots are coloured according to four features: i) strength
of trend, ii) strength of seasonality, iii) stability and iv) lumpiness. The
two features stability and lumpiness are calculated based on non-overlapping
windows of the time series. Then stability is measures based on the variance
of the means, while lumpiness is measured based on the variance of the
variances. The reason for selecting these features are they are very easy to
interpret. In addition to that the points are coloured according to their
province/ district name, training set corresponds to best forecasting method
and best forecasting method (forecasting method that gives the lowest MASE).

Figure 11: Model performance visualization for weekly province-wise series of
spatial hierarchical structure Figure 12: Model performance visualization for
monthly province-wise series of spatial hierarchical structure Figure 13:
Model performance visualization for quarterly province-wise series of spatial
hierarchical structure Figure 14: Model performance visualization for weekly
district-wise series of spatial hierarchical structure Figure 15: Model
performance visualization for monthly district-wise series of spatial
hierarchical structure Figure 16: Model performance visualization for
quarterly district-wise series of spatial hierarchical structure

According to Figure 11, only for Nothern province the best forecasts were
given by SNAIVE and average (AVG) method based on the training sets TS4 and
TS2 respectively. According to the feature visualization we can see the
strength of trend corresponds to the Nothern province is very low compared to
other series. Furthermore, the stability corresponds to the Nothern province
is considerably higher. According to Figure 12, Eastern province and
Sabaragamuwa province are located far away from the other provinces due to
differences in the features. The best forecasting method for Eastern province
is NAIVE approach. This could be due to differences in trend, seasonality and
lumpiness compared to other series. In Figure 11 to 16, for all series that
generates best forecasts with NAIVE or SNAIVE have low value for lumpiness,
high value for stability, strength of trend and strength of seasonality. In
addition to that in we see outlying stay points that are isolated from other
points tends to give best forecasts based on NAIVE, SNAIVE or Average method.

.

##  4 Conclusion

In this study, we have applied hierarchical time series forecasting approaches
to forecast dengue incidence in Sri Lanka. Hierarchical time series
forecasting approaches are extremely important to obtain optimal decisions
which nicely match with the decisions across the levels of hierarchical
structure. In hierarchical forecasting, first, we need to obtain forecasts of
each series without considering the hierarchical structure. This study obtain
the forecasts by ARIMA, ETS, NAIVE, SNAIVE, and average approaches which are
referred to as base forecasts. Then, these forecasts are adjusted in order to
preserve the coherency of forecasts. However, hierarchical time series
forecasting approaches do not provide the lowest MASE values in all situations
of spatial hierarchical series. Furthermore, the number of observations in the
training set does not impact the accuracy improvement. According to the
feature-based visualization of spatial hierarchical structure, SNAIVE and
NAIVE approaches are not the most accurate method if a series contains a high
lumpiness value. Furthermore, the hierarchical time series forecasting
approach indicates the best performance in all ranges of seasonality, trend,
stability, and lumpiness feature values. In the case of the high lumpiness
value, the average method also provides accurate performance. According to the
results of temporal hierarchical forecasting, the hierarchical time series
forecasting approach is the best approach for the years 2019 and 2020.
According to feature-based visualization we observe that the choice of the
best forecasting method depends of the features of the time series. We leave
the study of cross-temporal hierarchical forecasting as a future research. We
have developed a shiny app to disseminate the results of the analysis with the
general public. The app is available at
https://github.com/SamudraMadushani/DengueSriLankaApp. All data are shared
through an open source R package called DISC (DISease Counts) which is
available at https://github.com/SMART-Research/DISC. The maps are generated
using ceylon R package available at https://github.com/thiyangt/ceylon.

APPENDIX

We present the results corresponds to base and reconciled forecasts for four
different training and test sets in spatial hierarchical forecasting from
Table 3 to 14. Base forecasting approaches such as ARIMA, ETS, NAIVE, SNAIVE,
and Average are denoted by arm, snv, nve, ets, avg respectively. Hierarchical
time series forecasting approaches such as bottom up, top down, ordinary least
square estimator using variance scaling, structural scaling, one-step
covariance matrix when base forecasting approach is ARIMA are denoted by bup,
top, ols, mit, var, stc, and cov.

Table 3: MASE from training set: 2006-2019 for weekly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 3.97 | 2.14 | 4.07 | 4.49 | 0.90 | 3.96 | 3.97 | 3.95 | 3.85 | 3.92 | 3.93 | 3.10 | 3.90 | 4.49 | 4.36 | 3.51 | 3.75 | 3.91 | 1.31
Province
CN | 2.16 | 1.97 | 3.79 | 0.54 | 0.62 | 3.47 | 2.24 | 2.55 | 2.87 | 2.91 | 2.83 | 2.35 | 1.87 | 0.76 | 1.58 | 1.28 | 1.28 | 1.40 | 1.67
ET | 7.70 | 2.18 | 7.70 | 7.70 | 1.93 | 7.68 | 8.01 | 7.84 | 7.70 | 7.70 | 7.75 | 7.40 | 8.53 | 10.19 | 8.97 | 8.23 | 8.19 | 8.57 | 8.97
NC | 2.83 | 1.28 | 2.87 | 2.84 | 0.84 | 2.65 | 2.91 | 3.10 | 2.53 | 2.72 | 2.80 | 0.48 | 2.84 | 3.81 | 6.12 | 2.44 | 2.87 | 3.65 | 1.56
NW | 2.27 | 0.99 | 2.41 | 2.28 | 0.68 | 2.24 | 2.33 | 2.33 | 2.23 | 2.26 | 2.27 | 2.14 | 2.34 | 3.02 | 3.07 | 2.04 | 2.35 | 2.50 | 0.62
NT | 9.64 | 3.13 | 9.64 | 9.64 | 1.53 | 9.37 | 9.90 | 9.74 | 9.46 | 9.50 | 9.56 | 8.48 | 10.09 | 12.71 | 11.49 | 9.75 | 9.93 | 10.74 | 6.61
SB | 1.27 | 1.06 | 0.99 | 1.10 | 0.53 | 1.03 | 1.32 | 1.26 | 1.07 | 1.13 | 1.16 | 0.77 | 0.89 | 1.59 | 1.84 | 0.67 | 1.05 | 1.19 | 3.27
ST | 3.84 | 2.90 | 3.58 | 3.46 | 0.89 | 3.43 | 3.95 | 3.82 | 3.50 | 3.59 | 3.65 | 2.87 | 3.37 | 4.58 | 4.42 | 3.04 | 3.44 | 3.74 | 1.60
UVA | 1.58 | 1.21 | 2.56 | 2.67 | 0.73 | 2.65 | 1.61 | 2.12 | 2.08 | 2.17 | 2.15 | 1.28 | 2.65 | 3.51 | 4.94 | 2.40 | 2.69 | 3.22 | 0.40
WT | 3.23 | 2.15 | 3.27 | 2.53 | 0.86 | 3.24 | 3.32 | 3.25 | 3.17 | 3.24 | 3.24 | 2.24 | 3.27 | 3.39 | 2.96 | 2.92 | 3.10 | 2.98 | 0.33
Districts
KN | 2.85 | 1.95 | 3.05 | 0.65 | 0.64 | 2.85 | 1.79 | 2.45 | 2.18 | 2.23 | 2.57 | 1.59 | 0.65 | 0.51 | 0.60 | 0.51 | 0.52 | 0.57 | 1.25
MTL | 5.11 | 2.05 | 5.73 | 6.05 | 0.80 | 5.11 | 3.51 | 3.76 | 4.78 | 4.80 | 4.17 | 4.75 | 6.05 | 3.25 | 5.62 | 5.47 | 5.75 | 5.35 | 2.89
NE | 3.02 | 1.40 | 3.49 | 3.06 | 0.85 | 3.02 | 2.08 | 2.15 | 2.69 | 2.87 | 0.76 | 1.66 | 3.06 | 1.57 | 1.58 | 2.39 | 2.90 | 0.70 | 1.65
AM | 2.58 | 1.96 | 2.08 | 2.54 | 1.42 | 2.58 | 2.68 | 3.22 | 2.34 | 2.58 | 2.86 | 1.38 | 2.54 | 2.91 | 4.01 | 1.99 | 2.45 | 2.72 | 1.18
BAT | 5.42 | 1.46 | 5.53 | 5.42 | 1.99 | 5.42 | 5.65 | 5.50 | 5.45 | 5.43 | 5.46 | 5.41 | 5.42 | 6.58 | 5.65 | 5.27 | 5.20 | 5.44 | 7.75
KAL | 4.03 | 1.32 | 4.10 | 4.10 | 1.08 | 4.03 | 4.18 | 4.15 | 4.12 | 4.05 | 4.08 | 4.37 | 4.10 | 4.86 | 4.44 | 3.86 | 3.69 | 4.13 | 5.16
TRC | 12.56 | 3.56 | 12.56 | 15.31 | 2.53 | 12.56 | 13.04 | 12.69 | 12.57 | 12.58 | 12.62 | 11.87 | 15.31 | 18.15 | 15.67 | 14.96 | 14.97 | 15.34 | 12.63
APU | 2.63 | 1.17 | 2.80 | 2.84 | 0.78 | 2.63 | 2.88 | 2.93 | 2.51 | 2.71 | 2.74 | 0.31 | 2.84 | 3.78 | 5.17 | 2.38 | 2.86 | 3.41 | 1.90
POL | 2.25 | 1.29 | 2.49 | 2.36 | 0.82 | 2.25 | 2.48 | 2.92 | 2.17 | 2.30 | 2.47 | 0.83 | 2.36 | 3.14 | 6.71 | 2.15 | 2.37 | 3.40 | 0.75
KUR | 2.13 | 0.90 | 2.26 | 2.16 | 0.71 | 2.13 | 2.22 | 2.20 | 2.16 | 2.15 | 2.15 | 2.63 | 2.16 | 2.78 | 2.74 | 1.87 | 2.17 | 2.29 | 0.83
PUT | 2.32 | 1.09 | 2.54 | 2.52 | 0.62 | 2.32 | 2.40 | 2.41 | 2.22 | 2.32 | 2.34 | 1.31 | 2.52 | 3.23 | 3.41 | 2.22 | 2.53 | 2.72 | 0.32
JAF | 11.80 | 3.66 | 11.52 | 11.81 | 1.64 | 11.80 | 12.46 | 11.91 | 11.94 | 11.97 | 11.86 | 10.86 | 11.81 | 14.86 | 12.20 | 11.46 | 11.61 | 11.99 | 9.02
KIL | 11.16 | 2.90 | 10.22 | 11.16 | 1.53 | 11.16 | 11.80 | 12.80 | 11.15 | 11.18 | 12.03 | 11.28 | 11.16 | 14.06 | 17.20 | 10.49 | 11.14 | 13.98 | 3.89
MAN | 2.18 | 1.00 | 1.17 | 2.05 | 0.85 | 2.18 | 2.32 | 2.92 | 2.19 | 2.22 | 2.58 | 1.68 | 2.05 | 2.52 | 4.24 | 1.92 | 2.02 | 3.07 | 0.92
MUL | 4.59 | 1.48 | 3.40 | 4.58 | 0.89 | 4.59 | 4.86 | 6.35 | 4.47 | 4.61 | 5.52 | 3.32 | 4.58 | 5.72 | 10.66 | 4.01 | 4.56 | 7.42 | 0.68
VAV | 0.77 | 1.50 | 4.04 | 4.04 | 0.86 | 0.77 | 0.81 | 1.19 | 0.79 | 0.80 | 0.99 | 0.41 | 4.04 | 5.08 | 5.41 | 3.96 | 3.99 | 4.68 | 0.48
KEG | 1.04 | 0.81 | 1.27 | 0.82 | 0.46 | 1.04 | 1.30 | 1.25 | 1.09 | 1.11 | 1.17 | 1.56 | 0.82 | 1.40 | 1.70 | 0.66 | 0.94 | 1.11 | 1.80
RAT | 1.07 | 1.27 | 0.88 | 1.03 | 0.68 | 1.07 | 1.31 | 1.26 | 1.10 | 1.17 | 1.16 | 0.59 | 1.03 | 1.68 | 1.83 | 0.82 | 1.17 | 1.24 | 4.60
GAL | 5.94 | 3.75 | 6.16 | 5.43 | 0.96 | 5.94 | 6.85 | 6.26 | 6.01 | 6.12 | 6.13 | 4.82 | 5.43 | 7.35 | 6.25 | 5.17 | 5.50 | 5.72 | 3.94
HAM | 1.93 | 1.90 | 2.09 | 1.93 | 0.80 | 1.93 | 2.24 | 2.53 | 1.94 | 2.01 | 2.28 | 1.64 | 1.93 | 2.64 | 3.50 | 1.64 | 1.96 | 2.47 | 0.64
MAT | 1.22 | 2.24 | 1.31 | 1.59 | 0.76 | 1.22 | 1.41 | 1.53 | 1.32 | 1.38 | 1.40 | 1.15 | 1.59 | 2.16 | 2.41 | 1.23 | 1.67 | 1.88 | 0.56
BAD | 4.32 | 1.84 | 4.18 | 4.32 | 0.79 | 4.32 | 2.63 | 3.88 | 3.58 | 3.73 | 3.91 | 2.42 | 4.32 | 5.72 | 6.18 | 4.05 | 4.36 | 4.79 | 2.57
MON | 0.00 | 0.48 | 0.00 | 0.00 | 0.74 | 0.00 | 0.00 | 0.57 | 0.24 | 0.24 | 0.53 | 0.44 | 0.00 | 0.00 | 2.42 | 0.18 | 0.02 | 0.60 | 2.90
COL | 3.57 | 2.19 | 3.56 | 3.59 | 1.01 | 3.57 | 3.66 | 3.58 | 3.50 | 3.58 | 3.57 | 2.38 | 3.59 | 3.72 | 3.37 | 3.26 | 3.43 | 3.39 | 0.45
GAM | 2.25 | 1.75 | 2.46 | 2.46 | 0.66 | 2.25 | 2.31 | 2.26 | 2.20 | 2.26 | 2.25 | 1.58 | 2.46 | 2.55 | 2.21 | 2.14 | 2.28 | 2.23 | 0.26
KLU | 3.86 | 2.45 | 3.51 | 3.39 | 0.71 | 3.86 | 3.95 | 3.89 | 3.77 | 3.86 | 3.86 | 2.95 | 3.39 | 3.52 | 2.68 | 3.02 | 3.31 | 2.73 | 0.37

Table 4: MASE from training set: 2006-2019 for monthly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 2.25 | 2.18 | 4.35 | 4.11 | 0.85 | 2.50 | 2.25 | 2.40 | 4.51 | 2.94 | 2.79 | 4.14 | 4.53 | 4.11 | 4.24 | 5.73 | 4.81 | 4.63 | 3.66
Province
CN | 4.92 | 2.05 | 4.59 | 5.22 | 0.60 | 4.80 | 3.06 | 3.94 | 5.72 | 4.81 | 4.60 | 3.30 | 4.64 | 4.06 | 4.28 | 5.97 | 4.85 | 4.69 | 3.58
ET | 7.28 | 2.17 | 6.39 | 13.83 | 1.66 | 1.96 | 4.62 | 4.96 | 4.87 | 4.42 | 4.20 | 8.12 | 6.33 | 10.53 | 11.11 | 10.28 | 10.12 | 9.59 | 11.17
NC | 2.67 | 1.36 | 3.15 | 3.26 | 0.85 | 1.74 | 1.64 | 1.56 | 3.70 | 2.08 | 1.44 | 5.91 | 2.53 | 2.55 | 1.47 | 4.40 | 2.87 | 2.15 | 5.07
NW | 2.32 | 1.01 | 2.26 | 2.36 | 0.67 | 2.19 | 1.48 | 1.40 | 3.14 | 2.21 | 2.07 | 3.07 | 2.49 | 1.88 | 1.67 | 3.13 | 2.42 | 2.26 | 2.45
NT | 12.76 | 3.23 | 13.61 | 12.39 | 1.32 | 3.41 | 8.35 | 9.17 | 7.18 | 7.29 | 7.23 | 5.25 | 13.32 | 9.78 | 10.85 | 12.97 | 12.84 | 12.08 | 12.46
SB | 1.00 | 1.10 | 1.36 | 1.94 | 0.51 | 0.84 | 0.58 | 0.31 | 2.94 | 0.86 | 0.75 | 2.71 | 2.52 | 1.46 | 1.33 | 2.76 | 2.24 | 2.05 | 4.91
ST | 1.74 | 3.01 | 2.96 | 3.54 | 0.88 | 2.76 | 1.11 | 0.92 | 3.45 | 2.41 | 1.94 | 2.09 | 3.49 | 2.80 | 2.57 | 4.03 | 3.48 | 3.22 | 1.92
UVA | 2.48 | 1.28 | 2.77 | 2.47 | 0.74 | 1.11 | 1.59 | 0.64 | 3.20 | 1.60 | 1.25 | 2.21 | 2.17 | 1.92 | 0.65 | 2.98 | 2.26 | 1.81 | 2.88
WT | 2.09 | 2.23 | 3.56 | 3.66 | 0.84 | 2.08 | 1.28 | 1.81 | 4.06 | 1.97 | 2.01 | 3.79 | 3.64 | 2.94 | 3.42 | 4.70 | 3.50 | 3.58 | 3.50
Districts
KN | 3.96 | 2.03 | 3.63 | 3.86 | 0.62 | 3.96 | 2.44 | 3.57 | 4.87 | 3.98 | 3.87 | 2.06 | 3.86 | 3.28 | 3.70 | 5.15 | 4.12 | 3.89 | 2.73
MTL | 7.98 | 2.12 | 7.74 | 7.22 | 0.74 | 7.98 | 5.39 | 6.70 | 8.62 | 7.99 | 7.69 | 7.46 | 7.22 | 6.64 | 6.68 | 8.68 | 7.31 | 7.29 | 7.65
NE | 1.38 | 1.67 | 3.00 | 2.84 | 0.94 | 1.38 | 0.92 | 4.25 | 3.19 | 1.39 | 0.47 | 2.35 | 2.84 | 2.52 | 5.23 | 3.68 | 2.86 | 4.77 | 2.22
AM | 2.76 | 1.99 | 2.86 | 2.57 | 1.44 | 2.76 | 5.81 | 14.77 | 4.24 | 3.16 | 11.63 | 4.23 | 2.57 | 3.99 | 22.83 | 4.22 | 3.08 | 15.97 | 2.79
BAT | 1.65 | 1.37 | 4.20 | 4.20 | 1.82 | 1.65 | 4.09 | 3.11 | 2.90 | 2.77 | 2.66 | 4.40 | 4.20 | 7.47 | 6.99 | 7.19 | 6.48 | 6.10 | 8.35
KAL | 1.53 | 1.29 | 3.41 | 3.32 | 0.99 | 1.53 | 3.19 | 3.83 | 5.84 | 4.89 | 3.22 | 12.16 | 3.32 | 5.40 | 7.12 | 8.56 | 8.50 | 5.91 | 13.21
TRC | 2.56 | 3.65 | 10.98 | 10.97 | 2.32 | 2.56 | 5.00 | 5.01 | 5.07 | 4.99 | 4.39 | 7.21 | 10.97 | 17.80 | 14.87 | 13.98 | 14.70 | 13.63 | 11.09
APU | 1.41 | 1.23 | 3.30 | 2.60 | 0.79 | 1.41 | 1.32 | 0.93 | 3.86 | 1.81 | 1.18 | 7.05 | 2.60 | 2.62 | 1.02 | 4.43 | 3.00 | 2.33 | 4.81
POL | 2.22 | 1.47 | 2.46 | 2.09 | 0.89 | 2.22 | 2.14 | 2.69 | 2.91 | 2.41 | 1.85 | 2.80 | 2.09 | 2.18 | 3.06 | 3.80 | 2.26 | 1.69 | 5.03
KUR | 2.05 | 0.92 | 2.04 | 2.38 | 0.70 | 2.05 | 1.37 | 1.40 | 3.05 | 2.08 | 1.95 | 4.05 | 2.38 | 1.84 | 1.74 | 3.04 | 2.32 | 2.19 | 3.71
PUT | 2.33 | 1.11 | 2.52 | 2.58 | 0.61 | 2.33 | 1.60 | 1.36 | 3.19 | 2.35 | 2.19 | 1.51 | 2.58 | 1.88 | 1.56 | 3.17 | 2.51 | 2.29 | 1.50
JAF | 4.23 | 3.81 | 16.09 | 16.09 | 1.42 | 4.23 | 16.11 | 5.40 | 8.30 | 8.78 | 4.87 | 5.12 | 16.09 | 11.82 | 15.38 | 15.39 | 15.47 | 15.73 | 14.90
KIL | 10.77 | 3.38 | 13.54 | 10.73 | 1.43 | 10.77 | 131.98 | 40.97 | 13.27 | 11.24 | 30.92 | 16.47 | 10.73 | 7.73 | 4.70 | 12.94 | 10.69 | 4.29 | 13.84
MAN | 0.70 | 0.94 | 3.65 | 2.43 | 0.75 | 0.70 | 8.88 | 10.51 | 1.78 | 1.35 | 7.18 | 4.02 | 2.43 | 1.73 | 2.19 | 2.96 | 2.35 | 0.62 | 6.14
MUL | 2.05 | 1.60 | 6.12 | 5.63 | 0.96 | 2.05 | 26.20 | 34.57 | 3.62 | 2.25 | 23.74 | 4.53 | 5.63 | 4.12 | 9.40 | 6.50 | 5.61 | 3.47 | 3.82
VAV | 0.97 | 1.44 | 6.13 | 6.13 | 0.79 | 0.97 | 11.84 | 6.85 | 2.80 | 2.56 | 4.89 | 2.50 | 6.13 | 4.52 | 3.64 | 5.90 | 5.94 | 4.88 | 3.44
KEG | 0.60 | 0.84 | 1.35 | 1.56 | 0.49 | 0.60 | 0.39 | 0.28 | 2.35 | 0.61 | 0.51 | 3.92 | 1.56 | 1.01 | 0.50 | 2.33 | 1.44 | 1.14 | 1.05
RAT | 1.10 | 1.36 | 1.43 | 3.46 | 0.68 | 1.10 | 0.78 | 0.60 | 3.42 | 1.13 | 1.01 | 2.26 | 3.46 | 1.88 | 2.23 | 3.05 | 3.02 | 2.96 | 9.99
GAL | 4.00 | 3.97 | 4.17 | 4.42 | 0.95 | 4.00 | 1.63 | 2.52 | 4.18 | 3.72 | 3.33 | 2.39 | 4.42 | 3.53 | 3.68 | 4.85 | 4.41 | 4.20 | 2.53
HAM | 1.67 | 1.98 | 1.80 | 2.65 | 0.80 | 1.67 | 0.65 | 1.35 | 2.25 | 1.58 | 0.47 | 1.42 | 2.65 | 2.18 | 1.71 | 2.46 | 2.65 | 2.31 | 1.96
MAT | 1.81 | 2.27 | 2.06 | 2.63 | 0.77 | 1.81 | 0.71 | 0.29 | 2.98 | 1.28 | 1.13 | 1.93 | 2.63 | 2.09 | 1.89 | 3.61 | 2.61 | 2.42 | 2.48
BAD | 1.92 | 1.97 | 4.62 | 3.61 | 0.79 | 1.92 | 2.75 | 0.50 | 4.55 | 2.61 | 2.04 | 3.70 | 3.61 | 3.20 | 1.92 | 4.35 | 3.74 | 3.31 | 3.71
MON | 0.09 | 0.46 | 0.00 | 0.00 | 0.78 | 0.09 | 0.13 | 1.96 | 1.01 | 0.07 | 0.09 | 0.70 | 0.00 | 0.00 | 2.39 | 0.79 | 0.03 | 0.49 | 2.17
COL | 2.24 | 2.31 | 3.98 | 3.85 | 1.01 | 2.24 | 1.37 | 2.05 | 3.77 | 2.14 | 2.19 | 2.86 | 3.85 | 3.24 | 3.69 | 4.22 | 3.77 | 3.81 | 2.57
GAM | 1.64 | 1.80 | 2.98 | 3.06 | 0.64 | 1.64 | 1.05 | 1.42 | 4.03 | 1.50 | 1.58 | 4.66 | 3.06 | 2.38 | 2.88 | 4.72 | 2.84 | 3.01 | 4.63
KLU | 2.28 | 2.58 | 2.90 | 3.63 | 0.72 | 2.28 | 1.34 | 1.64 | 3.97 | 2.24 | 2.11 | 3.33 | 3.63 | 2.75 | 3.11 | 4.97 | 3.57 | 3.48 | 2.75

Table 5: MASE from training set: 2006-2019 for quarterly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 2.65 | 2.17 | 4.38 | 4.64 | 0.69 | 2.82 | 2.65 | 2.68 | 2.87 | 2.85 | 2.81 | 3.76 | 4.64 | 4.57 | 3.63 | 3.95 | 4.18
Province
CN | 1.84 | 1.94 | 5.41 | 5.55 | 0.31 | 1.99 | 1.72 | 1.67 | 1.99 | 1.92 | 1.84 | 5.07 | 6.19 | 5.89 | 4.81 | 5.31 | 5.53
ET | 1.80 | 2.19 | 3.44 | 2.30 | 2.03 | 1.82 | 1.45 | 1.43 | 1.80 | 1.78 | 1.53 | 2.08 | 2.42 | 2.53 | 2.23 | 2.16 | 2.33
NC | 2.68 | 1.44 | 2.68 | 2.83 | 0.81 | 2.43 | 2.36 | 1.67 | 2.52 | 2.53 | 2.31 | 2.62 | 3.17 | 4.74 | 2.34 | 2.71 | 3.41
NW | 1.83 | 0.99 | 2.24 | 2.47 | 0.51 | 1.45 | 1.71 | 1.51 | 1.63 | 1.61 | 1.59 | 2.58 | 2.75 | 2.94 | 2.33 | 2.56 | 2.68
NT | 2.66 | 3.36 | 11.15 | 6.25 | 1.18 | 3.34 | 2.27 | 2.29 | 3.13 | 3.06 | 2.74 | 3.06 | 6.91 | 6.74 | 3.93 | 4.00 | 5.33
SB | 0.99 | 0.97 | 1.86 | 2.20 | 0.24 | 0.72 | 0.85 | 0.69 | 0.83 | 0.80 | 0.80 | 1.72 | 2.50 | 2.51 | 2.08 | 1.93 | 2.12
ST | 2.68 | 3.00 | 3.66 | 4.45 | 0.73 | 2.20 | 2.45 | 2.30 | 2.32 | 2.34 | 2.35 | 4.32 | 4.95 | 4.97 | 4.17 | 4.39 | 4.64
UVA | 1.81 | 1.22 | 2.78 | 3.01 | 0.59 | 1.30 | 1.64 | 1.22 | 1.46 | 1.48 | 1.39 | 2.74 | 3.34 | 4.21 | 2.55 | 2.86 | 3.32
WT | 4.26 | 2.20 | 4.26 | 4.49 | 0.65 | 4.26 | 3.81 | 4.20 | 4.22 | 4.22 | 4.24 | 4.41 | 4.98 | 4.61 | 4.07 | 4.53 | 4.51
Districts
KN | 1.72 | 1.87 | 5.07 | 5.43 | 0.31 | 1.72 | 1.58 | 1.57 | 1.71 | 1.64 | 1.65 | 5.43 | 6.64 | 5.79 | 5.26 | 5.71 | 5.63
MTL | 2.94 | 2.18 | 6.74 | 3.89 | 0.61 | 2.94 | 2.70 | 2.45 | 3.00 | 2.92 | 2.72 | 3.89 | 4.67 | 5.14 | 3.41 | 4.00 | 4.58
NE | 1.25 | 1.84 | 3.21 | 3.61 | 0.93 | 1.25 | 0.98 | 2.27 | 1.21 | 1.25 | 0.78 | 3.61 | 4.44 | 9.13 | 2.94 | 3.64 | 6.65
AM | 2.60 | 2.07 | 2.60 | 2.60 | 1.44 | 2.60 | 2.15 | 3.06 | 2.45 | 2.60 | 2.14 | 2.60 | 3.05 | 5.81 | 2.59 | 2.61 | 4.10
BAT | 1.47 | 1.28 | 2.36 | 2.20 | 2.20 | 1.47 | 1.39 | 1.39 | 1.46 | 1.48 | 1.45 | 2.20 | 2.25 | 2.21 | 2.36 | 2.20 | 2.18
KAL | 1.33 | 1.18 | 1.92 | 1.35 | 0.90 | 1.33 | 1.10 | 0.79 | 1.27 | 1.29 | 1.07 | 1.35 | 1.58 | 1.70 | 1.55 | 1.43 | 1.55
TRC | 2.47 | 3.79 | 5.48 | 2.91 | 2.46 | 2.47 | 1.92 | 1.90 | 2.44 | 2.43 | 2.20 | 2.91 | 3.21 | 3.28 | 3.00 | 3.00 | 3.11
APU | 2.75 | 1.32 | 2.75 | 2.99 | 0.74 | 2.75 | 2.66 | 2.20 | 2.87 | 2.87 | 2.66 | 2.99 | 3.58 | 4.52 | 2.69 | 3.09 | 3.56
POL | 1.55 | 1.55 | 2.26 | 1.64 | 0.86 | 1.55 | 1.54 | 2.25 | 1.58 | 1.58 | 1.38 | 1.64 | 2.04 | 4.72 | 1.65 | 1.68 | 2.79
KUR | 0.90 | 0.89 | 1.82 | 2.11 | 0.54 | 0.90 | 1.08 | 0.94 | 1.14 | 1.10 | 1.01 | 2.11 | 2.24 | 2.40 | 1.63 | 2.08 | 2.18
PUT | 2.26 | 1.13 | 2.85 | 3.26 | 0.46 | 2.26 | 2.62 | 2.33 | 2.33 | 2.34 | 2.43 | 3.26 | 3.46 | 3.70 | 3.32 | 3.25 | 3.38
JAF | 4.24 | 4.27 | 14.50 | 3.89 | 1.37 | 4.24 | 2.55 | 3.91 | 3.93 | 3.90 | 4.05 | 3.89 | 8.72 | 4.95 | 5.10 | 5.25 | 4.53
KIL | 4.37 | 3.57 | 9.94 | 4.09 | 1.49 | 4.37 | 4.65 | 13.04 | 4.41 | 4.35 | 8.99 | 4.09 | 9.80 | 24.82 | 4.42 | 4.16 | 17.07
MAN | 0.39 | 0.80 | 1.93 | 0.56 | 0.56 | 0.39 | 0.58 | 3.53 | 0.38 | 0.40 | 2.30 | 0.56 | 0.71 | 6.38 | 0.61 | 0.57 | 4.02
MUL | 5.89 | 1.40 | 5.03 | 1.69 | 0.73 | 5.89 | 6.20 | 14.13 | 5.85 | 5.88 | 10.02 | 1.69 | 3.98 | 22.65 | 2.00 | 1.72 | 14.81
VAV | 0.74 | 1.35 | 4.02 | 1.19 | 0.70 | 0.74 | 0.83 | 2.46 | 0.72 | 0.68 | 1.73 | 1.19 | 2.83 | 4.89 | 1.47 | 1.49 | 3.51
KEG | 0.52 | 0.84 | 1.63 | 2.15 | 0.44 | 0.52 | 0.62 | 0.49 | 0.67 | 0.58 | 0.59 | 2.15 | 3.00 | 2.85 | 2.09 | 2.33 | 2.51
RAT | 0.95 | 1.19 | 2.09 | 1.29 | 0.36 | 0.95 | 1.11 | 0.91 | 1.00 | 1.04 | 1.04 | 1.29 | 1.80 | 2.01 | 1.99 | 1.43 | 1.58
GAL | 2.21 | 4.11 | 4.52 | 5.36 | 0.84 | 2.21 | 2.51 | 2.29 | 2.29 | 2.31 | 2.34 | 5.36 | 6.17 | 5.90 | 5.28 | 5.41 | 5.63
HAM | 2.38 | 1.99 | 2.65 | 3.59 | 0.65 | 2.38 | 2.59 | 2.53 | 2.40 | 2.43 | 2.61 | 3.59 | 4.08 | 4.60 | 3.95 | 3.64 | 4.09
MAT | 1.96 | 2.33 | 3.10 | 3.39 | 0.61 | 1.96 | 2.15 | 2.04 | 2.15 | 2.17 | 2.08 | 3.39 | 3.90 | 3.92 | 2.93 | 3.49 | 3.65
BAD | 1.70 | 1.97 | 4.60 | 4.53 | 0.68 | 1.70 | 2.21 | 1.48 | 1.92 | 1.96 | 1.78 | 4.53 | 5.53 | 5.77 | 4.21 | 4.69 | 5.02
MON | 0.63 | 0.51 | 0.12 | 0.12 | 0.80 | 0.63 | 0.71 | 1.16 | 0.70 | 0.69 | 0.73 | 0.12 | 0.14 | 1.70 | 0.12 | 0.17 | 0.75
COL | 4.73 | 2.31 | 4.73 | 4.72 | 0.81 | 4.73 | 4.23 | 4.69 | 4.70 | 4.70 | 4.72 | 4.72 | 5.38 | 4.86 | 4.14 | 4.84 | 4.80
GAM | 3.37 | 1.80 | 3.37 | 3.93 | 0.49 | 3.37 | 3.02 | 3.32 | 3.32 | 3.32 | 3.35 | 3.93 | 4.37 | 4.09 | 3.88 | 4.07 | 4.01
KLU | 4.52 | 2.66 | 4.52 | 3.98 | 0.50 | 4.52 | 4.02 | 4.37 | 4.50 | 4.51 | 4.47 | 3.98 | 4.57 | 4.43 | 3.66 | 4.03 | 4.22

Table 6: MASE from training set: 2006-2019 for yearly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 0.24 | 2.16 | 1.29 | 0.24 | 0.75 | 0.24 | 0.34 | 0.84 | 0.80 | 0.68 | 1.50 | 1.29 | 1.32 | 1.54 | 1.49 | 1.43
Province
CN | 0.10 | 1.91 | 0.92 | 0.10 | 0.10 | 0.52 | 0.68 | 0.02 | 0.12 | 0.30 | 0.87 | 0.74 | 0.72 | 0.89 | 0.88 | 0.83
ET | 1.68 | 0.52 | 0.18 | 2.12 | 1.97 | 2.37 | 2.79 | 1.84 | 1.92 | 2.28 | 0.28 | 0.47 | 0.55 | 0.19 | 0.26 | 0.38
NC | 0.33 | 1.14 | 0.72 | 0.33 | 0.33 | 0.12 | 2.23 | 0.40 | 0.32 | 0.33 | 1.40 | 0.58 | 0.11 | 1.23 | 1.20 | 0.84
NW | 0.58 | 1.09 | 0.67 | 0.58 | 0.58 | 0.18 | 0.01 | 0.62 | 0.56 | 0.43 | 1.56 | 0.57 | 0.78 | 1.35 | 1.32 | 1.07
NT | 3.12 | 4.06 | 3.11 | 0.32 | 2.46 | 1.18 | 1.48 | 2.74 | 2.69 | 2.00 | 3.12 | 2.67 | 2.61 | 3.17 | 3.11 | 2.85
SB | 0.21 | 1.03 | 0.24 | 0.21 | 0.21 | 0.18 | 0.35 | 0.25 | 0.19 | 0.06 | 1.13 | 0.15 | 0.36 | 0.92 | 0.90 | 0.64
ST | 0.42 | 3.16 | 2.60 | 0.42 | 1.29 | 0.01 | 0.03 | 1.14 | 1.08 | 0.62 | 2.38 | 2.32 | 2.33 | 2.46 | 2.43 | 2.41
UVA | 0.58 | 1.34 | 1.34 | 0.58 | 0.58 | 0.20 | 0.95 | 0.64 | 0.57 | 0.18 | 1.47 | 1.19 | 0.88 | 1.43 | 1.42 | 1.27
WT | 2.00 | 2.53 | 1.84 | 0.61 | 1.20 | 0.97 | 1.63 | 1.35 | 1.37 | 1.54 | 1.38 | 1.62 | 1.67 | 1.55 | 1.48 | 1.59
Districts
KN | 0.21 | 1.89 | 0.99 | 0.21 | 0.21 | 0.62 | 0.47 | 0.12 | 0.24 | 0.30 | 0.99 | 0.85 | 0.92 | 1.01 | 1.01 | 0.97
MTL | 0.21 | 2.04 | 0.21 | 0.21 | 0.21 | 0.19 | 0.72 | 0.28 | 0.20 | 0.11 | 0.21 | 0.14 | 0.03 | 0.24 | 0.21 | 0.14
NE | 0.34 | 1.36 | 1.75 | 0.34 | 0.34 | 0.22 | 3.88 | 0.41 | 0.34 | 1.12 | 1.75 | 1.57 | 0.66 | 1.74 | 1.75 | 1.45
AM | 0.90 | 0.25 | 0.15 | 0.90 | 0.90 | 1.15 | 3.52 | 0.81 | 0.89 | 1.88 | 0.15 | 0.24 | 1.00 | 0.09 | 0.14 | 0.48
BAT | 2.46 | 1.45 | 0.61 | 2.83 | 2.46 | 2.96 | 2.98 | 2.34 | 2.41 | 2.65 | 0.61 | 0.83 | 0.78 | 0.53 | 0.59 | 0.68
KAL | 0.52 | 0.20 | 0.72 | 0.52 | 0.52 | 0.79 | 1.15 | 0.40 | 0.47 | 0.76 | 0.72 | 0.59 | 0.52 | 0.81 | 0.74 | 0.65
TRC | 2.10 | 0.12 | 0.73 | 2.10 | 2.10 | 2.36 | 2.66 | 1.98 | 2.06 | 2.31 | 0.73 | 0.87 | 0.91 | 0.65 | 0.71 | 0.80
APU | 0.40 | 1.21 | 1.58 | 0.40 | 0.40 | 0.06 | 1.57 | 0.47 | 0.40 | 0.11 | 1.58 | 0.72 | 0.59 | 1.38 | 1.36 | 1.15
POL | 0.19 | 0.97 | 1.04 | 0.19 | 0.19 | 0.24 | 3.34 | 0.25 | 0.19 | 0.73 | 1.04 | 0.32 | 0.74 | 0.94 | 0.88 | 0.26
KUR | 0.57 | 0.95 | 1.34 | 0.57 | 0.57 | 0.17 | 0.11 | 0.61 | 0.56 | 0.45 | 1.34 | 0.45 | 0.72 | 1.13 | 1.10 | 0.95
PUT | 0.59 | 1.33 | 1.94 | 0.59 | 0.59 | 0.19 | 0.21 | 0.65 | 0.57 | 0.38 | 1.94 | 0.78 | 0.88 | 1.73 | 1.69 | 1.27
JAF | 3.11 | 4.81 | 3.92 | 0.47 | 3.11 | 1.60 | 2.84 | 3.43 | 3.39 | 2.98 | 3.92 | 3.38 | 3.78 | 3.97 | 3.91 | 3.84
KIL | 3.79 | 4.90 | 4.83 | 1.54 | 3.79 | 1.59 | 3.56 | 3.81 | 3.84 | 0.31 | 4.83 | 4.06 | 1.02 | 4.92 | 4.82 | 2.81
MAN | 0.03 | 0.37 | 0.03 | 0.03 | 0.03 | 0.24 | 1.72 | 0.12 | 0.07 | 0.80 | 0.03 | 0.05 | 0.87 | 0.05 | 0.03 | 0.45
MUL | 0.87 | 1.78 | 2.00 | 0.33 | 0.87 | 0.28 | 5.11 | 0.82 | 0.90 | 1.96 | 2.00 | 1.72 | 1.09 | 2.06 | 2.00 | 0.36
VAV | 0.30 | 1.68 | 0.30 | 0.30 | 0.30 | 0.04 | 0.79 | 0.48 | 0.41 | 0.21 | 0.30 | 0.19 | 0.26 | 0.31 | 0.30 | 0.00
KEG | 0.43 | 0.89 | 1.65 | 0.43 | 0.43 | 0.10 | 0.05 | 0.46 | 0.42 | 0.31 | 1.65 | 0.60 | 0.97 | 1.39 | 1.38 | 1.22
RAT | 0.09 | 1.13 | 0.41 | 0.09 | 0.09 | 0.52 | 0.70 | 0.02 | 0.10 | 0.25 | 0.41 | 0.41 | 0.43 | 0.29 | 0.24 | 0.12
GAL | 2.31 | 3.96 | 2.81 | 0.11 | 2.31 | 0.15 | 1.20 | 2.09 | 2.13 | 1.75 | 2.81 | 2.73 | 2.76 | 2.91 | 2.86 | 2.84
HAM | 0.54 | 2.39 | 1.44 | 0.54 | 0.54 | 0.22 | 1.63 | 0.51 | 0.40 | 0.56 | 1.44 | 1.40 | 1.36 | 1.47 | 1.46 | 1.50
MAT | 0.62 | 2.59 | 2.30 | 0.62 | 0.62 | 0.02 | 0.42 | 0.48 | 0.37 | 0.10 | 2.30 | 2.24 | 2.26 | 2.38 | 2.36 | 2.32
BAD | 0.30 | 1.79 | 1.78 | 0.30 | 0.30 | 0.08 | 0.97 | 0.37 | 0.30 | 0.03 | 1.78 | 1.42 | 1.30 | 1.72 | 1.73 | 1.62
MON | 0.85 | 0.58 | 0.85 | 0.85 | 0.85 | 0.53 | 0.79 | 0.89 | 0.85 | 0.43 | 0.85 | 0.73 | 0.23 | 0.85 | 0.82 | 0.64
COL | 1.62 | 2.63 | 1.91 | 0.76 | 1.62 | 1.33 | 1.92 | 1.68 | 1.78 | 1.85 | 1.91 | 2.22 | 2.12 | 2.02 | 1.98 | 2.06
GAM | 0.52 | 2.11 | 0.61 | 0.52 | 0.52 | 0.40 | 0.86 | 0.80 | 0.71 | 0.79 | 0.61 | 0.73 | 0.84 | 0.84 | 0.74 | 0.77
KLU | 1.58 | 2.92 | 1.60 | 0.28 | 1.58 | 1.27 | 2.58 | 1.57 | 1.64 | 2.37 | 1.60 | 1.90 | 2.28 | 1.72 | 1.64 | 2.09

Table 7: MASE from training set: 2016-2019 for weekly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 0.66 | 0.76 | 1.45 | 1.86 | 0.64 | 0.85 | 0.66 | 0.68 | 0.89 | 0.83 | 0.79 | 1.36 | 1.68 | 1.86 | 1.85 | 1.81 | 1.73 | 1.77 | 1.83
Province
CN | 0.45 | 0.77 | 1.49 | 2.12 | 0.50 | 0.45 | 0.32 | 0.29 | 0.50 | 0.44 | 0.38 | 1.04 | 1.90 | 2.24 | 2.17 | 2.08 | 2.00 | 2.06 | 2.34
ET | 1.33 | 0.75 | 2.64 | 3.29 | 0.85 | 0.86 | 1.05 | 1.03 | 1.04 | 1.04 | 0.99 | 1.01 | 2.94 | 3.48 | 3.35 | 3.07 | 3.09 | 3.19 | 2.76
NC | 0.58 | 0.53 | 1.19 | 1.17 | 0.58 | 0.59 | 0.46 | 0.19 | 0.68 | 0.58 | 0.39 | 1.90 | 1.17 | 1.24 | 1.63 | 1.23 | 1.18 | 1.32 | 2.03
NW | 0.56 | 0.37 | 0.90 | 0.90 | 0.53 | 0.60 | 0.45 | 0.42 | 0.61 | 0.57 | 0.54 | 0.94 | 1.12 | 0.95 | 1.07 | 1.08 | 1.03 | 1.04 | 1.18
NT | 4.00 | 1.47 | 4.52 | 5.76 | 1.15 | 4.04 | 3.18 | 3.55 | 4.00 | 4.00 | 3.76 | 3.87 | 5.85 | 6.07 | 6.05 | 5.80 | 5.82 | 5.99 | 4.06
SB | 0.43 | 0.41 | 0.38 | 0.85 | 0.45 | 0.42 | 0.32 | 0.28 | 0.49 | 0.40 | 0.38 | 1.45 | 0.25 | 0.91 | 0.75 | 0.65 | 0.50 | 0.57 | 0.45
ST | 0.69 | 1.03 | 1.27 | 1.62 | 0.63 | 1.26 | 0.54 | 0.64 | 1.10 | 1.04 | 0.90 | 1.72 | 1.23 | 1.71 | 1.64 | 1.43 | 1.38 | 1.47 | 1.40
UVA | 0.65 | 0.47 | 0.99 | 1.70 | 0.60 | 0.35 | 0.52 | 0.09 | 0.55 | 0.46 | 0.36 | 1.40 | 1.40 | 1.80 | 1.90 | 1.64 | 1.54 | 1.65 | 1.37
WT | 0.70 | 0.77 | 1.17 | 1.20 | 0.65 | 0.69 | 0.54 | 0.65 | 0.72 | 0.66 | 0.68 | 1.12 | 1.35 | 1.27 | 1.27 | 1.43 | 1.33 | 1.29 | 1.72
Districts
KN | 0.41 | 0.73 | 1.14 | 1.67 | 0.45 | 0.41 | 0.31 | 0.35 | 0.47 | 0.41 | 0.39 | 1.11 | 1.67 | 1.98 | 1.79 | 1.86 | 1.79 | 1.74 | 2.00
MTL | 0.63 | 0.97 | 2.70 | 2.75 | 0.64 | 0.63 | 0.52 | 0.37 | 0.65 | 0.63 | 0.52 | 0.78 | 2.75 | 3.18 | 3.23 | 2.85 | 2.78 | 3.02 | 3.79
NE | 0.76 | 0.77 | 1.92 | 1.72 | 0.74 | 0.76 | 0.61 | 0.73 | 0.82 | 0.76 | 0.29 | 1.45 | 1.72 | 2.01 | 3.79 | 1.82 | 1.74 | 2.91 | 1.33
AM | 0.73 | 0.79 | 0.84 | 1.01 | 0.75 | 0.73 | 0.81 | 1.47 | 0.82 | 0.77 | 1.27 | 1.77 | 1.01 | 1.18 | 2.86 | 1.12 | 1.07 | 2.13 | 1.16
BAT | 1.12 | 0.58 | 2.21 | 2.21 | 0.89 | 1.12 | 1.45 | 1.25 | 1.25 | 1.25 | 1.21 | 1.31 | 2.21 | 2.65 | 2.47 | 2.27 | 2.30 | 2.36 | 2.18
KAL | 0.54 | 0.42 | 1.30 | 1.30 | 0.58 | 0.54 | 0.66 | 0.68 | 0.76 | 0.80 | 0.65 | 0.22 | 1.30 | 1.53 | 1.61 | 1.46 | 1.50 | 1.48 | 0.80
TRC | 0.77 | 1.14 | 4.01 | 4.87 | 1.02 | 0.77 | 0.90 | 0.90 | 0.95 | 0.93 | 0.87 | 1.15 | 4.87 | 5.70 | 5.18 | 5.00 | 5.01 | 5.05 | 4.84
APU | 0.57 | 0.51 | 1.21 | 1.22 | 0.57 | 0.57 | 0.44 | 0.09 | 0.67 | 0.56 | 0.41 | 1.79 | 1.22 | 1.29 | 1.56 | 1.29 | 1.23 | 1.34 | 2.41
POL | 0.63 | 0.57 | 1.09 | 1.04 | 0.58 | 0.63 | 0.50 | 0.51 | 0.72 | 0.63 | 0.35 | 2.05 | 1.04 | 1.09 | 1.68 | 1.07 | 1.04 | 1.25 | 1.19
KUR | 0.62 | 0.37 | 0.92 | 1.32 | 0.56 | 0.62 | 0.46 | 0.46 | 0.64 | 0.59 | 0.57 | 1.02 | 1.32 | 1.11 | 1.27 | 1.27 | 1.20 | 1.25 | 1.30
PUT | 0.58 | 0.37 | 0.87 | 0.86 | 0.49 | 0.58 | 0.43 | 0.38 | 0.59 | 0.56 | 0.51 | 0.84 | 0.86 | 0.74 | 0.81 | 0.84 | 0.81 | 0.78 | 1.03
JAF | 4.65 | 1.55 | 4.88 | 6.30 | 1.14 | 4.65 | 3.67 | 4.52 | 4.60 | 4.60 | 4.58 | 4.42 | 6.30 | 6.54 | 6.35 | 6.23 | 6.26 | 6.33 | 4.46
KIL | 3.76 | 0.98 | 3.45 | 3.76 | 0.82 | 3.76 | 2.98 | 2.25 | 3.73 | 3.76 | 2.90 | 3.41 | 3.76 | 3.90 | 4.43 | 3.82 | 3.76 | 4.22 | 3.61
MAN | 0.67 | 0.61 | 0.72 | 1.46 | 0.67 | 0.67 | 0.58 | 0.82 | 0.61 | 0.67 | 0.38 | 0.41 | 1.46 | 1.52 | 1.99 | 1.43 | 1.45 | 1.81 | 0.53
MUL | 0.71 | 0.64 | 1.47 | 1.97 | 0.72 | 0.71 | 0.58 | 1.40 | 0.70 | 0.70 | 0.59 | 0.96 | 1.97 | 2.05 | 2.87 | 2.00 | 1.97 | 2.58 | 1.06
VAV | 0.92 | 1.09 | 2.93 | 3.58 | 0.90 | 0.92 | 0.74 | 0.28 | 0.93 | 0.91 | 0.53 | 1.38 | 3.58 | 3.70 | 3.88 | 3.56 | 3.57 | 3.79 | 2.34
KEG | 0.43 | 0.30 | 0.47 | 0.30 | 0.46 | 0.43 | 0.33 | 0.29 | 0.48 | 0.42 | 0.39 | 1.15 | 0.30 | 0.95 | 0.75 | 0.53 | 0.44 | 0.60 | 0.16
RAT | 0.48 | 0.56 | 0.39 | 0.30 | 0.48 | 0.48 | 0.40 | 0.36 | 0.55 | 0.46 | 0.44 | 1.81 | 0.30 | 0.83 | 0.74 | 0.81 | 0.61 | 0.56 | 0.98
GAL | 2.12 | 1.30 | 2.14 | 1.90 | 0.64 | 2.12 | 0.94 | 1.64 | 1.91 | 1.87 | 1.84 | 2.31 | 1.90 | 2.64 | 2.22 | 2.06 | 2.04 | 2.09 | 2.38
HAM | 0.67 | 0.79 | 0.86 | 0.80 | 0.67 | 0.67 | 0.27 | 0.44 | 0.60 | 0.56 | 0.10 | 0.94 | 0.80 | 1.13 | 1.53 | 0.93 | 0.86 | 1.24 | 1.21
MAT | 0.59 | 0.83 | 0.49 | 0.66 | 0.62 | 0.59 | 0.25 | 0.08 | 0.43 | 0.35 | 0.29 | 1.40 | 0.66 | 0.92 | 1.00 | 0.92 | 0.85 | 0.86 | 0.44
BAD | 0.62 | 0.77 | 1.75 | 2.49 | 0.62 | 0.62 | 0.92 | 0.37 | 0.89 | 0.77 | 0.63 | 2.06 | 2.49 | 3.19 | 2.93 | 2.84 | 2.69 | 2.71 | 2.33
MON | 0.00 | 0.17 | 0.00 | 0.00 | 0.58 | 0.00 | 0.00 | 0.27 | 0.10 | 0.06 | 0.03 | 0.48 | 0.00 | 0.00 | 0.47 | 0.08 | 0.05 | 0.24 | 0.12
COL | 0.83 | 0.90 | 1.47 | 1.91 | 0.80 | 0.83 | 0.66 | 0.80 | 0.86 | 0.81 | 0.82 | 1.35 | 1.91 | 1.78 | 1.84 | 1.95 | 1.88 | 1.86 | 2.26
GAM | 0.53 | 0.59 | 0.82 | 0.82 | 0.51 | 0.53 | 0.42 | 0.49 | 0.54 | 0.49 | 0.51 | 0.83 | 0.82 | 0.77 | 0.76 | 0.91 | 0.79 | 0.77 | 1.09
KLU | 0.68 | 0.84 | 1.21 | 1.17 | 0.57 | 0.68 | 0.53 | 0.59 | 0.73 | 0.67 | 0.65 | 1.16 | 1.17 | 1.09 | 0.99 | 1.26 | 1.16 | 1.04 | 1.79

Table 8: MASE from training set: 2016-2019 for monthly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 0.70 | 0.77 | 1.54 | 1.41 | 0.63 | 0.63 | 0.70 | 0.70 | 0.73 | 0.66 | 0.67 | 1.13 | 1.79 | 1.41 | 1.43 | 1.56 | 1.59 | 1.57 | 1.86
Province
CN | 0.69 | 0.80 | 1.80 | 2.28 | 0.50 | 0.72 | 0.70 | 0.72 | 0.78 | 0.71 | 0.72 | 1.22 | 2.20 | 2.12 | 2.12 | 2.10 | 2.20 | 2.15 | 2.15
ET | 0.71 | 0.74 | 2.17 | 1.48 | 0.75 | 0.54 | 0.73 | 0.70 | 0.62 | 0.61 | 0.64 | 1.28 | 2.02 | 1.42 | 1.43 | 1.71 | 1.73 | 1.62 | 1.41
NC | 0.58 | 0.55 | 1.27 | 1.62 | 0.58 | 0.59 | 0.60 | 0.71 | 0.66 | 0.59 | 0.64 | 1.08 | 1.63 | 1.50 | 1.14 | 1.48 | 1.62 | 1.36 | 1.76
NW | 0.57 | 0.38 | 0.85 | 0.84 | 0.52 | 0.58 | 0.58 | 0.59 | 0.63 | 0.58 | 0.58 | 0.69 | 0.95 | 0.79 | 0.76 | 0.86 | 0.92 | 0.84 | 1.20
NT | 1.18 | 1.51 | 6.36 | 4.32 | 1.07 | 0.93 | 1.11 | 1.18 | 1.06 | 1.04 | 1.10 | 1.85 | 6.23 | 3.95 | 4.28 | 4.49 | 4.62 | 4.96 | 4.72
SB | 0.46 | 0.42 | 0.52 | 1.00 | 0.46 | 0.26 | 0.47 | 0.40 | 0.41 | 0.33 | 0.37 | 1.07 | 0.77 | 0.96 | 0.79 | 0.90 | 0.88 | 0.82 | 1.89
ST | 0.65 | 1.05 | 1.04 | 1.37 | 0.64 | 0.67 | 0.67 | 0.69 | 0.71 | 0.67 | 0.68 | 0.96 | 1.52 | 1.31 | 1.26 | 1.50 | 1.47 | 1.36 | 1.98
UVA | 0.65 | 0.49 | 1.05 | 1.11 | 0.62 | 0.34 | 0.66 | 0.60 | 0.53 | 0.46 | 0.52 | 1.27 | 1.18 | 1.05 | 0.79 | 1.15 | 1.14 | 0.98 | 2.00
WT | 0.71 | 0.79 | 1.26 | 1.18 | 0.65 | 0.71 | 0.72 | 0.72 | 0.80 | 0.72 | 0.71 | 1.05 | 1.52 | 1.12 | 1.23 | 1.26 | 1.28 | 1.33 | 1.50
Districts
KN | 0.59 | 0.76 | 1.35 | 1.70 | 0.45 | 0.59 | 0.57 | 0.59 | 0.65 | 0.58 | 0.59 | 0.97 | 1.70 | 1.64 | 1.67 | 1.62 | 1.71 | 1.69 | 1.58
MTL | 1.24 | 0.99 | 3.60 | 4.25 | 0.62 | 1.24 | 1.21 | 1.24 | 1.30 | 1.23 | 1.24 | 2.17 | 4.25 | 4.10 | 4.11 | 4.08 | 4.25 | 4.17 | 4.23
NE | 0.77 | 0.86 | 1.55 | 1.59 | 0.78 | 0.77 | 0.77 | 0.89 | 0.84 | 0.77 | 0.81 | 1.62 | 1.59 | 1.55 | 1.40 | 1.62 | 1.59 | 1.30 | 2.89
AM | 0.70 | 0.76 | 1.10 | 0.99 | 0.74 | 0.70 | 0.81 | 1.38 | 0.69 | 0.69 | 1.06 | 0.93 | 0.99 | 0.71 | 2.65 | 0.87 | 0.94 | 1.58 | 1.60
BAT | 0.60 | 0.54 | 1.66 | 2.03 | 0.75 | 0.60 | 0.76 | 0.67 | 0.61 | 0.62 | 0.64 | 0.94 | 2.03 | 1.26 | 1.67 | 1.66 | 1.83 | 1.78 | 1.75
KAL | 0.54 | 0.41 | 1.07 | 1.06 | 0.56 | 0.54 | 0.66 | 0.65 | 0.66 | 0.63 | 0.59 | 0.95 | 1.06 | 0.73 | 0.67 | 0.62 | 0.64 | 0.79 | 0.84
TRC | 0.61 | 1.14 | 3.41 | 2.94 | 0.93 | 0.61 | 0.80 | 0.75 | 0.70 | 0.69 | 0.69 | 2.31 | 2.94 | 2.35 | 2.66 | 2.91 | 2.78 | 2.68 | 2.96
APU | 0.59 | 0.52 | 1.40 | 1.79 | 0.59 | 0.59 | 0.60 | 0.68 | 0.68 | 0.59 | 0.63 | 1.22 | 1.79 | 1.67 | 1.35 | 1.57 | 1.77 | 1.59 | 1.84
POL | 0.60 | 0.59 | 0.99 | 1.31 | 0.60 | 0.60 | 0.62 | 0.80 | 0.62 | 0.60 | 0.67 | 0.86 | 1.31 | 1.18 | 1.23 | 1.32 | 1.31 | 1.00 | 1.66
KUR | 0.59 | 0.38 | 0.84 | 1.00 | 0.56 | 0.59 | 0.60 | 0.60 | 0.65 | 0.59 | 0.59 | 0.60 | 1.00 | 0.80 | 0.83 | 0.95 | 0.99 | 0.90 | 1.08
PUT | 0.56 | 0.38 | 0.85 | 0.88 | 0.49 | 0.56 | 0.56 | 0.58 | 0.59 | 0.56 | 0.57 | 0.86 | 0.88 | 0.76 | 0.66 | 0.74 | 0.83 | 0.76 | 1.34
JAF | 0.96 | 1.60 | 6.74 | 6.73 | 1.07 | 0.96 | 1.13 | 0.99 | 1.08 | 1.08 | 0.98 | 1.57 | 6.73 | 4.21 | 6.23 | 4.57 | 4.68 | 6.40 | 4.78
KIL | 1.02 | 1.08 | 4.33 | 3.45 | 0.86 | 1.02 | 1.24 | 2.06 | 1.11 | 1.03 | 1.61 | 3.07 | 3.45 | 2.09 | 6.28 | 3.14 | 3.42 | 3.50 | 3.55
MAN | 0.60 | 0.57 | 2.21 | 1.41 | 0.61 | 0.60 | 0.73 | 1.32 | 0.64 | 0.61 | 1.01 | 1.77 | 1.41 | 0.82 | 4.32 | 0.99 | 1.36 | 2.47 | 0.41
MUL | 0.88 | 0.64 | 2.46 | 2.17 | 0.76 | 0.88 | 1.05 | 2.49 | 0.94 | 0.88 | 1.76 | 1.64 | 2.17 | 1.34 | 9.98 | 1.99 | 2.15 | 6.01 | 2.74
VAV | 1.50 | 1.07 | 4.55 | 4.71 | 0.86 | 1.50 | 1.74 | 1.90 | 1.54 | 1.51 | 1.77 | 2.90 | 4.71 | 3.48 | 3.72 | 4.67 | 4.66 | 3.39 | 5.33
KEG | 0.09 | 0.30 | 0.49 | 0.50 | 0.47 | 0.09 | 0.14 | 0.21 | 0.19 | 0.11 | 0.17 | 0.63 | 0.50 | 0.61 | 0.51 | 0.54 | 0.60 | 0.54 | 1.05
RAT | 0.52 | 0.59 | 0.62 | 1.20 | 0.52 | 0.52 | 0.91 | 0.69 | 0.72 | 0.63 | 0.65 | 1.96 | 1.20 | 1.42 | 1.16 | 1.37 | 1.27 | 1.20 | 3.03
GAL | 0.71 | 1.33 | 1.40 | 1.94 | 0.63 | 0.71 | 0.71 | 0.72 | 0.73 | 0.71 | 0.72 | 1.10 | 1.94 | 1.66 | 1.74 | 1.95 | 1.89 | 1.82 | 2.40
HAM | 0.66 | 0.80 | 0.73 | 1.24 | 0.69 | 0.66 | 0.66 | 0.69 | 0.69 | 0.66 | 0.67 | 0.43 | 1.24 | 1.01 | 0.77 | 1.13 | 1.19 | 0.94 | 1.08
MAT | 0.63 | 0.83 | 0.76 | 1.13 | 0.63 | 0.63 | 0.62 | 0.63 | 0.68 | 0.62 | 0.63 | 1.01 | 1.13 | 1.01 | 0.91 | 1.13 | 1.07 | 1.00 | 1.86
BAD | 0.61 | 0.80 | 1.89 | 2.11 | 0.63 | 0.61 | 1.17 | 0.84 | 0.86 | 0.78 | 0.77 | 2.19 | 2.11 | 1.88 | 1.73 | 2.11 | 2.07 | 1.93 | 3.41
MON | 0.00 | 0.16 | 0.00 | 0.00 | 0.59 | 0.00 | 0.00 | 0.25 | 0.08 | 0.04 | 0.17 | 0.28 | 0.00 | 0.00 | 0.50 | 0.13 | 0.07 | 0.22 | 0.29
COL | 0.85 | 0.91 | 1.56 | 1.97 | 0.78 | 0.85 | 0.86 | 0.85 | 0.94 | 0.86 | 0.85 | 0.97 | 1.97 | 1.40 | 1.74 | 1.58 | 1.69 | 1.82 | 1.25
GAM | 0.57 | 0.60 | 1.00 | 1.02 | 0.53 | 0.57 | 0.58 | 0.58 | 0.67 | 0.58 | 0.57 | 1.29 | 1.02 | 0.74 | 0.80 | 0.81 | 0.74 | 0.88 | 1.58
KLU | 0.61 | 0.86 | 0.97 | 1.49 | 0.57 | 0.61 | 0.62 | 0.62 | 0.68 | 0.61 | 0.61 | 0.90 | 1.49 | 1.25 | 0.99 | 1.47 | 1.46 | 1.07 | 1.70

Table 9: MASE for training set: 2016-2019 for quarterly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 0.65 | 0.77 | 1.55 | 1.92 | 0.52 | 0.60 | 0.65 | 0.65 | 0.68 | 0.62 | 0.63 | 1.44 | 1.92 | 1.87 | 1.49 | 1.54 | 1.66
Province
CN | 0.68 | 0.77 | 2.14 | 2.58 | 0.43 | 0.59 | 0.70 | 0.68 | 0.71 | 0.63 | 0.64 | 2.18 | 3.03 | 2.79 | 2.30 | 2.36 | 2.52
ET | 0.57 | 0.76 | 1.19 | 0.73 | 0.71 | 0.70 | 0.53 | 0.54 | 0.63 | 0.65 | 0.61 | 0.76 | 0.80 | 0.97 | 0.81 | 0.76 | 0.87
NC | 0.58 | 0.59 | 1.09 | 1.50 | 0.52 | 0.76 | 0.60 | 0.74 | 0.73 | 0.70 | 0.71 | 1.48 | 1.77 | 2.87 | 1.40 | 1.49 | 1.93
NW | 0.67 | 0.38 | 0.86 | 1.12 | 0.52 | 0.59 | 0.68 | 0.66 | 0.67 | 0.64 | 0.64 | 1.15 | 1.31 | 1.41 | 1.10 | 1.15 | 1.22
NT | 0.92 | 1.59 | 5.29 | 2.07 | 0.92 | 0.92 | 0.87 | 0.81 | 0.93 | 0.91 | 0.84 | 2.34 | 2.49 | 2.96 | 2.32 | 2.25 | 2.76
SB | 0.45 | 0.38 | 0.72 | 0.90 | 0.45 | 0.44 | 0.45 | 0.46 | 0.55 | 0.44 | 0.45 | 0.62 | 1.08 | 1.12 | 0.67 | 0.72 | 0.86
ST | 0.75 | 1.04 | 1.27 | 1.54 | 0.56 | 0.73 | 0.76 | 0.77 | 0.79 | 0.74 | 0.75 | 1.62 | 1.80 | 1.89 | 1.53 | 1.61 | 1.72
UVA | 0.65 | 0.46 | 1.05 | 1.30 | 0.59 | 0.28 | 0.67 | 0.59 | 0.51 | 0.42 | 0.49 | 0.86 | 1.52 | 1.98 | 0.99 | 1.04 | 1.35
WT | 0.69 | 0.78 | 1.51 | 1.90 | 0.62 | 0.72 | 0.71 | 0.71 | 0.77 | 0.72 | 0.71 | 1.61 | 2.22 | 1.91 | 1.70 | 1.77 | 1.79
Districts
KN | 0.61 | 0.71 | 1.92 | 2.36 | 0.41 | 0.61 | 0.74 | 0.65 | 0.73 | 0.66 | 0.64 | 2.36 | 3.28 | 2.63 | 2.50 | 2.58 | 2.51
MTL | 0.47 | 1.02 | 3.15 | 1.36 | 0.47 | 0.47 | 0.54 | 0.63 | 0.61 | 0.49 | 0.57 | 1.36 | 1.91 | 2.46 | 1.50 | 1.42 | 1.97
NE | 0.66 | 0.91 | 1.58 | 2.15 | 0.66 | 0.66 | 0.66 | 1.31 | 0.73 | 0.66 | 1.04 | 2.15 | 3.01 | 7.30 | 2.01 | 2.18 | 4.98
AM | 0.69 | 0.82 | 1.02 | 1.30 | 0.76 | 0.69 | 0.48 | 0.86 | 0.64 | 0.69 | 0.60 | 1.30 | 1.44 | 2.91 | 1.27 | 1.30 | 2.03
BAT | 0.91 | 0.53 | 0.98 | 0.91 | 0.91 | 0.91 | 0.70 | 0.79 | 0.82 | 0.86 | 0.84 | 0.91 | 0.89 | 0.88 | 0.88 | 0.90 | 0.89
KAL | 0.48 | 0.38 | 0.61 | 0.55 | 0.48 | 0.48 | 0.41 | 0.38 | 0.39 | 0.43 | 0.41 | 0.55 | 0.53 | 0.69 | 0.60 | 0.55 | 0.63
TRC | 0.93 | 1.22 | 1.77 | 0.93 | 0.93 | 0.93 | 0.79 | 0.81 | 0.90 | 0.88 | 0.86 | 0.93 | 0.98 | 1.09 | 0.96 | 0.94 | 1.02
APU | 0.81 | 0.56 | 1.17 | 1.66 | 0.52 | 0.81 | 0.64 | 0.79 | 0.76 | 0.74 | 0.77 | 1.66 | 1.98 | 2.71 | 1.60 | 1.67 | 2.01
POL | 0.64 | 0.62 | 0.91 | 1.08 | 0.53 | 0.64 | 0.49 | 0.61 | 0.63 | 0.61 | 0.57 | 1.08 | 1.31 | 3.08 | 0.98 | 1.08 | 1.74
KUR | 0.54 | 0.37 | 0.76 | 0.97 | 0.54 | 0.54 | 0.61 | 0.61 | 0.61 | 0.60 | 0.58 | 0.97 | 1.11 | 1.21 | 0.89 | 0.97 | 1.04
PUT | 0.66 | 0.39 | 0.98 | 1.37 | 0.48 | 0.66 | 0.77 | 0.74 | 0.74 | 0.69 | 0.71 | 1.37 | 1.55 | 1.66 | 1.36 | 1.37 | 1.45
JAF | 1.00 | 1.86 | 6.31 | 2.61 | 1.00 | 1.00 | 0.95 | 0.97 | 1.02 | 0.99 | 0.98 | 2.61 | 2.78 | 2.78 | 2.56 | 2.49 | 2.73
KIL | 0.74 | 1.12 | 3.11 | 2.10 | 0.74 | 0.74 | 0.70 | 0.66 | 0.70 | 0.74 | 0.61 | 2.10 | 2.24 | 4.34 | 2.11 | 2.10 | 3.61
MAN | 0.44 | 0.48 | 1.16 | 0.45 | 0.44 | 0.44 | 0.42 | 0.37 | 0.44 | 0.44 | 0.34 | 0.45 | 0.47 | 1.59 | 0.48 | 0.45 | 1.15
MUL | 0.62 | 0.55 | 1.96 | 0.62 | 0.62 | 0.62 | 0.63 | 0.88 | 0.73 | 0.62 | 0.82 | 0.62 | 0.66 | 3.45 | 0.75 | 0.62 | 2.52
VAV | 0.76 | 1.05 | 3.12 | 2.38 | 0.76 | 0.76 | 0.72 | 0.69 | 0.75 | 0.76 | 0.67 | 2.38 | 2.52 | 3.37 | 2.45 | 2.37 | 3.04
KEG | 0.45 | 0.31 | 0.60 | 0.73 | 0.46 | 0.45 | 0.46 | 0.48 | 0.52 | 0.46 | 0.46 | 0.73 | 1.29 | 1.15 | 0.68 | 0.85 | 0.93
RAT | 0.42 | 0.51 | 0.91 | 0.66 | 0.42 | 0.42 | 0.43 | 0.45 | 0.59 | 0.42 | 0.43 | 0.66 | 0.77 | 1.07 | 0.65 | 0.63 | 0.75
GAL | 0.92 | 1.33 | 1.46 | 2.24 | 0.48 | 0.92 | 0.96 | 0.94 | 0.95 | 0.92 | 0.93 | 2.24 | 2.49 | 2.45 | 2.14 | 2.23 | 2.32
HAM | 0.52 | 0.80 | 1.06 | 1.06 | 0.63 | 0.52 | 0.55 | 0.58 | 0.52 | 0.52 | 0.56 | 1.06 | 1.19 | 1.53 | 1.01 | 1.06 | 1.24
MAT | 0.62 | 0.86 | 1.14 | 1.18 | 0.62 | 0.62 | 0.63 | 0.65 | 0.72 | 0.63 | 0.63 | 1.18 | 1.31 | 1.41 | 1.10 | 1.16 | 1.27
BAD | 0.50 | 0.77 | 1.81 | 1.65 | 0.50 | 0.50 | 1.19 | 0.77 | 0.85 | 0.73 | 0.69 | 1.65 | 2.92 | 2.64 | 1.86 | 1.90 | 2.09
MON | 0.01 | 0.18 | 0.04 | 0.15 | 0.60 | 0.01 | 0.02 | 0.29 | 0.06 | 0.02 | 0.20 | 0.15 | 0.27 | 0.95 | 0.12 | 0.08 | 0.33
COL | 0.86 | 0.90 | 1.85 | 1.66 | 0.74 | 0.86 | 0.84 | 0.84 | 0.90 | 0.85 | 0.85 | 1.66 | 2.30 | 1.90 | 1.84 | 1.85 | 1.80
GAM | 0.55 | 0.62 | 1.15 | 1.45 | 0.52 | 0.55 | 0.53 | 0.53 | 0.59 | 0.54 | 0.54 | 1.45 | 2.00 | 1.69 | 1.50 | 1.62 | 1.59
KLU | 0.83 | 0.89 | 1.51 | 1.85 | 0.55 | 0.83 | 0.81 | 0.79 | 0.88 | 0.82 | 0.80 | 1.85 | 2.58 | 2.52 | 1.80 | 1.89 | 2.26

Table 10: MASE from training set: 2006-2018 for weekly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 1.29 | 1.44 | 1.34 | 1.19 | 1.62 | 1.34 | 1.29 | 1.30 | 1.33 | 1.34 | 1.33 | 1.27 | 1.35 | 1.19 | 1.21 | 1.36 | 1.34 | 1.30 | 1.84
Province
CN | 2.44 | 1.82 | 1.90 | 2.59 | 1.83 | 1.97 | 2.42 | 2.18 | 2.11 | 2.15 | 2.15 | 1.82 | 2.30 | 2.59 | 2.11 | 2.41 | 2.40 | 2.29 | 2.73
ET | 1.43 | 2.36 | 1.39 | 1.41 | 1.33 | 1.36 | 1.53 | 1.51 | 1.38 | 1.39 | 1.42 | 1.35 | 1.42 | 1.62 | 1.67 | 1.43 | 1.42 | 1.52 | 1.48
NC | 0.76 | 1.03 | 0.78 | 0.77 | 0.77 | 0.75 | 0.75 | 0.81 | 0.75 | 0.75 | 0.74 | 0.77 | 0.77 | 0.80 | 1.38 | 0.78 | 0.76 | 0.81 | 2.41
NW | 0.66 | 0.84 | 0.65 | 0.65 | 0.65 | 0.64 | 0.64 | 0.62 | 0.64 | 0.64 | 0.64 | 0.63 | 0.66 | 0.66 | 0.67 | 0.66 | 0.65 | 0.65 | 1.02
NT | 5.45 | 3.15 | 5.66 | 5.62 | 3.79 | 5.48 | 5.86 | 5.62 | 5.47 | 5.47 | 5.54 | 5.51 | 5.56 | 6.61 | 6.05 | 5.58 | 5.60 | 5.83 | 5.57
SB | 0.84 | 0.86 | 0.85 | 0.81 | 0.77 | 0.87 | 0.78 | 0.76 | 0.82 | 0.85 | 0.83 | 0.62 | 0.82 | 0.67 | 0.60 | 0.88 | 0.79 | 0.75 | 2.40
ST | 2.88 | 2.76 | 2.56 | 2.51 | 2.72 | 2.55 | 2.81 | 2.63 | 2.65 | 2.66 | 2.65 | 2.49 | 2.48 | 2.24 | 2.12 | 2.59 | 2.47 | 2.34 | 3.54
UVA | 0.82 | 0.83 | 0.71 | 0.71 | 0.75 | 0.70 | 0.91 | 0.88 | 0.73 | 0.72 | 0.75 | 0.95 | 0.70 | 0.69 | 0.90 | 0.72 | 0.70 | 0.69 | 1.39
WT | 1.39 | 1.49 | 1.38 | 1.41 | 1.66 | 1.38 | 1.31 | 1.37 | 1.36 | 1.37 | 1.38 | 1.28 | 1.39 | 1.22 | 1.36 | 1.39 | 1.37 | 1.38 | 1.75
Districts
KN | 2.02 | 1.78 | 1.87 | 2.46 | 1.88 | 2.02 | 2.48 | 2.10 | 2.18 | 2.23 | 2.09 | 1.81 | 2.46 | 2.64 | 2.38 | 2.55 | 2.57 | 2.45 | 2.67
MTL | 1.91 | 2.18 | 1.98 | 1.91 | 1.85 | 1.91 | 2.16 | 2.12 | 1.96 | 1.96 | 2.09 | 1.98 | 1.91 | 2.35 | 1.78 | 1.98 | 1.93 | 1.88 | 2.46
NE | 1.09 | 1.11 | 1.17 | 1.07 | 0.92 | 1.09 | 1.36 | 2.18 | 1.11 | 1.13 | 2.03 | 0.96 | 1.07 | 1.54 | 0.62 | 1.22 | 1.08 | 0.98 | 2.73
AM | 1.04 | 1.24 | 1.54 | 0.96 | 0.97 | 1.04 | 0.96 | 0.82 | 1.05 | 1.02 | 0.82 | 1.41 | 0.96 | 0.87 | 1.24 | 1.09 | 0.96 | 0.70 | 2.60
BAT | 1.34 | 3.15 | 1.32 | 1.34 | 1.19 | 1.34 | 1.59 | 1.44 | 1.37 | 1.37 | 1.38 | 1.41 | 1.34 | 1.57 | 1.48 | 1.34 | 1.34 | 1.40 | 1.47
KAL | 0.82 | 1.44 | 0.86 | 0.86 | 0.78 | 0.82 | 0.93 | 0.93 | 0.85 | 0.86 | 0.87 | 0.78 | 0.86 | 0.99 | 1.04 | 0.89 | 0.86 | 0.93 | 1.40
TRC | 1.83 | 2.02 | 1.93 | 1.94 | 1.86 | 1.83 | 1.90 | 1.90 | 1.85 | 1.85 | 1.86 | 1.84 | 1.94 | 2.08 | 2.09 | 1.95 | 1.95 | 2.00 | 1.88
APU | 0.79 | 1.09 | 0.84 | 0.79 | 0.77 | 0.79 | 0.79 | 0.88 | 0.78 | 0.78 | 0.80 | 0.81 | 0.79 | 0.86 | 1.15 | 0.79 | 0.79 | 0.85 | 2.63
POL | 0.71 | 0.87 | 0.67 | 0.67 | 0.71 | 0.71 | 0.69 | 0.67 | 0.72 | 0.72 | 0.65 | 0.76 | 0.67 | 0.67 | 1.74 | 0.69 | 0.67 | 0.70 | 1.58
KUR | 0.52 | 0.68 | 0.52 | 0.52 | 0.52 | 0.52 | 0.50 | 0.50 | 0.51 | 0.51 | 0.51 | 0.65 | 0.52 | 0.53 | 0.53 | 0.53 | 0.52 | 0.52 | 0.72
PUT | 0.86 | 1.10 | 0.84 | 0.85 | 0.84 | 0.86 | 0.86 | 0.82 | 0.87 | 0.86 | 0.86 | 1.22 | 0.85 | 0.85 | 0.88 | 0.86 | 0.84 | 0.84 | 1.51
JAF | 7.34 | 3.75 | 7.41 | 7.32 | 4.81 | 7.34 | 7.86 | 7.38 | 7.33 | 7.32 | 7.36 | 7.52 | 7.32 | 8.73 | 7.47 | 7.36 | 7.37 | 7.41 | 7.46
KIL | 4.04 | 4.08 | 5.95 | 4.24 | 3.43 | 4.04 | 4.33 | 4.77 | 3.97 | 4.04 | 4.38 | 3.53 | 4.24 | 5.10 | 6.97 | 4.12 | 4.25 | 5.71 | 3.62
MAN | 0.81 | 1.21 | 0.82 | 0.85 | 0.79 | 0.81 | 0.84 | 1.03 | 0.81 | 0.81 | 0.90 | 0.73 | 0.85 | 0.97 | 1.81 | 0.86 | 0.86 | 1.31 | 0.89
MUL | 1.45 | 1.81 | 1.37 | 1.45 | 1.43 | 1.45 | 1.49 | 1.99 | 1.43 | 1.45 | 1.71 | 1.38 | 1.45 | 1.52 | 3.58 | 1.41 | 1.45 | 2.38 | 1.60
VAV | 1.22 | 1.47 | 1.34 | 1.34 | 1.22 | 1.22 | 1.23 | 1.28 | 1.22 | 1.22 | 1.25 | 1.22 | 1.34 | 1.47 | 1.75 | 1.35 | 1.35 | 1.53 | 1.25
KEG | 0.57 | 0.60 | 0.58 | 0.57 | 0.56 | 0.57 | 0.52 | 0.52 | 0.55 | 0.56 | 0.55 | 0.56 | 0.57 | 0.50 | 0.47 | 0.60 | 0.56 | 0.54 | 1.21
RAT | 1.27 | 1.11 | 1.21 | 1.12 | 1.00 | 1.27 | 1.16 | 1.11 | 1.18 | 1.22 | 1.21 | 0.85 | 1.12 | 0.92 | 0.81 | 1.24 | 1.06 | 1.02 | 3.54
GAL | 4.23 | 4.24 | 4.13 | 4.05 | 3.98 | 4.23 | 4.39 | 4.33 | 4.40 | 4.36 | 4.35 | 4.68 | 4.05 | 3.88 | 3.69 | 4.19 | 4.05 | 3.92 | 5.53
HAM | 1.15 | 1.40 | 1.29 | 1.15 | 1.54 | 1.15 | 1.39 | 1.20 | 1.18 | 1.18 | 1.23 | 1.14 | 1.15 | 0.97 | 0.87 | 1.20 | 1.14 | 1.02 | 1.25
MAT | 1.69 | 1.93 | 1.68 | 1.67 | 1.90 | 1.69 | 1.86 | 1.70 | 1.73 | 1.75 | 1.72 | 1.51 | 1.67 | 1.50 | 1.49 | 1.74 | 1.66 | 1.60 | 2.65
BAD | 1.29 | 1.43 | 1.28 | 1.29 | 1.40 | 1.29 | 1.35 | 1.27 | 1.28 | 1.27 | 1.26 | 1.52 | 1.29 | 1.21 | 1.23 | 1.32 | 1.28 | 1.22 | 1.89
MON | 0.46 | 0.82 | 0.41 | 0.47 | 0.46 | 0.46 | 0.86 | 1.09 | 0.54 | 0.54 | 0.75 | 0.62 | 0.47 | 0.61 | 1.37 | 0.43 | 0.47 | 0.68 | 0.72
COL | 1.25 | 1.40 | 1.27 | 1.27 | 1.57 | 1.25 | 1.20 | 1.24 | 1.24 | 1.24 | 1.25 | 1.25 | 1.27 | 1.21 | 1.25 | 1.27 | 1.26 | 1.27 | 2.29
GAM | 1.27 | 1.30 | 1.24 | 1.29 | 1.39 | 1.27 | 1.23 | 1.27 | 1.25 | 1.27 | 1.27 | 1.12 | 1.29 | 1.18 | 1.26 | 1.26 | 1.27 | 1.28 | 0.92
KLU | 1.85 | 2.21 | 1.79 | 1.80 | 2.32 | 1.85 | 1.75 | 1.81 | 1.83 | 1.85 | 1.84 | 1.69 | 1.80 | 1.54 | 1.72 | 1.84 | 1.79 | 1.78 | 2.71

Table 11: MASE from training set: 2006-2018for monthly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 1.34 | 1.45 | 1.35 | 1.29 | 1.60 | 1.39 | 1.34 | 1.34 | 1.33 | 1.36 | 1.35 | 1.66 | 1.30 | 1.29 | 1.28 | 1.31 | 1.29 | 1.28 | 1.80
Province
CN | 1.87 | 1.83 | 1.82 | 1.68 | 1.88 | 1.87 | 1.88 | 1.87 | 1.91 | 1.87 | 1.87 | 2.45 | 1.80 | 1.71 | 1.74 | 1.76 | 1.72 | 1.73 | 2.19
ET | 1.60 | 2.48 | 1.55 | 1.55 | 1.31 | 2.02 | 1.55 | 1.65 | 1.85 | 1.82 | 1.81 | 2.35 | 1.53 | 1.34 | 1.33 | 1.51 | 1.54 | 1.44 | 1.32
NC | 0.84 | 1.03 | 0.72 | 0.76 | 0.76 | 0.71 | 0.85 | 0.91 | 0.74 | 0.74 | 0.76 | 0.85 | 0.74 | 0.58 | 0.51 | 0.61 | 0.64 | 0.53 | 1.34
NW | 0.58 | 0.84 | 0.64 | 0.75 | 0.63 | 0.65 | 0.58 | 0.60 | 0.58 | 0.62 | 0.61 | 0.58 | 0.72 | 0.70 | 0.70 | 0.74 | 0.73 | 0.72 | 1.20
NT | 4.49 | 3.03 | 5.18 | 4.15 | 3.69 | 3.50 | 4.46 | 4.31 | 3.86 | 3.89 | 4.00 | 3.83 | 5.16 | 3.77 | 4.02 | 4.53 | 4.59 | 4.44 | 4.10
SB | 0.64 | 0.81 | 0.74 | 0.71 | 0.73 | 0.67 | 0.65 | 0.66 | 0.65 | 0.66 | 0.65 | 1.06 | 0.75 | 0.70 | 0.72 | 0.78 | 0.72 | 0.72 | 1.25
ST | 2.69 | 2.82 | 2.55 | 2.58 | 2.76 | 2.40 | 2.73 | 2.65 | 2.56 | 2.48 | 2.54 | 2.67 | 2.54 | 2.70 | 2.76 | 2.56 | 2.56 | 2.63 | 2.76
UVA | 0.72 | 0.82 | 0.69 | 0.72 | 0.71 | 0.65 | 0.73 | 0.72 | 0.68 | 0.68 | 0.67 | 0.88 | 0.71 | 0.76 | 1.00 | 0.70 | 0.71 | 0.75 | 1.16
WT | 1.42 | 1.50 | 1.42 | 1.45 | 1.67 | 1.43 | 1.43 | 1.42 | 1.38 | 1.42 | 1.42 | 2.16 | 1.43 | 1.46 | 1.44 | 1.47 | 1.44 | 1.44 | 1.94
Districts
KN | 1.89 | 1.81 | 1.83 | 1.83 | 1.93 | 1.89 | 1.90 | 1.89 | 1.97 | 1.89 | 1.88 | 2.67 | 1.83 | 1.71 | 1.77 | 1.79 | 1.73 | 1.79 | 2.21
MTL | 1.91 | 2.24 | 1.87 | 1.87 | 1.87 | 1.91 | 1.93 | 1.94 | 1.92 | 1.91 | 1.92 | 2.03 | 1.87 | 1.96 | 2.19 | 1.92 | 1.88 | 2.07 | 2.30
NE | 0.98 | 1.05 | 1.09 | 1.02 | 0.98 | 0.98 | 0.99 | 1.00 | 0.99 | 0.98 | 0.96 | 1.13 | 1.02 | 1.17 | 2.31 | 1.07 | 1.03 | 1.82 | 1.45
AM | 1.05 | 1.09 | 0.87 | 0.82 | 0.98 | 1.05 | 1.19 | 2.90 | 1.26 | 1.09 | 2.08 | 1.47 | 0.82 | 0.86 | 1.47 | 0.77 | 0.82 | 1.06 | 1.13
BAT | 2.28 | 3.38 | 1.38 | 1.38 | 1.19 | 2.28 | 1.66 | 2.04 | 2.16 | 2.16 | 2.14 | 2.58 | 1.38 | 1.19 | 1.25 | 1.40 | 1.39 | 1.33 | 1.49
KAL | 1.33 | 1.44 | 0.84 | 0.81 | 0.73 | 1.33 | 1.00 | 1.06 | 1.15 | 1.04 | 1.18 | 2.07 | 0.81 | 0.71 | 0.67 | 0.73 | 0.81 | 0.74 | 1.02
TRC | 2.13 | 2.10 | 2.25 | 2.25 | 1.81 | 2.13 | 1.88 | 1.89 | 1.95 | 1.96 | 1.96 | 2.12 | 2.25 | 2.04 | 2.08 | 2.22 | 2.25 | 2.17 | 1.83
APU | 0.71 | 1.13 | 0.72 | 0.72 | 0.75 | 0.71 | 0.77 | 0.77 | 0.68 | 0.70 | 0.71 | 1.04 | 0.72 | 0.63 | 0.53 | 0.66 | 0.67 | 0.59 | 1.33
POL | 0.82 | 0.78 | 0.66 | 0.74 | 0.75 | 0.82 | 1.03 | 1.36 | 0.95 | 0.87 | 1.05 | 0.56 | 0.74 | 0.45 | 0.50 | 0.63 | 0.64 | 0.42 | 1.30
KUR | 0.50 | 0.67 | 0.48 | 0.59 | 0.48 | 0.50 | 0.53 | 0.54 | 0.57 | 0.51 | 0.51 | 0.45 | 0.59 | 0.65 | 0.65 | 0.62 | 0.64 | 0.64 | 1.16
PUT | 1.05 | 1.10 | 0.92 | 0.94 | 0.85 | 1.05 | 0.92 | 0.98 | 0.96 | 1.02 | 0.99 | 1.00 | 0.94 | 0.86 | 0.88 | 0.96 | 0.95 | 0.94 | 1.21
JAF | 4.44 | 3.65 | 6.90 | 6.90 | 4.75 | 4.44 | 5.60 | 4.68 | 4.86 | 4.88 | 4.59 | 5.14 | 6.90 | 4.97 | 6.52 | 6.17 | 6.25 | 6.68 | 5.65
KIL | 3.94 | 4.32 | 3.95 | 3.94 | 4.01 | 3.94 | 5.89 | 9.65 | 3.98 | 4.02 | 7.46 | 4.34 | 3.94 | 3.25 | 9.09 | 3.92 | 3.88 | 5.60 | 4.63
MAN | 0.73 | 1.13 | 0.84 | 0.76 | 0.73 | 0.73 | 0.96 | 2.40 | 0.87 | 0.86 | 1.68 | 0.76 | 0.76 | 0.62 | 2.32 | 0.63 | 0.63 | 1.41 | 0.76
MUL | 1.91 | 2.04 | 1.65 | 1.65 | 1.65 | 1.91 | 2.81 | 8.30 | 1.92 | 1.96 | 5.30 | 2.53 | 1.65 | 1.55 | 8.50 | 1.63 | 1.63 | 5.32 | 1.77
VAV | 1.15 | 1.49 | 1.29 | 1.29 | 1.20 | 1.15 | 1.35 | 1.83 | 1.30 | 1.36 | 1.49 | 1.87 | 1.29 | 1.10 | 1.60 | 1.09 | 1.07 | 1.15 | 2.35
KEG | 0.54 | 0.58 | 0.56 | 0.38 | 0.55 | 0.54 | 0.53 | 0.53 | 0.51 | 0.53 | 0.52 | 0.95 | 0.38 | 0.52 | 0.53 | 0.46 | 0.45 | 0.49 | 1.18
RAT | 0.78 | 1.04 | 0.95 | 1.36 | 1.00 | 0.78 | 0.75 | 0.76 | 0.77 | 0.76 | 0.76 | 1.28 | 1.36 | 1.08 | 1.33 | 1.34 | 1.33 | 1.34 | 1.60
GAL | 4.15 | 4.58 | 4.34 | 4.46 | 4.27 | 4.15 | 4.48 | 4.47 | 4.36 | 4.24 | 4.34 | 4.21 | 4.46 | 4.57 | 4.67 | 4.48 | 4.48 | 4.55 | 4.56
HAM | 1.16 | 1.46 | 1.10 | 1.07 | 1.55 | 1.16 | 1.48 | 1.52 | 1.22 | 1.18 | 1.35 | 1.48 | 1.07 | 0.92 | 0.98 | 1.09 | 1.05 | 0.98 | 1.65
MAT | 1.62 | 1.93 | 1.74 | 1.87 | 1.90 | 1.62 | 1.79 | 1.70 | 1.69 | 1.68 | 1.67 | 1.79 | 1.87 | 1.93 | 1.98 | 1.81 | 1.86 | 1.89 | 2.09
BAD | 1.27 | 1.50 | 1.42 | 1.46 | 1.45 | 1.27 | 1.41 | 1.34 | 1.33 | 1.32 | 1.31 | 1.46 | 1.46 | 1.52 | 1.72 | 1.41 | 1.47 | 1.50 | 2.06
MON | 0.45 | 0.80 | 0.51 | 0.46 | 0.44 | 0.45 | 0.36 | 0.30 | 0.41 | 0.43 | 0.37 | 1.05 | 0.46 | 0.42 | 0.45 | 0.41 | 0.45 | 0.41 | 0.54
COL | 1.36 | 1.42 | 1.37 | 1.41 | 1.60 | 1.36 | 1.37 | 1.36 | 1.32 | 1.36 | 1.36 | 2.06 | 1.41 | 1.61 | 1.50 | 1.47 | 1.44 | 1.47 | 1.81
GAM | 1.22 | 1.24 | 1.22 | 1.23 | 1.37 | 1.22 | 1.23 | 1.22 | 1.15 | 1.22 | 1.22 | 2.15 | 1.23 | 1.22 | 1.23 | 1.26 | 1.23 | 1.23 | 1.80
KLU | 2.09 | 2.28 | 1.93 | 2.06 | 2.40 | 2.09 | 2.05 | 1.99 | 2.12 | 2.08 | 2.00 | 2.21 | 2.06 | 2.04 | 1.94 | 2.00 | 2.01 | 1.95 | 2.41

Table 12: MASE from training set: 2006-2018 for quarterly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 1.46 | 1.52 | 1.46 | 1.43 | 1.73 | 1.46 | 1.46 | 1.46 | 1.41 | 1.45 | 1.46 | 1.55 | 1.43 | 1.43 | 1.41 | 1.50 | 1.46
Province
CN | 1.93 | 1.90 | 1.93 | 1.85 | 1.93 | 2.26 | 1.94 | 1.96 | 2.06 | 2.07 | 2.02 | 1.85 | 1.87 | 1.85 | 1.80 | 1.85 | 1.84
ET | 1.97 | 2.72 | 1.42 | 1.72 | 1.42 | 1.88 | 1.92 | 1.93 | 1.94 | 1.91 | 1.93 | 2.40 | 1.69 | 1.75 | 2.24 | 2.18 | 1.96
NC | 0.79 | 0.90 | 0.80 | 0.68 | 0.78 | 0.71 | 0.82 | 0.88 | 0.70 | 0.74 | 0.76 | 0.66 | 0.71 | 0.84 | 0.75 | 0.67 | 0.63
NW | 0.58 | 0.84 | 0.68 | 0.65 | 0.68 | 0.64 | 0.59 | 0.61 | 0.60 | 0.61 | 0.61 | 0.75 | 0.64 | 0.63 | 0.68 | 0.71 | 0.67
NT | 4.21 | 3.38 | 4.44 | 3.07 | 4.15 | 4.01 | 4.14 | 4.14 | 4.12 | 4.10 | 4.11 | 3.83 | 3.09 | 3.02 | 3.65 | 3.66 | 3.26
SB | 0.76 | 0.72 | 0.83 | 0.79 | 0.76 | 0.62 | 0.80 | 0.68 | 0.57 | 0.64 | 0.65 | 0.70 | 0.82 | 0.76 | 0.62 | 0.67 | 0.70
ST | 2.73 | 3.00 | 3.00 | 2.82 | 2.96 | 2.90 | 2.79 | 2.83 | 2.82 | 2.85 | 2.82 | 2.64 | 2.90 | 2.95 | 2.77 | 2.70 | 2.81
UVA | 0.60 | 0.79 | 0.74 | 0.59 | 0.69 | 0.58 | 0.63 | 0.66 | 0.55 | 0.58 | 0.59 | 0.76 | 0.64 | 0.94 | 0.80 | 0.69 | 0.76
WT | 1.54 | 1.53 | 1.54 | 1.46 | 1.78 | 1.58 | 1.59 | 1.56 | 1.53 | 1.57 | 1.56 | 1.64 | 1.49 | 1.50 | 1.53 | 1.58 | 1.54
Districts
KN | 2.50 | 1.84 | 1.93 | 1.86 | 1.99 | 2.50 | 2.22 | 2.36 | 2.26 | 2.26 | 2.40 | 1.86 | 1.89 | 1.86 | 1.78 | 1.86 | 1.86
MTL | 1.99 | 2.28 | 2.07 | 1.93 | 2.00 | 1.99 | 2.36 | 2.22 | 1.95 | 1.99 | 2.13 | 1.93 | 1.97 | 1.99 | 2.05 | 1.93 | 1.94
NE | 1.03 | 1.02 | 1.18 | 0.93 | 1.09 | 1.03 | 1.82 | 2.09 | 0.98 | 1.03 | 1.76 | 0.93 | 0.96 | 1.61 | 0.98 | 0.93 | 1.18
AM | 1.06 | 0.76 | 0.85 | 0.85 | 1.03 | 1.06 | 0.95 | 0.58 | 0.98 | 1.05 | 0.55 | 0.85 | 1.06 | 3.56 | 0.93 | 0.85 | 2.53
BAT | 1.94 | 3.80 | 1.39 | 3.13 | 1.33 | 1.94 | 2.01 | 1.98 | 2.01 | 1.97 | 1.97 | 3.13 | 1.93 | 2.72 | 3.04 | 2.97 | 2.86
KAL | 1.26 | 1.58 | 0.77 | 1.28 | 0.77 | 1.26 | 1.29 | 1.30 | 1.33 | 1.30 | 1.30 | 1.28 | 0.91 | 0.81 | 0.99 | 1.05 | 0.97
TRC | 2.07 | 2.04 | 1.88 | 2.19 | 1.96 | 2.07 | 2.10 | 2.12 | 2.10 | 2.10 | 2.11 | 2.19 | 1.92 | 2.03 | 2.12 | 1.96 | 1.89
APU | 0.80 | 1.08 | 0.79 | 0.69 | 0.79 | 0.80 | 0.89 | 0.93 | 0.79 | 0.84 | 0.84 | 0.69 | 0.69 | 0.70 | 0.84 | 0.69 | 0.61
POL | 0.48 | 0.72 | 0.79 | 0.65 | 0.72 | 0.48 | 0.62 | 0.72 | 0.49 | 0.49 | 0.55 | 0.65 | 0.69 | 1.13 | 0.60 | 0.65 | 0.70
KUR | 0.58 | 0.64 | 0.52 | 0.55 | 0.52 | 0.58 | 0.63 | 0.62 | 0.53 | 0.58 | 0.58 | 0.55 | 0.51 | 0.51 | 0.54 | 0.52 | 0.51
PUT | 1.09 | 1.13 | 1.01 | 1.10 | 0.91 | 1.09 | 1.01 | 1.05 | 1.07 | 1.07 | 1.05 | 1.10 | 0.95 | 0.94 | 1.01 | 1.06 | 1.00
JAF | 5.56 | 4.29 | 6.21 | 5.18 | 5.75 | 5.56 | 5.73 | 5.60 | 5.69 | 5.67 | 5.59 | 5.18 | 4.20 | 4.85 | 4.89 | 4.92 | 4.96
KIL | 4.47 | 5.42 | 4.19 | 4.49 | 4.91 | 4.47 | 4.68 | 5.30 | 4.46 | 4.48 | 5.10 | 4.49 | 3.17 | 6.58 | 4.40 | 4.47 | 4.77
MAN | 0.82 | 1.21 | 0.75 | 0.75 | 0.75 | 0.82 | 0.83 | 0.86 | 0.83 | 0.82 | 0.82 | 0.75 | 0.58 | 1.39 | 0.68 | 0.71 | 0.98
MUL | 2.04 | 2.28 | 1.88 | 1.88 | 1.88 | 2.04 | 2.12 | 2.72 | 2.03 | 2.05 | 2.56 | 1.88 | 1.47 | 4.97 | 2.04 | 1.86 | 3.48
VAV | 1.31 | 1.57 | 1.30 | 1.44 | 1.30 | 1.31 | 1.31 | 1.30 | 1.31 | 1.30 | 1.30 | 1.44 | 1.06 | 1.06 | 1.36 | 1.31 | 0.96
KEG | 0.63 | 0.57 | 0.59 | 0.63 | 0.59 | 0.63 | 0.77 | 0.79 | 0.58 | 0.66 | 0.71 | 0.63 | 0.69 | 0.72 | 0.65 | 0.67 | 0.69
RAT | 0.82 | 0.89 | 1.20 | 1.13 | 1.12 | 0.82 | 1.00 | 0.88 | 0.79 | 0.84 | 0.85 | 1.13 | 0.96 | 0.94 | 0.83 | 1.03 | 0.98
GAL | 4.67 | 5.08 | 5.08 | 4.79 | 4.75 | 4.67 | 4.55 | 4.61 | 4.64 | 4.63 | 4.60 | 4.79 | 4.96 | 5.08 | 4.88 | 4.84 | 4.96
HAM | 1.52 | 1.58 | 1.29 | 1.67 | 1.68 | 1.52 | 1.40 | 1.42 | 1.46 | 1.50 | 1.40 | 1.67 | 1.36 | 1.57 | 1.46 | 1.64 | 1.55
MAT | 2.05 | 2.06 | 2.09 | 1.91 | 2.03 | 2.05 | 1.96 | 2.00 | 1.92 | 1.98 | 1.99 | 1.91 | 2.08 | 2.14 | 2.04 | 1.99 | 2.04
BAD | 1.37 | 1.58 | 1.51 | 1.55 | 1.49 | 1.37 | 1.50 | 1.45 | 1.35 | 1.41 | 1.41 | 1.55 | 1.43 | 1.70 | 1.58 | 1.47 | 1.55
MON | 0.50 | 0.80 | 0.28 | 0.28 | 0.31 | 0.50 | 0.45 | 0.40 | 0.50 | 0.49 | 0.45 | 0.28 | 0.38 | 0.29 | 0.42 | 0.28 | 0.31
COL | 1.51 | 1.51 | 1.51 | 1.61 | 1.73 | 1.51 | 1.57 | 1.53 | 1.48 | 1.52 | 1.52 | 1.61 | 1.44 | 1.51 | 1.48 | 1.56 | 1.54
GAM | 1.50 | 1.28 | 1.32 | 1.48 | 1.45 | 1.50 | 1.49 | 1.46 | 1.39 | 1.48 | 1.47 | 1.48 | 1.34 | 1.37 | 1.39 | 1.41 | 1.40
KLU | 2.18 | 2.51 | 2.36 | 2.28 | 2.64 | 2.18 | 2.20 | 2.14 | 2.14 | 2.18 | 2.15 | 2.28 | 2.45 | 2.65 | 2.29 | 2.31 | 2.52

Table 13: MASE from training set: 2006-2018 for yearly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 2.15 | 1.55 | 1.24 | 2.15 | 1.31 | 2.15 | 2.12 | 1.51 | 1.55 | 1.84 | 1.48 | 1.24 | 1.27 | 1.44 | 1.44 | 1.38
Province
CN | 2.32 | 1.89 | 2.37 | 2.32 | 2.32 | 2.38 | 2.53 | 2.29 | 2.34 | 2.46 | 2.30 | 2.29 | 2.21 | 2.32 | 2.32 | 2.27
ET | 1.68 | 0.73 | 0.77 | 1.68 | 0.41 | 1.75 | 1.78 | 0.65 | 0.71 | 1.34 | 0.85 | 1.07 | 1.02 | 0.84 | 0.84 | 0.94
NC | 0.83 | 0.36 | 0.42 | 0.83 | 0.83 | 0.89 | 1.67 | 0.79 | 0.83 | 1.25 | 0.39 | 0.30 | 0.15 | 0.40 | 0.40 | 0.22
NW | 0.51 | 0.17 | 0.41 | 0.51 | 0.51 | 0.57 | 0.70 | 0.49 | 0.53 | 0.61 | 0.01 | 0.33 | 0.15 | 0.11 | 0.11 | 0.16
NT | 3.79 | 3.03 | 3.64 | 5.45 | 3.37 | 3.99 | 4.33 | 3.56 | 3.52 | 4.21 | 3.13 | 3.37 | 3.15 | 3.34 | 3.34 | 3.12
SB | 0.87 | 0.73 | 0.86 | 0.87 | 0.87 | 0.92 | 1.05 | 0.85 | 0.88 | 0.96 | 0.86 | 0.78 | 0.73 | 0.86 | 0.85 | 0.82
ST | 3.52 | 3.51 | 2.63 | 3.52 | 3.52 | 3.58 | 3.79 | 3.49 | 3.53 | 3.71 | 3.34 | 2.48 | 2.63 | 3.12 | 3.12 | 2.91
UVA | 0.79 | 0.56 | 0.47 | 0.79 | 0.79 | 0.85 | 1.30 | 0.76 | 0.80 | 1.05 | 0.07 | 0.37 | 0.00 | 0.16 | 0.16 | 0.16
WT | 2.20 | 1.82 | 1.35 | 2.20 | 0.65 | 2.27 | 1.87 | 1.10 | 1.13 | 1.47 | 1.63 | 1.18 | 1.38 | 1.52 | 1.51 | 1.47
Districts
KN | 2.42 | 1.92 | 2.43 | 2.42 | 2.42 | 2.48 | 2.52 | 2.40 | 2.45 | 2.49 | 2.43 | 2.42 | 2.39 | 2.45 | 2.45 | 2.41
MTL | 2.10 | 1.81 | 2.10 | 2.10 | 2.10 | 2.16 | 2.44 | 2.08 | 2.11 | 2.33 | 2.10 | 2.09 | 1.96 | 2.11 | 2.10 | 2.06
NE | 1.12 | 1.30 | 0.68 | 1.12 | 1.12 | 1.20 | 2.60 | 1.08 | 1.12 | 2.13 | 0.68 | 0.66 | 0.04 | 0.69 | 0.68 | 0.49
AM | 1.20 | 0.61 | 1.21 | 1.20 | 1.20 | 1.65 | 5.56 | 1.20 | 1.24 | 4.18 | 1.21 | 1.16 | 0.67 | 1.21 | 1.21 | 0.93
BAT | 2.19 | 3.32 | 3.19 | 1.75 | 2.19 | 0.85 | 1.13 | 1.84 | 1.83 | 1.46 | 3.19 | 3.57 | 3.32 | 3.18 | 3.18 | 3.26
KAL | 0.76 | 0.75 | 1.01 | 0.76 | 0.76 | 1.28 | 1.81 | 0.97 | 1.07 | 1.48 | 1.01 | 1.16 | 1.14 | 0.99 | 0.99 | 1.07
TRC | 2.23 | 1.55 | 1.20 | 2.23 | 2.23 | 2.69 | 3.22 | 2.41 | 2.47 | 2.90 | 1.20 | 1.09 | 1.08 | 1.21 | 1.21 | 1.14
APU | 0.82 | 0.17 | 0.64 | 0.82 | 0.82 | 0.88 | 1.45 | 0.78 | 0.82 | 1.14 | 0.64 | 0.56 | 0.23 | 0.65 | 0.65 | 0.51
POL | 0.82 | 0.70 | 0.09 | 0.82 | 0.82 | 0.88 | 2.01 | 0.78 | 0.82 | 1.42 | 0.09 | 0.21 | 0.86 | 0.08 | 0.08 | 0.33
KUR | 0.38 | 0.22 | 0.66 | 0.38 | 0.38 | 0.43 | 0.53 | 0.36 | 0.40 | 0.45 | 0.66 | 0.20 | 0.53 | 0.55 | 0.55 | 0.52
PUT | 0.75 | 0.09 | 1.09 | 0.75 | 0.75 | 0.80 | 1.00 | 0.71 | 0.76 | 0.87 | 1.09 | 1.22 | 1.30 | 1.24 | 1.23 | 1.32
JAF | 4.28 | 4.28 | 4.10 | 7.12 | 4.28 | 5.03 | 4.56 | 4.53 | 4.48 | 4.52 | 4.10 | 4.38 | 4.10 | 4.38 | 4.38 | 4.10
KIL | 2.01 | 0.51 | 0.48 | 6.67 | 2.01 | 3.09 | 7.95 | 2.04 | 2.04 | 7.15 | 0.48 | 0.97 | 0.61 | 0.52 | 0.52 | 0.43
MAN | 0.34 | 0.02 | 0.34 | 0.34 | 0.34 | 0.49 | 1.70 | 0.34 | 0.36 | 1.52 | 0.34 | 0.40 | 0.37 | 0.37 | 0.37 | 0.33
MUL | 1.56 | 0.84 | 0.43 | 1.54 | 1.56 | 1.79 | 6.52 | 1.63 | 1.58 | 5.85 | 0.43 | 0.23 | 0.33 | 0.41 | 0.41 | 0.48
VAV | 1.42 | 0.40 | 1.42 | 1.42 | 1.42 | 1.59 | 2.29 | 1.44 | 1.47 | 2.17 | 1.42 | 1.48 | 1.44 | 1.50 | 1.49 | 1.41
KEG | 0.48 | 0.54 | 0.73 | 0.48 | 0.48 | 0.53 | 0.64 | 0.47 | 0.50 | 0.56 | 0.73 | 0.68 | 0.62 | 0.72 | 0.72 | 0.69
RAT | 1.31 | 0.92 | 0.98 | 1.31 | 1.31 | 1.37 | 1.52 | 1.28 | 1.32 | 1.41 | 0.98 | 0.86 | 0.83 | 0.97 | 0.97 | 0.93
GAL | 5.78 | 6.10 | 5.39 | 5.78 | 5.78 | 5.84 | 6.04 | 5.75 | 5.79 | 5.96 | 5.39 | 4.32 | 4.69 | 5.17 | 5.17 | 4.96
HAM | 2.09 | 1.59 | 2.11 | 2.09 | 2.09 | 2.15 | 2.48 | 2.05 | 2.09 | 2.35 | 2.11 | 1.37 | 1.08 | 2.01 | 2.01 | 1.48
MAT | 2.34 | 2.32 | 2.22 | 2.34 | 2.34 | 2.39 | 2.54 | 2.32 | 2.35 | 2.47 | 2.22 | 1.50 | 1.71 | 1.95 | 1.95 | 1.91
BAD | 1.69 | 1.77 | 0.95 | 1.69 | 1.69 | 1.75 | 2.15 | 1.67 | 1.70 | 1.93 | 0.95 | 1.26 | 0.89 | 1.04 | 1.04 | 1.04
MON | 0.29 | 0.91 | 1.01 | 0.29 | 0.29 | 0.24 | 0.27 | 0.33 | 0.29 | 0.01 | 1.01 | 0.71 | 1.08 | 0.92 | 0.92 | 0.91
COL | 1.12 | 1.78 | 1.28 | 2.13 | 1.12 | 1.61 | 0.27 | 0.62 | 0.65 | 0.55 | 1.28 | 0.71 | 1.11 | 1.19 | 1.19 | 1.17
GAM | 1.78 | 1.53 | 1.74 | 1.78 | 1.78 | 2.31 | 2.74 | 2.24 | 2.30 | 2.42 | 1.74 | 1.50 | 1.54 | 1.59 | 1.59 | 1.61
KLU | 3.14 | 2.38 | 2.08 | 3.14 | 3.14 | 3.84 | 6.12 | 3.27 | 3.34 | 5.13 | 2.08 | 1.52 | 1.47 | 2.03 | 2.03 | 1.70

Table 14: MASE for training set: 2016-2018 for weekly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 0.34 | 0.39 | 0.37 | 0.32 | 0.37 | 0.36 | 0.34 | 0.34 | 0.35 | 0.35 | 0.35 | 0.64 | 0.32 | 0.32 | 0.32 | 0.30 | 0.32 | 0.32 | 0.46
Province
CN | 0.53 | 0.57 | 0.60 | 0.57 | 0.59 | 0.53 | 0.53 | 0.53 | 0.53 | 0.53 | 0.53 | 0.76 | 0.51 | 0.58 | 0.57 | 0.50 | 0.53 | 0.53 | 1.00
ET | 0.38 | 0.64 | 0.38 | 0.60 | 0.54 | 0.39 | 0.39 | 0.40 | 0.37 | 0.39 | 0.40 | 0.72 | 0.38 | 0.55 | 0.51 | 0.44 | 0.46 | 0.47 | 0.38
NC | 0.31 | 0.33 | 0.25 | 0.25 | 0.34 | 0.30 | 0.32 | 0.35 | 0.30 | 0.30 | 0.32 | 0.59 | 0.25 | 0.25 | 0.28 | 0.24 | 0.25 | 0.25 | 0.16
NW | 0.18 | 0.24 | 0.18 | 0.18 | 0.31 | 0.20 | 0.18 | 0.18 | 0.18 | 0.18 | 0.18 | 0.37 | 0.19 | 0.18 | 0.17 | 0.18 | 0.18 | 0.18 | 0.24
NT | 2.56 | 1.44 | 2.58 | 3.14 | 1.78 | 1.75 | 2.64 | 2.47 | 2.07 | 2.09 | 2.18 | 1.79 | 2.56 | 2.95 | 2.96 | 2.78 | 2.79 | 2.81 | 2.61
SB | 0.39 | 0.25 | 0.25 | 0.14 | 0.22 | 0.22 | 0.39 | 0.29 | 0.27 | 0.27 | 0.27 | 1.14 | 0.24 | 0.16 | 0.19 | 0.12 | 0.19 | 0.19 | 0.16
ST | 0.59 | 0.80 | 0.74 | 0.50 | 0.58 | 0.68 | 0.56 | 0.58 | 0.66 | 0.64 | 0.62 | 1.36 | 0.72 | 0.54 | 0.58 | 0.59 | 0.64 | 0.61 | 0.80
UVA | 0.26 | 0.24 | 0.21 | 0.20 | 0.31 | 0.22 | 0.27 | 0.27 | 0.23 | 0.23 | 0.24 | 0.63 | 0.19 | 0.20 | 0.19 | 0.20 | 0.19 | 0.19 | 0.41
WT | 0.37 | 0.41 | 0.38 | 0.37 | 0.38 | 0.37 | 0.35 | 0.36 | 0.36 | 0.36 | 0.37 | 0.40 | 0.33 | 0.38 | 0.36 | 0.32 | 0.34 | 0.35 | 0.70
Districts
KN | 0.49 | 0.53 | 0.56 | 0.46 | 0.54 | 0.49 | 0.49 | 0.49 | 0.50 | 0.50 | 0.49 | 0.60 | 0.46 | 0.53 | 0.48 | 0.45 | 0.48 | 0.47 | 1.00
MTL | 0.82 | 0.90 | 0.83 | 0.81 | 0.91 | 0.82 | 0.84 | 0.86 | 0.82 | 0.82 | 0.84 | 1.38 | 0.81 | 0.85 | 1.03 | 0.78 | 0.82 | 0.91 | 0.89
NE | 0.44 | 0.51 | 0.53 | 0.50 | 0.47 | 0.44 | 0.46 | 0.55 | 0.44 | 0.44 | 0.50 | 1.36 | 0.50 | 0.56 | 1.41 | 0.44 | 0.51 | 0.95 | 1.17
AM | 0.24 | 0.39 | 0.49 | 0.32 | 0.25 | 0.24 | 0.24 | 0.27 | 0.24 | 0.24 | 0.25 | 0.60 | 0.32 | 0.23 | 1.05 | 0.26 | 0.28 | 0.66 | 1.06
BAT | 0.46 | 1.11 | 0.46 | 0.46 | 0.61 | 0.46 | 0.46 | 0.47 | 0.44 | 0.46 | 0.47 | 0.92 | 0.46 | 0.73 | 0.56 | 0.51 | 0.51 | 0.52 | 0.48
KAL | 0.34 | 0.34 | 0.21 | 0.21 | 0.37 | 0.34 | 0.33 | 0.34 | 0.33 | 0.33 | 0.34 | 0.43 | 0.21 | 0.32 | 0.31 | 0.27 | 0.32 | 0.27 | 0.38
TRC | 0.50 | 0.50 | 0.47 | 0.47 | 0.55 | 0.50 | 0.50 | 0.49 | 0.56 | 0.51 | 0.49 | 1.44 | 0.47 | 0.57 | 0.54 | 0.52 | 0.52 | 0.52 | 0.61
APU | 0.34 | 0.37 | 0.29 | 0.27 | 0.37 | 0.34 | 0.35 | 0.37 | 0.34 | 0.34 | 0.35 | 0.34 | 0.27 | 0.27 | 0.28 | 0.26 | 0.27 | 0.27 | 0.36
POL | 0.26 | 0.30 | 0.23 | 0.23 | 0.29 | 0.26 | 0.27 | 0.34 | 0.25 | 0.26 | 0.28 | 1.06 | 0.23 | 0.23 | 0.31 | 0.23 | 0.23 | 0.23 | 0.60
KUR | 0.21 | 0.21 | 0.16 | 0.32 | 0.29 | 0.21 | 0.16 | 0.18 | 0.17 | 0.17 | 0.18 | 0.23 | 0.32 | 0.16 | 0.20 | 0.25 | 0.22 | 0.24 | 0.81
PUT | 0.22 | 0.28 | 0.22 | 0.22 | 0.33 | 0.22 | 0.22 | 0.22 | 0.22 | 0.22 | 0.23 | 0.54 | 0.22 | 0.24 | 0.31 | 0.23 | 0.24 | 0.27 | 0.51
JAF | 1.90 | 1.52 | 3.00 | 2.98 | 1.97 | 1.90 | 2.75 | 2.07 | 2.25 | 2.27 | 2.00 | 1.99 | 2.98 | 3.44 | 3.07 | 3.22 | 3.24 | 3.04 | 2.94
KIL | 1.24 | 1.15 | 1.68 | 1.23 | 0.93 | 1.24 | 2.41 | 4.21 | 1.36 | 1.38 | 2.88 | 0.94 | 1.23 | 1.44 | 3.22 | 1.36 | 1.31 | 2.31 | 1.55
MAN | 0.52 | 0.64 | 0.44 | 0.47 | 0.52 | 0.52 | 0.99 | 2.46 | 0.74 | 0.70 | 1.66 | 1.77 | 0.47 | 0.54 | 1.78 | 0.58 | 0.60 | 1.25 | 0.52
MUL | 0.66 | 0.65 | 0.49 | 0.52 | 0.69 | 0.66 | 1.15 | 4.39 | 0.72 | 0.75 | 2.77 | 1.57 | 0.52 | 0.54 | 2.95 | 0.56 | 0.55 | 1.87 | 0.54
VAV | 1.00 | 1.04 | 0.95 | 0.99 | 0.96 | 1.00 | 1.77 | 2.26 | 1.09 | 1.11 | 1.75 | 0.88 | 0.99 | 1.10 | 1.69 | 1.06 | 1.04 | 1.42 | 1.44
KEG | 0.24 | 0.16 | 0.16 | 0.16 | 0.22 | 0.24 | 0.28 | 0.34 | 0.28 | 0.28 | 0.32 | 0.63 | 0.16 | 0.11 | 0.13 | 0.11 | 0.14 | 0.13 | 0.09
RAT | 0.23 | 0.38 | 0.42 | 0.38 | 0.25 | 0.23 | 0.54 | 0.28 | 0.29 | 0.27 | 0.27 | 1.85 | 0.38 | 0.27 | 0.31 | 0.17 | 0.29 | 0.31 | 0.43
GAL | 1.28 | 1.26 | 1.23 | 1.21 | 0.98 | 1.28 | 1.21 | 1.16 | 1.26 | 1.22 | 1.20 | 2.17 | 1.21 | 1.07 | 1.08 | 1.12 | 1.13 | 1.10 | 1.45
HAM | 0.38 | 0.46 | 0.42 | 0.38 | 0.33 | 0.38 | 0.31 | 0.30 | 0.38 | 0.37 | 0.33 | 1.18 | 0.38 | 0.26 | 0.27 | 0.33 | 0.36 | 0.28 | 0.53
MAT | 0.44 | 0.57 | 0.50 | 0.50 | 0.44 | 0.44 | 0.36 | 0.40 | 0.42 | 0.42 | 0.41 | 0.70 | 0.50 | 0.37 | 0.42 | 0.36 | 0.43 | 0.44 | 0.40
BAD | 0.45 | 0.48 | 0.43 | 0.40 | 0.46 | 0.45 | 0.51 | 0.47 | 0.45 | 0.45 | 0.45 | 0.81 | 0.40 | 0.36 | 0.39 | 0.37 | 0.37 | 0.38 | 0.70
MON | 0.11 | 0.21 | 0.11 | 0.12 | 0.41 | 0.11 | 0.17 | 0.25 | 0.12 | 0.13 | 0.18 | 0.40 | 0.12 | 0.17 | 0.14 | 0.16 | 0.14 | 0.18 | 0.10
COL | 0.41 | 0.46 | 0.42 | 0.38 | 0.42 | 0.41 | 0.40 | 0.41 | 0.41 | 0.41 | 0.41 | 0.52 | 0.38 | 0.41 | 0.39 | 0.38 | 0.39 | 0.39 | 0.66
GAM | 0.29 | 0.33 | 0.31 | 0.37 | 0.31 | 0.29 | 0.28 | 0.29 | 0.29 | 0.29 | 0.29 | 0.28 | 0.37 | 0.38 | 0.39 | 0.35 | 0.38 | 0.38 | 0.95
KLU | 0.49 | 0.60 | 0.49 | 0.49 | 0.49 | 0.49 | 0.47 | 0.48 | 0.48 | 0.49 | 0.48 | 0.50 | 0.49 | 0.54 | 0.57 | 0.44 | 0.49 | 0.54 | 0.34

Table 15: MASE from training set: 2016-2018 for monthly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | cov | ebup | etop | eols | emit | evar | estc | ecov
SL | 0.36 | 0.40 | 0.37 | 0.37 | 0.37 | 0.38 | 0.36 | 0.36 | 0.36 | 0.37 | 0.37 | 0.35 | 0.31 | 0.37 | 0.37 | 0.41 | 0.33 | 0.34 | 0.45
Province
CN | 0.56 | 0.58 | 0.58 | 0.56 | 0.60 | 0.56 | 0.56 | 0.56 | 0.55 | 0.56 | 0.56 | 0.54 | 0.53 | 0.59 | 0.61 | 0.64 | 0.54 | 0.57 | 0.67
ET | 0.52 | 0.67 | 0.42 | 0.59 | 0.54 | 0.45 | 0.52 | 0.52 | 0.49 | 0.49 | 0.50 | 0.52 | 0.54 | 0.50 | 0.49 | 0.56 | 0.56 | 0.52 | 0.66
NC | 0.29 | 0.32 | 0.23 | 0.23 | 0.32 | 0.29 | 0.29 | 0.32 | 0.30 | 0.29 | 0.30 | 0.20 | 0.29 | 0.25 | 0.53 | 0.36 | 0.26 | 0.31 | 0.38
NW | 0.23 | 0.24 | 0.18 | 0.19 | 0.29 | 0.19 | 0.23 | 0.22 | 0.20 | 0.20 | 0.21 | 0.17 | 0.19 | 0.17 | 0.17 | 0.24 | 0.19 | 0.18 | 0.59
NT | 1.59 | 1.37 | 2.33 | 2.22 | 1.68 | 1.74 | 1.59 | 1.64 | 1.67 | 1.67 | 1.68 | 1.80 | 2.32 | 2.00 | 2.04 | 2.27 | 2.28 | 2.15 | 3.19
SB | 0.18 | 0.24 | 0.22 | 0.42 | 0.21 | 0.22 | 0.18 | 0.18 | 0.17 | 0.19 | 0.18 | 0.18 | 0.25 | 0.31 | 0.33 | 0.40 | 0.37 | 0.33 | 0.57
ST | 0.59 | 0.81 | 0.73 | 0.65 | 0.59 | 0.64 | 0.60 | 0.59 | 0.61 | 0.62 | 0.61 | 0.69 | 0.56 | 0.72 | 0.73 | 0.62 | 0.61 | 0.65 | 0.59
UVA | 0.26 | 0.24 | 0.20 | 0.25 | 0.29 | 0.21 | 0.26 | 0.25 | 0.22 | 0.22 | 0.23 | 0.18 | 0.18 | 0.21 | 0.28 | 0.25 | 0.21 | 0.20 | 0.39
WT | 0.38 | 0.41 | 0.39 | 0.39 | 0.39 | 0.38 | 0.38 | 0.38 | 0.36 | 0.38 | 0.38 | 0.37 | 0.38 | 0.43 | 0.40 | 0.48 | 0.39 | 0.39 | 0.44
Districts
KN | 0.52 | 0.54 | 0.54 | 0.51 | 0.55 | 0.52 | 0.52 | 0.52 | 0.51 | 0.52 | 0.52 | 0.51 | 0.51 | 0.56 | 0.54 | 0.62 | 0.53 | 0.52 | 0.62
MTL | 0.87 | 0.93 | 0.77 | 0.78 | 0.92 | 0.87 | 0.87 | 0.88 | 0.87 | 0.87 | 0.88 | 0.76 | 0.78 | 0.76 | 0.86 | 0.86 | 0.78 | 0.77 | 1.04
NE | 0.42 | 0.44 | 0.46 | 0.49 | 0.46 | 0.42 | 0.42 | 0.45 | 0.41 | 0.42 | 0.42 | 0.37 | 0.49 | 0.43 | 1.49 | 0.52 | 0.49 | 0.83 | 0.83
AM | 0.17 | 0.33 | 0.26 | 0.25 | 0.20 | 0.17 | 0.21 | 0.61 | 0.18 | 0.17 | 0.40 | 0.16 | 0.25 | 0.26 | 0.40 | 0.28 | 0.24 | 0.28 | 0.80
BAT | 0.65 | 1.19 | 0.49 | 0.57 | 0.63 | 0.65 | 0.76 | 0.70 | 0.67 | 0.67 | 0.68 | 0.70 | 0.57 | 0.52 | 0.53 | 0.60 | 0.59 | 0.56 | 0.93
KAL | 0.35 | 0.35 | 0.20 | 0.27 | 0.36 | 0.35 | 0.43 | 0.40 | 0.39 | 0.40 | 0.38 | 0.38 | 0.27 | 0.24 | 0.24 | 0.29 | 0.30 | 0.25 | 0.36
TRC | 0.54 | 0.51 | 0.54 | 0.73 | 0.55 | 0.54 | 0.53 | 0.46 | 0.49 | 0.48 | 0.48 | 0.43 | 0.73 | 0.68 | 0.70 | 0.74 | 0.75 | 0.72 | 1.11
APU | 0.32 | 0.38 | 0.24 | 0.24 | 0.35 | 0.32 | 0.32 | 0.34 | 0.32 | 0.32 | 0.33 | 0.20 | 0.24 | 0.24 | 0.47 | 0.32 | 0.23 | 0.27 | 0.46
POL | 0.24 | 0.24 | 0.20 | 0.41 | 0.27 | 0.24 | 0.24 | 0.30 | 0.25 | 0.24 | 0.26 | 0.25 | 0.41 | 0.28 | 0.69 | 0.43 | 0.41 | 0.38 | 0.45
KUR | 0.20 | 0.21 | 0.15 | 0.13 | 0.27 | 0.20 | 0.38 | 0.26 | 0.25 | 0.24 | 0.24 | 0.12 | 0.13 | 0.13 | 0.13 | 0.20 | 0.13 | 0.13 | 0.28
PUT | 0.28 | 0.29 | 0.24 | 0.26 | 0.32 | 0.28 | 0.29 | 0.24 | 0.26 | 0.27 | 0.25 | 0.25 | 0.26 | 0.23 | 0.22 | 0.30 | 0.26 | 0.25 | 0.99
JAF | 1.92 | 1.44 | 2.72 | 2.56 | 1.86 | 1.92 | 1.77 | 1.89 | 1.83 | 1.84 | 1.91 | 2.07 | 2.56 | 2.25 | 2.48 | 2.51 | 2.51 | 2.51 | 3.97
KIL | 1.03 | 1.14 | 1.04 | 1.03 | 0.96 | 1.03 | 0.91 | 0.76 | 1.03 | 1.02 | 0.84 | 1.02 | 1.03 | 0.95 | 1.58 | 1.06 | 1.03 | 1.11 | 1.73
MAN | 0.46 | 0.60 | 0.45 | 0.67 | 0.47 | 0.46 | 0.41 | 0.31 | 0.44 | 0.45 | 0.38 | 0.35 | 0.67 | 0.55 | 0.71 | 0.65 | 0.66 | 0.58 | 1.02
MUL | 0.73 | 0.68 | 0.55 | 0.55 | 0.77 | 0.73 | 0.63 | 0.48 | 0.73 | 0.72 | 0.57 | 0.56 | 0.55 | 0.53 | 1.86 | 0.57 | 0.55 | 1.17 | 1.00
VAV | 0.94 | 1.11 | 0.96 | 1.60 | 0.94 | 0.94 | 0.87 | 0.87 | 0.93 | 0.93 | 0.89 | 0.90 | 1.60 | 1.33 | 1.23 | 1.59 | 1.60 | 1.39 | 1.47
KEG | 0.25 | 0.16 | 0.15 | 0.10 | 0.22 | 0.25 | 0.23 | 0.16 | 0.19 | 0.21 | 0.18 | 0.10 | 0.10 | 0.19 | 0.18 | 0.27 | 0.22 | 0.16 | 0.28
RAT | 0.24 | 0.36 | 0.33 | 0.48 | 0.25 | 0.24 | 0.49 | 0.27 | 0.24 | 0.24 | 0.25 | 0.46 | 0.48 | 0.51 | 0.59 | 0.60 | 0.58 | 0.57 | 1.12
GAL | 0.99 | 1.30 | 1.23 | 1.26 | 1.00 | 0.99 | 0.95 | 0.95 | 0.97 | 0.98 | 0.97 | 1.11 | 1.26 | 1.32 | 1.42 | 1.33 | 1.32 | 1.33 | 1.23
HAM | 0.52 | 0.47 | 0.36 | 0.26 | 0.31 | 0.52 | 0.49 | 0.43 | 0.50 | 0.52 | 0.46 | 0.34 | 0.26 | 0.34 | 0.33 | 0.26 | 0.26 | 0.27 | 0.24
MAT | 0.44 | 0.57 | 0.51 | 0.34 | 0.44 | 0.44 | 0.40 | 0.41 | 0.41 | 0.42 | 0.42 | 0.50 | 0.34 | 0.50 | 0.45 | 0.37 | 0.36 | 0.39 | 0.40
BAD | 0.45 | 0.49 | 0.47 | 0.48 | 0.46 | 0.45 | 0.53 | 0.47 | 0.46 | 0.46 | 0.46 | 0.48 | 0.48 | 0.50 | 0.60 | 0.53 | 0.50 | 0.52 | 0.97
MON | 0.06 | 0.20 | 0.13 | 0.27 | 0.42 | 0.06 | 0.07 | 0.15 | 0.08 | 0.07 | 0.11 | 0.28 | 0.27 | 0.21 | 0.14 | 0.27 | 0.28 | 0.21 | 0.33
COL | 0.42 | 0.44 | 0.42 | 0.39 | 0.42 | 0.42 | 0.42 | 0.42 | 0.40 | 0.42 | 0.42 | 0.39 | 0.39 | 0.45 | 0.40 | 0.50 | 0.40 | 0.40 | 0.53
GAM | 0.30 | 0.32 | 0.31 | 0.31 | 0.31 | 0.30 | 0.30 | 0.30 | 0.28 | 0.30 | 0.30 | 0.29 | 0.31 | 0.35 | 0.33 | 0.42 | 0.33 | 0.32 | 0.42
KLU | 0.49 | 0.60 | 0.51 | 0.61 | 0.49 | 0.49 | 0.49 | 0.49 | 0.47 | 0.49 | 0.49 | 0.54 | 0.61 | 0.66 | 0.67 | 0.65 | 0.61 | 0.63 | 0.53

Table 16: MASE from training set: 2016-2018 for quarterly data | Base Forecasting | HTS based on ARIMA | HTS based on ETS
---|---|---|---
series | arm | snv | nve | ets | avg | bup | top | ols | mit | var | stc | ebup | etop | eols | emit | evar | estc
SL | 0.37 | 0.42 | 0.41 | 0.41 | 0.41 | 0.42 | 0.37 | 0.38 | 0.40 | 0.41 | 0.40 | 0.39 | 0.41 | 0.41 | 0.41 | 0.40 | 0.40
Province
CN | 0.59 | 0.62 | 0.63 | 0.65 | 0.63 | 0.59 | 0.55 | 0.56 | 0.59 | 0.59 | 0.58 | 0.54 | 0.69 | 0.66 | 0.65 | 0.58 | 0.60
ET | 0.56 | 0.77 | 0.40 | 1.95 | 0.56 | 0.77 | 0.59 | 0.64 | 0.69 | 0.71 | 0.69 | 0.48 | 1.34 | 1.42 | 0.73 | 0.74 | 0.92
NC | 0.35 | 0.29 | 0.26 | 0.29 | 0.34 | 0.25 | 0.33 | 0.27 | 0.25 | 0.28 | 0.23 | 0.23 | 0.23 | 0.22 | 0.19 | 0.25 | 0.22
NW | 0.28 | 0.25 | 0.20 | 0.20 | 0.31 | 0.23 | 0.29 | 0.23 | 0.22 | 0.23 | 0.24 | 0.21 | 0.20 | 0.21 | 0.20 | 0.20 | 0.21
NT | 1.95 | 1.59 | 2.08 | 1.95 | 1.95 | 1.94 | 1.98 | 1.99 | 1.94 | 1.95 | 1.97 | 1.95 | 1.95 | 1.93 | 1.93 | 1.95 | 1.93
SB | 0.21 | 0.22 | 0.25 | 0.40 | 0.21 | 0.22 | 0.21 | 0.18 | 0.22 | 0.20 | 0.20 | 0.21 | 0.42 | 0.41 | 0.40 | 0.24 | 0.31
ST | 1.00 | 0.86 | 0.86 | 0.85 | 0.62 | 0.78 | 0.98 | 0.83 | 0.85 | 0.84 | 0.84 | 0.79 | 0.91 | 0.95 | 0.92 | 0.81 | 0.85
UVA | 0.25 | 0.23 | 0.22 | 0.30 | 0.30 | 0.20 | 0.27 | 0.29 | 0.19 | 0.20 | 0.23 | 0.21 | 0.34 | 0.50 | 0.43 | 0.24 | 0.30
WT | 0.39 | 0.43 | 0.43 | 0.45 | 0.43 | 0.49 | 0.35 | 0.39 | 0.41 | 0.41 | 0.40 | 0.45 | 0.53 | 0.47 | 0.48 | 0.46 | 0.46
Districts
KN | 0.55 | 0.56 | 0.59 | 0.46 | 0.59 | 0.55 | 0.52 | 0.54 | 0.55 | 0.55 | 0.55 | 0.46 | 0.60 | 0.49 | 0.51 | 0.50 | 0.48
MTL | 0.91 | 0.95 | 0.86 | 1.20 | 0.98 | 0.91 | 0.94 | 0.98 | 0.88 | 0.91 | 0.94 | 1.20 | 1.11 | 1.78 | 1.35 | 1.22 | 1.50
NE | 0.46 | 0.41 | 0.47 | 0.45 | 0.46 | 0.46 | 0.50 | 0.81 | 0.45 | 0.46 | 0.57 | 0.45 | 0.64 | 2.72 | 0.60 | 0.47 | 1.63
AM | 0.21 | 0.23 | 0.26 | 0.26 | 0.21 | 0.21 | 0.19 | 0.84 | 0.22 | 0.21 | 0.47 | 0.26 | 0.46 | 5.31 | 0.31 | 0.23 | 3.39
BAT | 1.46 | 1.46 | 0.53 | 0.70 | 0.70 | 1.46 | 1.07 | 1.35 | 1.35 | 1.37 | 1.39 | 0.70 | 2.69 | 1.49 | 1.01 | 0.95 | 1.06
KAL | 0.33 | 0.39 | 0.19 | 0.19 | 0.33 | 0.33 | 0.27 | 0.24 | 0.25 | 0.27 | 0.27 | 0.19 | 0.39 | 0.81 | 0.55 | 0.62 | 0.50
TRC | 0.60 | 0.52 | 0.48 | 0.60 | 0.60 | 0.60 | 0.51 | 0.51 | 0.54 | 0.54 | 0.54 | 0.60 | 1.12 | 1.03 | 0.74 | 0.78 | 0.88
APU | 0.36 | 0.37 | 0.27 | 0.30 | 0.38 | 0.36 | 0.41 | 0.25 | 0.35 | 0.41 | 0.33 | 0.30 | 0.30 | 0.20 | 0.25 | 0.33 | 0.30
POL | 0.17 | 0.22 | 0.25 | 0.25 | 0.28 | 0.17 | 0.23 | 0.41 | 0.16 | 0.16 | 0.20 | 0.25 | 0.24 | 0.60 | 0.22 | 0.22 | 0.21
KUR | 0.18 | 0.21 | 0.17 | 0.18 | 0.28 | 0.18 | 0.27 | 0.16 | 0.16 | 0.16 | 0.16 | 0.18 | 0.19 | 0.21 | 0.19 | 0.18 | 0.18
PUT | 0.36 | 0.30 | 0.27 | 0.26 | 0.34 | 0.36 | 0.36 | 0.40 | 0.36 | 0.38 | 0.40 | 0.26 | 0.24 | 0.25 | 0.26 | 0.26 | 0.25
JAF | 2.47 | 1.86 | 2.69 | 2.47 | 2.47 | 2.47 | 2.51 | 2.49 | 2.46 | 2.47 | 2.48 | 2.47 | 2.47 | 2.47 | 2.45 | 2.47 | 2.47
KIL | 1.05 | 1.40 | 1.08 | 1.16 | 1.07 | 1.05 | 1.08 | 1.43 | 1.05 | 1.05 | 1.28 | 1.16 | 1.07 | 1.57 | 1.10 | 1.16 | 1.01
MAN | 0.50 | 0.64 | 0.40 | 0.50 | 0.50 | 0.50 | 0.54 | 0.79 | 0.51 | 0.51 | 0.65 | 0.50 | 0.43 | 0.50 | 0.56 | 0.50 | 0.38
MUL | 0.79 | 0.74 | 0.61 | 0.79 | 0.79 | 0.79 | 0.85 | 1.70 | 0.72 | 0.79 | 1.22 | 0.79 | 0.69 | 1.15 | 0.72 | 0.79 | 0.56
VAV | 1.04 | 1.26 | 1.05 | 1.07 | 1.07 | 1.04 | 1.06 | 1.11 | 1.03 | 1.04 | 1.08 | 1.07 | 1.05 | 1.02 | 1.05 | 1.07 | 1.02
KEG | 0.35 | 0.16 | 0.16 | 0.26 | 0.19 | 0.35 | 0.38 | 0.14 | 0.25 | 0.26 | 0.20 | 0.26 | 0.30 | 0.46 | 0.35 | 0.33 | 0.37
RAT | 0.28 | 0.31 | 0.41 | 0.44 | 0.28 | 0.28 | 1.04 | 0.35 | 0.32 | 0.28 | 0.29 | 0.44 | 0.61 | 0.41 | 0.61 | 0.43 | 0.39
GAL | 1.47 | 1.38 | 1.37 | 1.37 | 1.06 | 1.47 | 1.52 | 1.52 | 1.46 | 1.52 | 1.52 | 1.37 | 1.44 | 1.52 | 1.45 | 1.39 | 1.42
HAM | 0.57 | 0.51 | 0.42 | 0.22 | 0.34 | 0.57 | 0.74 | 0.66 | 0.53 | 0.58 | 0.66 | 0.22 | 0.41 | 0.39 | 0.24 | 0.22 | 0.25
MAT | 0.46 | 0.61 | 0.62 | 0.60 | 0.46 | 0.46 | 0.64 | 0.45 | 0.52 | 0.47 | 0.46 | 0.60 | 0.70 | 0.73 | 0.79 | 0.63 | 0.65
BAD | 0.46 | 0.50 | 0.48 | 0.48 | 0.46 | 0.46 | 0.61 | 0.55 | 0.45 | 0.47 | 0.49 | 0.48 | 0.62 | 0.77 | 0.73 | 0.53 | 0.57
MON | 0.10 | 0.21 | 0.07 | 0.07 | 0.42 | 0.10 | 0.08 | 0.26 | 0.09 | 0.09 | 0.12 | 0.07 | 0.05 | 0.22 | 0.10 | 0.07 | 0.05
COL | 0.44 | 0.47 | 0.48 | 0.48 | 0.48 | 0.44 | 0.59 | 0.42 | 0.43 | 0.43 | 0.43 | 0.48 | 0.56 | 0.48 | 0.45 | 0.48 | 0.48
GAM | 0.63 | 0.34 | 0.35 | 0.38 | 0.35 | 0.63 | 0.64 | 0.46 | 0.49 | 0.50 | 0.52 | 0.38 | 0.43 | 0.40 | 0.42 | 0.39 | 0.39
KLU | 0.82 | 0.67 | 0.63 | 0.66 | 0.52 | 0.82 | 0.71 | 0.51 | 0.73 | 0.78 | 0.57 | 0.66 | 0.73 | 0.71 | 0.75 | 0.66 | 0.67

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