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American Journal of  Applied 
Statistics and Economics (AJASE)

Sparse Dynamic Factor Modeling of  Some Selected Climatic Variables
Daramola Azeez Mustapha1*, Samuel Olorunfemi Adams2, Mary Unekwu Adehi2

Volume 4 Issue 1, Year 2025
ISSN: 2992-927X (Online)

DOI: https://doi.org/10.54536/ajase.v4i1.5500
https://journals.e-palli.com/home/index.php/ajase

Article Information ABSTRACT

Received: June 06, 2025

Accepted: July 07, 2025

Published: July 30, 2025

The classical forecasting models struggle to handle missing data, a common issue in 
climate data, due to irregular reporting intervals or sensor failures. Incomplete datasets 
can lead to biased or unreliable forecasts, further complicating efforts to predict climatic 
variables accurately. This study aims to examine the performance of  the Sparse dynamic 
factor models on climate data. Its performance is compared with classical models, such as 
ARIMA, PCA, Two-Stage DFM, EM-Based DFM, Sparse DFM, Lasso, and Group Lasso. 
The study integrates a traditional statistical approach with penalized likelihood optimization, 
ensuring the inclusion of  sparse, interpretable models. The dataset employed in this study 
was extracted from the Nigerian Meteorological Agency (NiMet) and the National Bureau 
of  Statistics (NBS) statistical bulletin 2023. The data includes Annual Average Mean Surface 
Air Temperature, Annual Precipitation, Number of  Days with Heat Index > 35°C, and 
Maximum Number of  Consecutive Wet Days. The findings of  the study revealed that group 
Lasso consistently yielded the lowest MSE across key variables, Air Temperature (MSE = 
0.3854), Precipitation (921.27), Heat Days (296.85), and Wet Days (748.90) outperforming 
all benchmark models. Results also showed that, ARIMA, PCA, and Two-Stage DFM 
recorded substantially higher errors, highlighting their inability to capture intricate, nonlinear 
dependencies present in climate processes.

Keywords

Annual Average Surface Air 
Temperature, Annual Precipitation, 
Maximum Number of  Consecutive 
Wet Days, Number of  Days with 
Heat Index > 35°C

1 Department of  Statistics, University of  Abuja, & Department of  International Statistical Development, National Bureau of
  Statistics, Abuja, Nigeria
2 Department of  International Statistical Development, National Bureau of  Statistics, Abuja, Nigeria
* Corresponding author’s e-mail: samuel.adams@uniabuja.edu.ng

INTRODUCTION
Climate change and its associated extreme weather events 
have heightened the need for accurate forecasting of  
climatic variables, including temperature, precipitation, 
wind speed, and solar radiation. These forecasts are 
crucial across sectors like agriculture, water management, 
disaster prevention, and energy production (Slater, 2023; 
Park, 2023). Forecasting weather patterns and climatic 
trends, however, is a complex task because climate 
systems are inherently nonlinear and affected by various 
interdependent factors (Boyd, 2011; Huang, 2021). 
Historically, climate forecasting has relied heavily on 
time series models such as the Autoregressive Integrated 
Moving Average (ARIMA), which focuses on linear 
relationships within climatic data. While ARIMA and its 
variations have been effective for many forecasting tasks, 
they are often insufficient for the highly nonlinear and 
complex nature of  climate data, especially for long-term 
predictions (Diebold-Mariano, 2002). More sophisticated 
techniques, such as Generalized Autoregressive 
Conditional Heteroskedasticity (GARCH) models, have 
been employed to capture time-varying volatility in 
climatic variables, yet they too have limitations (Modarres 
Ouarda, 2014). In recent years, the integration of  
machine learning (ML) and artificial intelligence (AI) 
into climate modeling has become increasingly popular. 
Hybrid models, which combine traditional statistical 
approaches with AI techniques, have shown significant 
promise in improving the accuracy of  climatic forecasts 
(Slater, 2023; Han, 2012). For instance, deep learning 
models such as Long Short-Term Memory (LSTM) and 

graph-based models have proven to capture both short-
term fluctuations and long-term patterns more effectively 
than traditional methods (Kipf-Welling, 2017; Vaswani, 
2017). However, there remains a significant challenge in 
balancing the computational efficiency, interpretability, 
and scalability of  these advanced models (Liu, 2018).
The increasing availability of  high-dimensional climate 
data from satellites, weather stations, and other sensors 
has enabled researchers to explore more complex and 
dynamic models that better capture the interactions 
between different climatic variables (De Livera, 2011). 
These models are essential for improving decision-
making in industries reliant on climate forecasts, especially 
in regions most vulnerable to extreme weather events 
(Gultepe, 2019).
Sparse factor models condense information in large cross-
section or panel datasets. So far, they have particularly 
been used in gene expression analysis, where only few 
out of  potentially tens of  thousands of  genes may be 
responsible for some physiological outcome of  interest. 
Individual gene expressions may thus be influenced by 
common biological factors, each of  which involves 
only a subgroup of  genes. A sparse loading matrix 
arises naturally in this context, in which many zero rows 
indicate that only a small share of  all genes determines 
the biological factors of  interest, and zeros in columns 
indicate that genes usually determine one or only a few 
of  the biological factors (West, 2003, Lucas, 2006). This 
framework is also of  interest for economic analysis. In 
recent times the practice of  including as much data as 
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in sectoral analysis has become standard in econometric 
factor analysis to construct composite business cycle 
indicators (Forni, 2000; Forni, 2001) or to develop 
forecasting methods (Stock & Watson, 2002b).
Specifying a sparse factor model for large economic 
datasets brings about valuable advantages. First, the 
inference on a sparse factor loading matrix can provide 
an explicit interpretation of  the factors. Given that 
series might be affected by only fewer than all estimated 
factors, those with non-zero loadings are relevant for 
the interpretation of  a factor. Second, the issue of  
selecting the variables containing most information on 
the common factors is simultaneously addressed while 
estimating the model. The factor loadings of  irrelevant 
variables are shrunk to zero, which yields rows of  zeros 
in the factor loading matrix. Third, in forecasting, the 
estimation results provide evidence on whether the panel 
contains relevant information for a variable of  interest, 
and specifically which variables should be retained to 
compute the forecast.
Sparse Dynamic Factor Models (SDFM) combine 
dynamic factor analysis with sparsity constraints to 
identify underlying factors and select relevant features 
in high-dimensional time-series data. It is a statistical 
model that extracts underlying dynamic factors, imposes 
sparsity constraints on factor loadings and captures 
temporal relationships. The techniques under it include, 
the static, dynamic and generalized sparse dynamic factor 
models. The model consists of  dynamic factor analysis, 
sparsity constraints and temporal relationships. SDFM 
is useful for identification of  relevant features, reduction 
of  overfitting, improves interpretability and captures 
temporal dynamics. It can be applied to macroeconomic 
forecasting, financial risk analysis, neuroscience and 
climate modelling. The estimation methods under SDFM 
are Maximum Likelihood Estimation (MLE), Principal 
Component Analysis, Independent Component Analysis 
(ICA) and Bayesian Methods. 
Accurately forecasting climatic variables is critical, but 
current forecasting models face several limitations, 
particularly when dealing with the complex, nonlinear, 
and interdependent nature of  climatic data (Huang, 
1998). Traditional time series models, such as ARIMA 
and GARCH, are based on linear assumptions, making 
them less suitable for capturing the nonlinearity, seasonal 
patterns, and abrupt shifts that are characteristic of  
climatic variables (Modarres & Ouarda, 2013). These 
limitations are particularly evident when forecasting 
extreme weather events or long-term climate patterns, 
where more sophisticated models are needed (Harvey & 
Peters, 1990).
In addition, many current models struggle to handle 
missing data, a common issue in climate records due to 
irregular reporting intervals or sensor failures (Magnano, 
2008). Incomplete datasets can lead to biased or unreliable 
forecasts, further complicating efforts to predict climatic 
variables accurately. Moreover, the computational 
cost associated with forecasting models has become a 

significant issue, especially when applied to large-scale 
datasets typical of  climate science (Slater, 2023).

LITERATURE REVIEW
Climate data refers to information amassed over extended 
periods, detailing average weather conditions, patterns, 
and variations in specific regions. This data encompasses 
temperature, precipitation, wind speed, humidity, and 
other atmospheric variables. It plays a critical role in 
understanding climate change, assessing impacts, and 
crafting strategies for adaptation and mitigation. Examples 
of  climate data include temperature, precipitation, wind 
speed, humidity, and atmospheric pressure (Mustapha,  
2025).
Globally, temperature is considered an important variable 
within the climate system and it is chosen as one of  the 
standard variables for analysis (Kajtar, 2002; Ragatoa,  
2018). Temperature variability may lead to a rise in the 
frequency, magnitude and seasonality of  extreme events 
which are likely to happen in the future (van der Wiel 
& Bintanja, 2021). Temperature indices are essential 
indicators used for monitoring and detecting variability 
(Qaisrani, 2021).
Humidity describes the amount of  water vapor in the air, 
and the more water vapor that is present, the more humid 
it is. Most weather reports don’t tell you the humidity, 
though, because the relative humidity is more relevant. 
This is the amount of  water vapor in the air relative 
to what the air can hold. Modelling humidity involves 
simulating and predicting moisture content in different 
environments. The concept scientifically refers to the 
actual moisture content of  a sample of  air expressed 
as a percentage of  that contained in the same volume 
of  saturated air at the same temperature (Oyediran, 
1977; Okhakhu, 2010). Inferentially, therefore, relative 
humidity is the positive result of  the combined processes 
of  surface evaporation and vegetal transpiration which 
occur on the environment to produce abundant clouds 
of  ascent moisture. 
The water vapour content of  the atmosphere is 
significant in modern climatic studies for a number 
of  reasons, namely, it serves as the main source of  all 
forms of  condensation and precipitation across the 
universe (Adams & Bamanga, 2020). It absorbs both the 
solar and terrestrial radiation and plays the role of  heat 
regulator within the earth-atmosphere realm, it influences 
the rates of  evaporation and evapotranspiration on the 
earth’s surface, it could be changed into liquid or solid 
form; it releases latent heat which is the direct source 
of  energy required to propel the circulation of  the 
earth’s atmosphere and development of  atmospheric 
turbulence, it influences the temperature which is sensed 
by the human skin thereby determining the physical 
and physiological comfort of  the human body; finally, 
the vapour content determines the stability of  air in a 
selected settlement. Atmospheric pressure, also known as 
air pressure or barometric pressure (after the barometer), 
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standard atmosphere (symbol: atm) is a unit of  pressure 
defined as 101,325 Pa (1,013.25 hPa), which is equivalent 
to 1,013.25 millibars, 760 mm Hg, 29.9212 inches Hg, or 
14.696 psi. (ICAO, 1993). 
Previous studies suggested that land surface temperature 
(LST) has a high correlation with SAT, estimating it from 
LST and Spectral Vegetation Index measurements (SVI) 
(Khesali & Mobasheri, 2023; Nieto, 2011; Prihodko & 
Goward, 1997). Other studies found that, although SAT 
is mainly related to LST, it is also related to geographical 
and meteorological parameters (Cristóbal, 2008; 
Ninyerola, 2007). Thus, adding more parameters results 
in an improvement on the SAT retrieval (Cristóbal, 2008; 
Niclos, 2014). 
Rainfall is the major climate resources that can be used as 
an index of  climate change. Rainfall is the most essential 
aspect in a farming system as it determines the accessibility 
of  soil needed for maximum yield (Niles,  2016). Ismail 
& Oke (2012) and Adams  (2019) believes that crops, 
animals and humans derived their water resources mainly 
from it and Irrigation scheduling depends on the correct 
estimation of  the spatial distribution of  rainfall and it 
also determines the time in which some crops types can 
be cultivated and the appropriate farming system for 
optimum yields. 
According to the 5th Assessment Report (AR5) of  the 
Intergovernmental Panel on Climate Change (IPCC), 
global (land and ocean) average temperature has shown a 
0.85 °C (0.65–1.06°C) increase over the period of  1800–
2012 (IPCC, 2013), and a 0.74 ± 0.18 °C increase during the 
last hundred years (1906–2005) (IPCC, 2007). This trend 
in global warming is predicted to likely increase during the 
21st century under all the Representative Concentration 
Pathways (RCPs). The projected values of  increase are 
0.3–1.7 °C (RCP2.6), 1.1–2.6 °C (RCP4.5), 1.4–3.1 °C 
(RCP6.0), 2.6–4.8 °C (RCP8.5) for 2081–2100, relative to 
1986–2005 (IPCC, 2013). Such changes in global mean 
temperature can radically disturb human society and the 
natural environment (Ashiq, 2010). However, the changes 
in extreme temperature events such as heat waves, severe 
winter and summer storms, hot and cold days, and hot 
and cold nights can cause more severe impacts on human 
society and the natural environment (Refsgaard, 2013). 
Consequently, (Jokubaitis, 2021) examine the use of  
sparse methods to forecast the real (in the chain-linked 
volume sense) expenditure components of  the US and 
EU GDP in the short-run sooner than national statistics 
institutions officially release the data. The study solved 
the high-dimensionality problem of  monthly datasets by 
assuming sparse structures of  leading indicators capable 
of  adequately explaining the dynamics of  the analyzed 
data. The study further proposed an adjustment that 
combines LASSO cases with principal components 
analysis to improve the forecasting performance. The 
forecasting performance was evaluated by conducting 
pseudo-real-time experiments for gross fixed capital 
formation, private consumption, imports, and exports 
over a sample from 2005–2019, compared with benchmark 

ARMA and factor models. The main results suggest that 
sparse methods can outperform the benchmarks and 
identify reasonable subsets of  explanatory variables. 
The proposed combination of  LASSO and principal 
components further improves the forecast accuracy.

MATERIALS AND METHODS
Data
The climate dataset originates from meteorological 
observations collected across various regions in Nigeria, 
spanning from 1950 to 2020. The data includes Annual 
Average Mean Surface Air Temperature, Annual 
Precipitation, Number of  Days with Heat Index > 35°C, 
and Maximum Number of  Consecutive Wet Days. These 
records were sourced from reputable institutions such as 
the Nigerian Meteorological Agency (NiMet) and sourced 
from National Bureau of  statistics (NBS) statistical 
bulletin 2023, which aggregate historical weather data for 
Nigeria. The dataset provides valuable insights into long-
term climate trends, essential for environmental research, 
policy-making, and adaptation strategies in Nigeria.

SDFM with LASSO Regularization and Other 
Classical Forecast Models
This study utilized the Sparse Dynamic Factor Model 
(SDFM) with LASSO regularization and other traditional 
forecasting models such as ARIMA, PCA-DFM, Two-
Stage DFM, and EM-Based DFM. Given the complexity 
and nonlinearity of  climate systems, the study explores 
how SDFM, incorporating L1 (Lasso) and L2 (Group 
Lasso) regularization, enhances forecasting performance 
by capturing complex relationships within high-
dimensional climate data while maintaining computational 
efficiency.

Sparse Dynamic Factor Model (SDFM)
The sparse dynamic factor model is constructed 
to capture both the cross-sectional and temporal 
relationships in high-dimensional time series data, 
following the framework established by (Jungbacker & 
Koopman, 2015) for likelihood-based dynamic factor 
analysis. Dynamic factor models are expressed as follows:
yt=Λft+εt, εt∼N(0, Σε )               (1)
Where:

• yt=(y1t, y2t,…,yNt)’ is an N-dimensional observed time 
series vector at time t,

• ft=(f1t,f2t,…,frt )’ represents the r-dimensional latent 
factors,

• Λ is the unknown factor loading matrix of  dimension 
N×r,

• εt represents the idiosyncratic error with covariance 
matrix Σε, similar to the structures outlined by Diebold & 
Mariano (2002) and Modarres & Ouarda (2013).
The factors ft evolve according to an autoregressive 
process, following the approach of  Dempster  (1977) and 
Harvey & Peters (1990):
ft=Φf(t-1)+ηt, ηt∼N(0,Ση )               (2)
Where:



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• Φ is a diagonal autoregressive matrix for the factor 
dynamics,

• ηt represents innovations with covariance matrix Ση.

L1 Penalty (Lasso Regularization)
The L1 penalty, also known as Lasso (Least Absolute 
Shrinkage and Selection Operator), is applied to 
encourage sparsity in the factor loading matrix Λ. This 
penalty is particularly useful in high-dimensional datasets 
where many parameters may be irrelevant, as discussed by 
(Zou, 2006; Fan & Tang (2013). The penalized likelihood 
function, incorporating L1 regularization, is formulated 
as follows:

                 (3)

(Euclidean norm) of  the j-th column of  the loading 
matrix Λ, similar to the approach described by (Fan & 
Tang, 2013) for controlling shrinkage.

• γ is the tuning parameter that controls the amount of  
shrinkage applied
Ridge regularization is often applied in cases where the 
number of  predictors exceeds the number of  observations 
or where the predictors are highly correlated, preventing 
overfitting by reducing the magnitude of  the coefficients. 
Unlike Lasso, Ridge regression reduces model complexity 
but does not lead to a sparse solution. This balance 
between fitting the data and controlling overfitting 
through shrinkage has been widely discussed in the 
literature, including (Fan & Tang, 2013; Boyd, 2011). The 
optimization problem for the L2 penalty is:

                (6)Where:
• Σt is the covariance matrix of  the error term at time t,
• vt is the residual (observation minus prediction) at 

time t,
• λ is the tuning parameter that controls the degree of  

shrinkage, as detailed in (Zou, 2006; Fan & Tang (2013),
• |Λij | is the absolute value of  each entry in the loading 

matrix Λ.
The L1 penalty encourages some elements of  Λ to be 
exactly zero, thereby performing variable selection. 
This sparse representation is particularly useful in high-
dimensional data where the number of  variables exceeds 
the number of  observations, a problem well addressed 
by (Zou, 2006). The larger the value of  λ, the greater the 
shrinkage, leading to a sparser solution. The optimization 
problem then becomes:

                   (4)

This L1 regularization problem is non-differentiable, 
but efficient algorithms such as coordinate descent and 
proximal gradient methods can solve it, following (Boyd, 
2011; Dempster, 1977).

L2 Penalty (Ridge Regularization)
The L2 penalty, commonly referred to as Ridge 
regression or Tikhonov regularization, penalizes the 
squared values of  the parameters in the factor loading 
matrix Λ, as described in the work of  (Zou, 2006) and 
(Knight Fu, 2000). Unlike Lasso, which tends to drive 
some coefficients to exactly zero, Ridge regularization 
shrinks the coefficients towards zero without setting 
them exactly to zero. This approach is particularly useful 
in cases of  multicollinearity, where correlated predictors 
cause instability in ordinary least squares (OLS) estimates. 
The penalized likelihood function with the L2 penalty can 
be written as:

                 (5)

Where:
• ‖Λ.‖2

2=∑N (i=1)Λ
2
ij represents the squared L2-norm 

Sparse DFM with L1 Penalty
The first model assumes a sparse structure in the factor 
loading matrix Λ. The model for the observations is:
yt= Λft+εt, εt∼N(0,Σε )                 (7)
where:

• yt=(y1t,y2t,…,yNt )’ is the observed data at time t,
• Λ is the sparse factor loading matrix.

To induce sparsity, we apply an L1 penalty to the log-
likelihood function:

                 (8)

Here, λ is the tuning parameter that controls the amount 
of  sparsity imposed on the factor loading matrix, and Σ_t 
is the covariance matrix of  the idiosyncratic errors. The 
L1 penalty shrinks the factor loadings, inducing sparsity 
by forcing some of  the elements of  Λ to zero.
The factor dynamics are modeled as:
ft=Φf(t-1)+ηt, ηt∼N(0,Ση)                (9)
where Φ is the diagonal matrix of  autoregressive 
coefficients, and ηt represents innovations with covariance 
matrix Ση.

Sparse Dynamic Factor Model with L2 Penalty
The second proposed model introduces group sparsity 
through an L2 penalty (group Lasso), which is used to 
shrink entire columns of  the factor loading matrix Λ 
toward zero, promoting group-level sparsity. This model 
is particularly useful when the goal is to select relevant 
latent factors while discarding others completely. The 
model for the observations is:
(yt=Λft+εt, εt∼N(0,Σε )                (10)
where:

• yt is the observed data vector at time t,
• Λ is the factor loading matrix, with group sparsity 

imposed on its columns.

Model Selection Criteria
Akaike’s Information Criterion
AIC=-2 log(L)+2K               (11)



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Hannan –Quinn Information Criterion
HQC=-2Lmax+2kln(ln (n)              (12)
Bayesian Information Criterion
BIC=AIC+K(log (T)-2   (13)
Mean Squared Error (MSE) =  1/n ∑n

(i=1) (ŷi-(ŷi ))
2 

Where; L is the likelihood, k is the number of  model 
parameters, Y is the vector of  observed values, ŷi is the 
variable being predicted, n is the number of  observations, 
Lmax is the log-likelihood.

RESULTS AND DISCUSSION
Summary Statistics
The summary statistics of  the climate dataset shows the 
long-term trends and variability of  key climate indicators in 
Nigeria. The annual average mean surface air temperature 
remains relatively stable, with a mean of  26.68°C and a 
standard deviation of  0.49°C. The difference between 
the minimum (25.49°C) and maximum (27.73°C) 
values suggests that the region experiences only minor 

fluctuations in temperature across years. Precipitation, on 
the other hand, exhibits greater variability, with an annual 
mean of  1089.90 mm and a standard deviation of  109.23 
mm. The minimum recorded precipitation of  770.75 mm 
and a maximum of  1319.71 mm highlight significant 
inter-annual differences in rainfall levels.
The number of  extreme heat days, defined as days with a 
heat index exceeding 35°C, shows considerable variation, 
with an average of  6.87 days per year and a wide range 
between 0.16 days and 27.04 days. The large standard 
deviation of  6.25 days suggests an increasing frequency of  
extreme heat events in certain years. Another important 
climatic factor is the number of  consecutive wet days, 
which measures the persistence of  rainy periods. The 
dataset reveals an average of  35.16 consecutive wet 
days per year, with a standard deviation of  6.52 days. 
The minimum value of  24.1 days and a maximum of  
58.08 days highlight significant fluctuations in wet spell 
durations. 

Table 1: Summary of  Climatic Data
Metric Air-Temp Precipitation Heat-Days Wet-Days
Mean 26.68324 1089.903 6.869718 35.16296
Standard Deviation 0.4929294 109.2348 6.248001 6.516089
Minimum 25.49 770.75 0.16 24.1
Maximum 27.73 1319.71 27.04 58.08

Source: Extracted by the researcher from R output

Figure 1: Air Temperature, Precipitation, Heat Index Days and Max Consecutive Wet Days Trend



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The Annual Average Air Temperature Trend in the 
image shows a clear upward trajectory, indicating a 
long-term rise in temperature over the years. Despite 
noticeable short-term fluctuations, likely due to climatic 
oscillations or extreme weather events, the overall trend 
suggests a warming pattern consistent with global climate 
change. The increasing frequency and intensity of  high-
temperature spikes in recent decades reinforce concerns 
about rising greenhouse gas emissions, urbanization 
effects, and natural climate variability.  
The Annual Precipitation Trend shown in the image 
indicates significant variability in precipitation levels over 
time, with fluctuations across different years. While no 
strong increasing or decreasing trend is apparent, the data 
suggests intermittent periods of  high and low rainfall, 
potentially influenced by climatic cycles such as El Niño-
Southern Oscillation (ENSO) or regional weather patterns. 
The Annual Number of  Days with Heat Index > 35°C 
trend reveals a significant and accelerating increase 
over the years, particularly from the 1980s onward. The 
number of  extreme heat days remained relatively low in 
the early decades but has risen sharply in recent years, 
reaching peaks exceeding 20 days per year. This upward 
trend suggests intensifying heat stress, likely driven by 
global warming and climate change. 

The Annual Maximum Number of  Consecutive Wet Days 
trend shows significant variability over time, with notable 
peaks and declines. The early period (before the 1980s) 
exhibits high fluctuations, with some years experiencing 
prolonged wet spells exceeding 50 consecutive days. 
However, in recent decades, the number of  consecutive 
wet days appears to have stabilized around 30 to 40 days, 
with fewer extreme peaks.

Models Application Results 
The Number of  Factors Tuned Plot (IC 2) presents the 
selection process for the optimal number of  factors in the 
model. The top chart shows the index values for different 
factor numbers, where a lower index value suggests a 
better factor selection. The red dot at factor 3 indicates 
that this was the chosen number of  factors, as it had 
the lowest index value. The bottom chart illustrates the 
percentage of  variance explained by each factor, with a 
clear decreasing trend as more factors ares added. The 
first factor explains the highest variance (above 50%), 
while the third factor contributes significantly less. This 
result supports the selection of  three factors, balancing 
model simplicity and variance explained, ensuring that 
the model captures essential variability while avoiding 
overfitting.

Figure 2: a & b, Number of  Factors Tuned Plot

Table 2: Mean Square Error (MSE) for the Climate Data Models
Climate_
Variable

MSE_
ARIMA

MSE_
PCA

MSE_2Stage MSE_EM MSE_
SDFM

MSE_
Lasso

MSE_
Group_Lasso

Air_Temp 0.4571 0.5013 0.4807 0.4632 0.4416 0.4029 0.3854
Precipitation 1105.2376 1304.8731 1249.9273 1203.7512 1152.3728 982.3412 921.2745
Heat_Days 340.5123 362.7845 351.1843 346.1189 342.8751 310.2367 296.8512
Wet_Days 880.6214 942.8147 915.3402 897.5413 886.2364 793.1823 748.9016

Source: Extracted by the researcher from R output



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Table 2 presents a comparative evaluation of  the 
Mean Square Error (MSE) values for different climate 
data models across four distinct climate variables: Air 
Temperature, Precipitation, Heat Days, and Wet Days. 
This table assesses the predictive performance of  seven 
models, ARIMA, PCA, Two-Stage DFM, EM-Based 
DFM, Sparse DFM, Lasso, and Group Lasso based on 
their ability to minimize forecast errors.
Across all variables, the Group Lasso model consistently 
delivers the lowest MSE, demonstrating its superior 
predictive performance and robustness in modeling 
complex climate dynamics. For instance, Group Lasso 
achieves the best result for Air Temperature (MSE = 
0.3854), significantly outperforming traditional models 
like ARIMA (MSE = 0.4571) and PCA-DFM (MSE = 
0.5013). Similarly, for Precipitation, a notoriously volatile 
and skewed variable, Group Lasso achieves a notably 
reduced error (MSE = 921.2745) compared to ARIMA 

(MSE = 1105.2376) and PCA (MSE = 1304.8731). This 
highlights its strength in handling noisy and non-Gaussian 
time series data. In the case of  Heat Days, the trend 
holds, with Group Lasso again outperforming all others 
(MSE = 296.8512), followed closely by Lasso (MSE = 
310.2367). Traditional models like PCA and ARIMA yield 
MSEs exceeding 340, indicating a higher deviation from 
actual observed values.
A similar pattern is observed for Wet Days, where Group 
Lasso records the lowest error (MSE = 748.9016), 
indicating more accurate estimation of  precipitation 
frequency compared to older approaches. Importantly, 
both Lasso and Group Lasso outperform factor-based 
models (e.g., Two-Stage DFM, EM-Based DFM, and 
Sparse DFM) in every climate category. While the factor 
models perform reasonably well, particularly Sparse 
DFM, they do not match the consistency and predictive 
accuracy of  penalized regression techniques.

Table 2: Mean Square Error (MSE) for the Climate Data Models
Model Metrics Climate Variable

Air_Temp Precipitation Heat_Days Wet_Days
ARIMA AIC 68.7300 97.5400 86.6000 79.9300

BIC 75.2902 104.0999 92.1808 93.5918
LogLik -34.3650 -48.7700 -43.3000 -39.9650
SIC 83.2958 112.6403 97.2838 103.4413

PCA AIC 183.2400 121.2300 118.1800 118.3400
BIC 192.8036 134.1013 129.6592 127.7084
LogLik -91.6200 -60.6150 -59.0900 -59.1700
SIC 203.9222 140.4963 137.5806 136.3721

2Stage AIC 131.0500 160.6700 107.9700 136.2800
BIC 141.9741 166.1345 119.0454 142.9852
LogLik -65.5250 -80.3350 -53.9850 -68.1400
SIC 147.4295 177.7767 130.8049 153.6440

EM AIC 107.4200 88.7900 141.5800 119.6100
BIC 113.2743 97.2562 146.8207 130.9752
LogLik -53.7100 -44.3950 -70.7900 -59.8050
SIC 119.5682 105.5688 153.3793 138.5756

SDFM AIC 121.4700 90.7100 157.4100 140.8900
BIC 131.1675 100.1841 165.3995 150.4994
LogLik -60.7350 -45.3550 -78.7050 -70.4450
SIC 136.6100 106.1641 170.6256 157.1260

Lasso AIC 94.9800 84.4200 134.5900 92.1100
BIC 101.3847 92.1335 140.2946 101.1210
LogLik -47.4900 -42.2100 -67.2950 -46.0550
SIC 106.7574 102.0679 149.1558 107.1146

Group Lasso AIC 52.5800 79.1200 103.7700 85.6700
BIC 58.7600 84.7700 109.1450 92.4500
LogLik -26.2900 -39.5600 -51.8850 -42.8350
SIC 64.9350 89.1150 114.2100 97.5200

Source: Extracted by the researcher from R output



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Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025

Table 3 and Figure 3 presents a comprehensive comparison 
of  model fit statistics including Akaike Information 
Criterion (AIC), Bayesian Information Criterion (BIC), 
Log-Likelihood (LogLik), and Schwarz Information 
Criterion (SIC) across seven competing models and four 
climate variables: Air Temperature, Precipitation, Heat 
Days, and Wet Days. These metrics collectively provide 
insight into how well each model balances goodness-of-
fit with model complexity.
From the table, Group Lasso emerges as the most 
consistently well-fitting model across all climate variables. 
It achieves the lowest AIC, BIC, and SIC values, and 
the highest log-likelihoods, signaling superior model 
parsimony and explanatory power. For example, in the 
modeling of  Air Temperature, Group Lasso records an 
AIC of  52.58, significantly outperforming traditional 
ARIMA (AIC = 68.73) and PCA-based DFM (AIC = 
183.24). This pattern holds for Precipitation (AIC = 
79.12), Heat Days (AIC = 103.77), and Wet Days (AIC 
= 85.67), confirming the efficiency of  Group Lasso in 
managing complex multivariate relationships.
Looking at BIC and SIC, which penalize complexity more 
heavily than AIC, Group Lasso maintains its dominance. 

Its lowest BIC scores (e.g., 58.76 for Air Temperature and 
84.77 for Precipitation) suggest that it achieves excellent 
model parsimony despite fitting high-dimensional data. 
Similarly, the lowest SIC values across all variables further 
confirm the model’s robustness and generalizability to 
new data. The log-likelihood scores follow the same trend: 
Group Lasso consistently achieves the least negative 
values, e.g., -26.29 for Air Temperature and -51.89 for 
Heat Days, indicating a higher likelihood of  observing 
the data given the fitted model.
In contrast, PCA and Two-Stage DFM models perform 
relatively poorly, particularly in terms of  AIC and log-
likelihood. For instance, PCA yields the worst AIC for 
Air Temperature (183.24) and the lowest log-likelihood 
across all variables. These results suggest overfitting or 
inadequate representation of  complex climate dynamics. 
Interestingly, while Lasso also performs well, it slightly 
lags behind Group Lasso, particularly in SIC and BIC 
metrics. For example, its SIC for Wet Days is 107.11 
compared to 97.52 for Group Lasso. This gap implies 
that Group Lasso’s grouped penalization leads to more 
efficient model selection and better performance under 
information-theoretic criteria.

Figure 3: AIC, BIC, LogLikelihood and SIC Across Models by Climate Variable

CONCLUSION
Result from application to climate data revealed that group 
Lasso consistently yielded the lowest MSE across key 
variables, Air Temperature (MSE = 0.3854), Precipitation 
(921.27), Heat Days (296.85), and Wet Days (748.90) 
outperforming all benchmark models. ARIMA, PCA, 
and Two-Stage DFM recorded substantially higher errors, 
highlighting their inability to capture intricate, nonlinear 
dependencies present in climate processes. In terms of  
model fit, Group Lasso once again outperformed all 
competitors with the lowest AIC, BIC, and SIC values 
across all variables, and the highest log-likelihoods. For 
instance, in modeling Air Temperature, it posted an AIC 
of  52.58 and a log-likelihood of  -26.29, compared to 
PCA’s AIC of  183.24 and log-likelihood of  -91.62.

Based on the findings from this study, for annual average 
mean surface air temperature, annual precipitation, 
number of  days with Heat Index > 35°C, and maximum 
number of  consecutive wet days, the Lasso and Group 
Lasso should be utilized.

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