Pa ge 1 Pa ge 70 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 available or using the highest possible disaggregation level Pa ge 71 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 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), is the pressure within the atmosphere of Earth. The Pa ge 72 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 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: Pa ge 73 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 • 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) Pa ge 74 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 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 Pa ge 75 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 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 Pa ge 76 https://journals.e-palli.com/home/index.php/ajase Am. J. Appl. Stat. Econ. 4(1) 70-79, 2025 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 Pa ge 77 https://journals.e-palli.com/home/index.php/ajase 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. REFERENCES Adams, S. O., & Bamanga, M. A. (2020). Modelling and forecasting seasonal behavior of rainfall in Abuja, Nigeria: A SARIMA approach. 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