Seasonal inflation dynamics: Insights from probabilistic and hybrid forecasting models

1. Introduction

Inflation is one of the most challenging and complex processes to define and quantify (Baciu, 2015). It is a process where money depreciates and the price index increases. It is a steady rise in the economy’s overall price level. Gross domestic product, money supply, exports, import prices, exchange rates, interest rates, fiscal policy, and more are all intimately correlated with the rate of inflation (Bashir et al., 2011). Significant shifts in the economy’s supply and demand can result in volatile inflation. In addition to making long-term planning challenging, the erratic and unexpected inflation rates deter both savings and investment (Baciu, 2015). The unpredictability of inflation has two different kinds of economic repercussions. The first impacts businesses and customers who have to make financial decisions based on projected future inflation. According to (Devereux, 1989), the uncertainty surrounding inflation in this situation influences financial markets by increasing interest rates, creating uncertainty about other factors crucial in economic decisions, and motivating businesses to allocate their resources to mitigate specific inflation-related risks. The second consequence occurs when the inflation rate deviates considerably from the forecast. A transfer of wealth from the creditor to the debtor occurs when inflation is better than anticipated. Because repayment is made with funds whose purchasing power is penalized by the inflation rate, the latter is preferred. Some contend that because inflation affects both the economy and society, it is not a desirable state of affairs (Khan and Gill, 2010). Both monetary and fiscal views were unable to definitively identify the causes of the rising inflation. Real income is randomly redistributed as a result of inflation’s effects on income distribution. A nation’s financial situation deteriorates when inflation raises the level of prices for goods, services, and other items. Therefore, it can be concluded that one of the main causes of rising commodity prices in every economy is inflation. The state of the economy is mostly determined by inflation (Suseno, 2009), thus it requires careful consideration from a variety of sources, particularly the monetary authorities in charge of managing inflation. A nation’s economic system and businesspeople in general cannot undervalue the rate of inflation. Annual inflation rate forecasts are very important for monetary policy as well as for planning in a variety of corporate and economic scenarios. Several studies have looked into ways to make inflation predictions better. For instance, (Stock and Watson, 1999) enhance their inflation projections by utilizing data from other economic indicators. Disaggregated data is used by (Hubrich, 2005). Additionally, (Camba and Kapetanios, 2004), (Hofmann, 2009), and (Angelini, et al., 2001) consider gleaning valuable data for inflation rate forecasts from huge datasets. Numerous studies have examined the topic of forecasting seasonal time series; (Franses, 1991; Paap et al., 1997), and for a more general discussion of seasonality (Hylleberg, 2014). A summary of the findings is given by (Osborn 2004). Forecasting is a technique that uses historical data to estimate future values. Among the time series data is information on inflation. It is possible to forecast future time data by modelling historical time data. In this study, we investigate the use of time series and probability models to anticipate inflation in the United States. To examine US inflation and short-term projections, we contrasted hybrid and non-hybrid models of time series and the probability distributions.

The majority of recent research on inflation forecasting still uses conventional econometric techniques like vector autoregression (VAR) and autoregressive integrated moving average (ARIMA) models, despite the wealth of prior research in this area. It is challenging for these conventional models, which often assume linear relationships, to explain for the seasonal and nonlinear variations commonly observed in real-world data. The complex patterns and anomalies that have influenced the historical evolution of inflation are, therefore, sometimes difficult for individuals to comprehend. It is also not often recognized how probabilistic and hybrid forecasting techniques might be used, particularly when studying market-driven seasonal inflation trends: expectations, cyclical economic swings, and changes in monetary policy. Prior studies have mostly examined quarterly or annual data, neglecting the short-term seasonal variations that may serve as a precursor to inflationary trends. The relative usefulness of complex hybrid models, such as ARIMA-ANN or ARIMA-TBATS, in forecasting inflation has also not been thoroughly examined in many studies; as a result, nothing is known here. Examine its performance by contrasting it with other machine learning methods, including ANN.

Additionally, there is not enough empirical data to evaluate the effects of combining learning-based and probabilistic models on the long-term stability and comprehensibility of inflation forecasts. By using both probabilistic and hybrid forecasting models to investigate seasonal inflation dynamics, the current study closes these gaps. This method increases the accuracy and adaptability of inflation forecasts while also bridging the gap between traditional statistical methods and modern machine learning techniques. Examining such complex modeling frameworks is essential since inflation prediction plays a major role in monetary policy, economic planning, and investment strategies. By identifying models that effectively capture seasonal and nonlinear inflation patterns, the work provides significant insights for researchers and policymakers seeking more reliable forecasting tools.

2. Objectives of the Study

This study’s primary objective is to apply probabilistic and hybrid forecasting techniques to analyze and forecast seasonal inflation patterns. The study’s goal is to identify the model that best captures the nonlinear, stochastic, and seasonal tendencies seen in inflation data.

The specific objectives of the study are as follows:

• Examine the seasonal variations and structural characteristics of the breakeven inflation rate using advanced time series analysis.

• Using probabilistic models to show the stochastic and probabilistic nature of inflation data, such as the inverse Weibull (IW), Weibull (W), exponentiated Weibull (EW), Kumaraswamy Weibull (KW), and exponential (E) models.

• Hybrid forecasting models such as ARIMA-TBATS, ARIMA-error, trend and seasonality (ARIMA-ETS), ARIMA-ANN, and error, trend and seasonality-artificial neural network-trigonometric seasonality Box-Cox transformation ARMA errors trend, and seasonal components (ARIMA-ETS-ANN-TBATS-STLM) should be developed and evaluated to improve prediction accuracy.

• To ascertain which method best captures seasonal and nonlinear inflation trends, the effectiveness of probabilistic, hybrid, and pure machine learning models is assessed.

• The anticipated accuracy of each model is evaluated using statistical performance measures such as mean absolute error (MAE), mean absolute percentage error (MAPE), mean absolute scaled error (MASE), and root mean square error (RMSE).

• To provide decision-makers and economists with relevant data that will enhance their understanding of inflation dynamics and facilitate the development of more astute, fact-based policies.

3. Data

The Federal Reserve Bank of St. Louis (United States) provided the 10-year breakeven inflation rate (T10YIEM) data used in this study. Evaluation of the data is available to the public at the repository at https://fred.stlouisfed.org/series/T10YIEM. Our data are seasonal, with a total of 257 observations. This dataset is categorized as a univariate time series, meaning that the only significant component is the 10-year breakeven inflation rate. This indicator provides crucial information about long-term pricing trends and is based on market-based forecasts of future inflation. The study focused on evaluating the prediction validity of several individual and hybrid time series models rather than incorporating any exogenous or explanatory elements. The R programming language was used for time series analysis and probability models. The models’ accuracy in representing and forecasting changes in inflation situations in the USA over time was assessed.

3.1 Pre-processing steps of the dataset

The data went through a number of processing stages before modeling to make sure it was suitable for analysis. After the dataset was examined for consistency and completeness, any missing values were filled in using linear interpolation to preserve continuity. Then, beginning in January 2003, the inflation series was transformed into a monthly time series object with a frequency of 12 to maintain its seasonal and temporal structure. Trends, seasonal patterns, and the necessity of differencing in models like ARIMA were identified by preliminary visualization and stationarity tests, such as the Augmented Dickey-Fuller (ADF) test.

Fig. 1 provides scatter plots for Lag-1 and Lag-2 values, with points color-coded by month. The observed linear pattern suggests strong temporal dependence, indicating that past inflation values significantly influence future values. The color-coded distribution further highlights seasonal variations, emphasizing that inflation trends exhibit distinct patterns across different months. In addition to time series analysis, the study employs visual tools to understand data distribution and variability. Additionally, Fig. 1 shows that a strong positive linear relationship between successive values in the time series is indicated by the lag plot, which displays $y_t$ versus $y_{t-1}$.

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Fig. 1.

(a) Lag 1 and lag 2 Scatter plots and (b) ACF and PACF plots of series.

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Fig. 2 shows that this plot is a combination of a histogram and a kernel density estimate (KDE) for the variable $y$ and the total time on test (TTT) plot; the descriptive statistics are completed. The histogram is in black bars, and the KDE is in a blue line. The shape of the density is left-skewed, and the TTT plot shows that the data are in an increasing trend.

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Fig. 2.

(a) Histogram with kernel density and (b) TTT plot

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A graphical approach to evaluate whether the failure rate of a dataset (often referred to as its hazard rate) is increasing, decreasing, or stable over time is represented by the TTT graph. Prior to selecting a statistical model (such as lognormal, Weibull, or exponential), it is beneficial to observe the fundamental pattern of life distribution.

Fig. 3 presents a violin plot, which combines density estimation with a box plot to display the distribution of inflation data. The black bar represents the interquartile range (IQR), and the white dot indicates the median. The shape of the violin plot suggests that while the central part of the data is densely packed, the distribution widens significantly toward the tails, reflecting variations in inflation trends over time. Additionally, Fig. 3 uses a boxplot to summarize key statistical properties of inflation data, illustrating the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values. The presence of whiskers and potential outliers helps in identifying seasonal variations and extreme fluctuations in inflation trends.

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Fig. 3.

(a) Violin plot and (b) box plot

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Table 1 shows the dataset’s intermediate value, where half of the data is above, and half is below. The value of the first quartile is 1.830, the third quartile is 2.360, and the minimum value is 1.06. 2.008 is the average of all the values in the dataset. 2.88 is the greatest value in the dataset. The information explains the data’s range, central tendency, and spread.

Table 1. Basic statistical measures of the seasonal inflation percentage in the United States.

Min	1st Qu	Median	Mean	3rd Qu	Max
1.06	1.830	2.19	2.088	2.360	2.88

4. Conceptual Framework And Definition of Variable

Within the conceptual framework, independent and dependent variables serve as the foundation for analyzing potential cause–and–effect relationships. As shown in Fig. 4, the independent variables considered are money supply (M3), exchange rate (ER), and gross domestic product (GDP). The dependent variable, represented as a percentage, is the inflation rate (IR).

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Fig. 4.

The model’s conceptual framework for the variables.

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The study’s conceptual underpinning is the idea that significant macroeconomic variables, such as the M3, ER, and GDP, affect inflation. The seasonal and directional tendencies of inflation are influenced by the interaction of various elements across time. It was hoped that this theoretical framework would be converted into quantitative models in order to reflect these processes through empirical investigation. Most accurate results were generated by the ANN model, even though a range of hybrid and probabilistic forecasting techniques were used. The fundamental idea is that traditional linear or somewhat mixed models are unable to adequately represent the complex and nonlinear character of inflation dynamics. This study supports the idea that representation should be egalitarian.

The theoretical hypothesis that nonlinear modelling techniques can better capture the seasonal and structural complexity of inflation trends is thus supported by the improved forecasting capabilities of the ANN model, which demonstrate the relationship between the conceptual framework and the empirical data.

5. Modeling Methodology

Time series forecasting techniques included ARIMA, Brown, Holt, Winters, ANN, ETS, ARFIMA, BATS, TBATS, Box-Cox parameter, Box-Cox transformation, Croston, structural, mean forecast, exponential smoothing forecasts, Theta, linear, and Bagged. These methods were chosen because of their efficacy in time series analysis. A range of forecasting techniques is used to analyze the trends of seasonal inflation, and the relative benefits of statistical, machine learning, and hybrid approaches are assessed.

6. Results and Discussion

Four common accuracy metrics were used to evaluate the forecasting ability of several time series and machine learning models for monthly inflation in the United States: RMSE, MAE, MAPE, and MASE. Table 2 presents the fitted models along with their forecasting accuracy measures for monthly inflation in the USA. All of these indicators show lower values for higher anticipated accuracy and model reliability. By far the most accurate predictions were produced by the ANN of all the models that were examined. The ANN’s remarkable capacity to capture the nonlinear, dynamic, and interactive nature of inflation, a process influenced by a variety of factors like shifts in energy costs, labor market conditions, global supply disruptions, and monetary policy actions, is demonstrated by this result. By dynamically learning complex connections within data, the ANN adjusts to structural and regime changes in the economy, unlike classic linear techniques that are predicated on set assumptions. With the lowest RMSE (0.1132), MAE (0.0809), MAPE (4.80%), and MASE (0.2298), the ANN model performs better than all other models. It is about 10–11% better than ARIMA and 5–7% better than hybrid models like ARIMA-ANN and ARIMA-ETS-ANN-TBATS. ANN offers the most reliable and accurate short-term inflation forecast, as seen by the larger errors of traditional models such as ARIMA, Holt-Winters, ETS, and ARFIMA.

The ARIMA–ANN hybrid model achieved the second-best results, demonstrating the advantages of combining the nonlinear learning capabilities of ANN with the linear forecasting framework of ARIMA. In a similar vein, the combination of ARIMA, ETS, ANN, TBATS, and STLM showed competitive performance, indicating that hybrid frameworks can successfully include complementary model strengths to simultaneously capture abnormal fluctuations, trends, and seasonality. These models can handle both short-term volatility and long-term persistence in inflation behavior due to the combination of machine learning and statistics.

Traditional techniques like ARIMA, ARFIMA, and Holt-Winters have a moderate but consistent accuracy, which reflects their ability to identify patterns and seasonal components, but their limited ability to adjust to abrupt changes in the economy. As inflation volatility rose, models with somewhat bigger errors, like ETS, structural, and theta, showed less response. Overfitting compromises a model’s out-of-sample performance by fitting the previous data too closely. Despite successfully capturing intricate seasonal patterns in this investigation, the BATS and TBATS models’ comparatively high MASE values raise the possibility of overfitting. Due to component redundancy and model complexity, the Bagged ensemble model also showed good in-sample performance but poorer generalizability.

Simpler methods like mean forecast, Croston’s model, and linear regression provide noticeably higher error levels, which highlights their inadequacies in capturing the dynamic and feedback-driven mechanisms of inflation. The poor performance of the Box–Cox transformation model shows that stabilizing variance alone is insufficient to explain asymmetric and nonlinear reactions to inflation. The computed transformation parameter (λ = 0.2059) provided no variance adjustment and had low predictive power.

The models with the lowest accuracy in simulating inflation were the Box-Cox transformation, Croston’s method, and linear regression. Because the Box–Cox method concentrates on variance stability rather than temporal dependence, it fails when inflation is affected by asymmetric or persistent shocks. Assuming that events are independent and uncommon, Croston’s method—which was developed for predicting intermittent demand—is unsuitable for continuous data, such as inflation, leading to skewed and unstable forecasts. Because linear regression assumes continuous relationships between variables, it ignores the nonlinearities and lag feedback effects that characterize real-world inflation.

Underfitting results from this rigidity, especially when the economy changes as a result of external shocks, policy adjustments, or shifts in commodity prices. These flaws demonstrate how static or oversimplified models fail to adequately represent the complex, time-varying nature of inflation. Inflation in the US is influenced by structural shifts, external disturbances, and shifting policy responses, all of which require dynamic model learning and adaptation. Because of their data-driven adaptability, ANN and hybrid techniques thus exhibit clear advantages over traditional frameworks in controlling such complicated dynamics.

There are important policy ramifications to the increased predictive capacity of ANN-based models. In order to forecast shifts in the structural regime and inflationary pressures, central banks can use these models as early warning systems. By exposing nonlinear links, they facilitate faster and more fact-based monetary policy adjustments. It is challenging to convey policies in a way that is understandable due to the restricted interpretability of neural networks, notwithstanding their accuracy. To get accuracy and transparency, it is feasible to combine ANN forecasts with interpretable statistical models, like ARIMA or ETS.

The necessity of finding a balance between generalizability and flexibility is highlighted by the overfitting observed in complex models like TBATS and Bagged Ensembles. Even while these models are capable of capturing complex seasonal and structural characteristics, if they are too complicated and replicate noise rather than actual trends, their usefulness for future forecasting may be reduced. For accurate inflation modeling, frameworks that balance responsiveness and stability must be adaptable yet cost-effective.

Overall, the results demonstrate that the naturally nonlinear inflation dynamics in the US are influenced by a number of interacting influences. Adaptive and hybrid models that account for this complexity, particularly ANN-based methods, yield the most accurate predictions. However, statistical or transformation-based models alone are insufficiently responsive to capture shifts in the structure and behavior of the economy. The aforementioned findings highlight how important it is to combine flexible, data-driven methodologies with easily understood frameworks in order to increase the precision and policy relevance of inflation forecasts.

The forecast for inflation of the next 10 months of inflation of the USA is shown in Fig. 5.

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Fig. 5.

ANN model forecast of U.S. inflation from June 2024 to March 2025, showing predicted values with 80% and 95% confidence intervals.

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The smooth fitted line of the ANN model closely resembles the real series in Fig. 5, demonstrating how well it captures the underlying inflation dynamics. The projection for June 2024–March 2025, based on the trained model, indicates a slow increase in inflation. The forecasted inflation rate, which is also displayed in Table 3, increases from roughly 2.32% in June 2024 to 2.37% in March 2025, suggesting modest inflationary pressure in the upcoming months.

Both the 80% and 95% prediction ranges are shown in the forecast graphic as shaded bands surrounding the forecast line. The degree of prediction uncertainty is reflected in these intervals. Toward the start of the forecast horizon (June–August 2024), the bands are comparatively narrow, suggesting strong model confidence in short-term forecasts. But as the time horizon grows toward early 2025, the gaps get wider, showing that forecast uncertainty rises with time. This is a typical occurrence in time series modeling because of compounding unpredictability and data noise.

The predicted time is visually separated from the historical data by the darkened grey area in the plot. Within this range, the projection is steady and does not alter suddenly, suggesting that the ANN model anticipates modest monthly rises in inflation.

All things considered, the ANN model produces accurate short-term predictions and realistic medium-term estimates. However, the increasing prediction intervals emphasize how crucial it is to exercise caution when evaluating longer-term projections because these expected values may deviate due to unforeseen structural changes, policy changes, or economic shocks.

7. Modelling Inflation Rate Data using Probability Models

Probability distributions play a crucial role in capturing variability, uncertainty, and the probabilistic nature of temporal processes in time series data, thereby supporting robust analysis, forecasting, and decision-making. In recent years, traditional lifetime distributions such as the Weibull and inverse Weibull have been among the most widely applied. These distributions typically exhibit left-skewed density behavior, consistent with data histograms that also show left skewness. In such cases, the left (negative) tail of the distribution is longer than the right, and the mean tends to be lower than the median, suggesting that negatively skewed distributions provide a suitable fit. To model these data, a range of distributions can be employed, including the W, EW, KW, E, and IW distributions. The flexibility of their hazard rate (HR) functions—ranging from monotone to non-monotone—has further driven their broad applicability across diverse fields.

Table 4 provides the maximum likelihood estimates (MLEs) of the model parameters with their standard errors (Ses), along with the corresponding information criteria values Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), and Hannan-Quinn information criterion (HQIC) for the competing distributions (W, EW, KW, E, and IW). From these results, it is evident that the W distribution outperforms its counterparts, as it records the lowest values across all information criteria. In addition, its parameter estimates show relatively small standard errors, indicating reliable estimation. The density plots further support these findings, showing that the W distribution offers the best fit to the observed data. Therefore, the Weibull distribution emerges as the most appropriate model for capturing the dynamics of U.S. seasonal inflation data.

Fig. 6 displays the fitted W, EW, KW, E, and IW distributions. While each distribution shows slight differences in tail behavior, peak height, and width, the W distribution provides the closest fit to the data and demonstrates superior performance compared to the others.

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Fig. 6.

Estimated density plots of USA seasonal inflation data.

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8. Discussion

This study examines and contrasts a variety of methods for predicting inflation in the United States, including probabilistic methods, stochastic linear and nonlinear time series models, and hybrid frameworks. Using monthly data from January 2003 to September 2024, the analysis evaluates how well different models—including ANNs—predict outcomes. Standard statistical criteria, such as MAE, MAPE, MASE, and RMSE, were used to assess model accuracy in order to guarantee a thorough comparison of forecasting capacity. The ANN showed the lowest error values (RMSE = 0.1132, MAE = 0.0809, MAPE = 4.7993, MASE = 0.2298) among all competing approaches, demonstrating its excellent ability to minimize forecast errors and accurately represent the nonlinear and changing nature of inflation dynamics.

The results confirm that ANNs perform better than traditional and hybrid methods, making them strong and trustworthy instruments for forecasting changes in inflation in intricate economic contexts.

Along with forecasting, the study uses probabilistic modeling to investigate the seasonal nature of U.S. inflation. Several probability distributions, such as the W, EW, KW, E, and IW distributions, were used to reflect the data’s inherent variability and cyclical character. These models’ appropriateness was evaluated using information-based criteria such as HQIC, AIC, CAIC, and BIC. The Weibull model, which fit the seasonal inflation data the best overall, had the lowest values across all criteria among the investigated distributions. The EW, KW, E, and IW distributions, on the other hand, yielded relatively larger values, indicating a weaker fit to the patterns in the data. These results show that the Weibull distribution is the best suitable option for capturing the probabilistic nature of U.S. inflation dynamics because of its exceptional adaptability to asymmetric and heavy-tailed features of inflation.

According to the overall evaluation based on forecasting accuracy metrics (RMSE, MAE, MAPE, and MASE) and information criteria (AIC, CAIC, BIC, and HQIC), the models with the lowest values across these benchmarks offer the most accurate and reliable depiction of inflation trends. These models provide important insights for economic forecasting and policymaking by improving predicted accuracy and advancing our grasp of the structural characteristics underpinning inflation.

A more thorough knowledge of inflation behavior can be obtained by combining probabilistic modeling with time series forecasting, especially when it comes to reflecting the risk and uncertainty of future price changes. Even though the main goal of time series models like ARIMA, ETS, and ANN is to find trends, seasonality, and temporal dependencies, they frequently provide deterministic predictions that might not accurately represent the spectrum of potential future events. Conversely, probabilistic models use the statistical distribution of past data to estimate the likelihood of different inflation scenarios, such as moderate inflation, disinflation, or abrupt price spikes, in order to quantify uncertainty.

This combination improves inflation analysis’s interpretive and predictive capabilities from an economic standpoint. While probabilistic models explain the distributional characteristics and tail behaviors that indicate volatility or the probability of extreme outcomes, time series projections offer point estimates that aid in tracking inflationary trends over time. For example, researchers can detect asymmetries in inflation dynamics—where positive or negative shocks may have varying magnitudes and persistence—by using the Weibull distribution’s flexibility in simulating skewness and heavy tails. These kinds of insights are essential for policymakers because they show the likelihood of deviations from goal levels in addition to the anticipated inflation trend.

Analysts can create reliable and useful projections for decision-making by integrating these two frameworks. By converting statistical results into risk-based metrics, the probabilistic viewpoint aids in the creation of monetary policies that take uncertainty into consideration. In order to inform interest rate adjustments and inflation-targeting tactics, central banks, for instance, can use the probability density functions to determine the possibility that inflation will surpass particular thresholds. Thus, by connecting statistical variability to practical policy implications, risk assessment, and strategic planning, the combination of probabilistic and time series modeling enhances not just technical performance but also the economic interpretation.

9. Conclusions

The primary aim of this study was to enhance inflation forecasting accuracy by combining advanced hybrid and non-hybrid time-series models with rigorous statistical methods. Compared with ARIMA (RMSE = 0.1268, MAPE = 5.38%) and ETS (RMSE = 0.1339, MAPE = 5.51%), the ANN model produced the lowest RMSE (0.1132), MAE (0.0809), MAPE (4.80%), and MASE (0.2298), confirming its superior predictive performance. In addition, model selection criteria—including AIC (233.9582), CAIC (234.0054), BIC (241.0563), and HQIC (236.8127)—indicated that the Weibull distribution was the most suitable probabilistic model for capturing seasonal inflation patterns and modeling the observed fluctuations.

The ANN-based forecasts reveal a steadily rising inflation trajectory with clear seasonal peaks, providing economists, financial analysts, and policymakers with actionable, data-driven insights for anticipating inflationary pressures and designing effective policy responses. By integrating modern computational approaches with traditional statistical techniques, this study establishes a robust forecasting framework that enhances methodological development while offering practical tools for economic decision-making.

Future research could extend the hybrid forecasting framework by incorporating deep-learning architectures such as LSTM (long short-term memory) or GRU (gated recurrent unit) to capture more complex nonlinear dynamics. Additional studies may also evaluate model performance using post-2024 data, compare forecasting accuracy across countries or historical periods, or integrate macroeconomic covariates to further assess model generalizability. Overall, this study makes a substantial contribution by advancing state-of-the-art forecasting methodology and delivering a practical, high-accuracy framework that supports real-world inflation monitoring and policy formulation.