Multivariate Time Series Models Research Articles

Package CovRegpy: Regularized covariance regression and forecasting in Python

Abstract This paper will outline the functionality available in the CovRegpy package which was written for actuarial practitioners, wealth managers, fund managers, and portfolio analysts in the language of Python 3.11. The objective is to develop a new class of covariance regression factor models for covariance forecasting, along with a library of portfolio allocation tools that integrate with this new covariance forecasting framework. The novelty is in two stages: the type of covariance regression model and factor extractions used to construct the covariates used in the covariance regression, along with a powerful portfolio allocation framework for dynamic multi-period asset investment management. The major contributions of package CovRegpy can be found on the GitHub repository for this library in the scripts: CovRegpy.py, CovRegpy_DCC.py, CovRegpy_RPP.py, CovRegpy_SSA.py, CovRegpy_SSD.py, and CovRegpy_X11.py. These six scripts contain implementations of software features including multivariate covariance time series models based on the regularized covariance regression (RCR) framework, dynamic conditional correlation (DCC) framework, risk premia parity (RPP) weighting functions, singular spectrum analysis (SSA), singular spectrum decomposition (SSD), and X11 decomposition framework, respectively. These techniques can be used sequentially or independently with other techniques to extract implicit factors to use them as covariates in the RCR framework to forecast covariance and correlation structures and finally apply portfolio weighting strategies based on the portfolio risk measures based on forecasted covariance assumptions. Explicit financial factors can be used in the covariance regression framework, implicit factors can be used in the traditional explicit market factor setting, and RPP techniques with long/short equity weighting strategies can be used in traditional covariance assumption frameworks. We examine, herein, two real-world case studies for actuarial practitioners. The first of these is a modification (demonstrating the regularization of covariance regression) of the original example from Hoff & Niu ((2012). Statistica Sinica, 22(2), 729–753) which modeled the covariance and correlative relationship that exists between forced expiratory volume (FEV) and age and FEV and height. We examine this within the context of making probabilistic predictions about mortality rates in patients with chronic obstructive pulmonary disease. The second case study is a more complete example using this package wherein we present a funded and unfunded UK pension example. The decomposition algorithm isolates high-, mid-, and low-frequency structures from FTSE 100 constituents over 20 years. These are used to forecast the forthcoming quarter’s covariance structure to weight the portfolio based on the RPP strategy. These fully funded pensions are compared against the performance of a fully unfunded pension using the FTSE 100 index performance as a proxy.

The development of a road safety policy index and its application in evaluating the effects of road safety policy

The paper presents an exploratory study of a road safety policy index developed for Norway. The index consists of ten road safety measures for which data on their use from 1980 to 2021 are available. The ten measures were combined into an index which had an initial value of 50 in 1980 and increased to a value of 185 in 2021. To assess the application of the index in evaluating the effects of road safety policy, negative binomial regression models and multivariate time series models were developed for traffic fatalities, fatalities and serious injuries, and all injuries. The coefficient for the policy index was negative, indicating the road safety policy has contributed to reducing the number of fatalities and injuries. The size of this contribution can be estimated by means of at least three estimators that do not always produce identical values. There is little doubt about the sign of the relationship: a stronger road safety policy (as indicated by index values) is associated with a larger decline in fatalities and injuries. A precise quantification is, however, not possible. Different estimators of effect, all of which can be regarded as plausible, yield different results.

Spatiotemporal Land Use/Land Cover Changes and Impact on Urban Thermal Environments: Analyzing Cool Island Intensity Variations

This study pioneers the comprehensive evaluation of the spatiotemporal evolution of land use/land cover (LULC) in Hangzhou city, introducing the novel water body shape index (WBSI) to analyze its seasonal impacts on the urban thermal environment and urban cool island (UCI) effects, uncovering distinct patterns of thermal regulation. It particularly investigates how distance gradients and the water body shape index (WBSI) influence land surface temperature (LST) in the urban core. The region’s climate, featuring hot summers and cold winters, highlights significant seasonal LST variations. Addressing a gap in existing UCI research, the analysis extends beyond the typical large-scale planning focus to include small-scale, high-resolution aspects. Employing remote sensing and geographic information system (GIS) analysis techniques, this study analyzes the seasonal dynamics in Hangzhou’s central urban area. High-resolution LST data, obtained through single-channel inversion and resolution enhancement algorithms, are crucial to this analysis. This study employs the maximum likelihood classification method to analyze land use and land cover changes from 1990 to 2020. This analysis reveals potential drivers of urban thermal environment changes, such as the expansion of residential and commercial areas and the reduction in green spaces. Different regions in LST data are delineated to assess the cool island effect, and the complexity of water body boundaries is quantified using the water body shape index. Spatial and temporal patterns of LST changes are investigated using multivariate regression and time-series analysis models. We identified significant changes in LULC over the past 30 years in Hangzhou, closely correlating with a continuous rise in LST. This observation underscores a clear finding: the strategic importance of blue–green infrastructure in mitigating urban heat, a novel insight that extends the current understanding of urban thermal dynamics. A clear and novel finding of this study is that the intensity of the cool island effect from large water bodies not only diminishes with distance but is intricately influenced by the complexity of their shapes, as quantified by the WBSI, whereas the complexity of their boundaries enhances this effect. Additionally, the regulatory role of the cool island effect is observed to vary seasonally, being most pronounced in summer and less so in autumn and winter, thereby demonstrating a positive impact. In conclusion, our findings innovatively highlight how the specific shapes of water bodies, quantified through the water body shape index (WBSI), emerge as critical, yet previously underappreciated, drivers in modulating the urban thermal environment. This underscores a new avenue for urban planning, advocating for the strategic design of water bodies within urban landscapes. It also finds that spatial factors and seasonal variations significantly affect the intensity of the cool island effect. These findings offer valuable evidence for urban planning and climate change adaptation, emphasizing balancing natural elements with the built environment in urban design.

Gaussian Process Vector Autoregressions and Macroeconomic Uncertainty

We develop a nonparametric multivariate time series model that remains agnostic on the precise relationship between a (possibly) large set of macroeconomic time series and their lagged values. The main building block of our model is a Gaussian process prior on the functional relationship that determines the conditional mean of the model, hence, the name of Gaussian process vector autoregression (GP-VAR). A flexible stochastic volatility specification is used to provide additional flexibility and control for heteroscedasticity. Markov chain Monte Carlo (MCMC) estimation is carried out through an efficient and scalable algorithm which can handle large models. The GP-VAR is used to analyze the effects of macroeconomic uncertainty, with a particular emphasis on time variation and asymmetries in the transmission mechanisms.

Multi-Scale Dynamic Graph Learning for Time Series Anomaly Detection (Student Abstract)

The success of graph neural networks (GNNs) has spurred numerous new works leveraging GNNs for modeling multivariate time series anomaly detection. Despite their achieved performance improvements, most of them only consider static graph to describe the spatial-temporal dependencies between time series. Moreover, existing works neglect the time and scale-changing structures of time series. In this work, we propose MDGAD, a novel multi-scale dynamic graph structure learning approach for time series anomaly detection. We design a multi-scale graph structure learning module that captures the complex correlations among time series, constructing an evolving graph at each scale. Meanwhile, an anomaly detector is used to combine bilateral prediction errors to detect abnormal data. Experiments conducted on two time series datasets demonstrate the effectiveness of MDGAD.

VECTOR AUTOREGRESSIVE WITH OUTLIER DETECTION ON RAINFALL AND WIND SPEED DATA

Vector Autoregressive (VAR) is a multivariate time series model that analyzes more than one variable where each variable in the model is endogenous. VAR is one of the models used in forecasting rainfall and wind speed. In observations of rainfall and wind speed, there are usually a series of events whose values are far from other observations or can be said to be outliers. The purpose of this study is to compare the VAR model on rainfall and wind speed data before and after outlier detection. This study uses secondary data, namely monthly data on rainfall and wind speed from 2019 to 2021. From the analysis results, the smallest AIC value obtained in the VAR model before outlier detection was 4.94, then the smallest AIC value in the VAR model after outlier detection was 0.25. Thus, it can be concluded that the best model is obtained in the VAR model after outlier detection seen from the smallest AIC value of the two VAR models.

Restored off‐channel pond habitats create thermal regime diversity and refuges within a Mediterranean‐climate watershed

Cool‐water habitats provide increasingly vital refuges for cold‐water fish living on the margins of their historical ranges; consequently, efforts to enhance or create cool‐water habitat are becoming a major focus of river restoration practices. However, the effectiveness of restoration projects for providing thermal refuge and creating diverse temperature regimes at the watershed scale remains unclear. In the Klamath River in northern California, the Karuk Tribe Fisheries Program, the Mid‐Klamath Watershed Council, and the U.S. Forest Service constructed a series of off‐channel ponds that recreate floodplain habitat and support juvenile coho salmon (Oncorhynchus kisutch) and steelhead (O. mykiss) along the Klamath River and its tributaries. We instrumented these ponds and applied multivariate autoregressive time series models of fine‐scale temperature data from ponds, tributaries, and the mainstem Klamath River to assess how off‐channel ponds contributed to thermal regime diversity and thermal refuge habitat in the Klamath riverscape. Our analysis demonstrated that ponds provide diverse thermal habitats that are significantly cooler than creek or mainstem river habitats, even during severe drought. Wavelet analysis of long‐term (10 years) temperature data indicated that thermal buffering (i.e. dampening of diel variation) increased over time but was disrupted by drought conditions in 2021. Our analysis demonstrates that in certain situations, human‐made off‐channel ponds can increase thermal diversity in modified riverscapes even during drought conditions, potentially benefiting floodplain‐dependent cold‐water species. Restoration actions that create and maintain thermal regime diversity and thermal refuges will become an essential tool to conserve biodiversity in climate‐sensitive watersheds.

Bivariate Time Series Analysis of Nigeria Gross Domestics Product and Communication Sector

The need to establish a relation between Gross Domestic Product and Communication sector was the major focus of this research. This research work investigated the contribution of Communication sector to the Gross Domestic product of Nigeria Economy. With the aid of ACF and PACF, ARIMA (1 1 1) was suggested for both variables. Alternative multivariate time series models used for the analysis were ARIMAV, MARDL and MARDL-MA models. The research has established interaction and interdependence between the two macroeconomic variables, and has also revealed that each of the variable has contributed significantly to each other at first time lag. The error variances of the bivariate time series model were derived for GDP and Communication sector. When comparing the three models for the two economic variables, ARIMAV model for Gross Domestic Product has the least error variance of 0.2183 making it the best model, while MARDL model for communication sector produced the least error variance of 0.0723, thereby indicating that MARDL model outperformed ARIMAV and MARDL-MA models for communication sector. Hence, this research has brought to focus the fact that performance of a model over another is predicated upon the nature of the economic data. That means there is no fixed multivariate time series model for a given macroeconomic data due to the dynamic nature of the time series.

Graph spatiotemporal process for multivariate time series anomaly detection with missing values

The detection of anomalies in multivariate time series data is crucial for various practical applications, including smart power grids, traffic flow forecasting, and industrial process control. However, real-world time series data is usually not well-structured, posting significant challenges to existing approaches: (1) The existence of missing values in multivariate time series data along variable and time dimensions hinders the effective modeling of interwoven spatial and temporal dependencies, resulting in important patterns being overlooked during model training; (2) Anomaly scoring with irregularly-sampled observations is less explored, making it difficult to use existing detectors for multivariate series without fully-observed values. In this work, we introduce a novel framework called GST-Pro, which utilizes a graph spatiotemporal process and anomaly scorer to tackle the aforementioned challenges in detecting anomalies on irregularly-sampled multivariate time series. Our approach comprises two main components. First, we propose a graph spatiotemporal process based on neural controlled differential equations. This process enables effective modeling of multivariate time series from both spatial and temporal perspectives, even when the data contains missing values. Second, we present a novel distribution-based anomaly scoring mechanism that alleviates the reliance on complete uniform observations. By analyzing the predictions of the graph spatiotemporal process, our approach allows anomalies to be easily detected. Our experimental results show that the GST-Pro method can effectively detect anomalies in time series data and outperforms state-of-the-art methods, regardless of whether there are missing values present in the data. Our code is available: https://github.com/huankoh/GST-Pro.

Welfare, egalitarianism, and polarization: the politics of noncontributory social programs

AbstractThis study analyzes the link between egalitarian ideals and the rise in party polarization in Congress. To demonstrate how philosophical differences over conceptions of fairness, equality, and justice help explain the recent growth in partisanship over the past few decades, I argue one overlooked explanatory factor which assists in capturing this ideological rift is noncontributory welfare spending. Recovering annual ideal point estimates between 1947 and 2018 that are comparable with annual federal spending, I use multivariate time series models and find convincing evidence which suggests welfare outlays have a strong short- and long-run effect on polarization. Moreover, analysis of the roll call record also shows when ideal point estimates are recovered by specific policy area, lawmakers exhibit higher levels of ideological separation on welfare compared to, among others, policies such as defense and transportation. Robustness checks confirm these findings also hold even when controlling for income inequality.

Machine Learning Data-Driven Residential Load Multi-Level Forecasting With Univariate and Multivariate Time Series Models Toward Sustainable Smart Homes

Machine Learning Data-Driven Residential Load Multi-Level Forecasting With Univariate and Multivariate Time Series Models Toward Sustainable Smart Homes

A multivariate heavy-tailed integer-valued GARCH process with EM algorithm-based inference

A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic (GARCH) process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the expectation-maximization (EM) algorithm are proposed for estimating the parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed process. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-tailedness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. Two empirical applications of our approach are provided. The first application concerns modeling bivariate count time series data on cannabis possession-related offenses in Australia, while the second one involves modeling intraday high-frequency financial transactions data from multiple holdings in the U.S. financial market.

Impact of Energy Efficiency and Financial Support on Green Upgrading in China’s Industrial Sector

The report from the 20th National Congress of China emphasizes the importance of focusing on the clean, low-carbon, and efficient use of energy, increasing financial support, and promoting green upgrading within the industrial sector. This paper, based on annual data, employs the entropy weight method to construct a comprehensive index reflecting the impact of green upgrading in industrial sectors. To delve deeper, it utilizes the DEA model to measure energy efficiency and its subdivision BCC model to break down energy efficiency into technical and scale efficiency. The financial support landscape is examined from the vantage points of both direct and indirect financing. Using a multivariate time series model, this paper thoroughly investigates the influence of energy efficiency and financial support on the green upgrading of the industrial sector. The findings reveal a significant positive impact of both energy efficiency and financial support on green upgrading in industrial industries. Notably, scale efficiency emerges as the primary driver of energy efficiency. Moreover, indirect financing proves to be more effective in promoting financial support than direct financing. The empirical results retain their robustness even after substituting explanatory variables. The study concludes by contextualizing the research findings within the current real-world scenario, offering practical insights, and proposing specific recommendations.

Bayesian vector heterogeneous autoregressive modelling

The Bayesian vector autoregressive (BVAR) model with the Minnesota prior proposed by Litterman [Litterman RB. Forecasting with bayesian vector autoregressions-five years of experience. J Business Economic Statist.1986;4(1):25–38.] has been very successful in multivariate time series modelling, providing better forecasting performance. However, the conventional Minnesota prior for BVAR depends only on the latest lag; in turn, it is not suitable for multivariate long memory time series forecasting. This study extends BVAR to (possibly) high-dimensional long memory time series. To this end, we incorporate a vector heterogeneous autoregressive (VHAR) structure to accommodate long-term persistence and impose priors on distant lags as well. Our proposed Bayesian VHAR (BVHAR) models, the so-called BVHAR-S and BVHAR-L, are easy to implement by using the Normal-inverse-Wishart prior and added dummy variable approach of [Bańbura M, Giannone D, Reichlin L. Large bayesian vector auto regressions. J Appl Econom. 2010;25(1):71–92.]. A supporting simulation study also proves posterior consistency. We further apply our models to nine CBOE Volatility Indices (VIXs) and show that our BVHAR models perform best in forecasting with the narrowest forecasting region.

Reduced-Rank Envelope Vector Autoregressive Model

The standard vector autoregressive (VAR) models suffer from overparameterization which is a serious issue for high-dimensional time series data as it restricts the number of variables and lags that can be incorporated into the model. Several statistical methods, such as the reduced-rank model for multivariate (multiple) time series (Velu, Reinsel, and Wichern; Reinsel and Velu; Reinsel, Velu, and Chen) and the Envelope VAR model (Wang and Ding), provide solutions for achieving dimension reduction of the parameter space of the VAR model. However, these methods can be inefficient in extracting relevant information from complex data, as they fail to distinguish between relevant and irrelevant information, or they are inefficient in addressing the rank deficiency problem. We put together the idea of envelope models into the reduced-rank VAR model to simultaneously tackle these challenges, and propose a new parsimonious version of the classical VAR model called the reduced-rank envelope VAR (REVAR) model. Our proposed REVAR model incorporates the strengths of both reduced-rank VAR and envelope VAR models and leads to significant gains in efficiency and accuracy. The asymptotic properties of the proposed estimators are established under different error assumptions. Simulation studies and real data analysis are conducted to evaluate and illustrate the proposed method.

MvTS-library: An open library for deep multivariate time series forecasting

Modeling multivariate time series has been a subject for a long time, which attracts the attention of scholars from many fields including economics, finance, traffic, etc. As the number of models increases, it is desired to design a unified framework to implement and evaluate these models. Based on Pytorch, we propose MvTS, an open library for multivariate time series forecasting. Through a highly modular design, MvTS systematically integrates the various stages of the whole process of model training. Currently, the library contains 33 models and 23 datasets, and is available at https://github.com/MTS-BenchMark/MvTS. Based on MvTS, we conduct extensive experiments on public datasets and demonstrate that the models reproduced by MvTS are effective and universally applicable to many other datasets. MvTS is a systematic, comprehensive, extensible, and easy-to-use multivariate time series forecasting library, and we believe it will contribute to the research of multivariate time series to some extent.

A new approach for time domain analysis of multivariate and functional time series

We apply the classical finite Fourier transform to construct an embedded functional process to a given multivariate time series model. The basic properties of the embedded functional process are presented. It appears that the embedded functional process is quite useful for the model building and prediction. We also provide a new method for approximating an infinite functional process with a finite functional process. The methodology is different from the classical one. Indeed, the active underlying frequencies are taken into consideration rather than the significant eigenvectors. The performance of our method is illustrated through simulations. Interestingly, in doing prediction, it appears that for our proposal the computation time is much shorter compared to existing ones.

Braking Sensor and Actuator Fault Diagnosis With Combined Model-Based and Data-Driven Pressure Estimation Methods

The braking system is significant for intelligent vehicles, which influences vehicle safety directly. However, under harsh working conditions, the inevitable health degradation and even failure of the braking system may cause severe safety issues. This paper proposes a novel braking actuator and sensor fault diagnosis scheme with combined model-based and data-driven pressure estimation methods. The model-based wheel cylinder pressure (WCP) estimator is established first based on the mathematical model of the hydraulic control unit (HCU) from the perspective of cause. The data-driven WCP estimator is then proposed based on the vehicle dynamics multivariate time series (MTS) model with the gated recurrent unit (GRU) from the perspective of effect. However, acquiring a large dataset is a practical challenge for real vehicle tests, so a novel data augmentation (DA) method called shifting is presented to enhance the model generalization ability. Next, fault detection, isolation, and identification are realized by comparing the threshold and the cumulative sum (CUSUM) of the residuals generated by the combined WCP estimation methods. The validation results show that the proposed data-driven method outperforms the traditional multilayer perceptron (MLP), long short-term memory (LSTM), and Transformer regarding accuracy and generalization. Vehicle tests simulating sensor and actuator faults validate the proposed fault diagnosis scheme.

Scaling law characteristics and spatiotemporal multicomponent analysis of syphilis from 2016 to 2022 in Zhejiang Province, China.

Syphilis has caused epidemics for hundreds of years, and the global syphilis situation remains serious. The reported incidence rate of syphilis in Zhejiang Province has ranked first in the province in terms of notifiable infectious diseases for many years and is the highest in China. This study attempts to use the scaling law theory to study the relationship between population size and different types of syphilis epidemics, while also exploring the main driving factors affecting the incidence of syphilis in different regions. Data on syphilis cases and affected populations at the county level were obtained from the China Disease Control and Prevention Information System. The scaling relationship between different stages of syphilis and population size was explained by scaling law. The trend of the incidence from 2016 to 2022 was tested by the joinpoint regression. The index of distance between indices of simulation and observation (DISO) was applied to evaluate the overall performance of joinpoint regression model. Furthermore, a multivariate time series model was employed to identify the main driving components that affected the occurrence of syphilis at the county level. The p value less than 0.05 or confidence interval (CI) does not include 0 represented statistical significance for all the tests. From 2016 to 2022, a total of 204,719 cases of syphilis were reported in Zhejiang Province, including 2 deaths, all of which were congenital syphilis. Latent syphilis accounted for 79.47% of total syphilis cases. The annual percent change (APCs) of all types of syphilis, including primary syphilis, secondary syphilis, tertiary syphilis, congenital syphilis and latent syphilis, were - 21.70% (p < 0.001, 95% CI: -26.70 to -16.30), -16.80% (p < 0.001, 95% CI: -20.30 to -13.30), -8.70% (p < 0.001, 95% CI: -11.30 to -6.00), -39.00% (p = 0.001, 95% CI: -49.30 to -26.60) and - 7.10% (p = 0.008, 95% CI: -11.20 to -2.80), respectively. The combined scaling exponents of primary syphilis, secondary syphilis, tertiary syphilis, congenital syphilis and latent syphilis based on the random effects model were 0.95 (95% CI: 0.88 to 1.01), 1.14 (95% CI: 1.12 to 1.16), 0.43 (95% CI: 0.37 to 0.49), 0.0264 (95% CI: -0.0047 to 0.0575) and 0.88 (95% CI: 0.82 to 0.93), respectively. The overall average effect values of the endemic component, spatiotemporal component and autoregressive component for all counties were 0.24, 0.035 and 0.72, respectively. The values of the autoregressive component for most counties were greater than 0.7. The endemic component of the top 10 counties with the highest values was greater than 0.34. Two counties with value of the spatiotemporal component higher than 0.1 were Xihu landscape county and Shengsi county. From 2016 to 2022, the endemic and autoregressive components of each county showed obvious seasonal changes. The scaling exponent had both temporal trend characteristics and significant heterogeneity in the association between each type of syphilis and population size. Primary syphilis and latent syphilis exhibited a linear pattern, secondary syphilis presented a superlinear pattern, and tertiary syphilis exhibited a sublinear pattern. This suggested that further prevention of infection and transmission among high-risk populations and improvement of diagnostic accuracy in underdeveloped areas is needed. The autoregressive components and the endemic components were the main driving factors that affected the occurrence of syphilis. Targeted prevention and control strategies must be developed based on the main driving modes of the epidemic in each county.

Forecasting Worldwide Temperature from Amazon Rainforest Deforestation Using a Long-Short Term Memory Model

Biosphere–atmosphere interactions are a critical component of the Earth’s climate system. Many of these interactions are currently contributing to temperature increases and accelerating global warming. One of the main factors responsible for this is land use and land cover changes; in particular, this work models the interaction between Amazon rainforest deforestation and global temperatures. A Long Short-Term Memory (LSTM) neural network is proposed to forecast temperature trends, including mean, average minimum, and average maximum temperatures, in 20 major cities worldwide. The Amazon rainforest, often referred to as the Earth’s “lungs”, plays a pivotal role in regulating global climate patterns. Over the past two decades, this region has experienced significant deforestation, largely due to human activities. We hypothesize that the extent of deforestation in the Amazon can serve as a valuable proxy for understanding and predicting temperature changes in distant urban centers. Using a dataset that tracks cumulative deforestation from 2001 to 2021 across 297 municipalities in the Amazon rainforest, a multivariate time series model was developed to forecast temperature trends worldwide up to 2030. The input data reveal a variety of behaviors, including complex deforestation patterns. Similarly, the forecasted temperature data showcases diverse trends. While some cities are expected to exhibit a steady temperature increase, others may experience gradual changes, while some cities may undergo drastic and rapid temperature shifts. Our findings contribute to a deeper understanding of the far-reaching impacts of deforestation on global climate patterns and underscore the importance of preserving vital ecosystems like the Amazon rainforest.