Introduction
This study involves a multivariate time series analysis mainly based on temperature data from Karnataka,
India. Using advanced statistical machine learning techniques, we examine the temporal trends, seasonal trends, and potential relationships in the temperature dataset Our findings reveal small-scale temperature variations in Karnataka, and provide insights which is valuable for meteorological studies and local climate change assessments. This review highlights the importance of temperature-focused research in understanding local climate change and identifies targeted strategies for climate resilience and adaptation in Karnataka. Arima model to be selected for best possible outcomes.

An analysis of temperature trends is essential for understanding climate variability, regional manifestations
and dynamics, as well as the implications of climate change. Karnataka, a state in the southern part of India, experiences, various climate patterns as a result of its geographical location, seasonal monsoons, and human-induced factors.Therefore, the current research intends to present a multivariate time series analysis of the temperature while only focusing on the available data from Karnataka to uncover trends, variation, regularity, and relationships within this meteorological variable, as it is crucial to all of the other variables. Temperature is an essential indicator of climate change and variation, as it may display long-term
trends, seasonal patterns, and short-term fluctuations. Therefore, it is possible to determine temporal patterns and seasonality and likely correlations within the datasets. For a climate researcher, policymaker, and stakeholder, it is important to know all the above patterns in order to develop an appropriate climate adaptive and a betterment strategy suitable to the climate challenged region. The analysis of the
multivariate time series by its dynamics allows us not only to see the general trend of temperature but also to assess the dependence of the temperature process on other meteorological parameters. Nevertheless, in this paper, we studied this pattern only within the bounds of one variable, as the most crucial and decisive, namely the air temperature. This work gives an idea of the dynamics of the temperature process, which helps to know the variability of the climate in the Karnataka region, which in turn will help the people and the government adapt and a better the measures appropriate for the climatic region.

Techniques Used for Computation
In our research on multivariate time series analysis focused on the temperature data of Karnataka for climate analysis, we have used various regression and forecasting models including Linear Regression, Lasso Regression, Ridge Regression, ANN, ensemble learning and ARIMA. We were aimed at determining out the most precise model for forecasting the temperature variations. Following the experiment and calculation, we discovered that the ARIMA model always gave superior results to the rest of the models. It delivered the most beneficial outcomes including temperature predictions varying between 25 and 27 which was quite close to the actual climate data.

Due to the quality of ARIMA model to fit the both linear and non-linear trends in temperature data, it is the most suitable for depicting the complex temporal patterns. This is even more critical in climate analysis where small variations and long-term trends are crucial for meticulous understanding of climate dynamics.
While the ANN ensemble learning method has the potential to capture very complex patterns, it was unable to deliver accurate temperatures forecasts in our case. The wide range of forecast from -15.0 to 15.3 shows that it does not have enough precision and reliability in predicting the actual temperature trend. On a related note, we saw that Lasso Regression and Ridge Regression may not provide better performance than Linear
Regression in some cases. Their performance resembles Linear Regression which suggest minimal gain in capturing the non-linear temperature changes in our data set.

To see if the model did well, we had to check the leftover errors to make sure it fits well and guesses right. Using the ARIMA model we set up, we then made guesses about future yearly temperatures for Karnataka from what we knew and assumed in our model.Putting the year's real temperatures, model fits, and future
temps on a chart let us check how well the ARIMA model did and what it thinks will happen next. We then figured out what the ARIMA model's outcomes and temperature guesses mean for Karnataka's weather patterns, changes, and what could happen because of these trends. The above graph results also include the data for all four divided parts of seasons which are summer, winter, monsoon and post monsoon respectively and the graphs represent the average of all the seasons temperature in the scale.

Identification of State Gap Present Currently
Although the climate is a multidimensional interconnected system, the focus of the research is on temperature variations in Karnataka using advanced predictive models, and only a little work on this aspect has been done. Unlike the current studies that basically use one or basic statistical methods which just cannot embrace the complex interdependencies of the climatic factors. Firstly, univariate approaches were popular in previous studies but these methods are ineffective as they do not capture the complex interdependencies between climatic factors. This is a major drawback, resulting in models that do not effectively predict the earth's temperature. In addition, there is no reliable validation technique available as the majority of the implemented models validate the results with incorrect patterns. The authors also made it clear that their study is gap filling, as it brings together the most competitive regression and machine learning models as well as the abbreviated ARIMA for a comprehensive exploration of temperature data of Karnataka.

Framework of the Evaluation Techniques
The framework of this study is based around a variety of statistical and deep learning methods and their ability to explain the model's accuracy and reliability. The agreement of the mean absolute error, root mean square error, mean absolute percentage error is solid since they yield the best accuracy in forecasting. Furthermore, another factor that is critically important for the reliability of the predictive models is when the model can be applied to other similar datasets, in other words, the technique should have generalizability. Another thing that we did is ACF (Autocorrelation Function) and PACF (Partial Autocorrelation Function) time-series-specific assessments of the data that are used to validate the lag dependencies. We perform comparative benchmarking against (Linear Regression, Lasso, Ridge) the basic models to prove the effectiveness of our model.

The Comparison between the models used
We have thoroughly analyzed the comparison of classical regression methods and such machine learning methods as Artificial Neural Networks (ANN) and ARIMA. It has been found that ARIMA is the best-fitting model, with a one-quarter mean-square error of 26.5, while the use of an ANN gives incorrect results, which range from -15.5 to 15.3. At the same time, both Lasso and Ridge regression models can be classified as one of the most similar to Linear Regression in terms of predictive power, but they are not able to significantly reinforce it. Through the side-by-side presentation of these models, a well-structured comparison has been built to make a clear and objective point about the good and bad aspects of each strategy.

The Drawbacks
The study is very promising even though the problem of limitations is not new to it. The dataset is both comprehensive and detailed, but it still may not capture the most severe weather changes because of the lack of an extended history of information. Furthermore, by ARIMA being a method that assumes stationary, the usage will probably have a harmful effect on the predictions of long-term trends. The ANN model, while having much to offer, needs to have its hyperparameters tuned and to be approached with the climatic variables, so they can lead to the improvement of its performance. Among the things that could be done are temperature data through satellites and methods of the ensemble type that will lessen the impact of these restrictions in the future.

Innovative Methods/Approaches
We, in our research, bring new learning to the ensemble. Thus, we, for example, have a connection between ARIMA and regression models wherein the features have been mechanically innovated to produce a model of better accuracy for the weather forecasts of Karnataka. We paid attention to many more aspects than conventional methods by adding some of the variables like humidity, wind speed, and air pressure to the basic one, which gave us a new method. Our model helps us visualize temperature changes comprehensively, thus making our regionally based climate prediction strategies better.

Possible Q/A's based on the topic

1. What are the main factors that affect temperature changes that occur in Karnataka?

Answer: Temperature variations in Karnataka are dependent on several environmental and climatic factors. Mentioned below are some factors that influence climate change and environmental factors on temperature of Karnataka:

Meteorological Variables: The amount of humidity, the wind speed, the occurrence of precipitation, and atmospheric pressure are some of the elements that have the most influence on temperature changes.

Geographical Factors: One of the most crucial things connected to the local factors is the altitude, the nearness of the coast, and the cover of the land which strictly impose the rules of temperature changes in all districts of the area.

Seasonal Patterns: The southwest monsoon, high summer temperatures, and the cooling of winter are the seasonal environmental phenomena that arise and disappear in the territory and thus have a significant effect on variation of temperature.

Anthropogenic Activities: The change of the natural surroundings in the process of the settlement process, wood being cut down with an ax, and punctuation of industry, by cities, are some of the results of human activity which not only control climate change in a certain area but also make it a new climate for the microzone and so affect also temperature trends.

2. What are the advantages of the multivariate time series approach in the context of temperature forecasting compared to the univariate models?

Answer: The basic univariate models (e.g., simple ARIMA) treat the temperature as a single variable thus having limited accuracy in prediction. The use of a multivariate approach, on the other hand, results in:
Manages numerous factors that have a significant impact on the prediction accuracy (e.g., humidity, air pressure, atmosphere wind), implementing the accuracy of the forecast. Detects the correlation between various meteorological elements, thus unmasking the meaning underlying the temperature alterations further.

Uses calculating models like Vector Autoregression (VAR) and Long Short-Term Memory (LSTM) which are effective for multiple time-dependent inputs thus reducing the error in prediction.The findings from our study reveal that ARIMA in combination with feature-engineered regression models is notably superior to univariate models in terms of accuracy and robustness.

3. What is the most suitable predictive model for forecasting temperature trends in Karnataka?
   Answer: Our research compared Linear Regression, Ridge, Lasso, ANN, and ARIMA to identify the best-performing model. The major results of our study include:
   ARIMA had the highest accuracy, with an error range of 25-27.
   ANN's results were not consistent with the set conditions, and a range of deviations was recorded (-15.0 to 15.3°C) due to insufficient training data.
   Lasso and Ridge Regression did not offer any significant improvement over traditional Linear Regression.

Conclusion: ARIMA, as long as the data itself is stationary, can be inferred as the most fitting model for the prediction of temperature variations in Karnataka.

Github Repository to access the code and the dataset as well https://github.com/XOSDX/Multivariate-time-series-analysis-on-the-basis-of-temperature-of-Karnataka-BEST-RESULT-ARIMA-MODEL

Best Results Obtained from the ARIMA model