Detrend

Detrend in time series refers to the process of removing a trend or long-term systematic variations from a given time series collection. This process leaves only short-term variations or noise. The trend analyzed can be linear or nonlinear and is used to extract underlying patterns.

The procedure removes trends from the data to prevent results with nonzero mean and trend terms. This is often necessary before obtaining meaningful spectrum results. As a result, detecting trends and detrending data is a popular and important aspect of data analysis. Its applications include climate data analyses, computing correlation functions, and performing spectral analysis.

Detrend in time series refers to the process of removing a trend from a time series, which is a change in the data mean over time. It is essentially removing a distortion from the data. It is essential to understand the data and its underlying structure for detrending. This procedure can be complex since one time series is dependent on another. Additionally, new components of its trend can be added through interaction with other time series. The complete time series must be detrended to address it, extract, and analyze residuals. These residuals or sample mistakes have a dependence structure and must exhibit acceptable qualities to be considered.

The generation of a detrended time series involves three major steps. They determine the overall trend of the data, convert it to a yearly trend, and then calculate the difference between the trend and individual data points to obtain residual values. This method is critical for avoiding time-consuming gaps and ensuring correct data representation. The residue left after this process is known as variability or fluctuation. However, it must be noted that there is a common constraint. The detrended series must have a zero-mean process for the specified period. 

Two common methods for detrended fluctuation analysis in time series data are the differencing method of detrending and the model fitting method of detrending. 

1. Differencing Method Of Detrending: The differencing method of detrending involves removing the trend by differencing. In this method, each observation in the new dataset is the difference between itself and the data’s previous observation. This process removes the trend by focusing on the changes between consecutive data points.
2. Model Fitting Method Of Detrending: The model fitting method of detrending involves detrending by model fitting. The method entails fitting a regression model to the data and calculating the difference between predicted and observed values. These differences provide detrended data. This method makes it easier to visualize a data's seasonal or cyclical trends. 

In Python detrend, when working with time series data, there are several methods available for detrended fluctuation analysis. 

• Moving Average Method: In the Moving Average method, the data is smoothed out by calculating the moving average over a specified window size. This helps in identifying the overall underlying trend in the data, providing a clearer picture of its behavior. We can obtain the detrended data by subtracting the data's moving average from the actual original data. 
• Polynomial Regression: Polynomial regression involves fitting a polynomial function to the time series data to capture the trend. The trend is extracted as data output. Subtracting this trend function from the original data yields the detrended data, allowing us to focus on other patterns or fluctuations.
• The Hodrick-Prescott Filter: The Hodrick-Prescott Filter is a commonly used method in macroeconomics. It separates time series data into its trend and cyclical components. It achieves this by minimizing the trend component's sum of the original data's squared deviations. 

Let us look into a few examples to understand the concept better. 

Example #1

Let's consider Dan, a statistician who wants to study climate change out of interest. He collects temperature data from various locations over decades, spanning from 1960 to 2020. He employs detrend in MATLAB, Stata, and R to test out various techniques. He uses statistical analysis techniques to understand the long-term trend of global warming by applying detrending methods like linear regression. After detrending the data, he observes a significant positive slope in the regression line, indicating an upward trend in global temperatures. 

A regression line showcasing a significant positive slope indicates an upward trend or relationship between the dependent variable (y) and the independent variable (x). Dan then examines residuals, which represent deviations from the long-term trend. By studying these residuals and conducting additional statistical analyses, he gains insights for further research. However, actual climate research requires more comprehensive data collection, sophisticated statistical methods, and collaboration across multiple scientific disciplines. 

Example #2

Proceedings of the National Academy of Sciences published an article on the trend, data detrending, and variability of nonlinear and non-stationary time series a while ago. The paper discussed finding the appropriate method of detrending. It was authored by Zhaohua Wu, Norden E. Huang, Steven R. Long, and Chung-Kang Peng. 

The article presents a coherent definition of a trend in nonlinear and non-stationary time series. It states it as a monotonic function that is intrinsically determined over a specific period, typically the data span. Furthermore, it emphasizes the need for an adaptive method to derive the trend and assumes the presence of a natural time scale. 

The article proposes the Empirical Mode Decomposition (EMD) method as a suitable approach for extracting different trends from a dataset. Once the trend is identified, the corresponding detrending process can be applied, enabling a natural understanding of data variability across different time scales.

Given below are a few reasons why the detrending of time series is essential:

• Unveils inherent trends and fluctuations within time series datasets.
• Detrending makes time series data stationary by removing the changing mean function over time.
• It serves as a preprocessing step before smoothing, enhancing the accuracy of subsequent analyses. It is the first step in spectral analysis.
• Detrending helps reveal underlying seasonal or cyclical patterns or subtrends in the data by removing the overall trend.
• Aids in detecting transient variations, irregularities, and recurring patterns.
• Improves prediction accuracy by focusing on residual data components.
• Enhances decision-making through detailed analysis of detrended data.
• Assists economists, scholars, and policymakers in comprehending and projecting economic trajectories.
• Augments the clarity and comprehensibility of time series in data analytics.
• Facilitates anomaly detection by pointing out abnormal variations (highs and lows) in the data.