Comparative Discussion Between Effect of Smoothing's

Gokul Saha

Abstract: Real-world data (signals) from fields like science, economics, and biology almost always include noise, which can make accurate interpretation difficult. Therefore, denoising using techniques like filtering or smoothing is a critical initial step in any analysis. This research specifically investigates how the process of smoothing (denoising) affects a signal's memory structure. Memory Structure refers to how past data points influence future ones. Homoscedasticity: The memory structure is uniform (constant memory length, n). Heteroscedasticity: The memory structure is non-uniform (memory length varies over time). When Simple Exponential Smoothing (SES) is applied to a signal that initially has a uniform (linear homoscedastic) memory of length $n$, the resulting smoothed signal preserves the homoscedastic structure. Effect on Memory Length: The smoothing process increases the memory length from n to n+1 except At the (n+1)-th step, the memory length remains n. Double Exponential Smoothing (DES) is particularly useful for analyzing signals that exhibit a trend (a gradual, long-term change). DES helps remove this trend (detrending) to allow for clearer analysis of the remaining components. The study reformulates the DES method using matrix equations to gain a deeper understanding of its mathematical properties. The effectiveness of DES is further tested using signals generated by autoregressive models of order 1 and 2 (AR (1) and AR (2)), which are standard mathematical models used to represent signals with linear homoscedastic memory. In essence, the study concludes that while Simple Exponential Smoothing preserves a signal's uniform memory, it subtly increases the memory length. It also sets up a matrix-based framework to analyze how Double Exponential Smoothing affects the memory structure of trended signals

Keywords: Signal Denoising, Memory Structure, Exponential Smoothing, Homoscedastic Signals, Autoregressive models

How to Cite?: Gokul Saha, "Comparative Discussion Between Effect of Smoothing's", Volume 14 Issue 12, December 2025, International Journal of Science and Research (IJSR), Pages: 1162-1168, https://www.ijsr.net/getabstract.php?paperid=SR251216064735, DOI: https://dx.doi.org/10.21275/SR251216064735