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Article.tex
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\documentclass{article}
\usepackage{arxiv}
\usepackage[utf8]{inputenc} % allow utf-8 input
\usepackage[T1]{fontenc} % use 8-bit T1 fonts
\usepackage{lmodern} % https://github.com/rstudio/rticles/issues/343
\usepackage{hyperref} % hyperlinks
\usepackage{url} % simple URL typesetting
\usepackage{booktabs} % professional-quality tables
\usepackage{amsfonts} % blackboard math symbols
\usepackage{nicefrac} % compact symbols for 1/2, etc.
\usepackage{microtype} % microtypography
\usepackage{graphicx}
\title{An analysis using simulation to compare several moving average
techniques for time series data.}
\author{
MD SAKIBUR HASAN
\\
Department of Statistics \\
Grand Valley State University \\
Campus Dr, Allendale, MI 49401, United States \\
\texttt{\href{mailto:[email protected]}{\nolinkurl{[email protected]}}} \\
\And
Bishal sarker
\\
Department of Applied Statistics \\
University of Dhaka \\
Dhaka, Bangladesh \\
\texttt{\href{mailto:[email protected]}{\nolinkurl{[email protected]}}} \\
}
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\usepackage{threeparttablex}
\usepackage[normalem]{ulem}
\usepackage{makecell}
\usepackage{xcolor}
\begin{document}
\maketitle
\begin{abstract}
There are many smoothing techniques available, and selecting the
appropriate Technique is a very important issue to achieve good
smoothing performance. This study intends to compare different types of
moving average smoothing techniques. The study compares the simple
moving average (SMA), exponential moving average (EMA), double
exponential moving average (DEMA), weighted moving average (WMA), and
zero-lag exponential moving average (ZLEMA).Some error measures are mean
error, mean absolute error, and mean square error. are calculated for
the above smoothing techniques to compare the smoothing accuracy of
these methods.The simulated data used for the study came from six
different statistical distributions.The study assists in determining the
best moving average smoothing method for any time series data.
\end{abstract}
\keywords{
SMA
\and
WMA
\and
ZLEMA
\and
DEMP
\and
Moving average
\and
EMA
}
\hypertarget{introduction}{%
\section{Introduction:}\label{introduction}}
\hypertarget{background-of-the-study}{%
\subsection{Background of The Study}\label{background-of-the-study}}
Data for time series is gathered from various points throughout time. As
a result, the data set has a great deal of variety. So, a technique
known as smoothing is employed to lessen these variations. Techniques
for removing noise from a time series of data include smoothing
techniques. It aids in determining the datset's trend. When data is
compiled, any volatility or other types of noise can be removed or
reduced. Data smoothing is the term for this. Data smoothing is based on
the notion that it can recognize simpler changes to assist in the
prediction of various trends and patterns. It serves as a tool for
statisticians or traders who must examine a lot of data, which is
frequently challenging.
\hypertarget{objective-of-the-study}{%
\subsection{Objective of The Study}\label{objective-of-the-study}}
Among all others smoothing methods moving average methods is the oldest
and simplest smoothing methods. The main object of that study is
comparing different types of moving average smoothing techniques such as
simple moving average(SMA),exponentially-weighted moving average (EWMA),
weighted moving average (WMA), double exponential moving
average(DEMA),Hull moving average(HMA), Zero lag exponential moving
average(ZLEMA) etc. Among them the syudy compares SMA, EWMA, WMA,
DEMA,HMA and ZLEMA.
\hypertarget{literature-review}{%
\section{Literature review}\label{literature-review}}
Raudys, Lenčiauskas, and Malčius (2013) smoothed financial data using
the moving average.
Ivanovski, Milenkovski, and Narasanov (2018) extrapolated the number of
tourists using the moving average. The study aids in making wise
decisions for the future.
Hameed (2015) compares every smoothing method currently in use to
forecast future demand for private universities in Bangladesh. They
contrast different kinds of currently used smoothing techniques and
discover that Holt's method provides the optimum accuracy for their
work.
For the purpose of early detection of infectious disease outbreaks, Yang
et al. (2018) conducts simulation-based studies on the comparison of
statistical and time series forecasting techniques. Here, various
approaches are discussed in an effort to use simulation to produce the
greatest results.
Sinaga and Irawati (2020) studied about medical disposable supply demand
forecasting by moving average and exponential moving average method.
For predicting power load, Karim and Alwi (2013) employed exponential
smoothing and moving average, and exponential moving average
outperformed moving average.
Fong et al. (2020) tried to find an accurate early forecasting model
from Small dataset of 2019-nCoV Novel Coronavirus outbreak.
\hypertarget{methodology}{%
\section{Methodology}\label{methodology}}
\hypertarget{simple-moving-average-sma}{%
\subsection{Simple moving average
(SMA)}\label{simple-moving-average-sma}}
By averaging several different subsets of the entire data set, a simple
moving average (SMA) statistical technique is utilized to examine data
points. Because it is calculated by averaging a predetermined number of
successive data points with equal weight for each, it is referred to as
being ``simple.'' The SMA is used to highlight long-term trends or
patterns in the data and to smooth out short-term volatility in the
data.The formula for SMA is given below,
\[
{T_t = \frac{\sum_{i=-m}^{m} Y_{t+i}}{k}}
\]
here k is the number of order and \(Y_i\) is the observation.
m=\(\frac{k-1}{2}\) is the half width of a moving average as the number
of points. For example, a arithmetic moving average of 3 ordered at time
t is \(T_t=\frac{Y_{t-1}+Y_t+Y_{t+1}}{3}\).
Simple Moving Average (SMA) and Exponential Moving Average (EMA) both
measure trend direction over time. SMA only determines an average, but
EMA gives more weight to data that is more recent.
\hypertarget{exponentially-moving-average-ema}{%
\subsection{Exponentially moving average
(EMA)}\label{exponentially-moving-average-ema}}
Simple Moving Average (SMA) and Exponential Moving Average (EMA) both
measure trend direction over time. SMA only determines an average, but
EMA gives more weight to data that is more recent. \[
EMA = C – P \frac{2}{(n+1)} + P
\]
where C and P are current data point and an exponential moving average
of the previous period (simple average used for the first period)
respectively. THe formula is given below,
\hypertarget{weighted-moving-average}{%
\subsection{Weighted moving average}\label{weighted-moving-average}}
When calculating the weighted moving average, recent data points are
given more weighting than historical data points. When added together,
the weights' total value ought to be 100\%, or 1. The weighting factor
used to calculate the WMA is determined by the period selected for the
indicator. For example, a 5 period WMA would be calculated as follows:
\[
WMA = \frac{(5P_{1} + 4P_2 + 3P_3 + 2P_4 + 1P_5 )} {(5 + 4+ 3 + 2 + 1)}
\]
Where, \(P_1\)= current price \(P_2\) = price one bar ago and so on.
\hypertarget{zlema}{%
\subsection{ZLEMA}\label{zlema}}
John Ehlers and Ric Way created the Zero Lag Exponential Moving Average
(ZLEMA) indicator. The goal is to get rid of the inherent lag that all
averages and other trend following indicators have.
This is what the ZLEMA aims to accomplish by tracking recent prices more
closely than historical prices, much like a standard EMA but with an
even greater emphasis on recent prices. A moving average with less lag
and good smoothing is the end product.
The benefits of using zero lag moving averages are as follows:
\begin{itemize}
\tightlist
\item
A moving average exponential without lag that is more responsive to
current price changes.
\item
The indicator is applicable across all instruments and timeframes.
\item
The indicator may be used as a signal or as a filter for signals.
\end{itemize}
Formula are given below ,
\[
\alpha = \frac{2}{n+1}
\] \[
Z_{t} = (1-\alpha)Z_t + \alpha (T_t - T_{t-\frac{n-1}{2}})
\]
Where , n is the number of period. \(\alpha\) represents the lag and
\(Z_t\) represents the ZLEMA moving average at t time points.
\hypertarget{dema}{%
\subsection{DEMA}\label{dema}}
A level component and a trend component are used in double exponential
smoothing at each period. Two weights, also known as smoothing
parameters, are used in double exponential smoothing to update the
components at each time.The formula is given below,
\[
L_{t} = \alpha Y_{t} + (1-\alpha)(L_{t-1} + T_{t-1})
\] \[
T_t = \gamma (L_{t}–L_{t–1}) + (1-\gamma) T_{t–1}
\] \[
\hat Y= L_{t–1} + T_{t–1}
\] Here , \(L_t\) level at time t.\\
\(\alpha\) weight for the level.\\
\(T_t\) trend at time t\\
\(\gamma\) weight for the trend\\
\(Y_t\) data value at time t\\
\(\hat Y_t\) fitted value, or one-step-ahead forecast, at time t\\
\hypertarget{mean-absolute-error-mae}{%
\subsection{Mean absolute error (MAE)}\label{mean-absolute-error-mae}}
Mean absolute error (MAE), which measures errors between fitted and
observed values, is a statistical concept. Comparisons of expected
against observed data, subsequent time against initial time, and one
measuring technique against an alternate measurement technique are a few
examples of Y vs X. The MAE is calculated by dividing the total absolute
errors by the sample size.
\[
MAE = \frac{\sum_{i=1}^{n} |Y_i - \hat Y_i|}{n}=\frac{\sum_{i=1}^{n}|e_i|}{n}
\]
Here \(\hat Y_i\) is the predicted value and \(Y_i\) is the true value.
\hypertarget{mean-square-error-mse}{%
\subsection{Mean square error (MSE)}\label{mean-square-error-mse}}
The mean square error (MSE), which is derived as the total of the
squared discrepancies between the forecasts and actual values, divided
by the number of data points, is a metric for the average difference
between actual and predicted values in a dataset. \[
MSE = \frac{\sum_{i}^{n} (\hat Yi - Yi)^2}{n}
\]
Here \(\hat Y_i\) is the prediction and \(Y_i\) is the true value.
\hypertarget{analysis}{%
\section{Analysis}\label{analysis}}
Here 100000 values are generated from six different distribution. Data
are generated from Normal distribution with \(N ~(\mu=50,\sigma=5)\),
Poisson \(P~(\lambda = 10)\), Gamma distribution
\(Gamma ~(\alpha = 10, \beta = 2)\), T distribution \(T_{10}\),
exponential distribution \(exp(\lambda = 5)\) and weibull distribution
\(weib~(\alpha=10, \beta =5)\). After generating the data, transform it
as a time series data. A comparison table among all the moving average
techniques used for that study is given below.
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-9}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Normal Distribution} & \multicolumn{3}{c}{Poisson Distribution} & \multicolumn{3}{c}{Weibull Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 2.830 & 12.579 & SMA & 1.776 & 5.033 & SMA & 0.320 & 0.165\\
EMA & 1.634 & 4.192 & EMA & 1.030 & 1.675 & EMA & 0.185 & 0.055\\
DEMA & 0.746 & 0.873 & DEMA & 0.470 & 0.349 & DEMA & 0.084 & 0.011\\
WMA & 1.887 & 5.591 & WMA & 1.184 & 2.237 & WMA & 0.213 & 0.073\\
ZLEMA & 1.885 & 5.585 & ZLEMA & 1.187 & 2.224 & ZLEMA & 0.214 & 0.073\\
\bottomrule
\end{tabular}
\end{table}
\hypertarget{smoothing-comparison-table-for-time-period-2-and-100000-values}{%
\subsection{Smoothing comparison table for time period 2 and 100000
values}\label{smoothing-comparison-table-for-time-period-2-and-100000-values}}
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-10}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Gamma Distribution} & \multicolumn{3}{c}{T Distribution} & \multicolumn{3}{c}{Exponential Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 1.776 & 5.033 & SMA & 0.619 & 0.626 & SMA & 0.100 & 0.020\\
EMA & 1.030 & 1.675 & EMA & 0.358 & 0.209 & EMA & 0.059 & 0.007\\
DEMA & 0.470 & 0.349 & DEMA & 0.163 & 0.043 & DEMA & 0.027 & 0.001\\
WMA & 1.184 & 2.237 & WMA & 0.413 & 0.278 & WMA & 0.067 & 0.009\\
ZLEMA & 1.187 & 2.224 & ZLEMA & 0.415 & 0.278 & ZLEMA & 0.069 & 0.009\\
\bottomrule
\end{tabular}
\end{table}
\hypertarget{smoothing-comparison-table-for-5-time-period-and-10000-values.}{%
\subsection{Smoothing comparison table for 5 time period and 10000
values.}\label{smoothing-comparison-table-for-5-time-period-and-10000-values.}}
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-12}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Normal Distribution} & \multicolumn{3}{c}{Poisson Distribution} & \multicolumn{3}{c}{Weibull Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 3.562 & 19.830 & SMA & 2.250 & 8.039 & SMA & 4.689 & 34.957\\
EMA & 2.912 & 13.270 & EMA & 1.845 & 5.365 & EMA & 3.584 & 20.613\\
DEMA & 2.233 & 1.011 & DEMA & 1.413 & 3.144 & DEMA & 2.735 & 12.083\\
WMA & 3.036 & 14.391 & WMA & 1.916 & 5.805 & WMA & 3.723 & 22.277\\
ZLEMA & 2.617 & 10.742 & ZLEMA & 1.652 & 4.275 & ZLEMA & 3.197 & 16.354\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-12}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Gamma Distribution} & \multicolumn{3}{c}{T Distribution} & \multicolumn{3}{c}{Exponential Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 0.786 & 1.001 & SMA & 0.129 & 0.031 & SMA & 0.401 & 0.256\\
EMA & 0.640 & 0.665 & EMA & 0.105 & 0.020 & EMA & 0.328 & 0.171\\
DEMA & 0.492 & 0.390 & DEMA & 0.080 & 0.012 & DEMA & 0.251 & 0.101\\
WMA & 0.667 & 0.720 & WMA & 0.109 & 0.022 & WMA & 0.341 & 0.185\\
ZLEMA & 0.577 & 0.534 & ZLEMA & 0.096 & 0.016 & ZLEMA & 0.293 & 0.137\\
\bottomrule
\end{tabular}
\end{table}
\hypertarget{smoothing-comparison-table-for-7-time-period-and-1000-values.}{%
\subsection{Smoothing comparison table for 7 time period and 1000
values.}\label{smoothing-comparison-table-for-7-time-period-and-1000-values.}}
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-13}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Normal Distribution} & \multicolumn{3}{c}{Poisson Distribution} & \multicolumn{3}{c}{Weibull Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 3.600 & 20.884 & SMA & 2.363 & 8.665 & SMA & 4.739 & 35.694\\
EMA & 3.110 & 15.570 & EMA & 2.025 & 6.412 & EMA & 3.997 & 25.331\\
DEMA & 2.576 & -27833.008 & DEMA & 1.659 & 4.379 & DEMA & 3.307 & 17.418\\
WMA & 3.202 & 16.454 & WMA & 2.081 & 6.783 & WMA & 4.107 & 26.764\\
ZLEMA & 2.826 & 12.946 & ZLEMA & 1.813 & 5.273 & ZLEMA & 3.640 & 21.278\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[H]
\caption{\label{tab:unnamed-chunk-13}Smoothing comparison table}
\centering
\begin{tabular}[t]{lrrlrrlrr}
\toprule
\multicolumn{3}{c}{Gamma Distribution} & \multicolumn{3}{c}{T Distribution} & \multicolumn{3}{c}{Exponential Distribution} \\
\cmidrule(l{3pt}r{3pt}){1-3} \cmidrule(l{3pt}r{3pt}){4-6} \cmidrule(l{3pt}r{3pt}){7-9}
TYPE & MAE & MSE & TYPE & MAE & MSE & TYPE & MAE & MSE\\
\midrule
SMA & 0.823 & 1.111 & SMA & 0.130 & 0.032 & SMA & 0.425 & 0.275\\
EMA & 0.717 & 0.844 & EMA & 0.113 & 0.024 & EMA & 0.364 & 0.205\\
DEMA & 0.604 & 0.585 & DEMA & 0.093 & 0.015 & DEMA & 0.299 & 0.137\\
WMA & 0.740 & 0.895 & WMA & 0.116 & 0.025 & WMA & 0.374 & 0.216\\
ZLEMA & 0.676 & 0.720 & ZLEMA & 0.105 & 0.020 & ZLEMA & 0.330 & 0.169\\
\bottomrule
\end{tabular}
\end{table}
\hypertarget{findings-and-conclusion}{%
\section{Findings and conclusion}\label{findings-and-conclusion}}
From the smoothing comparison table, it is seen that for any datasets
and any time period, DEMA has the lowest mean square error among all the
moving average smoothing techniques. After DEMA, EMA has the lowest mean
square error. Among all the moving average smoothing techniques used for
the study, SMA has the highest MSE value. SMA is the simplest and most
straightforward method for smoothing, but it performs the worst. DEMA
has complex smoothing techniques, but it performs far better than other
techniques.
\hypertarget{reference}{%
\section*{Reference}\label{reference}}
\addcontentsline{toc}{section}{Reference}
\hypertarget{refs}{}
\begin{CSLReferences}{1}{0}
\leavevmode\vadjust pre{\hypertarget{ref-fong2020finding}{}}%
Fong, Simon James, Gloria Li, Nilanjan Dey, Rubén González Crespo, and
Enrique Herrera-Viedma. 2020. {``Finding an Accurate Early Forecasting
Model from Small Dataset: A Case of 2019-Ncov Novel Coronavirus
Outbreak.''} \emph{arXiv Preprint arXiv:2003.10776}.
\leavevmode\vadjust pre{\hypertarget{ref-hameed2015smoothing}{}}%
Hameed, Haifaa Hussein. 2015. {``Smoothing Techniques for Time Series
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