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| Boilerpipe Text | Exponential smoothing
or
exponential moving average (EMA)
is a
rule of thumb
technique for
smoothing
time series
data using the exponential
window function
. Whereas in the
simple moving average
the past observations are
weighted
equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data.
Exponential smoothing is one of many
window functions
commonly applied to smooth data in
signal processing
, acting as
low-pass filters
to remove high-frequency
noise
. This method is preceded by
Poisson
's use of recursive exponential window functions in convolutions from the 19th century, as well as
Kolmogorov and Zurbenko's use of recursive moving averages
from their studies of turbulence in the 1940s.
The raw data sequence is often represented by
beginning at time
, and the output of the exponential smoothing algorithm is commonly written as
, which may be regarded as a best estimate of what the next value of
will be. When the sequence of observations begins at time
, the simplest form of exponential smoothing is given by the following formulas:
[
1
]
where
is the
smoothing factor
, and
. If
is substituted into
continuously so that the formula of
is fully expressed in terms of
, then
exponentially decaying weighting factors
on each raw data
is revealed, showing how exponential smoothing is named.
The simple exponential smoothing is not able to predict what would be observed at
based on the raw data up to
, while the
double exponential smoothing
and
triple exponential smoothing
can be used for the prediction due to the presence of
as the sequence of best estimates of the linear trend.
Basic (simple) exponential smoothing
[
edit
]
The use of the exponential window function is first attributed to
Poisson
[
2
]
as an extension of a numerical analysis technique from the 17th century, and later adopted by the
signal processing
community in the 1940s. Here, exponential smoothing is the application of the exponential, or Poisson,
window function
. Exponential smoothing was suggested in the statistical literature without citation to previous work by
Robert Goodell Brown
in 1956,
[
3
]
and expanded by
Charles C. Holt
in 1957.
[
4
]
The formulation below, which is the one commonly used, is attributed to Brown and is known as "Brownās simple exponential smoothing".
[
5
]
All the methods of Holt, Winters, and Brown may be seen as a simple application of
recursive filtering
, first found in the 1940s
[
2
]
to convert
finite impulse response
(FIR) filters to
infinite impulse response
filters.
The simplest form of exponential smoothing is given by the formula:
where
is the
smoothing factor
, with
. In other words, the smoothed statistic
is a simple weighted average of the current
observation
and the previous smoothed statistic
. Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available. The term
smoothing factor
applied to
here is something of a misnomer, as larger values of
actually reduce the level of smoothing, and in the limiting case with
= 1 the smoothing output series is just the current observation. Values of
close to 1 have less of a smoothing effect and give greater weight to recent changes in the data, while values of
closer to 0 have a greater smoothing effect and are less responsive to recent changes. In the limiting case with
= 0, the output series is just flat or a constant as the observation
at the beginning of the smoothening process
.
The method for choosing
must be decided by the modeler. Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to
optimize
the value of
. For example, the
method of least squares
might be used to determine the value of
for which the sum of the quantities
is minimized.
[
6
]
Unlike some other smoothing methods, such as the simple moving average, this technique does not require any minimum number of observations to be made before it begins to produce results. In practice, however, a "good average" will not be achieved until several samples have been averaged together; for example, a constant signal will take approximately
stages to reach 95% of the actual value. To accurately reconstruct the original signal without information loss, all stages of the exponential moving average must also be available, because older samples decay in weight exponentially. This is in contrast to a simple moving average, in which some samples can be skipped without as much loss of information due to the constant weighting of samples within the average. If a known number of samples will be missed, one can adjust a weighted average for this as well, by giving equal weight to the new sample and all those to be skipped.
This simple form of exponential smoothing is also known as an
exponentially weighted moving average
(EWMA). Technically it can also be classified as an
autoregressive integrated moving average
(ARIMA) (0,1,1) model with no constant term.
[
7
]
The
time constant
of an exponential moving average is the amount of time for the smoothed response of a
unit step function
to reach
of the final signal. The relationship between this time constant,
, and the smoothing factor,
, is given by the following formula:
, thus
where
is the sampling time interval of the discrete time implementation. If the sampling time is fast compared to the time constant (
) then, by using
the Taylor expansion of the exponential function
,
, thus
Choosing the initial smoothed value
[
edit
]
Note that in the definition above,
(the initial output of the exponential smoothing algorithm) is being initialized to
(the initial raw data or observation). Because exponential smoothing requires that, at each stage, we have the previous forecast
, it is not obvious how to get the method started. We could assume that the initial forecast is equal to the initial value of demand; however, this approach has a serious drawback. Exponential smoothing puts substantial weight on past observations, so the initial value of demand will have an unreasonably large effect on early forecasts. This problem can be overcome by allowing the process to evolve for a reasonable number of periods (10 or more) and using the average of the demand during those periods as the initial forecast. There are many other ways of setting this initial value, but it is important to note that the smaller the value of
, the more sensitive your forecast will be on the selection of this initial smoother value
.
[
8
]
[
9
]
For every exponential smoothing method, we also need to choose the value for the smoothing parameters. For simple exponential smoothing, there is only one smoothing parameter (
Ī±
), but for the methods that follow there are usually more than one smoothing parameter.
There are cases where the smoothing parameters may be chosen in a subjective manner ā the forecaster specifies the value of the smoothing parameters based on previous experience. However, a more robust and objective way to obtain values of the unknown parameters included in any exponential smoothing method is to estimate them from the observed data.
The unknown parameters and the initial values for any exponential smoothing method can be estimated by minimizing the
sum of squared errors
(SSE). The errors are specified as
for
(the one-step-ahead within-sample forecast errors) where
and
are a variable to be predicted at
and a variable as the prediction result at
(based on the previous data or prediction), respectively. Hence, we find the values of the unknown parameters and the initial values that minimize
[
10
]
Unlike the regression case (where we have formulae to directly compute the regression coefficients which minimize the SSE) this involves a non-linear minimization problem, and we need to use an
optimization
tool to perform this.
"Exponential" naming
[
edit
]
The name
exponential smoothing
is attributed to the use of the exponential function as the filter
impulse response
in the
convolution
.
By direct substitution of the defining equation for simple exponential smoothing back into itself we find that
In other words, as time passes the smoothed statistic
becomes the weighted average of a greater and greater number of the past observations
, and the weights assigned to previous observations are proportional to the terms of the geometric progression
A
geometric progression
is the discrete version of an
exponential function
, so this is where the name for this smoothing method originated according to
Statistics
lore.
Comparison with moving average
[
edit
]
Exponential smoothing and moving average have similar defects of introducing a lag relative to the input data. While this can be corrected by shifting the result by half the window length for a symmetrical kernel, such as a moving average or gaussian, this approach is not possible for exponential smoothing since it is an
IIR filter
and therefore has an asymmetric kernel and frequency-dependent
group delay
. This means each constituent frequency is shifted by a different amount and therefore, there is no single number of samples that can be used to shift the output signal to account for the lag.
Both filters also both have roughly the same distribution of forecast error when
Ī±
= 2/(
k
Ā +Ā 1) where
k
is the number of past data points in consideration of moving average. They differ in that exponential smoothing takes into account all past data, whereas moving average only takes into account
k
past data points. Computationally speaking, they also differ in that moving average requires that the past
k
data points, or the data point at lag
k
Ā +Ā 1 plus the most recent forecast value, to be kept, whereas exponential smoothing only needs the most recent forecast value to be kept.
[
11
]
In the
signal processing
literature, the use of non-causal (symmetric) filters is commonplace, and the exponential
window function
is broadly used in this fashion, but a different terminology is used: exponential smoothing is equivalent to a first-order
infinite-impulse response
(IIR) filter and moving average is equivalent to a
finite impulse response filter
with equal weighting factors.
Double exponential smoothing (Holt linear)
[
edit
]
Simple exponential smoothing does not do well when there is a
trend
in the data.
[
1
]
In such situations, several methods were devised under the name "double exponential smoothing" or "second-order exponential smoothing," which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing".
The basic idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series exhibiting some form of trend. This slope component is itself updated via exponential smoothing.
One method works as follows:
[
12
]
Again, the raw data sequence of observations is represented by
, beginning at time
. We use
to represent the smoothed value for time
, and
is our best estimate of the trend at time
. The output of the algorithm is now written as
, an estimate of the value of
at time
based on the raw data up to time
. Double exponential smoothing is given by the formulas
and for
by
where
(
) is the
data smoothing factor
, and
(
) is the
trend smoothing factor
.
To forecast beyond
is given by the following approximation:
.
Setting the initial value
is a matter of preference. An option other than the one listed above is
for some
.
Note that
F
0
is undefined (there is no estimation for time 0), and according to the definition
F
1
=
s
0
+
b
0
, which is well defined, thus further values can be evaluated.
A second method, referred to as either Brown's linear exponential smoothing (LES) or Brown's double exponential smoothing, has only one smoothing factor,
:
[
13
]
where
a
t
, the estimated level at time
t
, and
b
t
, the estimated trend at time
t
, are given by
Triple exponential smoothing (HoltāWinters)
[
edit
]
Triple exponential smoothing applies exponential smoothing three times, which is commonly used when there are three high frequency signals to be removed from a
time series
under study. There are different types of seasonality: 'multiplicative' and 'additive' in nature. Multiplicative signifies that the seasonal effect size depends on the current level, while additive seasonal effects do not.
If in
every
summer we sell 10,000 more balls of ice cream than we do in winter, then seasonality is
additive
. However if we sell 6 times more ice cream in summer than in winter, that means that the difference in sales amounts moves together with the level itself: in years with overall larger consumption, the difference between summer and winter is larger and vice versa. The effect is thus
multiplicative
. Multiplicative seasonality can be represented as a constant factor, additive seasonality as an absolute amount.
[
14
]
Triple exponential smoothing was first suggested by Holt's student, Peter Winters, in 1960 after reading a signal processing book from the 1940s on exponential smoothing.
[
15
]
Holt's novel idea was to repeat filtering an odd number of times greater than 1 and less than 5, which was popular with scholars of previous eras.
[
15
]
While recursive filtering had been used previously, it was applied twice and four times to coincide with the
Hadamard conjecture
, while triple application required more than double the operations of singular convolution. The use of a triple application is considered a
rule of thumb
technique, rather than one based on theoretical foundations and has often been over-emphasized by practitioners. Suppose we have a sequence of observations
beginning at time
with a cycle of seasonal change of length
.
The method calculates a trend line for the data as well as seasonal indices that weight the values in the trend line based on where that time point falls in the cycle of length
.
Let
represent the smoothed value of the constant part for time
,
is the sequence of best estimates of the linear trend that are superimposed on the seasonal changes, and
is the sequence of seasonal correction factors. We wish to estimate
at every time
mod
in the cycle that the observations take on. As a rule of thumb, a minimum of two full seasons (or
periods) of historical data is needed to initialize a set of seasonal factors.
The output of the algorithm is again written as
, an estimate of the value of
at time
based on the raw data up to time
. Triple exponential smoothing with multiplicative seasonality is given by the formulas
[
1
]
where
(
) is the
data smoothing factor
,
(
) is the
trend smoothing factor
, and
(
) is the
seasonal change smoothing factor
.
The general formula for the initial trend estimate
is
.
Setting the initial estimates for the seasonal indices
for
is a bit more involved. If
is the number of complete cycles present in your data, then
where
.
Note that
is the average value of
in the
cycle of your data.
This results in
Triple exponential smoothing with additive seasonality is given by
[
citation needed
]
Implementations in statistics packages
[
edit
]
R
: the HoltWinters function in the stats package
[
16
]
and ets function in the forecast package
[
17
]
(a more complete implementation, generally resulting in a better performance
[
18
]
).
Python
: the holtwinters module of the statsmodels package allow for simple, double and triple exponential smoothing.
IBM
SPSS
includes Simple, Simple Seasonal, Holt's Linear Trend, Brown's Linear Trend, Damped Trend, Winters' Additive, and Winters' Multiplicative in the Time-Series modeling procedure within its Statistics and Modeler statistical packages. The default Expert Modeler feature evaluates all seven exponential smoothing models and ARIMA models with a range of nonseasonal and seasonal
p
,
d
, and
q
values, and selects the model with the lowest
Bayesian Information Criterion
statistic.
Stata
: tssmooth command
[
19
]
LibreOffice
5.2
[
20
]
Microsoft Excel
2016
[
21
]
Julia
: TrendDecomposition.jl package
[
22
]
implements simple and double exponential smoothing and Holts-Winters forecasting procedure.
Autoregressive moving average model
(ARMA)
Errors and residuals in statistics
Moving average
^
a
b
c
"NIST/SEMATECH e-Handbook of Statistical Methods"
. NIST
. Retrieved
23 May
2010
.
^
a
b
Oppenheim, Alan V.; Schafer, Ronald W. (1975).
Digital Signal Processing
.
Prentice Hall
. p.Ā 5.
ISBN
Ā
0-13-214635-5
.
^
Brown, Robert G. (1956).
Exponential Smoothing for Predicting Demand
. Cambridge, Massachusetts: Arthur D. Little Inc. p.Ā 15.
^
Holt, Charles C.
(1957). "Forecasting Trends and Seasonal by Exponentially Weighted Averages".
Office of Naval Research Memorandum
.
52
.
reprinted in
Holt, Charles C.
(JanuaryāMarch 2004). "Forecasting Trends and Seasonal by Exponentially Weighted Averages".
International Journal of Forecasting
.
20
(1):
5ā
10.
doi
:
10.1016/j.ijforecast.2003.09.015
.
^
Brown, Robert Goodell (1963).
Smoothing Forecasting and Prediction of Discrete Time Series
. Englewood Cliffs, NJ: Prentice-Hall.
^
"NIST/SEMATECH e-Handbook of Statistical Methods, 6.4.3.1. Single Exponential Smoothing"
. NIST
. Retrieved
5 July
2017
.
^
Nau, Robert.
"Averaging and Exponential Smoothing Models"
. Retrieved
26 July
2010
.
^
"Production and Operations Analysis" Nahmias. 2009.
^
Äisar, P., & Äisar, S. M. (2011). "Optimization methods of EWMA statistics."
Acta Polytechnica Hungarica
, 8(5), 73ā87. Page 78.
^
7.1 Simple exponential smoothing | Forecasting: Principles and Practice
.
^
Nahmias, Steven; Olsen, Tava Lennon.
Production and Operations Analysis
(7thĀ ed.). Waveland Press. p.Ā
53
.
ISBN
Ā
9781478628248
.
^
"6.4.3.3. Double Exponential Smoothing"
.
itl.nist.gov
. Retrieved
25 September
2011
.
^
"Averaging and Exponential Smoothing Models"
.
duke.edu
. Retrieved
25 September
2011
.
^
Kalehar, Prajakta S.
"Time series Forecasting using HoltāWinters Exponential Smoothing"
(PDF)
. Retrieved
23 June
2014
.
^
a
b
Winters, P. R. (April 1960). "Forecasting Sales by Exponentially Weighted Moving Averages".
Management Science
.
6
(3):
324ā
342.
doi
:
10.1287/mnsc.6.3.324
.
^
"R: HoltāWinters Filtering"
.
stat.ethz.ch
. Retrieved
5 June
2016
.
^
"ets {forecast} | inside-R | A Community Site for R"
.
inside-r.org
. Archived from
the original
on 16 July 2016
. Retrieved
5 June
2016
.
^
"Comparing HoltWinters() and ets()"
.
Hyndsight
. 29 May 2011
. Retrieved
5 June
2016
.
^
tssmooth
in Stata manual
^
"LibreOffice 5.2: Release Notes ā the Document Foundation Wiki"
.
^
"Excel 2016 Forecasting Functions | Real Statistics Using Excel"
.
^
TrendDecomposition.jl
Julia implementation of exponential smoothing and Holt-Winters forecasting procedure
Lecture notes on exponential smoothing (Robert Nau, Duke University)
Data Smoothing
by Jon McLoone,
The Wolfram Demonstrations Project
The HoltāWinters Approach to Exponential Smoothing: 50 Years Old and Going Strong
by Paul Goodwin (2010)
Foresight: The International Journal of Applied Forecasting
Algorithms for Unevenly Spaced Time Series: Moving Averages and Other Rolling Operators
by Andreas Eckner |
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## Contents
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- [1 Basic (simple) exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing#Basic_\(simple\)_exponential_smoothing)
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- [1\.1 Time constant](https://en.wikipedia.org/wiki/Exponential_smoothing#Time_constant)
- [1\.2 Choosing the initial smoothed value](https://en.wikipedia.org/wiki/Exponential_smoothing#Choosing_the_initial_smoothed_value)
- [1\.3 Optimization](https://en.wikipedia.org/wiki/Exponential_smoothing#Optimization)
- [1\.4 "Exponential" naming](https://en.wikipedia.org/wiki/Exponential_smoothing#"Exponential"_naming)
- [1\.5 Comparison with moving average](https://en.wikipedia.org/wiki/Exponential_smoothing#Comparison_with_moving_average)
- [2 Double exponential smoothing (Holt linear)](https://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing_\(Holt_linear\))
- [3 Triple exponential smoothing (HoltāWinters)](https://en.wikipedia.org/wiki/Exponential_smoothing#Triple_exponential_smoothing_\(Holt%E2%80%93Winters\))
- [4 Implementations in statistics packages](https://en.wikipedia.org/wiki/Exponential_smoothing#Implementations_in_statistics_packages)
- [5 See also](https://en.wikipedia.org/wiki/Exponential_smoothing#See_also)
- [6 Notes](https://en.wikipedia.org/wiki/Exponential_smoothing#Notes)
- [7 External links](https://en.wikipedia.org/wiki/Exponential_smoothing#External_links)
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From Wikipedia, the free encyclopedia
Generates a forecast of future values of a time series
**Exponential smoothing** or **exponential moving average (EMA)** is a [rule of thumb](https://en.wikipedia.org/wiki/Rule_of_thumb "Rule of thumb") technique for [smoothing](https://en.wikipedia.org/wiki/Smoothing "Smoothing") [time series](https://en.wikipedia.org/wiki/Time_series "Time series") data using the exponential [window function](https://en.wikipedia.org/wiki/Window_function "Window function"). Whereas in the [simple moving average](https://en.wikipedia.org/wiki/Simple_moving_average "Simple moving average") the past observations are [weighted](https://en.wikipedia.org/wiki/Weight_function "Weight function") equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data.
Exponential smoothing is one of many [window functions](https://en.wikipedia.org/wiki/Window_functions "Window functions") commonly applied to smooth data in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing"), acting as [low-pass filters](https://en.wikipedia.org/wiki/Low-pass_filter "Low-pass filter") to remove high-frequency [noise](https://en.wikipedia.org/wiki/Noise "Noise"). This method is preceded by [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "SimĆ©on Denis Poisson")'s use of recursive exponential window functions in convolutions from the 19th century, as well as [Kolmogorov and Zurbenko's use of recursive moving averages](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Zurbenko_filter "KolmogorovāZurbenko filter") from their studies of turbulence in the 1940s.
The raw data sequence is often represented by { x t } {\\textstyle \\{x\_{t}\\}}  beginning at time t \= 0 {\\textstyle t=0} , and the output of the exponential smoothing algorithm is commonly written as { s t } {\\textstyle \\{s\_{t}\\}} , which may be regarded as a best estimate of what the next value of x {\\textstyle x}  will be. When the sequence of observations begins at time t \= 0 {\\textstyle t=0} , the simplest form of exponential smoothing is given by the following formulas:[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1)
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{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\s\_{t}&=\\alpha x\_{t}+(1-\\alpha )s\_{t-1},\\quad t\>0\\end{aligned}}}
where Ī± {\\textstyle \\alpha }  is the *smoothing factor*, and 0 \< Ī± \< 1 {\\textstyle 0\<\\alpha \<1} . If s t ā 1 {\\textstyle s\_{t-1}}  is substituted into s t {\\textstyle s\_{t}}  continuously so that the formula of s t {\\textstyle s\_{t}}  is fully expressed in terms of { x t } {\\textstyle \\{x\_{t}\\}} , then [exponentially decaying weighting factors](https://en.wikipedia.org/wiki/Exponential_smoothing#"Exponential"_naming) on each raw data x t {\\textstyle x\_{t}}  is revealed, showing how exponential smoothing is named.
The simple exponential smoothing is not able to predict what would be observed at t \+ m {\\textstyle t+m}  based on the raw data up to t {\\textstyle t} , while the [double exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing_\(Holt_linear\)) and [triple exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing#Triple_exponential_smoothing_\(Holt%E2%80%93Winters\)) can be used for the prediction due to the presence of b t {\\displaystyle b\_{t}}  as the sequence of best estimates of the linear trend.
## Basic (simple) exponential smoothing
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=1 "Edit section: Basic (simple) exponential smoothing")\]
The use of the exponential window function is first attributed to [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "SimĆ©on Denis Poisson")[\[2\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Oppenheim,_Alan_V._1975_5-2) as an extension of a numerical analysis technique from the 17th century, and later adopted by the [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") community in the 1940s. Here, exponential smoothing is the application of the exponential, or Poisson, [window function](https://en.wikipedia.org/wiki/Window_function "Window function"). Exponential smoothing was suggested in the statistical literature without citation to previous work by [Robert Goodell Brown](https://en.wikipedia.org/wiki/Robert_Goodell_Brown "Robert Goodell Brown") in 1956,[\[3\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-3) and expanded by [Charles C. Holt](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") in 1957.[\[4\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-4) The formulation below, which is the one commonly used, is attributed to Brown and is known as "Brownās simple exponential smoothing".[\[5\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-5) All the methods of Holt, Winters, and Brown may be seen as a simple application of [recursive filtering](https://en.wikipedia.org/wiki/Recursive_filter "Recursive filter"), first found in the 1940s[\[2\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Oppenheim,_Alan_V._1975_5-2) to convert [finite impulse response](https://en.wikipedia.org/wiki/Finite_impulse_response "Finite impulse response") (FIR) filters to [infinite impulse response](https://en.wikipedia.org/wiki/Infinite_impulse_response "Infinite impulse response") filters.
The simplest form of exponential smoothing is given by the formula:
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{\\displaystyle s\_{t}=\\alpha x\_{t}+(1-\\alpha )s\_{t-1}\\,,}
where Ī± {\\displaystyle \\alpha }  is the *smoothing factor*, with 0 ā¤ Ī± ā¤ 1 {\\displaystyle 0\\leq \\alpha \\leq 1} . In other words, the smoothed statistic s t {\\displaystyle s\_{t}}  is a simple weighted average of the current [observation](https://en.wikipedia.org/wiki/Random_variate "Random variate") x t {\\displaystyle x\_{t}}  and the previous smoothed statistic s t ā 1 {\\displaystyle s\_{t-1}} . Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available. The term *smoothing factor* applied to Ī± {\\displaystyle \\alpha }  here is something of a misnomer, as larger values of Ī± {\\displaystyle \\alpha }  actually reduce the level of smoothing, and in the limiting case with Ī± {\\displaystyle \\alpha }  = 1 the smoothing output series is just the current observation. Values of Ī± {\\displaystyle \\alpha }  close to 1 have less of a smoothing effect and give greater weight to recent changes in the data, while values of Ī± {\\displaystyle \\alpha }  closer to 0 have a greater smoothing effect and are less responsive to recent changes. In the limiting case with Ī± {\\displaystyle \\alpha }  = 0, the output series is just flat or a constant as the observation x 0 {\\textstyle x\_{0}}  at the beginning of the smoothening process t \= 0 {\\textstyle t=0} .
The method for choosing Ī± {\\displaystyle \\alpha }  must be decided by the modeler. Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to *optimize* the value of Ī± {\\displaystyle \\alpha } . For example, the [method of least squares](https://en.wikipedia.org/wiki/Least_squares "Least squares") might be used to determine the value of Ī± {\\displaystyle \\alpha }  for which the sum of the quantities ( s t ā x t \+ 1 ) 2 {\\displaystyle (s\_{t}-x\_{t+1})^{2}}  is minimized.[\[6\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST6431-6)
Unlike some other smoothing methods, such as the simple moving average, this technique does not require any minimum number of observations to be made before it begins to produce results. In practice, however, a "good average" will not be achieved until several samples have been averaged together; for example, a constant signal will take approximately 3 / Ī± {\\displaystyle 3/\\alpha }  stages to reach 95% of the actual value. To accurately reconstruct the original signal without information loss, all stages of the exponential moving average must also be available, because older samples decay in weight exponentially. This is in contrast to a simple moving average, in which some samples can be skipped without as much loss of information due to the constant weighting of samples within the average. If a known number of samples will be missed, one can adjust a weighted average for this as well, by giving equal weight to the new sample and all those to be skipped.
This simple form of exponential smoothing is also known as an [exponentially weighted moving average](https://en.wikipedia.org/wiki/Moving_average#Exponential_moving_average "Moving average") (EWMA). Technically it can also be classified as an [autoregressive integrated moving average](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average "Autoregressive integrated moving average") (ARIMA) (0,1,1) model with no constant term.[\[7\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-7)
### Time constant
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=2 "Edit section: Time constant")\]
The [time constant](https://en.wikipedia.org/wiki/Time_constant "Time constant") of an exponential moving average is the amount of time for the smoothed response of a [unit step function](https://en.wikipedia.org/wiki/Unit_step_function "Unit step function") to reach 1 ā 1 / e ā 63\.2 % {\\displaystyle 1-1/e\\approx 63.2\\,\\%}  of the final signal. The relationship between this time constant, Ļ {\\displaystyle \\tau } , and the smoothing factor, Ī± {\\displaystyle \\alpha } , is given by the following formula:
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{\\displaystyle \\alpha =1-e^{-\\Delta T/\\tau }}
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{\\displaystyle \\tau =-{\\frac {\\Delta T}{\\ln(1-\\alpha )}}}
where Ī T {\\displaystyle \\Delta T}  is the sampling time interval of the discrete time implementation. If the sampling time is fast compared to the time constant (Ī T āŖ Ļ {\\displaystyle \\Delta T\\ll \\tau } ) then, by using [the Taylor expansion of the exponential function](https://en.wikipedia.org/wiki/Taylor_series#Examples "Taylor series"),
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{\\displaystyle \\alpha \\approx {\\frac {\\Delta T}{\\tau }}}
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{\\displaystyle \\tau \\approx {\\frac {\\Delta T}{\\alpha }}}
### Choosing the initial smoothed value
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=3 "Edit section: Choosing the initial smoothed value")\]
Note that in the definition above, s 0 {\\displaystyle s\_{0}}  (the initial output of the exponential smoothing algorithm) is being initialized to x 0 {\\displaystyle x\_{0}}  (the initial raw data or observation). Because exponential smoothing requires that, at each stage, we have the previous forecast s t ā 1 {\\displaystyle s\_{t-1}} , it is not obvious how to get the method started. We could assume that the initial forecast is equal to the initial value of demand; however, this approach has a serious drawback. Exponential smoothing puts substantial weight on past observations, so the initial value of demand will have an unreasonably large effect on early forecasts. This problem can be overcome by allowing the process to evolve for a reasonable number of periods (10 or more) and using the average of the demand during those periods as the initial forecast. There are many other ways of setting this initial value, but it is important to note that the smaller the value of Ī± {\\displaystyle \\alpha } , the more sensitive your forecast will be on the selection of this initial smoother value s 0 {\\displaystyle s\_{0}} .[\[8\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-9)
### Optimization
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=4 "Edit section: Optimization")\]
For every exponential smoothing method, we also need to choose the value for the smoothing parameters. For simple exponential smoothing, there is only one smoothing parameter (*Ī±*), but for the methods that follow there are usually more than one smoothing parameter.
There are cases where the smoothing parameters may be chosen in a subjective manner ā the forecaster specifies the value of the smoothing parameters based on previous experience. However, a more robust and objective way to obtain values of the unknown parameters included in any exponential smoothing method is to estimate them from the observed data.
The unknown parameters and the initial values for any exponential smoothing method can be estimated by minimizing the [sum of squared errors](https://en.wikipedia.org/wiki/Sum_of_squared_errors_of_prediction "Sum of squared errors of prediction") (SSE). The errors are specified as e t \= y t ā y ^ t ā£ t ā 1 {\\textstyle e\_{t}=y\_{t}-{\\hat {y}}\_{t\\mid t-1}}  for t \= 1 , ā¦ , T {\\textstyle t=1,\\ldots ,T}  (the one-step-ahead within-sample forecast errors) where y t {\\textstyle y\_{t}}  and y ^ t ā£ t ā 1 {\\textstyle {\\hat {y}}\_{t\\mid t-1}}  are a variable to be predicted at t {\\displaystyle t}  and a variable as the prediction result at t {\\displaystyle t}  (based on the previous data or prediction), respectively. Hence, we find the values of the unknown parameters and the initial values that minimize
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{\\displaystyle {\\text{SSE}}=\\sum \_{t=1}^{T}(y\_{t}-{\\hat {y}}\_{t\\mid t-1})^{2}=\\sum \_{t=1}^{T}e\_{t}^{2}}
[\[10\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-otexts.org-10)
Unlike the regression case (where we have formulae to directly compute the regression coefficients which minimize the SSE) this involves a non-linear minimization problem, and we need to use an [optimization](https://en.wikipedia.org/wiki/Mathematical_optimization "Mathematical optimization") tool to perform this.
### "Exponential" naming
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=5 "Edit section: \"Exponential\" naming")\]
The name *exponential smoothing* is attributed to the use of the exponential function as the filter [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") in the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution").
By direct substitution of the defining equation for simple exponential smoothing back into itself we find that
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{\\displaystyle {\\begin{aligned}s\_{t}&=\\alpha x\_{t}+(1-\\alpha )s\_{t-1}\\\\\[3pt\]&=\\alpha x\_{t}+\\alpha (1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}s\_{t-2}\\\\\[3pt\]&=\\alpha \\left\[x\_{t}+(1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}x\_{t-2}+(1-\\alpha )^{3}x\_{t-3}+\\cdots +(1-\\alpha )^{t-1}x\_{1}\\right\]+(1-\\alpha )^{t}x\_{0}.\\end{aligned}}}
![{\\displaystyle {\\begin{aligned}s\_{t}&=\\alpha x\_{t}+(1-\\alpha )s\_{t-1}\\\\\[3pt\]&=\\alpha x\_{t}+\\alpha (1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}s\_{t-2}\\\\\[3pt\]&=\\alpha \\left\[x\_{t}+(1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}x\_{t-2}+(1-\\alpha )^{3}x\_{t-3}+\\cdots +(1-\\alpha )^{t-1}x\_{1}\\right\]+(1-\\alpha )^{t}x\_{0}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e0741e7160744aaff5c671b60bc05f0be172fb)
In other words, as time passes the smoothed statistic s t {\\displaystyle s\_{t}}  becomes the weighted average of a greater and greater number of the past observations s t ā 1 , ā¦ , s t ā n , ā¦ {\\displaystyle s\_{t-1},\\ldots ,s\_{t-n},\\ldots } , and the weights assigned to previous observations are proportional to the terms of the geometric progression
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{\\displaystyle 1,(1-\\alpha ),(1-\\alpha )^{2},\\ldots ,(1-\\alpha )^{n},\\ldots }
A [geometric progression](https://en.wikipedia.org/wiki/Geometric_progression "Geometric progression") is the discrete version of an [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function"), so this is where the name for this smoothing method originated according to [Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") lore.
### Comparison with moving average
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=6 "Edit section: Comparison with moving average")\]
Exponential smoothing and moving average have similar defects of introducing a lag relative to the input data. While this can be corrected by shifting the result by half the window length for a symmetrical kernel, such as a moving average or gaussian, this approach is not possible for exponential smoothing since it is an [IIR filter](https://en.wikipedia.org/wiki/IIR_filter "IIR filter") and therefore has an asymmetric kernel and frequency-dependent [group delay](https://en.wikipedia.org/wiki/Group_delay "Group delay"). This means each constituent frequency is shifted by a different amount and therefore, there is no single number of samples that can be used to shift the output signal to account for the lag.
Both filters also both have roughly the same distribution of forecast error when *Ī±* = 2/(*k* + 1) where *k* is the number of past data points in consideration of moving average. They differ in that exponential smoothing takes into account all past data, whereas moving average only takes into account *k* past data points. Computationally speaking, they also differ in that moving average requires that the past *k* data points, or the data point at lag *k* + 1 plus the most recent forecast value, to be kept, whereas exponential smoothing only needs the most recent forecast value to be kept.[\[11\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-11)
In the [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") literature, the use of non-causal (symmetric) filters is commonplace, and the exponential [window function](https://en.wikipedia.org/wiki/Window_function "Window function") is broadly used in this fashion, but a different terminology is used: exponential smoothing is equivalent to a first-order [infinite-impulse response](https://en.wikipedia.org/wiki/Infinite-impulse_response "Infinite-impulse response") (IIR) filter and moving average is equivalent to a [finite impulse response filter](https://en.wikipedia.org/wiki/Finite_impulse_response_filter "Finite impulse response filter") with equal weighting factors.
## Double exponential smoothing (Holt linear)
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=7 "Edit section: Double exponential smoothing (Holt linear)")\]
Simple exponential smoothing does not do well when there is a [trend](https://en.wikipedia.org/wiki/Trend_estimation "Trend estimation") in the data.[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1) In such situations, several methods were devised under the name "double exponential smoothing" or "second-order exponential smoothing," which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing". The basic idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series exhibiting some form of trend. This slope component is itself updated via exponential smoothing.
One method works as follows:[\[12\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-12)
Again, the raw data sequence of observations is represented by x t {\\displaystyle x\_{t}} , beginning at time t \= 0 {\\displaystyle t=0} . We use s t {\\displaystyle s\_{t}}  to represent the smoothed value for time t {\\displaystyle t} , and b t {\\displaystyle b\_{t}}  is our best estimate of the trend at time t {\\displaystyle t} . The output of the algorithm is now written as F t \+ m {\\displaystyle F\_{t+m}} , an estimate of the value of x t \+ m {\\displaystyle x\_{t+m}}  at time m \> 0 {\\displaystyle m\>0}  based on the raw data up to time t {\\displaystyle t} . Double exponential smoothing is given by the formulas
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{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\b\_{0}&=x\_{1}-x\_{0}\\\\\\end{aligned}}}
and for t \> 0 {\\displaystyle t\>0}  by
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{\\displaystyle {\\begin{aligned}s\_{t}&=\\alpha x\_{t}+(1-\\alpha )(s\_{t-1}+b\_{t-1})\\\\b\_{t}&=\\beta (s\_{t}-s\_{t-1})+(1-\\beta )b\_{t-1}\\\\\\end{aligned}}}
where Ī± {\\displaystyle \\alpha }  (0 ā¤ Ī± ā¤ 1 {\\displaystyle 0\\leq \\alpha \\leq 1} ) is the *data smoothing factor*, and Ī² {\\displaystyle \\beta }  (0 ā¤ Ī² ā¤ 1 {\\displaystyle 0\\leq \\beta \\leq 1} ) is the *trend smoothing factor*.
To forecast beyond x t {\\displaystyle x\_{t}}  is given by the following approximation:
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{\\displaystyle F\_{t+m}=s\_{t}+m\\cdot b\_{t}}
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Setting the initial value b {\\displaystyle b}  is a matter of preference. An option other than the one listed above is x n ā x 0 n {\\textstyle {\\frac {x\_{n}-x\_{0}}{n}}}  for some n {\\displaystyle n} .
Note that *F*0 is undefined (there is no estimation for time 0), and according to the definition *F*1\=*s*0\+*b*0, which is well defined, thus further values can be evaluated.
A second method, referred to as either Brown's linear exponential smoothing (LES) or Brown's double exponential smoothing, has only one smoothing factor, Ī± {\\displaystyle \\alpha } :[\[13\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-13)
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{\\displaystyle {\\begin{aligned}s'\_{0}&=x\_{0}\\\\s''\_{0}&=x\_{0}\\\\s'\_{t}&=\\alpha x\_{t}+(1-\\alpha )s'\_{t-1}\\\\s''\_{t}&=\\alpha s'\_{t}+(1-\\alpha )s''\_{t-1}\\\\F\_{t+m}&=a\_{t}+mb\_{t},\\end{aligned}}}
where *a**t*, the estimated level at time *t*, and *b**t*, the estimated trend at time *t*, are given by
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{\\displaystyle {\\begin{aligned}a\_{t}&=2s'\_{t}-s''\_{t}\\\\\[5pt\]b\_{t}&={\\frac {\\alpha }{1-\\alpha }}(s'\_{t}-s''\_{t}).\\end{aligned}}}
![{\\displaystyle {\\begin{aligned}a\_{t}&=2s'\_{t}-s''\_{t}\\\\\[5pt\]b\_{t}&={\\frac {\\alpha }{1-\\alpha }}(s'\_{t}-s''\_{t}).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/496559e07c278001d2b3e1cf46743c3c4f9cae7c)
## Triple exponential smoothing (HoltāWinters)
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=8 "Edit section: Triple exponential smoothing (HoltāWinters)")\]
Triple exponential smoothing applies exponential smoothing three times, which is commonly used when there are three high frequency signals to be removed from a [time series](https://en.wikipedia.org/wiki/Time_series "Time series") under study. There are different types of seasonality: 'multiplicative' and 'additive' in nature. Multiplicative signifies that the seasonal effect size depends on the current level, while additive seasonal effects do not.
If in *every* summer we sell 10,000 more balls of ice cream than we do in winter, then seasonality is *additive*. However if we sell 6 times more ice cream in summer than in winter, that means that the difference in sales amounts moves together with the level itself: in years with overall larger consumption, the difference between summer and winter is larger and vice versa. The effect is thus *multiplicative*. Multiplicative seasonality can be represented as a constant factor, additive seasonality as an absolute amount.[\[14\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-14)
Triple exponential smoothing was first suggested by Holt's student, Peter Winters, in 1960 after reading a signal processing book from the 1940s on exponential smoothing.[\[15\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Winters_324%E2%80%93342-15) Holt's novel idea was to repeat filtering an odd number of times greater than 1 and less than 5, which was popular with scholars of previous eras.[\[15\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Winters_324%E2%80%93342-15) While recursive filtering had been used previously, it was applied twice and four times to coincide with the [Hadamard conjecture](https://en.wikipedia.org/wiki/Hadamard_conjecture "Hadamard conjecture"), while triple application required more than double the operations of singular convolution. The use of a triple application is considered a [rule of thumb](https://en.wikipedia.org/wiki/Rule_of_thumb "Rule of thumb") technique, rather than one based on theoretical foundations and has often been over-emphasized by practitioners. Suppose we have a sequence of observations x t , {\\displaystyle x\_{t},}  beginning at time t \= 0 {\\displaystyle t=0}  with a cycle of seasonal change of length L {\\displaystyle L} .
The method calculates a trend line for the data as well as seasonal indices that weight the values in the trend line based on where that time point falls in the cycle of length L {\\displaystyle L} .
Let s t {\\displaystyle s\_{t}}  represent the smoothed value of the constant part for time t {\\displaystyle t} , b t {\\displaystyle b\_{t}}  is the sequence of best estimates of the linear trend that are superimposed on the seasonal changes, and c t {\\displaystyle c\_{t}}  is the sequence of seasonal correction factors. We wish to estimate c t {\\displaystyle c\_{t}}  at every time t {\\displaystyle t} mod L {\\displaystyle L}  in the cycle that the observations take on. As a rule of thumb, a minimum of two full seasons (or 2 L {\\displaystyle 2L}  periods) of historical data is needed to initialize a set of seasonal factors.
The output of the algorithm is again written as F t \+ m {\\displaystyle F\_{t+m}} , an estimate of the value of x t \+ m {\\displaystyle x\_{t+m}}  at time t \+ m \> 0 {\\displaystyle t+m\>0}  based on the raw data up to time t {\\displaystyle t} . Triple exponential smoothing with multiplicative seasonality is given by the formulas[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1)
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{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\\[5pt\]s\_{t}&=\\alpha {\\frac {x\_{t}}{c\_{t-L}}}+(1-\\alpha )(s\_{t-1}+b\_{t-1})\\\\\[5pt\]b\_{t}&=\\beta (s\_{t}-s\_{t-1})+(1-\\beta )b\_{t-1}\\\\\[5pt\]c\_{t}&=\\gamma {\\frac {x\_{t}}{s\_{t}}}+(1-\\gamma )c\_{t-L}\\\\\[5pt\]F\_{t+m}&=(s\_{t}+mb\_{t})c\_{t-L+1+(m-1){\\bmod {L}}},\\end{aligned}}}
![{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\\[5pt\]s\_{t}&=\\alpha {\\frac {x\_{t}}{c\_{t-L}}}+(1-\\alpha )(s\_{t-1}+b\_{t-1})\\\\\[5pt\]b\_{t}&=\\beta (s\_{t}-s\_{t-1})+(1-\\beta )b\_{t-1}\\\\\[5pt\]c\_{t}&=\\gamma {\\frac {x\_{t}}{s\_{t}}}+(1-\\gamma )c\_{t-L}\\\\\[5pt\]F\_{t+m}&=(s\_{t}+mb\_{t})c\_{t-L+1+(m-1){\\bmod {L}}},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cee3212f28cf23aad40563caab52147a225fd2a8)
where Ī± {\\displaystyle \\alpha }  (0 ā¤ Ī± ā¤ 1 {\\displaystyle 0\\leq \\alpha \\leq 1} ) is the *data smoothing factor*, Ī² {\\displaystyle \\beta }  (0 ā¤ Ī² ā¤ 1 {\\displaystyle 0\\leq \\beta \\leq 1} ) is the *trend smoothing factor*, and Ī³ {\\displaystyle \\gamma }  (0 ā¤ Ī³ ā¤ 1 {\\displaystyle 0\\leq \\gamma \\leq 1} ) is the *seasonal change smoothing factor*.
The general formula for the initial trend estimate b {\\displaystyle b}  is
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{\\displaystyle {\\begin{aligned}b\_{0}&={\\frac {1}{L}}\\left({\\frac {x\_{L+1}-x\_{1}}{L}}+{\\frac {x\_{L+2}-x\_{2}}{L}}+\\cdots +{\\frac {x\_{L+L}-x\_{L}}{L}}\\right)\\end{aligned}}}
.
Setting the initial estimates for the seasonal indices c i {\\displaystyle c\_{i}}  for i \= 1 , 2 , ā¦ , L {\\displaystyle i=1,2,\\ldots ,L}  is a bit more involved. If N {\\displaystyle N}  is the number of complete cycles present in your data, then
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{\\displaystyle c\_{i}={\\frac {1}{N}}\\sum \_{j=1}^{N}{\\frac {x\_{L(j-1)+i}}{A\_{j}}}\\quad {\\text{for }}i=1,2,\\ldots ,L}
where
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{\\displaystyle A\_{j}={\\frac {\\sum \_{k=1}^{L}x\_{L(j-1)+k}}{L}}\\quad {\\text{for }}j=1,2,\\ldots ,N}
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Note that A j {\\displaystyle A\_{j}}  is the average value of x {\\displaystyle x}  in the j th {\\displaystyle j^{\\text{th}}}  cycle of your data.
This results in
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{\\displaystyle c\_{i}={\\frac {1}{N}}\\sum \_{j=1}^{N}{\\frac {x\_{L(j-1)+i}}{{\\frac {1}{L}}\\sum \_{k=1}^{L}x\_{L(j-1)+k}}}}
Triple exponential smoothing with additive seasonality is given by \[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
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{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\s\_{t}&=\\alpha (x\_{t}-c\_{t-L})+(1-\\alpha )(s\_{t-1}+b\_{t-1})\\\\b\_{t}&=\\beta (s\_{t}-s\_{t-1})+(1-\\beta )b\_{t-1}\\\\c\_{t}&=\\gamma (x\_{t}-s\_{t-1}-b\_{t-1})+(1-\\gamma )c\_{t-L}\\\\F\_{t+m}&=s\_{t}+mb\_{t}+c\_{t-L+1+(m-1){\\bmod {L}}}.\\\\\\end{aligned}}}
## Implementations in statistics packages
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=9 "Edit section: Implementations in statistics packages")\]
- [R](https://en.wikipedia.org/wiki/R_\(programming_language\) "R (programming language)"): the HoltWinters function in the stats package[\[16\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-16) and ets function in the forecast package[\[17\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-17) (a more complete implementation, generally resulting in a better performance[\[18\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-18)).
- [Python](https://en.wikipedia.org/wiki/Python_\(programming_language\) "Python (programming language)"): the holtwinters module of the statsmodels package allow for simple, double and triple exponential smoothing.
- IBM [SPSS](https://en.wikipedia.org/wiki/SPSS "SPSS") includes Simple, Simple Seasonal, Holt's Linear Trend, Brown's Linear Trend, Damped Trend, Winters' Additive, and Winters' Multiplicative in the Time-Series modeling procedure within its Statistics and Modeler statistical packages. The default Expert Modeler feature evaluates all seven exponential smoothing models and ARIMA models with a range of nonseasonal and seasonal *p*, *d*, and *q* values, and selects the model with the lowest [Bayesian Information Criterion](https://en.wikipedia.org/wiki/Bayesian_Information_Criterion "Bayesian Information Criterion") statistic.
- [Stata](https://en.wikipedia.org/wiki/Stata "Stata"): tssmooth command[\[19\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-19)
- [LibreOffice](https://en.wikipedia.org/wiki/LibreOffice "LibreOffice") 5.2[\[20\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-20)
- [Microsoft Excel](https://en.wikipedia.org/wiki/Microsoft_Excel "Microsoft Excel") 2016[\[21\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-21)
- [Julia](https://en.wikipedia.org/wiki/Julia_\(programming_language\) "Julia (programming language)"): TrendDecomposition.jl package[\[22\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-22) implements simple and double exponential smoothing and Holts-Winters forecasting procedure.
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=10 "Edit section: See also")\]
- [Autoregressive moving average model](https://en.wikipedia.org/wiki/Autoregressive_moving_average_model "Autoregressive moving average model") (ARMA)
- [Errors and residuals in statistics](https://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics "Errors and residuals in statistics")
- [Moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average")
## Notes
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=11 "Edit section: Notes")\]
1. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-1) [***c***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-2)
["NIST/SEMATECH e-Handbook of Statistical Methods"](http://www.itl.nist.gov/div898/handbook/). NIST. Retrieved 23 May 2010.
2. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Oppenheim,_Alan_V._1975_5_2-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Oppenheim,_Alan_V._1975_5_2-1)
Oppenheim, Alan V.; Schafer, Ronald W. (1975). *Digital Signal Processing*. [Prentice Hall](https://en.wikipedia.org/wiki/Prentice_Hall "Prentice Hall"). p. 5. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-214635-5](https://en.wikipedia.org/wiki/Special:BookSources/0-13-214635-5 "Special:BookSources/0-13-214635-5")
.
3. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-3)**
Brown, Robert G. (1956). [*Exponential Smoothing for Predicting Demand*](http://legacy.library.ucsf.edu/tid/dae94e00;jsessionid=104A0CEFFA31ADC2FA5E0558F69B3E1D.tobacco03). Cambridge, Massachusetts: Arthur D. Little Inc. p. 15.
4. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-4)**
[Holt, Charles C.](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") (1957). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". *Office of Naval Research Memorandum*. **52**.
reprinted in
[Holt, Charles C.](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") (JanuaryāMarch 2004). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". *[International Journal of Forecasting](https://en.wikipedia.org/wiki/International_Journal_of_Forecasting "International Journal of Forecasting")*. **20** (1): 5ā10\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.ijforecast.2003.09.015](https://doi.org/10.1016%2Fj.ijforecast.2003.09.015).
5. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-5)**
Brown, Robert Goodell (1963). [*Smoothing Forecasting and Prediction of Discrete Time Series*](http://babel.hathitrust.org/cgi/pt?id=mdp.39015004514728;view=1up;seq=1). Englewood Cliffs, NJ: Prentice-Hall.
6. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST6431_6-0)**
["NIST/SEMATECH e-Handbook of Statistical Methods, 6.4.3.1. Single Exponential Smoothing"](http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm). NIST. Retrieved 5 July 2017.
7. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-7)**
Nau, Robert. ["Averaging and Exponential Smoothing Models"](http://www.duke.edu/~rnau/411avg.htm). Retrieved 26 July 2010.
8. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-8)** "Production and Operations Analysis" Nahmias. 2009.
9. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-9)** Äisar, P., & Äisar, S. M. (2011). "Optimization methods of EWMA statistics." *Acta Polytechnica Hungarica*, 8(5), 73ā87. Page 78.
10. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-otexts.org_10-0)**
[*7\.1 Simple exponential smoothing \| Forecasting: Principles and Practice*](https://www.otexts.org/fpp/7/1).
11. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-11)**
Nahmias, Steven; Olsen, Tava Lennon. *Production and Operations Analysis* (7th ed.). Waveland Press. p. [53](https://books.google.com/books?id=SIsoBgAAQBAJ&pg=PA53). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781478628248](https://en.wikipedia.org/wiki/Special:BookSources/9781478628248 "Special:BookSources/9781478628248")
.
12. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-12)**
["6.4.3.3. Double Exponential Smoothing"](http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc433.htm). *itl.nist.gov*. Retrieved 25 September 2011.
13. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-13)**
["Averaging and Exponential Smoothing Models"](http://www.duke.edu/~rnau/411avg.htm). *duke.edu*. Retrieved 25 September 2011.
14. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-14)**
Kalehar, Prajakta S. ["Time series Forecasting using HoltāWinters Exponential Smoothing"](http://www.it.iitb.ac.in/~praj/acads/seminar/04329008_ExponentialSmoothing.pdf) (PDF). Retrieved 23 June 2014.
15. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Winters_324%E2%80%93342_15-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Winters_324%E2%80%93342_15-1)
Winters, P. R. (April 1960). "Forecasting Sales by Exponentially Weighted Moving Averages". *[Management Science](https://en.wikipedia.org/wiki/Management_Science:_A_Journal_of_the_Institute_for_Operations_Research_and_the_Management_Sciences "Management Science: A Journal of the Institute for Operations Research and the Management Sciences")*. **6** (3): 324ā342\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1287/mnsc.6.3.324](https://doi.org/10.1287%2Fmnsc.6.3.324).
16. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-16)**
["R: HoltāWinters Filtering"](https://stat.ethz.ch/R-manual/R-patched/library/stats/html/HoltWinters.html). *stat.ethz.ch*. Retrieved 5 June 2016.
17. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-17)**
["ets {forecast} \| inside-R \| A Community Site for R"](https://web.archive.org/web/20160716153135/http://www.inside-r.org/packages/cran/forecast/docs/ets). *inside-r.org*. Archived from [the original](http://www.inside-r.org/packages/cran/forecast/docs/ets) on 16 July 2016. Retrieved 5 June 2016.
18. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-18)**
["Comparing HoltWinters() and ets()"](http://robjhyndman.com/hyndsight/estimation2/). *Hyndsight*. 29 May 2011. Retrieved 5 June 2016.
19. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-19)** [tssmooth](https://www.stata.com/help.cgi?tssmooth) in Stata manual
20. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-20)**
["LibreOffice 5.2: Release Notes ā the Document Foundation Wiki"](https://wiki.documentfoundation.org/ReleaseNotes/5.2#New_spreadsheet_functions).
21. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-21)**
["Excel 2016 Forecasting Functions \| Real Statistics Using Excel"](http://www.real-statistics.com/time-series-analysis/basic-time-series-forecasting/excel-2016-forecasting-functions/).
22. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-22)** [TrendDecomposition.jl](https://sdbrinkmann.github.io/TrendDecomposition.jl/stable/) Julia implementation of exponential smoothing and Holt-Winters forecasting procedure
## External links
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=12 "Edit section: External links")\]
- [Lecture notes on exponential smoothing (Robert Nau, Duke University)](http://people.duke.edu/~rnau/411avg.htm)
- [Data Smoothing](http://demonstrations.wolfram.com/DataSmoothing/) by Jon McLoone, [The Wolfram Demonstrations Project](https://en.wikipedia.org/wiki/The_Wolfram_Demonstrations_Project "The Wolfram Demonstrations Project")
- [The HoltāWinters Approach to Exponential Smoothing: 50 Years Old and Going Strong](https://foresight.forecasters.org/product/foresight-issue-19/) by Paul Goodwin (2010) [Foresight: The International Journal of Applied Forecasting](https://en.wikipedia.org/wiki/Foresight:_The_International_Journal_of_Applied_Forecasting "Foresight: The International Journal of Applied Forecasting")
- [Algorithms for Unevenly Spaced Time Series: Moving Averages and Other Rolling Operators](https://web.archive.org/web/20150623211648/http://www.eckner.com/papers/ts_alg.pdf) by Andreas Eckner
| [v](https://en.wikipedia.org/wiki/Template:Statistics "Template:Statistics") [t](https://en.wikipedia.org/wiki/Template_talk:Statistics "Template talk:Statistics") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Statistics "Special:EditPage/Template:Statistics")[Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") | |
|---|---|
| [Outline](https://en.wikipedia.org/wiki/Outline_of_statistics "Outline of statistics") [Index](https://en.wikipedia.org/wiki/List_of_statistics_articles "List of statistics articles") | |
| [Descriptive statistics](https://en.wikipedia.org/wiki/Descriptive_statistics "Descriptive statistics") | |
| | |
| [Continuous data](https://en.wikipedia.org/wiki/Continuous_probability_distribution "Continuous probability distribution") | |
| | |
| [Center](https://en.wikipedia.org/wiki/Central_tendency "Central tendency") | [Mean](https://en.wikipedia.org/wiki/Mean "Mean") [Arithmetic](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean") [Arithmetic-Geometric](https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean "Arithmeticāgeometric mean") [Contraharmonic](https://en.wikipedia.org/wiki/Contraharmonic_mean "Contraharmonic mean") [Cubic](https://en.wikipedia.org/wiki/Cubic_mean "Cubic mean") [Generalized/power](https://en.wikipedia.org/wiki/Generalized_mean "Generalized mean") [Geometric](https://en.wikipedia.org/wiki/Geometric_mean "Geometric mean") [Harmonic](https://en.wikipedia.org/wiki/Harmonic_mean "Harmonic mean") [Heronian](https://en.wikipedia.org/wiki/Heronian_mean "Heronian mean") [Heinz](https://en.wikipedia.org/wiki/Heinz_mean "Heinz mean") [Lehmer](https://en.wikipedia.org/wiki/Lehmer_mean "Lehmer mean") [Median](https://en.wikipedia.org/wiki/Median "Median") [Mode](https://en.wikipedia.org/wiki/Mode_\(statistics\) "Mode (statistics)") |
| [Dispersion](https://en.wikipedia.org/wiki/Statistical_dispersion "Statistical dispersion") | [Average absolute deviation](https://en.wikipedia.org/wiki/Average_absolute_deviation "Average absolute deviation") [Coefficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation "Coefficient of variation") [Interquartile range](https://en.wikipedia.org/wiki/Interquartile_range "Interquartile range") [Percentile](https://en.wikipedia.org/wiki/Percentile "Percentile") [Range](https://en.wikipedia.org/wiki/Range_\(statistics\) "Range (statistics)") [Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation "Standard deviation") [Variance](https://en.wikipedia.org/wiki/Variance#Sample_variance "Variance") |
| [Shape](https://en.wikipedia.org/wiki/Shape_of_the_distribution "Shape of the distribution") | [Central limit theorem](https://en.wikipedia.org/wiki/Central_limit_theorem "Central limit theorem") [Moments](https://en.wikipedia.org/wiki/Moment_\(mathematics\) "Moment (mathematics)") [Kurtosis](https://en.wikipedia.org/wiki/Kurtosis "Kurtosis") [L-moments](https://en.wikipedia.org/wiki/L-moment "L-moment") [Skewness](https://en.wikipedia.org/wiki/Skewness "Skewness") |
| [Count data](https://en.wikipedia.org/wiki/Count_data "Count data") | [Index of dispersion](https://en.wikipedia.org/wiki/Index_of_dispersion "Index of dispersion") |
| Summary tables | [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Frequency distribution](https://en.wikipedia.org/wiki/Frequency_distribution "Frequency distribution") [Grouped data](https://en.wikipedia.org/wiki/Grouped_data "Grouped data") |
| [Dependence](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Pearson product-moment correlation](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient "Pearson correlation coefficient") [Rank correlation](https://en.wikipedia.org/wiki/Rank_correlation "Rank correlation") [Kendall's Ļ](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient "Kendall rank correlation coefficient") [Spearman's Ļ](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient "Spearman's rank correlation coefficient") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") |
| [Graphics](https://en.wikipedia.org/wiki/Statistical_graphics "Statistical graphics") | [Bar chart](https://en.wikipedia.org/wiki/Bar_chart "Bar chart") [Biplot](https://en.wikipedia.org/wiki/Biplot "Biplot") [Box plot](https://en.wikipedia.org/wiki/Box_plot "Box plot") [Control chart](https://en.wikipedia.org/wiki/Control_chart "Control chart") [Correlogram](https://en.wikipedia.org/wiki/Correlogram "Correlogram") [Fan chart](https://en.wikipedia.org/wiki/Fan_chart_\(statistics\) "Fan chart (statistics)") [Forest plot](https://en.wikipedia.org/wiki/Forest_plot "Forest plot") [Histogram](https://en.wikipedia.org/wiki/Histogram "Histogram") [Pie chart](https://en.wikipedia.org/wiki/Pie_chart "Pie chart") [QāQ plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot "QāQ plot") [Radar chart](https://en.wikipedia.org/wiki/Radar_chart "Radar chart") [Run chart](https://en.wikipedia.org/wiki/Run_chart "Run chart") [Scatter plot](https://en.wikipedia.org/wiki/Scatter_plot "Scatter plot") [Stem-and-leaf display](https://en.wikipedia.org/wiki/Stem-and-leaf_display "Stem-and-leaf display") [Violin plot](https://en.wikipedia.org/wiki/Violin_plot "Violin plot") |
| [Statistical data processing](https://en.wikipedia.org/wiki/Data_preprocessing "Data preprocessing") | |
| | |
| [Transformations](https://en.wikipedia.org/wiki/Data_transformation_\(statistics\) "Data transformation (statistics)") | [Data transformation](https://en.wikipedia.org/wiki/Data_transformation_\(statistics\) "Data transformation (statistics)") [Log transformation](https://en.wikipedia.org/w/index.php?title=Log_transformation&action=edit&redlink=1 "Log transformation (page does not exist)") [Power transform](https://en.wikipedia.org/wiki/Power_transform "Power transform") [BoxāCox transformation](https://en.wikipedia.org/wiki/Box%E2%80%93Cox_transformation "BoxāCox transformation") [YeoāJohnson transformation](https://en.wikipedia.org/wiki/Yeo%E2%80%93Johnson_transformation "YeoāJohnson transformation") [Variance-stabilizing transformation](https://en.wikipedia.org/wiki/Variance-stabilizing_transformation "Variance-stabilizing transformation") [Anscombe transform](https://en.wikipedia.org/wiki/Anscombe_transform "Anscombe transform") [Fisher transformation](https://en.wikipedia.org/wiki/Fisher_transformation "Fisher transformation") |
| [Scaling and normalization](https://en.wikipedia.org/wiki/Feature_scaling "Feature scaling") | [Feature scaling](https://en.wikipedia.org/wiki/Feature_scaling "Feature scaling") [Normalization](https://en.wikipedia.org/wiki/Normalization_\(statistics\) "Normalization (statistics)") [Standardization (z-score)](https://en.wikipedia.org/wiki/Standard_score "Standard score") [Mināmax normalization](https://en.wikipedia.org/w/index.php?title=Min%E2%80%93max_normalization&action=edit&redlink=1 "Mināmax normalization (page does not exist)") [Unit vector normalization](https://en.wikipedia.org/w/index.php?title=Unit_vector_normalization&action=edit&redlink=1 "Unit vector normalization (page does not exist)") |
| Data cleaning | [Data cleaning](https://en.wikipedia.org/wiki/Data_cleaning "Data cleaning") [Outlier](https://en.wikipedia.org/wiki/Outlier "Outlier") [Winsorizing](https://en.wikipedia.org/wiki/Winsorizing "Winsorizing") [Truncation](https://en.wikipedia.org/wiki/Truncation_\(statistics\) "Truncation (statistics)") [Missing data](https://en.wikipedia.org/wiki/Missing_data "Missing data") |
| Data reduction | [Dimensionality reduction](https://en.wikipedia.org/wiki/Dimensionality_reduction "Dimensionality reduction") [Principal component analysis](https://en.wikipedia.org/wiki/Principal_component_analysis "Principal component analysis") [Factor analysis](https://en.wikipedia.org/wiki/Factor_analysis "Factor analysis") |
| Time-series preprocessing | [Differencing](https://en.wikipedia.org/wiki/Differencing "Differencing") [Detrending](https://en.wikipedia.org/wiki/Detrending "Detrending") [Seasonal adjustment](https://en.wikipedia.org/wiki/Seasonal_adjustment "Seasonal adjustment") [Stationarity transformation](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") |
| [Data collection](https://en.wikipedia.org/wiki/Data_collection "Data collection") | |
| | |
| [Study design](https://en.wikipedia.org/wiki/Design_of_experiments "Design of experiments") | [Effect size](https://en.wikipedia.org/wiki/Effect_size "Effect size") [Missing data](https://en.wikipedia.org/wiki/Missing_data "Missing data") [Optimal design](https://en.wikipedia.org/wiki/Optimal_design "Optimal design") [Population](https://en.wikipedia.org/wiki/Statistical_population "Statistical population") [Replication](https://en.wikipedia.org/wiki/Replication_\(statistics\) "Replication (statistics)") [Sample size determination](https://en.wikipedia.org/wiki/Sample_size_determination "Sample size determination") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Statistical power](https://en.wikipedia.org/wiki/Statistical_power "Statistical power") |
| [Survey methodology](https://en.wikipedia.org/wiki/Survey_methodology "Survey methodology") | [Sampling](https://en.wikipedia.org/wiki/Sampling_\(statistics\) "Sampling (statistics)") [Cluster](https://en.wikipedia.org/wiki/Cluster_sampling "Cluster sampling") [Stratified](https://en.wikipedia.org/wiki/Stratified_sampling "Stratified sampling") [Opinion poll](https://en.wikipedia.org/wiki/Opinion_poll "Opinion poll") [Questionnaire](https://en.wikipedia.org/wiki/Questionnaire "Questionnaire") [Standard error](https://en.wikipedia.org/wiki/Standard_error "Standard error") |
| [Controlled experiments](https://en.wikipedia.org/wiki/Experiment "Experiment") | [Blocking](https://en.wikipedia.org/wiki/Blocking_\(statistics\) "Blocking (statistics)") [Factorial experiment](https://en.wikipedia.org/wiki/Factorial_experiment "Factorial experiment") [Interaction](https://en.wikipedia.org/wiki/Interaction_\(statistics\) "Interaction (statistics)") [Random assignment](https://en.wikipedia.org/wiki/Random_assignment "Random assignment") [Randomized controlled trial](https://en.wikipedia.org/wiki/Randomized_controlled_trial "Randomized controlled trial") [Randomized experiment](https://en.wikipedia.org/wiki/Randomized_experiment "Randomized experiment") [Scientific control](https://en.wikipedia.org/wiki/Scientific_control "Scientific control") |
| Adaptive designs | [Adaptive clinical trial](https://en.wikipedia.org/wiki/Adaptive_clinical_trial "Adaptive clinical trial") [Stochastic approximation](https://en.wikipedia.org/wiki/Stochastic_approximation "Stochastic approximation") [Up-and-down designs](https://en.wikipedia.org/wiki/Up-and-Down_Designs "Up-and-Down Designs") |
| [Observational studies](https://en.wikipedia.org/wiki/Observational_study "Observational study") | [Cohort study](https://en.wikipedia.org/wiki/Cohort_study "Cohort study") [Cross-sectional study](https://en.wikipedia.org/wiki/Cross-sectional_study "Cross-sectional study") [Natural experiment](https://en.wikipedia.org/wiki/Natural_experiment "Natural experiment") [Quasi-experiment](https://en.wikipedia.org/wiki/Quasi-experiment "Quasi-experiment") |
| [Statistical inference](https://en.wikipedia.org/wiki/Statistical_inference "Statistical inference") | |
| | |
| [Statistical theory](https://en.wikipedia.org/wiki/Statistical_theory "Statistical theory") | [Population](https://en.wikipedia.org/wiki/Population_\(statistics\) "Population (statistics)") [Statistic](https://en.wikipedia.org/wiki/Statistic "Statistic") [Probability distribution](https://en.wikipedia.org/wiki/Probability_distribution "Probability distribution") [Sampling distribution](https://en.wikipedia.org/wiki/Sampling_distribution "Sampling distribution") [Order statistic](https://en.wikipedia.org/wiki/Order_statistic "Order statistic") [Empirical distribution](https://en.wikipedia.org/wiki/Empirical_distribution_function "Empirical distribution function") [Density estimation](https://en.wikipedia.org/wiki/Density_estimation "Density estimation") [Statistical model](https://en.wikipedia.org/wiki/Statistical_model "Statistical model") [Model specification](https://en.wikipedia.org/wiki/Model_specification "Model specification") [L*p* space](https://en.wikipedia.org/wiki/Lp_space "Lp space") [Parameter](https://en.wikipedia.org/wiki/Statistical_parameter "Statistical parameter") [location](https://en.wikipedia.org/wiki/Location_parameter "Location parameter") [scale](https://en.wikipedia.org/wiki/Scale_parameter "Scale parameter") [shape](https://en.wikipedia.org/wiki/Shape_parameter "Shape parameter") [Parametric family](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") [Likelihood](https://en.wikipedia.org/wiki/Likelihood_function "Likelihood function") [(monotone)](https://en.wikipedia.org/wiki/Monotone_likelihood_ratio "Monotone likelihood ratio") [Locationāscale family](https://en.wikipedia.org/wiki/Location%E2%80%93scale_family "Locationāscale family") [Exponential family](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Completeness](https://en.wikipedia.org/wiki/Completeness_\(statistics\) "Completeness (statistics)") [Sufficiency](https://en.wikipedia.org/wiki/Sufficient_statistic "Sufficient statistic") [Statistical functional](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [U](https://en.wikipedia.org/wiki/U-statistic "U-statistic") [V](https://en.wikipedia.org/wiki/V-statistic "V-statistic") [Optimal decision](https://en.wikipedia.org/wiki/Optimal_decision "Optimal decision") [loss function](https://en.wikipedia.org/wiki/Loss_function "Loss function") [Efficiency](https://en.wikipedia.org/wiki/Efficiency_\(statistics\) "Efficiency (statistics)") [Statistical distance](https://en.wikipedia.org/wiki/Statistical_distance "Statistical distance") [divergence](https://en.wikipedia.org/wiki/Divergence_\(statistics\) "Divergence (statistics)") [Asymptotics](https://en.wikipedia.org/wiki/Asymptotic_theory_\(statistics\) "Asymptotic theory (statistics)") [Robustness](https://en.wikipedia.org/wiki/Robust_statistics "Robust statistics") |
| [Frequentist inference](https://en.wikipedia.org/wiki/Frequentist_inference "Frequentist inference") | |
| | |
| [Point estimation](https://en.wikipedia.org/wiki/Point_estimation "Point estimation") | [Estimating equations](https://en.wikipedia.org/wiki/Estimating_equations "Estimating equations") [Maximum likelihood](https://en.wikipedia.org/wiki/Maximum_likelihood "Maximum likelihood") [Method of moments](https://en.wikipedia.org/wiki/Method_of_moments_\(statistics\) "Method of moments (statistics)") [M-estimator](https://en.wikipedia.org/wiki/M-estimator "M-estimator") [Minimum distance](https://en.wikipedia.org/wiki/Minimum_distance_estimation "Minimum distance estimation") [Unbiased estimators](https://en.wikipedia.org/wiki/Bias_of_an_estimator "Bias of an estimator") [Mean-unbiased minimum-variance](https://en.wikipedia.org/wiki/Minimum-variance_unbiased_estimator "Minimum-variance unbiased estimator") [RaoāBlackwellization](https://en.wikipedia.org/wiki/Rao%E2%80%93Blackwell_theorem "RaoāBlackwell theorem") [LehmannāScheffĆ© theorem](https://en.wikipedia.org/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem "LehmannāScheffĆ© theorem") [Median unbiased](https://en.wikipedia.org/wiki/Median-unbiased_estimator "Median-unbiased estimator") [Plug-in](https://en.wikipedia.org/wiki/Plug-in_principle "Plug-in principle") |
| [Interval estimation](https://en.wikipedia.org/wiki/Interval_estimation "Interval estimation") | [Confidence interval](https://en.wikipedia.org/wiki/Confidence_interval "Confidence interval") [Pivot](https://en.wikipedia.org/wiki/Pivotal_quantity "Pivotal quantity") [Likelihood interval](https://en.wikipedia.org/wiki/Likelihood_interval "Likelihood interval") [Prediction interval](https://en.wikipedia.org/wiki/Prediction_interval "Prediction interval") [Tolerance interval](https://en.wikipedia.org/wiki/Tolerance_interval "Tolerance interval") [Resampling](https://en.wikipedia.org/wiki/Resampling_\(statistics\) "Resampling (statistics)") [Bootstrap](https://en.wikipedia.org/wiki/Bootstrapping_\(statistics\) "Bootstrapping (statistics)") [Jackknife](https://en.wikipedia.org/wiki/Jackknife_resampling "Jackknife resampling") |
| [Testing hypotheses](https://en.wikipedia.org/wiki/Statistical_hypothesis_testing "Statistical hypothesis testing") | [1- & 2-tails](https://en.wikipedia.org/wiki/One-_and_two-tailed_tests "One- and two-tailed tests") [Power](https://en.wikipedia.org/wiki/Power_\(statistics\) "Power (statistics)") [Uniformly most powerful test](https://en.wikipedia.org/wiki/Uniformly_most_powerful_test "Uniformly most powerful test") [Permutation test](https://en.wikipedia.org/wiki/Permutation_test "Permutation test") [Randomization test](https://en.wikipedia.org/wiki/Randomization_test "Randomization test") [Multiple comparisons](https://en.wikipedia.org/wiki/Multiple_comparisons "Multiple comparisons") |
| [Parametric tests](https://en.wikipedia.org/wiki/Parametric_statistics "Parametric statistics") | [Likelihood-ratio](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Score/Lagrange multiplier](https://en.wikipedia.org/wiki/Score_test "Score test") [Wald](https://en.wikipedia.org/wiki/Wald_test "Wald test") |
| [Specific tests](https://en.wikipedia.org/wiki/List_of_statistical_tests "List of statistical tests") | |
| | |
| [*Z*\-test (normal)](https://en.wikipedia.org/wiki/Z-test "Z-test") [Student's *t*\-test](https://en.wikipedia.org/wiki/Student%27s_t-test "Student's t-test") [*F*\-test](https://en.wikipedia.org/wiki/F-test "F-test") | |
| [Goodness of fit](https://en.wikipedia.org/wiki/Goodness_of_fit "Goodness of fit") | [Chi-squared](https://en.wikipedia.org/wiki/Chi-squared_test "Chi-squared test") [*G*\-test](https://en.wikipedia.org/wiki/G-test "G-test") [KolmogorovāSmirnov](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test "KolmogorovāSmirnov test") [AndersonāDarling](https://en.wikipedia.org/wiki/Anderson%E2%80%93Darling_test "AndersonāDarling test") [Lilliefors](https://en.wikipedia.org/wiki/Lilliefors_test "Lilliefors test") [JarqueāBera](https://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test "JarqueāBera test") [Normality (ShapiroāWilk)](https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test "ShapiroāWilk test") [Likelihood-ratio test](https://en.wikipedia.org/wiki/Likelihood-ratio_test "Likelihood-ratio test") [Model selection](https://en.wikipedia.org/wiki/Model_selection "Model selection") [Cross validation](https://en.wikipedia.org/wiki/Cross-validation_\(statistics\) "Cross-validation (statistics)") [AIC](https://en.wikipedia.org/wiki/Akaike_information_criterion "Akaike information criterion") [BIC](https://en.wikipedia.org/wiki/Bayesian_information_criterion "Bayesian information criterion") |
| [Rank statistics](https://en.wikipedia.org/wiki/Rank_statistics "Rank statistics") | [Sign](https://en.wikipedia.org/wiki/Sign_test "Sign test") [Sample median](https://en.wikipedia.org/wiki/Sample_median "Sample median") [Signed rank (Wilcoxon)](https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test "Wilcoxon signed-rank test") [HodgesāLehmann estimator](https://en.wikipedia.org/wiki/Hodges%E2%80%93Lehmann_estimator "HodgesāLehmann estimator") [Rank sum (MannāWhitney)](https://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test "MannāWhitney U test") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_statistics "Nonparametric statistics") [anova](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [1-way (KruskalāWallis)](https://en.wikipedia.org/wiki/Kruskal%E2%80%93Wallis_test "KruskalāWallis test") [2-way (Friedman)](https://en.wikipedia.org/wiki/Friedman_test "Friedman test") [Ordered alternative (JonckheereāTerpstra)](https://en.wikipedia.org/wiki/Jonckheere%27s_trend_test "Jonckheere's trend test") [Van der Waerden test](https://en.wikipedia.org/wiki/Van_der_Waerden_test "Van der Waerden test") |
| [Bayesian inference](https://en.wikipedia.org/wiki/Bayesian_inference "Bayesian inference") | [Bayesian probability](https://en.wikipedia.org/wiki/Bayesian_probability "Bayesian probability") [prior](https://en.wikipedia.org/wiki/Prior_probability "Prior probability") [posterior](https://en.wikipedia.org/wiki/Posterior_probability "Posterior probability") [Credible interval](https://en.wikipedia.org/wiki/Credible_interval "Credible interval") [Bayes factor](https://en.wikipedia.org/wiki/Bayes_factor "Bayes factor") [Bayesian estimator](https://en.wikipedia.org/wiki/Bayes_estimator "Bayes estimator") [Maximum posterior estimator](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation "Maximum a posteriori estimation") |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | |
| | |
| [Correlation](https://en.wikipedia.org/wiki/Correlation_and_dependence "Correlation and dependence") | [Pearson product-moment](https://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient "Pearson product-moment correlation coefficient") [Partial correlation](https://en.wikipedia.org/wiki/Partial_correlation "Partial correlation") [Confounding variable](https://en.wikipedia.org/wiki/Confounding "Confounding") [Coefficient of determination](https://en.wikipedia.org/wiki/Coefficient_of_determination "Coefficient of determination") |
| [Regression analysis](https://en.wikipedia.org/wiki/Regression_analysis "Regression analysis") | [Errors and residuals](https://en.wikipedia.org/wiki/Errors_and_residuals "Errors and residuals") [Regression validation](https://en.wikipedia.org/wiki/Regression_validation "Regression validation") [Mixed effects models](https://en.wikipedia.org/wiki/Mixed_model "Mixed model") [Simultaneous equations models](https://en.wikipedia.org/wiki/Simultaneous_equations_model "Simultaneous equations model") [Multivariate adaptive regression splines (MARS)](https://en.wikipedia.org/wiki/Multivariate_adaptive_regression_splines "Multivariate adaptive regression splines")  [Template:Least squares and regression analysis](https://en.wikipedia.org/wiki/Template:Least_squares_and_regression_analysis "Template:Least squares and regression analysis") |
| [Linear regression](https://en.wikipedia.org/wiki/Linear_regression "Linear regression") | [Simple linear regression](https://en.wikipedia.org/wiki/Simple_linear_regression "Simple linear regression") [Ordinary least squares](https://en.wikipedia.org/wiki/Ordinary_least_squares "Ordinary least squares") [General linear model](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Bayesian regression](https://en.wikipedia.org/wiki/Bayesian_linear_regression "Bayesian linear regression") |
| Non-standard predictors | [Nonlinear regression](https://en.wikipedia.org/wiki/Nonlinear_regression "Nonlinear regression") [Nonparametric](https://en.wikipedia.org/wiki/Nonparametric_regression "Nonparametric regression") [Semiparametric](https://en.wikipedia.org/wiki/Semiparametric_regression "Semiparametric regression") [Isotonic](https://en.wikipedia.org/wiki/Isotonic_regression "Isotonic regression") [Robust](https://en.wikipedia.org/wiki/Robust_regression "Robust regression") [Homoscedasticity and Heteroscedasticity](https://en.wikipedia.org/wiki/Homoscedasticity_and_heteroscedasticity "Homoscedasticity and heteroscedasticity") |
| [Generalized linear model](https://en.wikipedia.org/wiki/Generalized_linear_model "Generalized linear model") | [Exponential families](https://en.wikipedia.org/wiki/Exponential_family "Exponential family") [Logistic (Bernoulli)](https://en.wikipedia.org/wiki/Logistic_regression "Logistic regression") / [Binomial](https://en.wikipedia.org/wiki/Binomial_regression "Binomial regression") / [Poisson regressions](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") |
| [Partition of variance](https://en.wikipedia.org/wiki/Partition_of_sums_of_squares "Partition of sums of squares") | [Analysis of variance (ANOVA, anova)](https://en.wikipedia.org/wiki/Analysis_of_variance "Analysis of variance") [Analysis of covariance](https://en.wikipedia.org/wiki/Analysis_of_covariance "Analysis of covariance") [Multivariate ANOVA](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom_\(statistics\) "Degrees of freedom (statistics)") |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") / [multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") / [time-series](https://en.wikipedia.org/wiki/Time_series "Time series") / [survival analysis](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
| [Categorical](https://en.wikipedia.org/wiki/Categorical_variable "Categorical variable") | [Cohen's kappa](https://en.wikipedia.org/wiki/Cohen%27s_kappa "Cohen's kappa") [Contingency table](https://en.wikipedia.org/wiki/Contingency_table "Contingency table") [Graphical model](https://en.wikipedia.org/wiki/Graphical_model "Graphical model") [Log-linear model](https://en.wikipedia.org/wiki/Poisson_regression "Poisson regression") [McNemar's test](https://en.wikipedia.org/wiki/McNemar%27s_test "McNemar's test") [CochranāMantelāHaenszel statistics](https://en.wikipedia.org/wiki/Cochran%E2%80%93Mantel%E2%80%93Haenszel_statistics "CochranāMantelāHaenszel statistics") |
| [Multivariate](https://en.wikipedia.org/wiki/Multivariate_statistics "Multivariate statistics") | [Regression](https://en.wikipedia.org/wiki/General_linear_model "General linear model") [Manova](https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance "Multivariate analysis of variance") [Principal components](https://en.wikipedia.org/wiki/Principal_component_analysis "Principal component analysis") [Canonical correlation](https://en.wikipedia.org/wiki/Canonical_correlation "Canonical correlation") [Discriminant analysis](https://en.wikipedia.org/wiki/Linear_discriminant_analysis "Linear discriminant analysis") [Cluster analysis](https://en.wikipedia.org/wiki/Cluster_analysis "Cluster analysis") [Classification](https://en.wikipedia.org/wiki/Statistical_classification "Statistical classification") [Structural equation model](https://en.wikipedia.org/wiki/Structural_equation_modeling "Structural equation modeling") [Factor analysis](https://en.wikipedia.org/wiki/Factor_analysis "Factor analysis") [Multivariate distributions](https://en.wikipedia.org/wiki/Multivariate_distribution "Multivariate distribution") [Elliptical distributions](https://en.wikipedia.org/wiki/Elliptical_distribution "Elliptical distribution") [Normal](https://en.wikipedia.org/wiki/Multivariate_normal_distribution "Multivariate normal distribution") |
| [Time-series](https://en.wikipedia.org/wiki/Time_series "Time series") | |
| | |
| General | [Decomposition](https://en.wikipedia.org/wiki/Decomposition_of_time_series "Decomposition of time series") [Trend](https://en.wikipedia.org/wiki/Trend_estimation "Trend estimation") [Stationarity](https://en.wikipedia.org/wiki/Stationary_process "Stationary process") [Seasonal adjustment](https://en.wikipedia.org/wiki/Seasonal_adjustment "Seasonal adjustment") [Exponential smoothing]() [Cointegration](https://en.wikipedia.org/wiki/Cointegration "Cointegration") [Structural break](https://en.wikipedia.org/wiki/Structural_break "Structural break") [Granger causality](https://en.wikipedia.org/wiki/Granger_causality "Granger causality") |
| Specific tests | [DickeyāFuller](https://en.wikipedia.org/wiki/Dickey%E2%80%93Fuller_test "DickeyāFuller test") [Johansen](https://en.wikipedia.org/wiki/Johansen_test "Johansen test") [Q-statistic (LjungāBox)](https://en.wikipedia.org/wiki/Ljung%E2%80%93Box_test "LjungāBox test") [DurbināWatson](https://en.wikipedia.org/wiki/Durbin%E2%80%93Watson_statistic "DurbināWatson statistic") [BreuschāGodfrey](https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey_test "BreuschāGodfrey test") |
| [Time domain](https://en.wikipedia.org/wiki/Time_domain "Time domain") | [Autocorrelation (ACF)](https://en.wikipedia.org/wiki/Autocorrelation "Autocorrelation") [partial (PACF)](https://en.wikipedia.org/wiki/Partial_autocorrelation_function "Partial autocorrelation function") [Cross-correlation (XCF)](https://en.wikipedia.org/wiki/Cross-correlation "Cross-correlation") [ARMA model](https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model "Autoregressiveāmoving-average model") [ARIMA model (BoxāJenkins)](https://en.wikipedia.org/wiki/Box%E2%80%93Jenkins_method "BoxāJenkins method") [Autoregressive conditional heteroskedasticity (ARCH)](https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity "Autoregressive conditional heteroskedasticity") [Vector autoregression (VAR)](https://en.wikipedia.org/wiki/Vector_autoregression "Vector autoregression") ([Autoregressive model (AR)](https://en.wikipedia.org/wiki/Autoregressive_model "Autoregressive model")) |
| [Frequency domain](https://en.wikipedia.org/wiki/Frequency_domain "Frequency domain") | [Spectral density estimation](https://en.wikipedia.org/wiki/Spectral_density_estimation "Spectral density estimation") [Fourier analysis](https://en.wikipedia.org/wiki/Fourier_analysis "Fourier analysis") [Least-squares spectral analysis](https://en.wikipedia.org/wiki/Least-squares_spectral_analysis "Least-squares spectral analysis") [Wavelet](https://en.wikipedia.org/wiki/Wavelet "Wavelet") [Whittle likelihood](https://en.wikipedia.org/wiki/Whittle_likelihood "Whittle likelihood") |
| [Survival](https://en.wikipedia.org/wiki/Survival_analysis "Survival analysis") | |
| | |
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Exponential smoothing
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| Readable Markdown | **Exponential smoothing** or **exponential moving average (EMA)** is a [rule of thumb](https://en.wikipedia.org/wiki/Rule_of_thumb "Rule of thumb") technique for [smoothing](https://en.wikipedia.org/wiki/Smoothing "Smoothing") [time series](https://en.wikipedia.org/wiki/Time_series "Time series") data using the exponential [window function](https://en.wikipedia.org/wiki/Window_function "Window function"). Whereas in the [simple moving average](https://en.wikipedia.org/wiki/Simple_moving_average "Simple moving average") the past observations are [weighted](https://en.wikipedia.org/wiki/Weight_function "Weight function") equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied procedure for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is often used for analysis of time-series data.
Exponential smoothing is one of many [window functions](https://en.wikipedia.org/wiki/Window_functions "Window functions") commonly applied to smooth data in [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing"), acting as [low-pass filters](https://en.wikipedia.org/wiki/Low-pass_filter "Low-pass filter") to remove high-frequency [noise](https://en.wikipedia.org/wiki/Noise "Noise"). This method is preceded by [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "SimĆ©on Denis Poisson")'s use of recursive exponential window functions in convolutions from the 19th century, as well as [Kolmogorov and Zurbenko's use of recursive moving averages](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Zurbenko_filter "KolmogorovāZurbenko filter") from their studies of turbulence in the 1940s.
The raw data sequence is often represented by  beginning at time , and the output of the exponential smoothing algorithm is commonly written as , which may be regarded as a best estimate of what the next value of  will be. When the sequence of observations begins at time , the simplest form of exponential smoothing is given by the following formulas:[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1)
where  is the *smoothing factor*, and . If  is substituted into  continuously so that the formula of  is fully expressed in terms of , then [exponentially decaying weighting factors](https://en.wikipedia.org/wiki/Exponential_smoothing#"Exponential"_naming) on each raw data  is revealed, showing how exponential smoothing is named.
The simple exponential smoothing is not able to predict what would be observed at  based on the raw data up to , while the [double exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing#Double_exponential_smoothing_\(Holt_linear\)) and [triple exponential smoothing](https://en.wikipedia.org/wiki/Exponential_smoothing#Triple_exponential_smoothing_\(Holt%E2%80%93Winters\)) can be used for the prediction due to the presence of  as the sequence of best estimates of the linear trend.
## Basic (simple) exponential smoothing
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=1 "Edit section: Basic (simple) exponential smoothing")\]
The use of the exponential window function is first attributed to [Poisson](https://en.wikipedia.org/wiki/Sim%C3%A9on_Denis_Poisson "SimĆ©on Denis Poisson")[\[2\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Oppenheim,_Alan_V._1975_5-2) as an extension of a numerical analysis technique from the 17th century, and later adopted by the [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") community in the 1940s. Here, exponential smoothing is the application of the exponential, or Poisson, [window function](https://en.wikipedia.org/wiki/Window_function "Window function"). Exponential smoothing was suggested in the statistical literature without citation to previous work by [Robert Goodell Brown](https://en.wikipedia.org/wiki/Robert_Goodell_Brown "Robert Goodell Brown") in 1956,[\[3\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-3) and expanded by [Charles C. Holt](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") in 1957.[\[4\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-4) The formulation below, which is the one commonly used, is attributed to Brown and is known as "Brownās simple exponential smoothing".[\[5\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-5) All the methods of Holt, Winters, and Brown may be seen as a simple application of [recursive filtering](https://en.wikipedia.org/wiki/Recursive_filter "Recursive filter"), first found in the 1940s[\[2\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Oppenheim,_Alan_V._1975_5-2) to convert [finite impulse response](https://en.wikipedia.org/wiki/Finite_impulse_response "Finite impulse response") (FIR) filters to [infinite impulse response](https://en.wikipedia.org/wiki/Infinite_impulse_response "Infinite impulse response") filters.
The simplest form of exponential smoothing is given by the formula:
where  is the *smoothing factor*, with . In other words, the smoothed statistic  is a simple weighted average of the current [observation](https://en.wikipedia.org/wiki/Random_variate "Random variate")  and the previous smoothed statistic . Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available. The term *smoothing factor* applied to  here is something of a misnomer, as larger values of  actually reduce the level of smoothing, and in the limiting case with  = 1 the smoothing output series is just the current observation. Values of  close to 1 have less of a smoothing effect and give greater weight to recent changes in the data, while values of  closer to 0 have a greater smoothing effect and are less responsive to recent changes. In the limiting case with  = 0, the output series is just flat or a constant as the observation  at the beginning of the smoothening process .
The method for choosing  must be decided by the modeler. Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to *optimize* the value of . For example, the [method of least squares](https://en.wikipedia.org/wiki/Least_squares "Least squares") might be used to determine the value of  for which the sum of the quantities  is minimized.[\[6\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST6431-6)
Unlike some other smoothing methods, such as the simple moving average, this technique does not require any minimum number of observations to be made before it begins to produce results. In practice, however, a "good average" will not be achieved until several samples have been averaged together; for example, a constant signal will take approximately  stages to reach 95% of the actual value. To accurately reconstruct the original signal without information loss, all stages of the exponential moving average must also be available, because older samples decay in weight exponentially. This is in contrast to a simple moving average, in which some samples can be skipped without as much loss of information due to the constant weighting of samples within the average. If a known number of samples will be missed, one can adjust a weighted average for this as well, by giving equal weight to the new sample and all those to be skipped.
This simple form of exponential smoothing is also known as an [exponentially weighted moving average](https://en.wikipedia.org/wiki/Moving_average#Exponential_moving_average "Moving average") (EWMA). Technically it can also be classified as an [autoregressive integrated moving average](https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average "Autoregressive integrated moving average") (ARIMA) (0,1,1) model with no constant term.[\[7\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-7)
The [time constant](https://en.wikipedia.org/wiki/Time_constant "Time constant") of an exponential moving average is the amount of time for the smoothed response of a [unit step function](https://en.wikipedia.org/wiki/Unit_step_function "Unit step function") to reach  of the final signal. The relationship between this time constant, , and the smoothing factor, , is given by the following formula:
, thus 
where  is the sampling time interval of the discrete time implementation. If the sampling time is fast compared to the time constant () then, by using [the Taylor expansion of the exponential function](https://en.wikipedia.org/wiki/Taylor_series#Examples "Taylor series"),
, thus 
### Choosing the initial smoothed value
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=3 "Edit section: Choosing the initial smoothed value")\]
Note that in the definition above,  (the initial output of the exponential smoothing algorithm) is being initialized to  (the initial raw data or observation). Because exponential smoothing requires that, at each stage, we have the previous forecast , it is not obvious how to get the method started. We could assume that the initial forecast is equal to the initial value of demand; however, this approach has a serious drawback. Exponential smoothing puts substantial weight on past observations, so the initial value of demand will have an unreasonably large effect on early forecasts. This problem can be overcome by allowing the process to evolve for a reasonable number of periods (10 or more) and using the average of the demand during those periods as the initial forecast. There are many other ways of setting this initial value, but it is important to note that the smaller the value of , the more sensitive your forecast will be on the selection of this initial smoother value .[\[8\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-9)
For every exponential smoothing method, we also need to choose the value for the smoothing parameters. For simple exponential smoothing, there is only one smoothing parameter (*Ī±*), but for the methods that follow there are usually more than one smoothing parameter.
There are cases where the smoothing parameters may be chosen in a subjective manner ā the forecaster specifies the value of the smoothing parameters based on previous experience. However, a more robust and objective way to obtain values of the unknown parameters included in any exponential smoothing method is to estimate them from the observed data.
The unknown parameters and the initial values for any exponential smoothing method can be estimated by minimizing the [sum of squared errors](https://en.wikipedia.org/wiki/Sum_of_squared_errors_of_prediction "Sum of squared errors of prediction") (SSE). The errors are specified as  for  (the one-step-ahead within-sample forecast errors) where  and  are a variable to be predicted at  and a variable as the prediction result at  (based on the previous data or prediction), respectively. Hence, we find the values of the unknown parameters and the initial values that minimize
[\[10\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-otexts.org-10)
Unlike the regression case (where we have formulae to directly compute the regression coefficients which minimize the SSE) this involves a non-linear minimization problem, and we need to use an [optimization](https://en.wikipedia.org/wiki/Mathematical_optimization "Mathematical optimization") tool to perform this.
### "Exponential" naming
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=5 "Edit section: \"Exponential\" naming")\]
The name *exponential smoothing* is attributed to the use of the exponential function as the filter [impulse response](https://en.wikipedia.org/wiki/Impulse_response "Impulse response") in the [convolution](https://en.wikipedia.org/wiki/Convolution "Convolution").
By direct substitution of the defining equation for simple exponential smoothing back into itself we find that
![{\\displaystyle {\\begin{aligned}s\_{t}&=\\alpha x\_{t}+(1-\\alpha )s\_{t-1}\\\\\[3pt\]&=\\alpha x\_{t}+\\alpha (1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}s\_{t-2}\\\\\[3pt\]&=\\alpha \\left\[x\_{t}+(1-\\alpha )x\_{t-1}+(1-\\alpha )^{2}x\_{t-2}+(1-\\alpha )^{3}x\_{t-3}+\\cdots +(1-\\alpha )^{t-1}x\_{1}\\right\]+(1-\\alpha )^{t}x\_{0}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08e0741e7160744aaff5c671b60bc05f0be172fb)
In other words, as time passes the smoothed statistic  becomes the weighted average of a greater and greater number of the past observations , and the weights assigned to previous observations are proportional to the terms of the geometric progression
A [geometric progression](https://en.wikipedia.org/wiki/Geometric_progression "Geometric progression") is the discrete version of an [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function"), so this is where the name for this smoothing method originated according to [Statistics](https://en.wikipedia.org/wiki/Statistics "Statistics") lore.
### Comparison with moving average
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=6 "Edit section: Comparison with moving average")\]
Exponential smoothing and moving average have similar defects of introducing a lag relative to the input data. While this can be corrected by shifting the result by half the window length for a symmetrical kernel, such as a moving average or gaussian, this approach is not possible for exponential smoothing since it is an [IIR filter](https://en.wikipedia.org/wiki/IIR_filter "IIR filter") and therefore has an asymmetric kernel and frequency-dependent [group delay](https://en.wikipedia.org/wiki/Group_delay "Group delay"). This means each constituent frequency is shifted by a different amount and therefore, there is no single number of samples that can be used to shift the output signal to account for the lag.
Both filters also both have roughly the same distribution of forecast error when *Ī±* = 2/(*k* + 1) where *k* is the number of past data points in consideration of moving average. They differ in that exponential smoothing takes into account all past data, whereas moving average only takes into account *k* past data points. Computationally speaking, they also differ in that moving average requires that the past *k* data points, or the data point at lag *k* + 1 plus the most recent forecast value, to be kept, whereas exponential smoothing only needs the most recent forecast value to be kept.[\[11\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-11)
In the [signal processing](https://en.wikipedia.org/wiki/Signal_processing "Signal processing") literature, the use of non-causal (symmetric) filters is commonplace, and the exponential [window function](https://en.wikipedia.org/wiki/Window_function "Window function") is broadly used in this fashion, but a different terminology is used: exponential smoothing is equivalent to a first-order [infinite-impulse response](https://en.wikipedia.org/wiki/Infinite-impulse_response "Infinite-impulse response") (IIR) filter and moving average is equivalent to a [finite impulse response filter](https://en.wikipedia.org/wiki/Finite_impulse_response_filter "Finite impulse response filter") with equal weighting factors.
## Double exponential smoothing (Holt linear)
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=7 "Edit section: Double exponential smoothing (Holt linear)")\]
Simple exponential smoothing does not do well when there is a [trend](https://en.wikipedia.org/wiki/Trend_estimation "Trend estimation") in the data.[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1) In such situations, several methods were devised under the name "double exponential smoothing" or "second-order exponential smoothing," which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing". The basic idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series exhibiting some form of trend. This slope component is itself updated via exponential smoothing.
One method works as follows:[\[12\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-12)
Again, the raw data sequence of observations is represented by , beginning at time . We use  to represent the smoothed value for time , and  is our best estimate of the trend at time . The output of the algorithm is now written as , an estimate of the value of  at time  based on the raw data up to time . Double exponential smoothing is given by the formulas
and for  by
where  () is the *data smoothing factor*, and  () is the *trend smoothing factor*.
To forecast beyond  is given by the following approximation:
.
Setting the initial value  is a matter of preference. An option other than the one listed above is  for some .
Note that *F*0 is undefined (there is no estimation for time 0), and according to the definition *F*1\=*s*0\+*b*0, which is well defined, thus further values can be evaluated.
A second method, referred to as either Brown's linear exponential smoothing (LES) or Brown's double exponential smoothing, has only one smoothing factor, :[\[13\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-13)
where *a**t*, the estimated level at time *t*, and *b**t*, the estimated trend at time *t*, are given by
![{\\displaystyle {\\begin{aligned}a\_{t}&=2s'\_{t}-s''\_{t}\\\\\[5pt\]b\_{t}&={\\frac {\\alpha }{1-\\alpha }}(s'\_{t}-s''\_{t}).\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/496559e07c278001d2b3e1cf46743c3c4f9cae7c)
## Triple exponential smoothing (HoltāWinters)
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=8 "Edit section: Triple exponential smoothing (HoltāWinters)")\]
Triple exponential smoothing applies exponential smoothing three times, which is commonly used when there are three high frequency signals to be removed from a [time series](https://en.wikipedia.org/wiki/Time_series "Time series") under study. There are different types of seasonality: 'multiplicative' and 'additive' in nature. Multiplicative signifies that the seasonal effect size depends on the current level, while additive seasonal effects do not.
If in *every* summer we sell 10,000 more balls of ice cream than we do in winter, then seasonality is *additive*. However if we sell 6 times more ice cream in summer than in winter, that means that the difference in sales amounts moves together with the level itself: in years with overall larger consumption, the difference between summer and winter is larger and vice versa. The effect is thus *multiplicative*. Multiplicative seasonality can be represented as a constant factor, additive seasonality as an absolute amount.[\[14\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-14)
Triple exponential smoothing was first suggested by Holt's student, Peter Winters, in 1960 after reading a signal processing book from the 1940s on exponential smoothing.[\[15\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Winters_324%E2%80%93342-15) Holt's novel idea was to repeat filtering an odd number of times greater than 1 and less than 5, which was popular with scholars of previous eras.[\[15\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-Winters_324%E2%80%93342-15) While recursive filtering had been used previously, it was applied twice and four times to coincide with the [Hadamard conjecture](https://en.wikipedia.org/wiki/Hadamard_conjecture "Hadamard conjecture"), while triple application required more than double the operations of singular convolution. The use of a triple application is considered a [rule of thumb](https://en.wikipedia.org/wiki/Rule_of_thumb "Rule of thumb") technique, rather than one based on theoretical foundations and has often been over-emphasized by practitioners. Suppose we have a sequence of observations  beginning at time  with a cycle of seasonal change of length .
The method calculates a trend line for the data as well as seasonal indices that weight the values in the trend line based on where that time point falls in the cycle of length .
Let  represent the smoothed value of the constant part for time ,  is the sequence of best estimates of the linear trend that are superimposed on the seasonal changes, and  is the sequence of seasonal correction factors. We wish to estimate  at every time mod  in the cycle that the observations take on. As a rule of thumb, a minimum of two full seasons (or  periods) of historical data is needed to initialize a set of seasonal factors.
The output of the algorithm is again written as , an estimate of the value of  at time  based on the raw data up to time . Triple exponential smoothing with multiplicative seasonality is given by the formulas[\[1\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-NIST-1)
![{\\displaystyle {\\begin{aligned}s\_{0}&=x\_{0}\\\\\[5pt\]s\_{t}&=\\alpha {\\frac {x\_{t}}{c\_{t-L}}}+(1-\\alpha )(s\_{t-1}+b\_{t-1})\\\\\[5pt\]b\_{t}&=\\beta (s\_{t}-s\_{t-1})+(1-\\beta )b\_{t-1}\\\\\[5pt\]c\_{t}&=\\gamma {\\frac {x\_{t}}{s\_{t}}}+(1-\\gamma )c\_{t-L}\\\\\[5pt\]F\_{t+m}&=(s\_{t}+mb\_{t})c\_{t-L+1+(m-1){\\bmod {L}}},\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cee3212f28cf23aad40563caab52147a225fd2a8)
where  () is the *data smoothing factor*,  () is the *trend smoothing factor*, and  () is the *seasonal change smoothing factor*.
The general formula for the initial trend estimate  is
.
Setting the initial estimates for the seasonal indices  for  is a bit more involved. If  is the number of complete cycles present in your data, then
where
.
Note that  is the average value of  in the  cycle of your data.
This results in
Triple exponential smoothing with additive seasonality is given by \[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\]
## Implementations in statistics packages
\[[edit](https://en.wikipedia.org/w/index.php?title=Exponential_smoothing&action=edit§ion=9 "Edit section: Implementations in statistics packages")\]
- [R](https://en.wikipedia.org/wiki/R_\(programming_language\) "R (programming language)"): the HoltWinters function in the stats package[\[16\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-16) and ets function in the forecast package[\[17\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-17) (a more complete implementation, generally resulting in a better performance[\[18\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-18)).
- [Python](https://en.wikipedia.org/wiki/Python_\(programming_language\) "Python (programming language)"): the holtwinters module of the statsmodels package allow for simple, double and triple exponential smoothing.
- IBM [SPSS](https://en.wikipedia.org/wiki/SPSS "SPSS") includes Simple, Simple Seasonal, Holt's Linear Trend, Brown's Linear Trend, Damped Trend, Winters' Additive, and Winters' Multiplicative in the Time-Series modeling procedure within its Statistics and Modeler statistical packages. The default Expert Modeler feature evaluates all seven exponential smoothing models and ARIMA models with a range of nonseasonal and seasonal *p*, *d*, and *q* values, and selects the model with the lowest [Bayesian Information Criterion](https://en.wikipedia.org/wiki/Bayesian_Information_Criterion "Bayesian Information Criterion") statistic.
- [Stata](https://en.wikipedia.org/wiki/Stata "Stata"): tssmooth command[\[19\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-19)
- [LibreOffice](https://en.wikipedia.org/wiki/LibreOffice "LibreOffice") 5.2[\[20\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-20)
- [Microsoft Excel](https://en.wikipedia.org/wiki/Microsoft_Excel "Microsoft Excel") 2016[\[21\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-21)
- [Julia](https://en.wikipedia.org/wiki/Julia_\(programming_language\) "Julia (programming language)"): TrendDecomposition.jl package[\[22\]](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_note-22) implements simple and double exponential smoothing and Holts-Winters forecasting procedure.
- [Autoregressive moving average model](https://en.wikipedia.org/wiki/Autoregressive_moving_average_model "Autoregressive moving average model") (ARMA)
- [Errors and residuals in statistics](https://en.wikipedia.org/wiki/Errors_and_residuals_in_statistics "Errors and residuals in statistics")
- [Moving average](https://en.wikipedia.org/wiki/Moving_average "Moving average")
1. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-1) [***c***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST_1-2)
["NIST/SEMATECH e-Handbook of Statistical Methods"](http://www.itl.nist.gov/div898/handbook/). NIST. Retrieved 23 May 2010.
2. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Oppenheim,_Alan_V._1975_5_2-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Oppenheim,_Alan_V._1975_5_2-1)
Oppenheim, Alan V.; Schafer, Ronald W. (1975). *Digital Signal Processing*. [Prentice Hall](https://en.wikipedia.org/wiki/Prentice_Hall "Prentice Hall"). p. 5. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0-13-214635-5](https://en.wikipedia.org/wiki/Special:BookSources/0-13-214635-5 "Special:BookSources/0-13-214635-5")
.
3. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-3)**
Brown, Robert G. (1956). [*Exponential Smoothing for Predicting Demand*](http://legacy.library.ucsf.edu/tid/dae94e00;jsessionid=104A0CEFFA31ADC2FA5E0558F69B3E1D.tobacco03). Cambridge, Massachusetts: Arthur D. Little Inc. p. 15.
4. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-4)**
[Holt, Charles C.](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") (1957). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". *Office of Naval Research Memorandum*. **52**.
reprinted in
[Holt, Charles C.](https://en.wikipedia.org/wiki/Charles_C._Holt "Charles C. Holt") (JanuaryāMarch 2004). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". *[International Journal of Forecasting](https://en.wikipedia.org/wiki/International_Journal_of_Forecasting "International Journal of Forecasting")*. **20** (1): 5ā10\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.ijforecast.2003.09.015](https://doi.org/10.1016%2Fj.ijforecast.2003.09.015).
5. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-5)**
Brown, Robert Goodell (1963). [*Smoothing Forecasting and Prediction of Discrete Time Series*](http://babel.hathitrust.org/cgi/pt?id=mdp.39015004514728;view=1up;seq=1). Englewood Cliffs, NJ: Prentice-Hall.
6. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-NIST6431_6-0)**
["NIST/SEMATECH e-Handbook of Statistical Methods, 6.4.3.1. Single Exponential Smoothing"](http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm). NIST. Retrieved 5 July 2017.
7. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-7)**
Nau, Robert. ["Averaging and Exponential Smoothing Models"](http://www.duke.edu/~rnau/411avg.htm). Retrieved 26 July 2010.
8. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-8)** "Production and Operations Analysis" Nahmias. 2009.
9. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-9)** Äisar, P., & Äisar, S. M. (2011). "Optimization methods of EWMA statistics." *Acta Polytechnica Hungarica*, 8(5), 73ā87. Page 78.
10. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-otexts.org_10-0)**
[*7\.1 Simple exponential smoothing \| Forecasting: Principles and Practice*](https://www.otexts.org/fpp/7/1).
11. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-11)**
Nahmias, Steven; Olsen, Tava Lennon. *Production and Operations Analysis* (7th ed.). Waveland Press. p. [53](https://books.google.com/books?id=SIsoBgAAQBAJ&pg=PA53). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9781478628248](https://en.wikipedia.org/wiki/Special:BookSources/9781478628248 "Special:BookSources/9781478628248")
.
12. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-12)**
["6.4.3.3. Double Exponential Smoothing"](http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc433.htm). *itl.nist.gov*. Retrieved 25 September 2011.
13. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-13)**
["Averaging and Exponential Smoothing Models"](http://www.duke.edu/~rnau/411avg.htm). *duke.edu*. Retrieved 25 September 2011.
14. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-14)**
Kalehar, Prajakta S. ["Time series Forecasting using HoltāWinters Exponential Smoothing"](http://www.it.iitb.ac.in/~praj/acads/seminar/04329008_ExponentialSmoothing.pdf) (PDF). Retrieved 23 June 2014.
15. ^ [***a***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Winters_324%E2%80%93342_15-0) [***b***](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-Winters_324%E2%80%93342_15-1)
Winters, P. R. (April 1960). "Forecasting Sales by Exponentially Weighted Moving Averages". *[Management Science](https://en.wikipedia.org/wiki/Management_Science:_A_Journal_of_the_Institute_for_Operations_Research_and_the_Management_Sciences "Management Science: A Journal of the Institute for Operations Research and the Management Sciences")*. **6** (3): 324ā342\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1287/mnsc.6.3.324](https://doi.org/10.1287%2Fmnsc.6.3.324).
16. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-16)**
["R: HoltāWinters Filtering"](https://stat.ethz.ch/R-manual/R-patched/library/stats/html/HoltWinters.html). *stat.ethz.ch*. Retrieved 5 June 2016.
17. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-17)**
["ets {forecast} \| inside-R \| A Community Site for R"](https://web.archive.org/web/20160716153135/http://www.inside-r.org/packages/cran/forecast/docs/ets). *inside-r.org*. Archived from [the original](http://www.inside-r.org/packages/cran/forecast/docs/ets) on 16 July 2016. Retrieved 5 June 2016.
18. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-18)**
["Comparing HoltWinters() and ets()"](http://robjhyndman.com/hyndsight/estimation2/). *Hyndsight*. 29 May 2011. Retrieved 5 June 2016.
19. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-19)** [tssmooth](https://www.stata.com/help.cgi?tssmooth) in Stata manual
20. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-20)**
["LibreOffice 5.2: Release Notes ā the Document Foundation Wiki"](https://wiki.documentfoundation.org/ReleaseNotes/5.2#New_spreadsheet_functions).
21. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-21)**
["Excel 2016 Forecasting Functions \| Real Statistics Using Excel"](http://www.real-statistics.com/time-series-analysis/basic-time-series-forecasting/excel-2016-forecasting-functions/).
22. **[^](https://en.wikipedia.org/wiki/Exponential_smoothing#cite_ref-22)** [TrendDecomposition.jl](https://sdbrinkmann.github.io/TrendDecomposition.jl/stable/) Julia implementation of exponential smoothing and Holt-Winters forecasting procedure
- [Lecture notes on exponential smoothing (Robert Nau, Duke University)](http://people.duke.edu/~rnau/411avg.htm)
- [Data Smoothing](http://demonstrations.wolfram.com/DataSmoothing/) by Jon McLoone, [The Wolfram Demonstrations Project](https://en.wikipedia.org/wiki/The_Wolfram_Demonstrations_Project "The Wolfram Demonstrations Project")
- [The HoltāWinters Approach to Exponential Smoothing: 50 Years Old and Going Strong](https://foresight.forecasters.org/product/foresight-issue-19/) by Paul Goodwin (2010) [Foresight: The International Journal of Applied Forecasting](https://en.wikipedia.org/wiki/Foresight:_The_International_Journal_of_Applied_Forecasting "Foresight: The International Journal of Applied Forecasting")
- [Algorithms for Unevenly Spaced Time Series: Moving Averages and Other Rolling Operators](https://web.archive.org/web/20150623211648/http://www.eckner.com/papers/ts_alg.pdf) by Andreas Eckner |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Exponential_smoothing s443 |