Exponential smoothing 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 timeseries data.
Exponential smoothing is one of many window functions commonly applied to smooth data in signal processing, acting as lowpass 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 formulas:^{[1]}
where is the smoothing factor, and .
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Transcription
Welcome to this forecasting tutorial on Exponential Smoothing. We will be calculating forecasts using exponential smoothing, we will also be calculating the mean squared error for the forecasts. We will be calculating exponentially smoothed forecast for these sales values collected over a 7week period. The actual sales values will be represented by At. Note that some authors use Yt to represent actual values. Forecast values on the other hand are generally represented by Ft. The Exponential Smoothing Method uses the formula Ft+1 = Ft + α(At – Ft) where alpha is a value between 0 and 1, referred to as the smoothing constant. This means that the forecast for the current period is obtained by adding the forecast in the last period to a fraction of the error from the last period. To make calculations easier, this formula can be rewritten as Ft+1 = αAt + (1  α)Ft That is, the new forecast equals the alpha times the last actual value plus (1 –alpha) times the last forecast value. We will be using this second formula for our calculations. Note that both formulas will give the same result. It is just a little easier to use the second one. So our first objective is to calculate exponential smoothing forecasts data using α = 0.2. Since alpha = 0.2, then 1 minus alpha will be 1  0.2 which equals 0.8. The formula then becomes Ft+1 = 0.2 time the actual values + 0.8 times the forecast values. Since the forecast requires both actual and forecast values from the last period, we are sometimes given a forecast value for the first period. If no forecast value for the first period is given, then we assume F1 = A1. That is, the first forecast value is assumed to be the first actual value. So F1 = 39. We then calculate F2 as 0.2(39) + 0.8(39) which again will always give the same value, 39. As a result, we just usually assume that F2 = A1 and not bother calculating it at all. That is, if F1 is not given, we simply start by copying whatever value we have in A1 into F2. This means that our first real use of the formula begins with F3. So F3 is going to be .2 times 44 + .8 times 39, which gives 40. Consequently, F4 will be 0.2(40) + 0.8(40.00) which gives 40. For F5, we have 0.2(45) + 0.8(40) which gives 41. For F6, 0.2(38) + 0.8(41.00) which gives 40.4. F7 equals 0.2(43) + 0.8(40.40) and that gives 40.92. We can also forecast week 8 as 0.2(39) + 0.8(40.92) which gives 40.54. Next we compute the mean squared error. To obtain the mean squared error, MSE, we first obtain the forecast errors, square them, and then find the mean or average of the squared errors. When computing errors for exponential smoothing forecasts, we do not calculate an error for period 1 unless otherwise stated. So the error for week 2 is 44  39 which is 5. For week 3, it is 40 minus 40 which gives 0. For week 4, it is 45 minus 40 which is 5. For week 5, it is 38 minus 41 which gives 3. For week 6 it is 2.6, and for week 7 it is 1.92. Next we square the errors. Starting with week 2, 5 squared is 25 Zero squared is zero 3 squared is 9 2.6 squared is 6.76 and 1.92 squared is 3.69. Since we only have 6 periods with errors, we add up these squared errors and divide by 6 to obtain a Mean squared error of 11.58. And that’s how to calculate forecasts and MSE with exponential smoothing. Please leave a question or comment below and thanks for watching.
Contents
Background
Exponential smoothing is based on the use of window functions to smooth time series data.
Basic exponential smoothing
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 first suggested in the statistical literature without citation to previous work by Robert Goodell Brown in 1956,^{[3]} and then 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 “Brown’s 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 FIR filters to IIR filters.
The simplest form of exponential smoothing is given by the formula:
 .
where α is the smoothing factor, and 0 < α < 1. In other words, the smoothed statistic s_{t'} is a simple weighted average of the current observation x_{t} and the previous smoothed statistic s_{t−1}. The term smoothing factor applied to α here is something of a misnomer, as larger values of α actually reduce the level of smoothing, and in the limiting case with α = 1 the output series is just the current observation. Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available.
Values of α close to one have less of a smoothing effect and give greater weight to recent changes in the data, while values of α closer to zero have a greater smoothing effect and are less responsive to recent changes. There is no formally correct procedure for choosing α. Sometimes the statistician's judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to optimize the value of α. For example, the method of least squares might be used to determine the value of α 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 3/α 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]}
Time Constant
The time constant of an exponential moving average is the amount of time for the smoothed response of a unit set function to reach of the original signal. The relationship between this time constant, , and the smoothing factor, , is given by the formula:
Where is the sampling time interval of the discrete time implementation. If the sampling time is fast compared to the time constant () then
Choosing the initial smoothed value
Note that in the definition above, s_{0} is being initialized to x_{0}. 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 α, the more sensitive your forecast will be on the selection of this initial smoother value s_{1}.^{[8]}
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 (α), but for the methods that follow there is usually more than one smoothing parameter.
There are cases where the smoothing parameters may be chosen in a subjective manner — the forecaster specifies the value of the smoothing parameters based on previous experience. However, a more robust and objective way to obtain values for 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 t=1,...,T (the onestepahead withinsample forecast errors). Hence we find the values of the unknown parameters and the initial values that minimize
^{[9]}
Unlike the regression case (where we have formulae to directly compute the regression coefficients which minimize the SSE) this involves a nonlinear minimization problem and we need to use an optimization tool to perform this.
“Exponential” naming
The name 'exponential smoothing' is attributed to the use of the exponential window function during convolution. It is no longer attributed to Holt, Winters & Brown.
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 s_{t} becomes the weighted average of a greater and greater number of the past observations x_{t−n}, and the weights assigned to previous observations are in general proportional to the terms of the geometric progression {1, (1 − α), (1 − α)^{2}, (1 − α)^{3}, ...}. 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
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, it is unclear how appropriate this would be for exponential smoothing. They also both have roughly the same distribution of forecast error when α = 2/(k+1). 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.^{[10]}
In the signal processing literature, the use of noncausal (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 firstorder Infinite Impulse Response or IIR filter and moving average is equivalent to a Finite Impulse Response or FIR filter with equal weighting factors.
Double exponential smoothing
Simple exponential smoothing does not do well when there is a trend in the data, which is inconvenient.^{[1]} In such situations, several methods were devised under the name "double exponential smoothing" or "secondorder exponential smoothing.", which is the recursive application of an exponential filter twice, thus being termed "double exponential smoothing". This nomenclature is similar to quadruple exponential smoothing, which also references its recursion depth.^{[11]} 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, sometimes referred to as "HoltWinters double exponential smoothing" works as follows:^{[12]}
Again, the raw data sequence of observations is represented by {x_{t}}, beginning at time t = 0. We use {s_{t}} to represent the smoothed value for time t, and {b_{t}} is our best estimate of the trend at time t. The output of the algorithm is now written as F_{t+m}, an estimate of the value of x at time t + m for m > 0 based on the raw data up to time t. Double exponential smoothing is given by the formulas
And for t > 1 by
where α is the data smoothing factor, 0 < α < 1, and β is the trend smoothing factor, 0 < β < 1.
To forecast beyond x_{t}
Setting the initial value b_{0} is a matter of preference. An option other than the one listed above is (x_{n}  x_{0})/n for some n > 1.
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 works as follows.^{[13]}
where a_{t}, the estimated level at time t and b_{t}, the estimated trend at time t are:
Triple exponential smoothing
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, much like addition and multiplication are basic operations in mathematics.
If every month of December we sell 10,000 more apartments than we do in November the seasonality is additive in nature. However, if we sell 10% more apartments in the summer months than we do in the winter months the seasonality is multiplicative in nature. Multiplicative seasonality can be represented as a constant factor, not 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 overemphasized by practitioners.
Suppose we have a sequence of observations {x_{t}}, beginning at time t = 0 with a cycle of seasonal change of length 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.
{s_{t}} represents the smoothed value of the constant part for time t. {b_{t}} represents the sequence of best estimates of the linear trend that are superimposed on the seasonal changes. {c_{t}} is the sequence of seasonal correction factors. c_{t} is the expected proportion of the predicted trend at any time t mod L in the cycle that the observations take on. As a rule of thumb, a minimum of two full seasons (or 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}, an estimate of the value of x at time t+m, m>0 based on the raw data up to time t. Triple exponential smoothing with multiplicative seasonality is given by the formulas^{[1]}
where α is the data smoothing factor, 0 < α < 1, β is the trend smoothing factor, 0 < β < 1, and γ is the seasonal change smoothing factor, 0 < γ < 1.
The general formula for the initial trend estimate b_{0} is:
Setting the initial estimates for the seasonal indices c_{i} for i = 1,2,...,L is a bit more involved. If N is the number of complete cycles present in your data, then:
where
Note that A_{j} is the average value of x in the jth cycle of your data.
Triple exponential smoothing with additive seasonality is given by:
Implementations in statistics packages
 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]}).
 IBM SPSS includes Simple, Simple Seasonal, Holt's Linear Trend, Brown's Linear Trend, Damped Trend, Winters' Additive, and Winters' Multiplicative in the TimeSeries 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]}
See also
 Autoregressive moving average model (ARMA)
 Errors and residuals in statistics
 Moving average
 Continued fraction
Notes
 ^ ^{a} ^{b} ^{c} "NIST/SEMATECH eHandbook of Statistical Methods". NIST. Retrieved 23 May 2010.
 ^ ^{a} ^{b} Oppenheim, Alan V.; Schafer, Ronald W. (1975). Digital Signal Processing. Prentice Hall. p. 5. ISBN 0132146355.
 ^ Brown, Robert G. (1956). Exponential Smoothing for Predicting Demand. Cambridge, Massachusetts: Arthur D. Little Inc. p. 15.
 ^ Holt, Charles C. (1957). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". Office of Naval Research Memorandum. 52. reprinted in Holt, Charles C. (January – March 2004). "Forecasting Trends and Seasonal by Exponentially Weighted Averages". International Journal of Forecasting. 20 (1): 5–10.
 ^ Brown, Robert Goodell (1963). Smoothing Forecasting and Prediction of Discrete Time Series. Englewood Cliffs, NJ: PrenticeHall.
 ^ "NIST/SEMATECH eHandbook 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.
 ^ https://www.otexts.org/fpp/7/1
 ^ Nahmias, Steven. Production and Operations Analysis (6th ed.). ISBN 0073377856.^{[page needed]}
 ^ "Model: SecondOrder Exponential Smoothing". SAP AG. Retrieved 23 January 2013.
 ^ "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 HoltWinters 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): 324–342. doi:10.1287/mnsc.6.3.324.
 ^ "R: HoltWinters Filtering". stat.ethz.ch. Retrieved 5 June 2016.
 ^ "ets {forecast}  insideR  A Community Site for R". www.insider.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
 ^ https://wiki.documentfoundation.org/ReleaseNotes/5.2#New_spreadsheet_functions
 ^ http://www.realstatistics.com/timeseriesanalysis/basictimeseriesforecasting/excel2016forecastingfunctions/
External links
 Lecture notes on exponential smoothing (Robert Nau, Duke University)
 Data Smoothing by Jon McLoone, The Wolfram Demonstrations Project
 The HoltWinters 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