Exponential smoothing

Exponential smoothing is a technique that can be applied to time series data, either to produce smoothed data for presentation, or to make forecasts. The time series data themselves are a sequence of observations. The observed phenomenon may be an essentially random process, or it may be an orderly, but noisy, process. Whereas in the simple moving average the past observations are weighted equally, exponential smoothing assigns exponentially decreasing weights over time.

Exponential smoothing is commonly applied to financial market and economic data, but it can be used with any discrete set of repeated measurements. The raw data sequence is often represented by {x_t}, and the output of the exponential smoothing algorithm is commonly written as {s_t}, which may be regarded as a best estimate of what the next value of x will be. When the sequence of observations begins at time t = 0, the simplest form of exponential smoothing is given by the formulas^[1]

$\begin{align} s_1& = x_0\\ s_{t}& = \alpha x_{t-1} + (1-\alpha)s_{t-1}, t>1 \end{align}$

where α is the smoothing factor, and 0 < α < 1.

Background

The simple moving average

Intuitively, the simplest way to smooth a time series is to calculate a simple, or unweighted, moving average. The smoothed statistic s_t is then just the mean of the last k observations:

$s_t = \frac{1}{k} \, \sum_{n=0}^{k-1} x_{t-n} = \frac{x_t + x_{t-1} + x_{t-2} + \cdots + x_{t-k+1}}{k} = s_{t-1} + \frac{x_t - x_{t-k}}{k},$

where the choice of an integer k > 1 is arbitrary. A small value of k will have less of a smoothing effect and be more responsive to recent changes in the data, while a larger k will have a greater smoothing effect, and produce a more pronounced lag in the smoothed sequence. One disadvantage of this technique is that it cannot be used on the first k −1 terms of the time series.

The weighted moving average

A slightly more intricate method for smoothing a raw time series {x_t} is to calculate a weighted moving average by first choosing a set of weighting factors

$\lbrace w_1, w_2,\dots,w_k \rbrace$ such that $\sum_{n=1}^k w_n = 1$

and then using these weights to calculate the smoothed statistics {s_t}:

$s_t = \sum_{n=1}^k w_n x_{t+1-n} = w_1x_t + w_2x_{t-1} + \cdots + w_kx_{t-k+1}.$

In practice the weighting factors are often chosen to give more weight to the most recent terms in the time series and less weight to older data. Notice that this technique has the same disadvantage as the simple moving average technique (i.e., it cannot be used until at least k observations have been made), and that it entails a more complicated calculation at each step of the smoothing procedure. In addition to this disadvantage, if the data from each stage of the averaging is not available for analysis, it may be difficult if not impossible to reconstruct a changing signal accurately (because older samples may be given less weight). If the number of stages missed is known however, the weighting of values in the average can be adjusted to give equal weight to all missed samples to avoid this issue.

The exponential moving average

Exponential smoothing was first suggested by Charles C. Holt in 1957,^[2] although the formulation below, which is the one commonly used, is attributed to Brown and is known as "Brown's simple exponential smoothing".^[3]

The simplest form of exponential smoothing is given by the formulae:

$\begin{align} s_1& = x_0\\ s_t& = \alpha x_{t-1} + (1-\alpha)s_{t-1} = s_{t-1} + \alpha (x_{t-1} - s_{t-1}), t>1 \, \end{align}$

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 previous observation x_t-1 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. In the limiting case with α = 1 the output series is just the same as the original series. 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 (s_n-1 − x_n-1)² is minimized.

Unlike some other smoothing methods, 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.^[4]

Why is it "exponential"?

By direct substitution of the defining equation for simple exponential smoothing back into itself we find that

$\begin{align} s_t& = \alpha x_{t-1} + (1-\alpha)s_{t-1}\\[3pt] & = \alpha x_{t-1} + \alpha (1-\alpha)x_{t-2} + (1 - \alpha)^2 s_{t-2}\\[3pt] & = \alpha \left[x_{t-1} + (1-\alpha)x_{t-2} + (1-\alpha)^2 x_{t-3} + (1-\alpha)^3 x_{t-4} + \cdots \right] + (1-\alpha)^{t-1} x_0. \end{align}$

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.

Comparison with moving average

Exponential smoothing and moving average are similar in that they both assume a stationary, not trending, time series, therefore lagging behind the trend if one exists. 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. Technically speaking, they also differ in that moving average requires that the past k data points be kept, whereas exponential smoothing only needs the most recent forecast value to be kept.^[5]

Double exponential smoothing

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".

One method, sometimes referred to as "Holt-Winters double exponential smoothing"^[6] works as follows:^[7]

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, m>0 based on the raw data up to time t. Double exponential smoothing is given by the formulas

$\begin{align} s_0& = x_0\\ s_{t}& = \alpha x_{t} + (1-\alpha)(s_{t-1} + b_{t-1})\\ b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\ F_{t+m}& = s_t + mb_t, \end{align}$

where α is the data smoothing factor, 0 < α < 1, β is the trend smoothing factor, 0 < β < 1, and b₀ is taken as (x_n-1 - x₀)/(n - 1) for some n > 1. Note that F₀ is undefined (there is no estimation for time 0), and according to the definition F₁=s₀+b₀, 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.^[8]

$\begin{align} s'_0& = x_0\\ s''_0& = x_0\\ s'_{t}& = \alpha x_{t} + (1-\alpha)s'_{t-1}\\ s''_{t}& = \alpha s'_{t} + (1-\alpha)s''_{t-1}\\ F_{t+m}& = a_t + mb_t, \end{align}$

where a_t, the estimated level at time t and b_t, the estimated trend at time t are:

$\begin{align} a_t& = 2s'_t - s''_t\\ b_t& = \frac \alpha {1-\alpha} (s'_t - s''_t). \end{align}$

Triple exponential smoothing

Triple exponential smoothing takes into account seasonal changes as well as trends. It was first suggested by Holt's student, Peter Winters, in 1960.^[9]

The sequence of observations is again represented by {x_t}, beginning at time t = 0. {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 for time t. L is the period of time of one cycle of seasonal change. 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 is given by the formulas^[1]

$\begin{align} s_0& = x_0\\ s_{t}& = \alpha \frac{x_{t}}{c_{t-L}} + (1-\alpha)F_t\\ b_{t}& = \beta (s_t - s_{t-1}) + (1-\beta)b_{t-1}\\ c_{t}& = \gamma \frac{x_{t}}{s_{t}}+(1-\gamma)c_{t-L}\\ F_{t+m}& = (s_t + mb_t)c_{(t+m) \pmod L}, \end{align}$

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.

See also

• Moving average
• Autoregressive moving average model (ARMA)
• Autoregressive integrated moving average (ARIMA)
• Errors and residuals in statistics

Notes

1. ^ ^{a b c} "NIST/SEMATECH e-Handbook of Statistical Methods". NIST. http://www.itl.nist.gov/div898/handbook/. Retrieved 2010-05-23. 
2. ^ 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. doi:10.1016/S1097-2765(00)00071-X. PMID 11030352. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V92-4BJVV07-3&_user=1535420&_coverDate=03%2F31%2F2004&_rdoc=1&_fmt=high&_orig=gateway&_origin=gateway&_sort=d&_docanchor=&view=c&_searchStrId=1755927165&_rerunOrigin=google&_acct=C000053610&_version=1&_urlVersion=0&_userid=1535420&md5=0e881c06b0512cbf976a92268d48edc4&searchtype=a. 
3. ^ Brown, Robert Goodell (1963). Smoothing Forecasting and Prediction of Discrete Time Series. Englewood Cliffs, NJ: Prentice-Hall. 
4. ^ "Averaging and Exponential Smoothing Models". http://www.duke.edu/~rnau/411avg.htm. Retrieved 26 July 2010. 
5. ^ Nahmias, Steven. Production and Operations Analysis (6th edition ed.). ISBN 0073377856. ^{[page needed]}
6. ^ Prajakta S. Kalekar. "Time series Forecasting using Holt-Winters Exponential Smoothing" (PDF). http://www.it.iitb.ac.in/~praj/acads/seminar/04329008_ExponentialSmoothing.pdf. 
7. ^ "6.4.3.3. Double Exponential Smoothing". itl.nist.gov. http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc433.htm. Retrieved 25 September 2011. 
8. ^ "Averaging and Exponential Smoothing Models". duke.edu. http://www.duke.edu/~rnau/411avg.htm. Retrieved 25 September 2011. 
9. ^ 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. http://mansci.journal.informs.org/cgi/content/abstract/6/3/324. 

External links

• Notes for a statistics class (Decision 411) at Duke University
• Data Smoothing by Jon McLoone, The Wolfram Demonstrations Project.
• The Holt-Winters Approach to Exponential Smoothing: 50 Years Old and Going Strong by Paul Goodwin (2010) Foresight: The International Journal of Applied Forecasting