Exponential Smoothing Equations
Exponential smoothing was initially introduced in the statistical literature without considering the past work done by Robert Goodell Brown in 1956 and then further expanded by Charles C. Holt in 1957. Exponential smoothing is a reliable principle for smoothing time series data through the exponential window function. The controlling input of the exponential smoothing calculation is stated as the smoothing factor or the smoothing constant.
Forecast of the weighted averages of past observations are introduced using exponential smoothing methods, with the weights breaking down exponentially as the observations get formed. In other words, the more the latest the observation the higher the corresponding weight.
As we are aware of the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It can be easily applied for making determinations on the basis of prior assumptions by the user, such as seasonality. Exponential smoothing is primarily used for time-series data analysis.
Exponential Smoothing Formula
The exponential smoothing formula is derived by:
st = θxt+(1 – θ)st-1= st-1+ θ(xt – st-1)
Here,
st is a former smoothed statistic, it is the simple weighted average of present observation xt
st-1 is former smoothed statistic
θ is smoothing factor of data; 0 < θ < 1
t is time period
If the value of the smoothing factor is greater, then the level of smoothing will be minimized. Value of α nearer to 1 minimum smoothing effect and offer higher weights to recent changes in the data, while the value of θ nearer to zero has higher smoothing effect and is less responsive to recent changes.
There is no precise method of choosing θ. Accurate factors are selected on the basis of the statistician's judgments or else a statistical technique may be used to optimize the value of θ. For example, the method of least squares can be used to estimate the value of θ for which the sum of the quantities is diminished.
Exponential Smoothing Methods
The three exponential smoothing methods to determine exponential smoothing. They are:
Simple or single exponential smoothing
Double exponential smoothing
Triple exponential smoothing
Single Exponential Smoothing
If the data which is obtained has no trend and no seasonal pattern, then the single exponential smoothing method for forecasting the time series is primarily used. This method makes use of weighted moving averages with exponentially decreasing weights.
The single exponential smoothing method formula is given by:
st = θxt+(1 – θ)st-1 = st-1 + θ(xt – st-1)
Double Exponential Smoothing
The double exponential smoothing method is also known as Holt's trend corrected or second-order exponential smoothing. This method is primarily used to forecast the time series when the data has a linear trend and no seasonal pattern. The motive of double exponential smoothing is to introduce a term considering the possibility of a series indicating some form of trend. This slope component is itself reformed through exponential smoothing.
The double exponential smoothing formula is derived by:
S1 = y1
B1 = y1-y0
For t>1,
st = θyt + (1 – θ)(st-1 + bt-1)
βt = β(st – st-1) + (1 – β)bt-1
Here,
St is smoothed statistic, it is the simple weighted average of present observation yt
st-1 = former smoothed statistic
θ = smoothing factor of data; 0 < θ < 1
t = time period
bt = accurate estimation of trend at time t
β = trend smoothing factor; 0 < β <1
Triple Exponential Smoothing
In the triple exponential smoothing method, exponential smoothing is used thrice. This method is primarily used to forecast the time series when the data has both linear trend and seasonal patterns.This method is also known as holt-Winters exponential smoothing.
The triple exponential smoothing formula is derived by:
s\[_{0}\] = x\[_{0}\]
s\[_{t}\] = α\[\frac{x_{t}}{c_{t-L}}\] + (1 - α)(s\[_{t-1}\] + b\[_{t-1}\])
b\[_{t}\] = β(s\[_{t}\] - s\[_{t-1}\] + (1 - β)b\[_{t-1}\]
c\[_{t}\] = γ\[\frac{x_{t}}{s_{t}}\] + (1 - γ)c\[_{t-L}\]
Here,
st = smoothed statistic, it is the simple weighted average of present observation xt
st-1 = previous smoothed statistic
α = smoothing factor of data; 0 < α < 1
t = time period
bt = accurate estimation of trend at time t
β = trend smoothing factor; 0 < β <1
ct = sequence of seasonal error-free factors at time t
γ = seasonal variation smoothing factor; 0 < γ < 1
Solved Examples
1. The Sales of Books in a Bookstall for the Last 10 Months is Given Below in Tabulated Form. Calculate the Simple Exponential Smoothing Estimating α =0.3 for the Below Data.
Solution
Quiz Time
1. The Use of Smoothing Technique is Accurate When
The primary source of variation is random behavior
Seasonality is includes
Data exhibit a strong trend
All the above are accurate
2. Times Series Data is Classified in
Two components
Three components
Four components
Five components
FAQs on Exponential Smoothing
1. What is meant by exponential smoothing in time series analysis?
Exponential smoothing is a time series forecasting method for univariate data that can be extended to support data with a systematic trend or seasonal component. It assigns exponentially decreasing weights to past observations, meaning more recent data points are given more importance in the forecast than older data points. This makes it particularly effective for short-term forecasting.
2. What is the formula for simple exponential smoothing?
The formula for simple exponential smoothing (SES) is used for data without a trend or seasonality. The forecast for the next period (St+1) is calculated as:
St+1 = αYt + (1 - α)St
Where:
- St+1 is the forecast for the next period.
- α (alpha) is the smoothing constant (0 ≤ α ≤ 1).
- Yt is the actual observed value in the current period t.
- St is the forecast for the current period t.
3. What is the role of the smoothing constant (alpha) in exponential smoothing?
The smoothing constant, alpha (α), determines how much weight is given to the most recent observation versus older observations. Its value ranges from 0 to 1.
- A high alpha (e.g., 0.8) makes the forecast highly responsive to recent changes, as it gives more weight to the latest actual data.
- A low alpha (e.g., 0.2) results in a smoother forecast that is less influenced by recent random fluctuations, as it gives more weight to past forecasts.
4. When should you use exponential smoothing for forecasting?
Exponential smoothing is best used for short-term forecasting when dealing with time series data. The specific type of exponential smoothing to use depends on the data's characteristics:
- Use Simple Exponential Smoothing when the data is stationary (no trend or seasonality).
- Use Double Exponential Smoothing (Holt's method) when the data has a clear trend but no seasonality.
- Use Triple Exponential Smoothing (Holt-Winters method) when the data exhibits both a trend and seasonality.
5. What are the key advantages of using the exponential smoothing method?
The main advantages of exponential smoothing are:
- Simplicity: The models are relatively easy to understand and computationally inexpensive, requiring only a few data points to start.
- Accuracy for Short-Term Forecasts: It often provides highly accurate forecasts for the near future, especially when the underlying patterns are stable.
- Emphasis on Recent Data: It intuitively gives more importance to recent events, which is often a realistic assumption for many business and economic time series.
6. What are the limitations of exponential smoothing?
Despite its benefits, exponential smoothing has some limitations:
- Lagging Forecasts: The models tend to lag behind actual data, especially when there are sharp turns or shifts in the pattern.
- Parameter Selection: Finding the optimal values for the smoothing constants (alpha, beta, gamma) can be subjective and may require trial and error.
- Difficulty with Long-Term Forecasts: While excellent for the short term, its accuracy diminishes significantly for long-range forecasting, especially for simple models that cannot project trends far into the future.
7. How does Double Exponential Smoothing improve upon the simple method?
Double Exponential Smoothing, or Holt's method, improves upon simple smoothing by explicitly accounting for trends in the data. It uses two smoothing constants: alpha (α) for the level and beta (β) for the trend. By forecasting both the level and the slope of the data, it can generate more accurate predictions for time series that are consistently increasing or decreasing, a task where simple exponential smoothing would systematically lag behind.
8. Which forecasting model is better: Exponential Smoothing or ARIMA?
Neither model is universally better; the choice depends on the data. Exponential Smoothing models are simpler, more intuitive, and perform well for data with clear trend and seasonality patterns. They are essentially weighted averages of past observations. ARIMA (AutoRegressive Integrated Moving Average) models are more complex and powerful. They can model more intricate time series structures by using correlations between past values (autocorrelation). While ARIMA can be more accurate for complex data, it requires more statistical expertise to identify, fit, and validate the model.
