Time Series mining-prediction algorithm-three exponential smoothing (holt-winters)-Three exponential smoothing algorithm can save the trend and seasonal information of time series data well

From:http://www.cnblogs.com/kemaswill/archive/2013/04/01/2993583.html

In the time series, we need to predict the following trend based on the current data of the time series, and the three exponential smoothing (Triple/three Order exponential smoothing,holt-winters) algorithm can predict the time series well.

Time series data generally have the following characteristics: 1. Trend (Trend) 2. Seasonal (seasonality).

Trends describe the overall movement of the time series, such as the overall ascent or overall decline. The time series shown is the overall ascent:

Seasonality describes cyclical fluctuations in data, such as periods of years or weeks, such as:

The three-times exponential smoothing algorithm predicts time series with both trend and seasonality, based on an exponential smoothing and two exponential smoothing algorithm.

The exponential smoothing algorithm is based on the following recursive relationships:

si=αxi+ (1-α) si-1

where α is a smoothing parameter , SI is the smoothing value of the previous I data, the value is [0,1],α closer to 1, the smoothed value is closer to the current time of the data value, the less smooth the data, α closer to 0, the smoothed value closer to the previous I data smoothing value, the more smooth data, The value of alpha can often be tried several times to achieve optimal results.

The formula for predicting an exponential smoothing algorithm is: Xi+h=si, where I is the coordinates of the current last data record, that is, the predicted time series is a straight line, which does not reflect the trend and seasonality of the time series.

The two-times exponential smoothing preserves the trend's information so that the predicted time series can contain trends in previous data. Two-times exponential smoothing represents a smoothed trend by adding a new variable t:

si=αxi+ (1-α) (si-1+ti-1)

Ti=? (si-si-1) + (1-?) Ti-1

Two exponential smoothing prediction formula for Xi+h=si+hti two times exponential smoothing is a diagonal line.

Three exponential smoothing retains seasonal information on the basis of two exponential smoothing, making it possible to predict seasonal time series. Three times the exponential smoothing adds a new parameter p to indicate the trend after smoothing.

Three exponential smoothing there are cumulative and multiplicative two methods, the following is the cumulative three-time exponential smoothing

Si=α (xi-pi-k) + (1-α) (si-1+ti-1)

Ti=? (si-si-1) + (1-?) Ti-1

Pi=γ (Xi-si) + (1-γ) Pi-k where K is the period

The prediction formula for the cumulative three exponential smoothing is: xi+h=si+hti+pi-k+ (h mod k) Note: The charm of the data P88 there are errors, as amended by Wikipedia.

The following is a three-time exponential smoothing of the multiplicative:

si=αxi/pi-k+ (1-α) (si-1+ti-1)

Ti=? (si-si-1) + (1-?) Ti-1

pi=γxi/si+ (1-γ) pi-k where K is the period

The prediction formula for multiplicative three exponential smoothing is: xi+h= (Si+hti) pi-k+ (h mod k) Note: The charm of the data P88 there is an error, as amended by Wikipedia.

The values of α,?,γ are located between [0,1] and can be tested several times to achieve optimal results.

S,t,p initial value selection for the overall effect of the algorithm is not particularly large, the usual value is s0=x0,t0=x1-x0, the cumulative p=0, when the multiplicative p=1.

We use DataMarket's international Airline passengers data to test the performance of the cumulative and multiplicative three exponential smoothing algorithms, which record the number of passengers per month on international routes:

The effect of predicting using the cumulative three exponential smoothing: where red is the source time series, Blue is the predicted time series, and the Α,?,γ value is 0.45,0.2,0.95:

To predict the effect of multiplicative three exponential smoothing, the value of Α,?,γ is 0.4,0.05,0.9:

We can see that three times exponential smoothing algorithm can save the trend and seasonal information of time series data well, and it is better to international the smoothing exponent algorithm on Airline passengers dataset.

Reference documents:

[1]. Data charm: Data analysis based on open source tools

[2]. datamarket:international Airline passengers

[3]. Wikipedia:exponential Smoothing

Time Series mining-prediction algorithm-three exponential smoothing (holt-winters)-Three exponential smoothing algorithm can save the trend and seasonal information of time series data well