1) Moving average and smoothing techniques are used to reduce random variations in time series data and reveal underlying trends. 2) There are four smoothing techniques in XLMiner: exponential, moving average, double exponential, and Holt-Winters. Exponential and moving average are simple techniques that should not be used for seasonal data. 3) Holt-Winters smoothing introduces a third parameter to account for seasonality and is suitable for trend and seasonal data. It has three models: multiplicative, additive, and no trend.

1. TUGAS TAMBAHAN
Moving Average or Smoothing Techniques
Dibuat oleh Alam Afrizal
2201802424

2. WHAT ARE MOVING AVERAGE OR
SMOOTHINGTECHNIQUES?
When data collected over time displays random
variation, smoothing techniques can be used to reduce or
cancel the effect of these variations.When properly applied,
these techniques smooth out the random variation in the
time series data to reveal underlying trends.
XLMiner features four different smoothing
techniques: Exponential, MovingAverage, Double
Exponential, and Holt-Winters. Exponential and Moving
Average are relatively simple smoothing techniques and
should not be performed on data sets involving seasonality.
Double Exponential and Holt-Winters are more advanced
techniques that can be used on data sets involving
seasonality.

3. EXPONENTIAL SMOOTHING
Exponential Smoothing is one of the more popular smoothing techniques due to its
flexibility, ease in calculation, and good performance. Exponential Smoothing uses a simple average
calculation to assign exponentially decreasing weights starting with the most recent observations.
New observations are given relatively more weight in the average calculation than older
observations.The Exponential Smoothing tool uses the following formulas.
S0= x0
St = αxt-1 + (1-α)st-1, t > 0
where
original observations are denoted by {xt} starting at t = 0
α is the smoothing factor which lies between 0 and 1
Exponential Smoothing should only be used when the data set contains no seasonality.The forecast
is a constant value that is the smoothed value of the last observation.

4. MOVING AVERAGE SMOOTHING
In Moving Average Smoothing, each observation is assigned an equal weight, and each
observation is forecasted by using the average of the previous observation(s). Using the time
series X1, X2, X3, ....., Xt, this smoothing technique predicts Xt+k as follows :
St = Average (xt-k+1, xt-k+2, ....., xt), t= k, k+1, k+2, ...N
where, k is the smoothing parameter.
XLMiner allows a parameter value between 2 and t-1 where t is the number of
observations in the data set. Note that when choosing this parameter, a large parameter value
will oversmooth the data, while a small parameter value will undersmooth the data.The past
three observations will predict the future observations. As with Exponential Smoothing, this
technique should not be applied when seasonality is present in the data set.

5. DOUBLE EXPONENTIAL SMOOTHING
Double Exponential Smoothing can be defined as the recursive application of an
exponential filter twice in a time series. Double Exponential Smoothing should not be used when
the data includes seasonality.This technique introduces a second equation that includes a trend
parameter; thus, this technique should be used when a trend is inherent in the data set, but not
used when seasonality is present. Double Exponential Smoothing is defined by the following
formulas.
St = At + Bt , t = 1,2,3,..., N
Where, At = axt + (1- a) St-1 0< a <= 1
Bt = b (At - At-1) + (1 - b ) Bt-1 0< b <= 1
The forecast equation is: Xt+k = At + K Bt , K = 1, 2, 3, ...
where, a denotes the Alpha parameter, and b denotes the trend parameters.These two
parameters can be entered manually.

6. XLMiner includes an optimize feature that will choose the best values for alpha and trend
parameters based on the Forecasting Mean Squared Error. If the trend parameter is 0, then this
technique is equivalent to the Exponential Smoothing technique. (However, results may not be
identical due to different initialization methods for these two techniques.)

7. HOLT-WINTERS SMOOTHING
HoltWinters Smoothing introduces a third parameter (g) to account for seasonality (or
periodicity) in a data set.The resulting set of equations is called the Holt-Winters method, after
the names of the inventors.The Holt-Winters method can be used on data sets involving trend
and seasonality (a, b , g).Values for all three parameters can range between 0 and 1.
The following three models associated with this method.

8. Multiplicative: Xt = (At+ Bt)* St +et where At and Bt are previously calculated initial estimates. St is the
average seasonal factor for the tth season.
At = axt/St-p + (1-a)(At-1 + Bt-1)
Bt = b(At + At-1) + (1 - b)Bt-1
St = gxt/At + (1 - g)St-p
Additive: Xt = (At+ Bt) +St + et
NoTrend: b = 0, so, Xt = A * St +et
Holt-Winters smoothing is similar to Exponential Smoothing if b
and g = 0, and is similar to Double Exponential Smoothing if g = 0.