# Proposed Modification of Holt’s Method for Short Term Forecasting

DOI : 10.17577/IJERTV6IS110225

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#### Proposed Modification of Holt’s Method for Short Term Forecasting

Md. Hasibul Haque Department of Mathematics

Khulna University of Engineering & Technology Khulna-9203, Bangladesh

A R M Jalal Uddin Jamali

Department of Mathematics

Khulna University of Engineering & Technology Khulna-9203, Bangladesh

Mohammad Babul Hasan Department of Mathematics University of Dhaka

Abstract Forecasting has long been our part of life. It was centered to weather forecasting only till 19th century but in 20th century it gets new dimension in the business planning. Since 1950s a lot of research works has been carried out on business forecasting and is continuing today to improve the existing forecasting methods and develop a new method or model. This article deals with such an existing method namely Holts method (or sometimes called Holt-Winters method) to forecast the time series data containing trends or linear trends but no seasonality. It is noted that this method used only the observed (real) data to predict data for all the next periods ahead (3 to 5 ) but it does not take into consideration the most recent inter trends relation. We know that recent (last few periods) data have more significant effect rather than far old data on forecast. Exploiting this idea in this research works a modification is proposed to estimate future data. In the proposed modified approach, we take into account the recent available data (may be real or predicted) as weight parameter along with previous trend to forecast the next period outcome. We expect that our modified forecasts can be a better approximation or give the best upper or lower limit of the forecast depending on the nature of last few data.

Keywords Forecasting, Trends, seasonality, Holts method

1. INTRODUCTION

We all make and use forecasts every now and then, both in our jobs and everyday life. Armstrong [1] defined forecasting as the prediction of an actual value in a future time period. Makridakis et al. [2] stated that forecasting supplies information of what may occur in the future. And therefore, it is used to estimate when an event is likely to happen so that we can take necessary actions.

In business, forecasting is the basis for budgeting, planning capacity, sales, production and inventory, personnel, purchasing etc. which affects decisions and activities throughout an organization [3]. Business forecasting is used not only in predicting demand but also that of profits, revenues, costs, productivity changes, raw materials, interest rates, movement of key economic indicators(e.g., GDP, inflation, government borrowing ) and prices of stocks and bonds. Though computers and sophisticated mathematical models are used in forecasting they are not exact science rather successful forecasting requires a proper blending of art and science. So in this modern age of business competitiveness is everywhere and to survive in such competitive world market business

organization needs to predict the business involved future events as precisely as possible. To serve this purpose they have to use some mathematical model to predict the future outcomes based on the historical data available to them. The sequence of historical data collected at uniform time intervals is called time series [4]. The time intervals may be in hour(s), day(s), week(s), month(s), quarter year or year(s).

Holts (linear exponential smoothing [5]) method performs well for the time series where only trends [6] exist but no seasonality. Its extended version called Holt-Winters method which is also a univariate method is used for the time series where trends and seasonality both exists [7]. Holts method is easy than some other method such as ARIMA [8].

2. EXISTING HOLTS METHOD Exponential or single exponential [5] method does not work

well if the time series data contains trends or seasonality. To overcome the In that case several methods were developed to overcome the difficulties involving errors in forecasting and usually they are referred to double exponential smoothing method. One of the methods is named "Holt-Winters double exponential smoothing" or only Holts Method. This method works as follows:

We suppose that the raw data sequence of observations is represented by {X t}, beginning at time t = 0. We use {St} to represent the smoothed value for time t, and {Bt} is our best estimate of the trend at time t. The output of the algorithm is now written as Ft + m , an estimate of the value of Xt at time t+m, m>0 based on the raw data up to time t. Double exponential smoothing is given by the formulas:

St = Xt+(1- )(St-1+Bt-1)

Bt = (St-St-1) + (1- ) Bt-1

Where and are smoothing constants such that 0 < , < 1; Xt denotes observed data whereas Bt indicates tend value at time t and St be smoothed value at time t. Now we need the initial value of St, Bt and for t >1 they have the following form:

S0=X0 and B0 = (Xn-X0)/n (3)

And the h-step forecast by this method is given by the following equation:

Ft,h St h Bt

(4)

TABLE II. SMOOTHING DATA

 T Year X t St Bt Ft-1,1 0 1991 591 591.00 54.00 1 1992 620 627.50 41.75 645.00 2 1993 699 690.08 56.33 669.25 3 1994 781 770.62 73.28 746.40 4 1995 891 876.87 96.36 843.90 5 1996 993 987.07 106.05 973.23 6 1997 1111 1105.63 114.81 1093.12 7 1998 1149 1170.43 79.80 1220.44 8 1999 1301 1285.77 104.68 1250.24 9 2000 1440 1425.13 128.96 1390.45 10 2001 1661 1628.93 181.34 1554.09 11 2002 1770 1782.08 161.61 1810.27 12 2003 1851 1878.81 116.19 1943.69 13 2004 1954 1966.30 96.10 1995.00 14 2005 2023 2034.82 76.80 2062.40 15 2006 2079 2088.78 60.81 2111.62 16 2007 2146 2147.08 59.05 2149.60 17 2008 2430 2362.84 16875 2206.13 18 2009 2746 2681.68 273.81 2531.59 19 2010 3069 3034.95 329.43 2955.49 20 2011 3649 3563.61 468.90 3364.38 21 2012 4159 4121.05 530.88 4032.51 22 2013 4686 4675.78 547.57 4651.93

Here Ft,h denotes the forecasting value determined for t+h period at period t based on available data for the first t period data. The following section numerically illustrates how this method functions.

3. NUMERICAL EXAMPLE

Let us consider the following time series data. We are to forecast the value for the next five periods.

TABLE I. TIME SERIES DATA

 Period Value Period Value Period Value 1991 591 1999 1301 2007 2146 1992 620 2000 1440 2008 2430 1993 699 2001 1661 2009 2746 1994 781 2002 1770 2010 3069 1995 891 2003 1851 2011 3649 1996 993 2004 1954 2012 4159 1997 1111 2005 2023 2013 4686 1998 1149 2006 2079

Fig. 1. Graph of observed data with trend line

At first, we plot the data to see if the trend exists but no seasonality. From the Fig. 1, we observe that there exist trends only but no seasonality. So we can apply Holts method to forecast. It is noted that a trend line plotted on the same axis is fitted well to the observed data.

According to the Halts for the initialization we set S0 = X0 = 591 which is the first observed data and

It is observed that the observed values and estimated values obtained by the method are almost identical. So the values =

0.7 and = 0.7 are perfect enough for the instance considered.

Now the goal that is to forecast for the next five future years (namely year 2014, 2015, 2016, 2017 and 2018), with

period T = 23, 24, 25, 26 and 27 respectively from the last

period 22 (i.e. year 2013). Then the forecast for h =1, 2, 3, 4 and 5 based on 22nd period smoothed and trend value by (4) which are accomplished below:

F23=F22,1=S22+1*B22=4675.78+1*547.57=5223.35 F24=F22,2=S22+2*B22=4675.78+2*547.57=5770.92 F25=F22,3=S22+3*B22=4675.78+3*547.57=6318.49

B X 2 X 0 699 591 54

0 2 2

After testing several combination of different values of these two parameters to find very closer smoothed or fitted values, we set = 0.7 and = 0.7 for this numerical instance.

Now using the formulas of Holts approach we have obtained the trends values. The details numerical results are shown in the Table II.

F26=F22,4=S22+4*B22=4675.78+4*547.57=6866.06

F27=F22,1=S22+5*B22=4675.78+5*547.57=7413.63

The forecasted values are displayed in the Table III. It is observed that for the predicted values of the years 2014, 2015, 2016, 2017, and 2018, the smooth value and trends value of based year remain constant for all the cases.

TABLE III. FORECAST VALUE OBTAINED BY THE HOLTS METHOD

 Year Forecasted Value 2014 5223.35 2015 5770.92 2016 6318.49 2017 6866.06 2018 7413.63
4. OUR PROPOSED MODIFIED METHOD

It is observed in Holts method that to obtain the smoothed values they used available immediate observed values and immediate previous trends. But for forecasting any far years they only used just last smoothing observed value (in the above example that was 22nd) as their base and added the consecutive multiple of the corresponding trend of the base period (here 22nd).

It might be assumed that the recent available data have more significant effect on future prediction rather than far old data. By exploiting this idea we want to develop a forecast model based on Holts approach. However before formulation the proposed modification, it is better to analyze the estimated values obtained by the Holts method of the given instance.

At first we have compared the actual values and estimated value for recent periods namely year 2008 2013. The comparison is shown in the Fig. 2. It is observed that the actual values are always greater than estimated values for all the period (recent) considered.

Fig. 2. Comparison between actual and estimated values

In order to measure the relative error between actual value and estimated value obtained by Holts method for these recent periods, we set the following formula

Relative error = (actual value estimated value)/ actual value Ã—100.

Fig. 3. Relative percentage error in 1-step estimated values of Holts

method

Moreover, we have noticed that there is a gradual change in the trend which is the increment compared to the immediate last trend value. The numerical example in Section III has the following trend values for the last four periods:

B19 = 329.43; B20 = 468.9; B21 = 530.88; B22 = 547.57

Now we want to observe the inter trend relation of the existing method which is shown in Table IV.

TABLE IV. MOST RECENT INTER-TREND RELATIONS

 Trend Values Difference D t = (Bt-Bt-1) Remarks B19 =329.43 ——— ——– B20 = 468.9 D20=139.47 B19+ D20 B21=530.88 D21=61.98 B20+ D21 B22 =547.57 D22=16.69 B21+ D22

Thus, when we make forecast according to (4) based on the current period smoothed data we incorporate here only the current trend but it is obviously true that we do not take the changes in the trend into consideration. It is seen from Table IV that trends do not remain fixed rather it changes from the previous trend value by some amount and so it is convincible that in future this gradual change also remain in the time series data and to forecast adding this change in the model (4) is reasonable.

From this analysis it may be concluded that the forecasting value by Holts method based on the last smoothed value (corresponding to the observed value) contain a significant error. To reduce this error we want to modify (4). The proposed modified equation of (4) is given as follows:

The relative errors are plotted against the years (periods) which given in the Fig. 3. It is noticed in the figure that there

where,

Ft,h St h (Bt Dt h )

(5)

are significant errors.

D t = (Bt-Bt-1) (6)

Dt + h = Harmonic Mean of (Dt+h-1, Dt+h-2, Dt+ h-3) (7)

Here (7) is formulated by using recent trend values. So (5) provides the forecasts where trend is updatedby adding the additional parameter by (7) in each and every time of forecasting. Thus it adds more weight to the most recent trend than the far old data.

TABLE V. UPDATED DIFFERENCE IN FUTURE TRENDS

 Updated Difference in Trends Value D23 HM (D20 ,D21 ,D22) =36.05 D24 HM (D21 ,D22 ,D23) =28.90 D25 HM (D22 ,D23 ,D24) =24.54 D26 HM (D23 ,D24 ,D25) =29.10 D27 HM (D24 ,D25 ,D26) =27.34

*HM means Harmonic mean

TABLE VI. FORECASTS COMPARISON TABLE

 Year/ Period Old Modified 2014=F23=F22,1 5223.35 5259.4 2015=F24=F22,2 5770.92 5828.72 2016=F25=F22,3 6318.49 6392.11 2017=F26=F22,4 6866.06 6982.46 2018=F27=F22,5 7413.63 7550.33

Now we have implemented this modified formula to the numerical example given in section III. So modified forecasting values obtained by our proposed modification (5) which is displayed in the Table VI. Now we have compared the modified forecasting with Holts forecasting given in the Table VI.

Fig. 3. Comparison between Holts and Modified Forecasts

It is observed that our modified forecasts have a bit greater value than the Holts forecasts. To observe the relatve increase in forecasts by our proposed method relative to Holts method we use the following formula:

Relative increase = (Modified value Holts value)/ Modified value Ã—100.

Relative percentage increase of the proposed method over Holts method is shown in the Fig. 4. It is observed in the figure that the increase of predicted values obtained by our proposed method is not much larger. So this increment of forecasts by (5) is very much resonable and it should be much closer to the real value than the the forecasts by Holts method.

Fig. 4. Relative percentage changes in new forecasting

5. CONCLUSION

To forecasts for any periods, Holts method uses the last smoothed observed value. On the other hand, our proposed modified method uses the last smoothed observed value along with most recent estimated trend values to weight the most recent estimated trend values over far old data to forecast. Numerical experiments suggest that the estimated values obtained by the proposed method are much closer to the real values than that of Holts method. It is also expected that the forecasts values obtained by modified method will be closer to the real values.

REFERENCES

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