# A Study On Quantitative Forecasting Methods Computer Science Essay

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It is an activity about trying predict future events based from analysis of historical data. There are different types of forecasting techniques: Qualitative methods are used when data is little and vague and Quantitative Methods when there is enough historical informa- tion available. The following section will consider quantitative forecast method as there is enough data available and trends are regular.

In the paper, (?), general forecasting techniques are investigated on a real real cellular network and even tried to combine several forecasting techniques using weights so as to adjust and penalize extreme values. Therefore making the new combine forecast method to achieve improved forecast value compared to other trend projection forecast. These weights depends on absolute forecasting error. Different Equations were obtained based on the trend projection used for each location investigated. It was noted that four forecast- ing methods were combined and did come up with forecasting values as close as to other techniques. However results showed that these combined forecast did not predict values as closed as for linear regression technique. The results is such that combining forecast does not always provide the best prediction and at times simpler forecast method without weight can gear towards more optimize forecast.

1.2 Quantitative Forecasting Methods

Such methods involves the use of mathematical techniques, when the system where the forecast is used should already exist plus and enough past data is available. Examples where such methods are used are for forecasting sales, predicting traffic intensity on network.

1.2.1 Moving Average

It is known as smoothing approach as it smoothen irregular data. It is mostly used when there is little or no trend is visible in the data with irregular pattern, this technique requires huge amount of data to be able to predict. This method is based on arithmetic means which

Pn

will use previous data to evaluate new value using average. M oving average =

i=1 xi

n

where x is previous data and n the number of terms to consider. This equation can also

1 Pn

be expressed as time variable: M oving averaget = n

i=1 x(t âˆ’ i) where t is the time

variable and n is the number of terms to consider backwards in time.

1.2.2 Trend Projection

It is a statistical technique to help analysing data by making use of trend line. Some ex- ample of trend projection methods are linear trend based on least mean square, exponential trends,and polynomial trends. Each trend equation are given in simplified form in the com- ing pages. This technique helps by fitting a trend line on the existing data in trying to get the best fit which will there fore be the model trend projection. This technique is used when trend pattern can be seen and historical data is available.

1.2.3 Linear Regression

It is similar as the trend projection method, it is used when changed occur in one or more unrelated variables, thus this can be used to predict future outcomes. Example of such is linear regression analysis.

1.3 Forecasting using Trend Projection and Linear Regression

These forecasting methods try to minimize the square error between the actual measured values and the trend line 's values. Let Yc is the data according to trend line while Yi is the

measured data which have been collected and i vary from 1 to n.

n

X(Yi âˆ’ Yc)2 = minimum value

i=1

1.3.1 Least Mean Square Method

This method ensure linear relationship with the collected data, such is used to have predic- tions not far in the future. Trend line equation for linear line is as follows: Yc = a + bX Where X is a time series and a, b are constants. Thus using general equation and the linear equation, the end result will help to generate the constant values. Let Yc is the data accord- ing to trend line while Yi is the measured data which have been collected and i vary from 1 to n.

n n

X(Yi âˆ’ Yc)2 = X(Yi âˆ’ (a + bX ))2 = 0

i=1

i=1

The value of a, b are found by partial differentiation, The simplified equation for least mean square is as below:

a = Yi

where Yi is the mean value of Yi

Pn

Pnb = i=1 xYi i=1 X 2

1.3.2 Exponential Form

Trend line equation for exponential is as follows:Yc = a + bX . Log is applied on both sides to help simplifying

log Yc = log a + (log b)X

To find the value of a, b the simplified equation below is utilized:

Pnlog a = i=1 log Yi

n

Pn

Pnlog b = i=1 x log Yi i=1 x2

1.3.3 Second Order Polynomial Form

Trend line equation for second order polynomial is as follows:Yc = a + bX + cX 2 To find the value of a, b, c the following equations are used:

Pn Pn 2

a = i=1 Yi âˆ’ c i=1 xi

n

Pn

Pnb = i=1 xYi i=1 x2

n Pn 2

Pn 2 Pn

Pnc = i=1 x Yi âˆ’ i=1 x i=1 xYi

n Pni=1 x4 âˆ’ (

i=1 x2)2

1.3.4 Linear Regression

The original linear regression formula is:

Yi = Î²0 + Î²1X +

where Î²0 is Y intercept, Î²1 is the slope of the line and is random error from the process of data collecting or calculation. Î²0 and Î²1 are parameters called "regression coefficients" which are calculated from the collected data and these can be interchanged by b0 and b1. Therefore the new equation is:Y = b0 + b1X + e Based on following assumptions:

â€¢ Average of e is 0

â€¢ Variance of e is constant

â€¢ e is normal distribution

â€¢ e is independent to each other

Thus the new simplified linear regression equation is as follows:

xiYË† = b0 + b1 Xli

where YË†

is the estimated value of Yi , b0 is the regression coefficient of Î²0 and b1 is the

regression coefficient of Î²1. The simplified equations for obtaining values of b0 and b1 is given below:

b0 =

Pn

i=1

Yi âˆ’ b1 Pn xi

i=1n

(nb1 =

Pn

i=1

xi yi âˆ’ Pn

Pn

i=1

yi )

n Pn

i âˆ’ (i=1 x2

Pn

i=1i=1

xi )2

1.4 Classification of Cells with Similar Configuration

The forecasting using the four methods described above will be utilized on the data cap- tured for the study of traffic distribution. For each cell the weekly peak traffic is obtain and used when applying the methods described above. Each cell will have a trend pro- jection equation for each forecasting method, therefore causing 19 equations per method. However, as there are many cells with similar configuration to minimize the number of equations, the weekly average traffic will be employed instead of individual weekly peaks. The average does remove bias as different cells have dissimilar traffic pattern, therefore the average helps to provide a standard value and minimizing the number of trend equations from 19 to 4 per forecasting method. There is only one cell for Class D, thus the average traffic will still be the maximum traffic. The four Classification groups are as described in table 1.

Class Type

Number of cells

Class A: 2 TRX 14 TCH

12

Class B: 2 TRX 24 TCH

3

Class C: 3 TRX 33 TCH

3

Class D: 4 TRX 53 TCH

1

Table 1: Group Classification of Cell with Similar Configuration

The figure 1 is used to illustrate how several weekly peaks for cells with similar configura- tion of 14 traffic channel is combined to the weekly average forming class A.

Figure 1: Graph of Weekly Max and Average Traffic Intensity for Class A - Figure shows

Traffic Intensity for Several Cells with 14TCH Configuration.

The figure 2 is used to illustrate how several weekly peaks for cells with similar configura-

tion of 24 traffic channel is combined to the weekly average forming class B.

Figure 2: Graph of Weekly Max and Average Traffic Intensity for Class B - Figure shows

Traffic Intensity for Several Cells with 24TCH Configuration.

The figure 3 is used to illustrate how several weekly peaks for cells with similar configura- tion of 33 traffic channel is combined to the weekly average forming class C.

Figure 3: Graph of Weekly Max and Average Traffic Intensity for Class C - Figure shows

Traffic Intensity for Several Cells with 33TCH Configuration.

1.5 Model Preparation

Traffic data collected for several weeks for different classification and configuration of hardware is used to form trend lines. The following trend projection techniques will be used: Least Mean Square, Exponential, Second Order Polynomial and Linear Regression. The equations obtained from each individual forecasting techniques are displayed in the tables 2, 3, 4, 5.

1.5.1 Linear Trend Experiment

For each class type, the corresponding linear trend is plotted together with the weekly traf- fic. This provides a visual comparison which helps to evaluate how close are the forecast from the actual value based on least mean square method. The linear trend has been gener- ated from the least mean square method 1.3.1, the equations for each class of cells is listed in the table 2.

Class Type

Least Mean Square

Class A: 2 TRX 14 TCH

Y = 0.2049x + 8.3661

Class B: 2 TRX 24 TCH

Y = 0.3480x + 14.2040

Class C: 3 TRX 33 TCH

Y = 0.4459x + 18.1721

Class D: 4 TRX 53 TCH

Y = 0.9803x + 39.9686

Table 2: Forecasting Least Mean Square Trend Equation for Each Classification Group

Observation for linear trend method: based on the previous graphs it can be concluded that for the network and cells in the class type least mean square forecast is largely over estimated.

Figure 4: Least Mean Square Trend Graph for Class A - Figure shows actual traffic intensity together with corresponding linear forecast

Figure 5: Least Mean Square Trend Graph for Class B - Figure shows actual traffic intensity together with corresponding linear forecast

1.5.2 Exponential Trend Experiment

For each class type, the corresponding exponential trend is plotted together with the weekly traffic. This provides a comparison view which helps to evaluate how close are the forecast from the actual value based on exponential method. The exponential trend has been gen- erated from the exponential method 1.3.2, the equations for each class type is given in the table 3.

Observation for exponential method: The forecast value is slightly over estimated, however it does happen that the forecast is nearly similar to the actual value.

Figure 6: Least Mean Square Trend Graph for Class C - Figure shows actual traffic inten- sity together with corresponding linear forecast

Figure 7: Least Mean Square Trend Graph for Class D - Figure shows actual traffic inten- sity together with corresponding linear forecast

1.5.3 Second Order Polynomial Trend Experiment

Class Type

Exponential

Class A: 2 TRX 14 TCH

y = 8.1080e0.0030x

Class B: 2 TRX 24 TCH

y = 13.5800e0.0050x

Class C: 3 TRX 33 TCH

y = 16.8680e0.00950x

Class D: 4 TRX 53 TCH

y = 37.6000e0.0080x

Table 3: Forecasting Exponential Trend Equation for Each Classification Group

Figure 8: Exponential Trend Graph for Class A - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 9: Exponential Trend Graph for Class B - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 10: Exponential Trend Graph for Class C - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 11: Exponential Trend Graph for Class D - Figure shows actual traffic intensity together with corresponding exponential forecast

Class Type

Second Order Polynomial

Class A: 2 TRX 14 TCH

y = âˆ’0.010x2 + 0.1930x + 7.6980

Class B: 2 TRX 24 TCH

y = âˆ’0.018x2 + 0.3480x + 12.9000

Class C: 3 TRX 33 TCH

y = âˆ’0.044x2 + 0.8263x + 15.165

Class D: 4 TRX 53 TCH

y = âˆ’0.025x2 + 0.6960x + 36.580

Table 4: Forecasting Second Order Polynomial Trend Equation for Each Classification

Group

For each class type, the corresponding second order polynomial trend is plotted together with the weekly traffic. This provides a way to compare visually thus helping to evalu- ate how correct the forecast are from the actual value based on second order polynomial method. The polynomial trend has been generated from the second order polynomial equa- tion 1.3.3 in the table 4.

Figure 12: Second Order Polynomial Trend Graph for Class A - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 13: Second Order Polynomial Trend Graph for Class B - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 14: Second Order Polynomial Trend Graph for Class C - Figure shows actual traffic intensity together with corresponding exponential forecast

Figure 15: Second Order Polynomial Trend Graph for Class D - Figure shows actual traffic intensity together with corresponding exponential forecast

Observation for second order polynomial method: It can be seen that the forecast is under estimated and is far away from the actual values plus mostly does come up with negative values.

1.5.4 Linear Regression Trend Experiment

Class Type

Linear Regression

Class A: 2 TRX 14 TCH

Y = 0.0316x + 7.0854

Class B: 2 TRX 24 TCH

Y = 0.0770x + 11.0836

Class C: 3 TRX 33 TCH

Y = 0.1662x + 11.4391

Class D: 4 TRX 53 TCH

Y = 0.3156x + 27.1884

Table 5: Forecasting Linear Regression Trend Equation for Each Classification Group

The comparison figures below help to evaluate the correctness of forecast from the actual value based on linear regression method. The linear trend has been generated from the linear regression method equations 1.3.4, the model equations obtained are listed in the table 5, these are used to plot the linear regression trend against the real traffic intensity for each class type defined.

Figure 16: Linear regression Trend Graph for Class A - Figure shows actual traffic intensity together with corresponding linear regression forecast

Figure 17: Linear regression Trend Graph for Class B - Figure shows actual traffic intensity together with corresponding linear regression forecast

Observation for linear regression: It can be seen that the forecast is quite close to the actual values.

Figure 18: Linear regression Trend Graph for Class C - Figure shows actual traffic intensity together with corresponding linear regression forecast

Figure 19: Linear regression Trend Graph for Class D - Figure shows actual traffic intensity together with corresponding linear regression forecast

1.5.5 Mean Absolute Percentage Error (MAPE)

1.6 Conclusion based on results

Following the experiments above, it was noted that the Mean Absolute Percentage Error

?? for linear regression forecasting technique is significantly less compared to the other techniques. However combining several techniques using weights will not necessarily help towards providing a more accurate forecast. The Methods do have some Errors and com- bining them could minimize extreme errors but these will not be precise as they are just

Figure 20: Mean Absolute Percentage Error for Each Forecasting Technique - Figure shows the mean percentage error for the set of forecast selected.

reducing absolute errors using some weights. Even if using the Linear Regression com- bined with some other techniques will only divert the forecast. Therefore Linear Regres- sion Forecast Technique is the best fit for the network under study. (?) investigates and indicates similar results for other situation.