It is an activity about seeking predict hereafter events based from analysis of historical informations. There are different types of calculating techniques: Qualitative methods are employed when informations is small and obscure and Quantitative Methods when there is adequate historical information available. The undermentioned subdivision will analyze quantitative prognosis method as there is adequate informations available and tendencies are regular. Mention for this subdivision is made from ( ? ? ? ) .

1.2 Related Work Forecast

? ) investigated about general prediction techniques for a existent cellular web and even tried to unite several calculating techniques utilizing weights so as to set and punish utmost values. Therefore doing the new combine prognosis method to accomplish improved prognosis value compared to other tendency projection prognosis. These weights depends on absolute prediction mistake. Different Equations were obtained based on the tendency projection implemented for each location investigated. It was noted that four calculating methods were combined and did come up with calculating values every bit near as to other techniques.

However consequences showed that these combined prognosis did non foretell values every bit closed as for additive arrested development technique. The consequences is such that uniting prognosis does non ever supply the best anticipation and at times simpler prognosis method without weight can pitch towards more optimize prognosis.

1.3 Quantitative Forecasting Methods

Such methods involves the application of mathematical techniques, when the system where the prognosis is exercised must already be plus adequate yesteryear informations shall be available. Exam- ples where such methods are employed are for calculating gross revenues, foretelling traffic strength on web.

1.3.1 Traveling Average

It is known as smoothing attack as it smoothen irregular informations. It is largely applied when there is small or no tendency is seeable in the information with irregular form, this technique requires immense sum of informations to be able to foretell. This method is based on arithmetic agencies which will cover with old informations to measure new value utilizing mean. M oving mean =

N

Ten eleven

i=1

N

where ten is old informations and n the figure of footings accounted. This equation can

besides be expressed as clip variable: M oving mean = 1

T N

N

Ten

i=1

ten ( t a?’ I ) where T is the clip

variable and n is the figure of footings to see backwards in clip.

1.3.2 Trend Projection

It is a statistical technique to assist analyzing informations by the support of tendency line. Some exam- ple of tendency projection methods are additive tendency based on least average square, exponential tendencies, and multinomial tendencies. Each tendency equation are given in simplified signifier in the com- ing pages. This technique helps by suiting a tendency line on the bing informations in seeking to acquire

the best tantrum which will there forward be the theoretical account tendency projection. This technique is acceptable when tendency form can be seen and historical information is available.

1.3.3 Linear Arrested development

It is similar as the tendency projection method, but alterations can happen in one or more unrelated variables, therefore this can be accounted to foretell future results ( ? ) . Example of such is additive arrested development analysis.

1.4 Forecasting utilizing Trend Projection and Linear Re- gression

These prediction methods try to minimise the square mistake between the existent measured values and the tendency line ‘s values. Let Yc is the informations harmonizing to tendency line while Yi is the measured information which have been collected and one vary from 1 to n. In this subdivision ten is the clip based which is the hebdomad of the twelvemonth and Y is the traffic strength, therefore yc is the prognosis traffic strength and Lolo is the mensural traffic strength at the hebdomad eleven.

N

Ten ( Yi a?’ Yc ) 2 = minimal value

i=1

1.4.1 Least Mean Square Method

This method ensures additive relationship with the collected informations, such anticipations are merely valid for values non far in the close hereafter. Trend line equation for additive line is as follows: Yc = a + bX Where X is a clip series and a, B are invariables. Therefore utilizing general equation and the additive equation, the terminal consequence will assist to bring forth the changeless values. Let Yc is the informations harmonizing to tendency line while Yi is the measured information which have been collected

and one vary from 1 to Ns.

Ns Ns

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

i=1

i=1

The value of a, B are found by partial distinction, The simplified equation for least average

square is as below:

a = Lolo

where Yi is the average value of Yi

N

Ten xYi

B =i=1

N

Ten x2

i=1

1.4.2 Exponential Form

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

log Yc = log a + ( log B ) Ten

To happen the value of a, b the simplified equation below is utilised:

log a =

N

Ten log Yi

i=1

N

log B =

N

Ten x log Yi

i=1

N

Ten x2

i=1

1.4.3 Second Order Polynomial Form

Trend line equation for 2nd order multinomial is as follows: Yc = a + bX + cX 2 To happen the value of a, B, c the undermentioned equations are needed:

Ns Ns

nine Yi a?’ c X x2

a = i=1

i=1

N

N

Ten xYi

B =i=1

N

xX 2

I

i=1

n N N

n X x2 Yi a?’ X x2 X xi Yi

I

degree Celsiuss = i=1

N

I

i=1

N

i=1

! 2

in X x4 a?’

i=1

X 2

eleven

i=1

1.4.4 Linear Arrested development

The original additive arrested development expression is:

Yi = I?0 + I?1X +

where I?0 is Y intercept, I?1 is the incline of the line and is random mistake from the procedure of informations roll uping or computation. I?0 and I?1 are parametric quantities called “ arrested development coefficients ” which are calculated from the collected informations and these can be interchanged by b0 and b1. Therefore the new equation is: Y = b0 + b1X + vitamin E Based on following premises:

aˆ? Average of vitamin E is 0

aˆ? Variance of vitamin E is changeless

aˆ? vitamin E is normal distribution

aˆ? vitamin E is independent to each other

Therefore the new simplified additive arrested development equation is as follows:

YE† = b0 + b1 Xli

where YE†

is the estimated value of Yi, b0 is the arrested development coefficient of I?0 and b1 is the

arrested development coefficient of I?1. The simplified equations for obtaining values of b0 and b1 is given below:

Ns Ns

Ten Yi a?’ b1 X eleven

b0 =

i=1

N

i=1

N

Ns Ns

( n X xi yi a?’ X xi X Lolo )

b1 =

i=1

N

i=1

N

i=1

in X x2 a?’ ( X xi ) 2

i=1

i=1

1.5 Categorization of Cells with Similar Configuration

The prediction utilizing the four methods described above will be utilized on the informations cap- tured for the survey of traffic distribution. For each cell the hebdomadal extremum traffic is obtained and selected when using the methods described above. Each cell will hold a tendency pro- jection equation for each prediction method, hence doing 19 equations per method. However, as there are many cells with similar constellation to minimise the figure of equations, the hebdomadal norm traffic will be employed alternatively of single hebdomadal extremums. The norm does take bias as different cells have dissimilar traffic form, therefore the norm helps to supply a standard value and minimising the figure of tendency equations from 19 to 4 per calculating method. There is merely one cell for Class D, therefore the mean traffic will still be the maximal traffic. The four Categorization groups are as described in Table? ? .

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 1.1: Group Classification of Cell with Similar Configuration

The figure? ? is used to exemplify how several hebdomadal extremums for cells with similar configu- ration of 14 traffic channel is combined to the hebdomadal norm organizing category A.

Figure 1.1: Graph of Weekly Max and Average Traffic Intensity for Class A – Traffic Inten- sity for Several Cells with 14TCH Configuration.

The figure? ? is used to exemplify how several hebdomadal extremums for cells with similar configu-

ration of 24 traffic channel is combined to the hebdomadal norm organizing category B.

Figure 1.2: Graph of Weekly Max and Average Traffic Intensity for Class B – Traffic Inten- sity for Several Cells with 24TCH Configuration.

The figure? ? is used to exemplify how several hebdomadal extremums for cells with similar configu- ration of 33 traffic channel is combined to the hebdomadal norm organizing category C.

Figure 1.3: Graph of Weekly Max and Average Traffic Intensity for Class C – Traffic Inten- sity for Several Cells with 33TCH Configuration.

1.6 Model Preparation

Traffic informations collected for several hebdomads for different location and constellation of hard- ware is taken into history to organize tendency lines. The undermentioned tendency projection techniques considered are: Least Mean Square, Exponential, Second Order Polynomial and Linear Re- gression. The equations obtained from each single prediction techniques are displayed in the Tables? ? , ? ? , ? ? , ? ? .

1.6.1 Linear Trend Experiment

For each category type, the corresponding additive tendency is plotted together with the hebdomadal traf- fic. This provides a ocular comparing which helps to measure how close are the prognosis from the existent value based on least average square method. The additive tendency has been gener- ated from the least average square method? ? , the equations for each category of cells is listed in the Table? ? .

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 1.2: Prediction Least Mean Square Trend Equation for Each Classification Group

Figure 1.4: Least Mean Square Trend Graph for Class A – Actual traffic strength together

with matching additive prognosis

Figure 1.5: Least Mean Square Trend Graph for Class B – Actual traffic strength together with matching additive prognosis

Figure 1.6: Least Mean Square Trend Graph for Class C – Actual traffic strength together

with matching additive prognosis

Figure 1.7: Least Mean Square Trend Graph for Class D – Actual traffic strength together with matching additive prognosis

Observation for additive tendency method: based on the old graphs it can be concluded that for the web and cells in the category type least average square prognosis is mostly over estimated.

1.6.2 Exponential 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 1.3: Forecasting Exponential Trend Equation for Each Classification Group

For each category type, the corresponding exponential tendency is plotted together with the hebdomadal traffic. This provides a comparing position which helps to measure how close are the prognosis from the existent value based on exponential method. The exponential tendency has been gener- ated from the exponential method? ? , the equations for each category type is given in the Table

## ? ? .

Figure 1.8: Exponential Trend Graph for Class A – Actual traffic strength together with matching exponential prognosis

Figure 1.9: Exponential Trend Graph for Class B – Actual traffic strength together with

matching exponential prognosis

Figure 1.10: Exponential Trend Graph for Class C – Actual traffic strength together with matching exponential prognosis

Figure 1.11: Exponential Trend Graph for Class D – Actual traffic strength together with matching exponential prognosis

Observation for exponential method: The prognosis value is somewhat over estimated, nevertheless it does go on that the prognosis is about similar to the existent value.

1.6.3 Second Order Polynomial Trend Experiment

Class Type

Second Order Polynomial

Class A: 2 TRX 14 TCH

Y = a?’0.010×2 + 0.1930x + 7.6980

Class B: 2 TRX 24 TCH

Y = a?’0.018×2 + 0.3480x + 12.9000

Class C: 3 TRX 33 TCH

Y = a?’0.044×2 + 0.8263x + 15.165

Class D: 4 TRX 53 TCH

Y = a?’0.025×2 + 0.6960x + 36.580

Table 1.4: Forecasting Second Order Polynomial Trend Equation for Each Categorization

Group

For each category type, the corresponding 2nd order multinomial tendency is plotted together with the hebdomadal traffic. This provides a manner to compare visually therefore assisting to evalu- Ate how correct the prognosis are from the existent value based on 2nd order multinomial method. The multinomial tendency has been generated from the 2nd order multinomial equa- tion? ? in the Table? ? .

Figure 1.12: Second Order Polynomial Trend Graph for Class A – Actual traffic strength together with matching exponential prognosis

Figure 1.13: Second Order Polynomial Trend Graph for Class B – Actual traffic strength

together with matching exponential prognosis

Figure 1.14: Second Order Polynomial Trend Graph for Class C – Actual traffic strength together with matching exponential prognosis

Figure 1.15: Second Order Polynomial Trend Graph for Class D – Actual traffic strength together with matching exponential prognosis

Observation for 2nd order multinomial method: It can be seen that the prognosis is under estimated and is far off from the existent values plus largely does come up with negative values.

1.6.4 Linear Regression Trend Experiment

Class Type

Linear Arrested development

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 1.5: Prediction Linear Regression Trend Equation for Each Classification Group

The comparing figures below aid to measure the rightness of prognosis from the existent value based on additive arrested development method. The additive tendency has been generated from the additive arrested development method equations? ? , the theoretical account equations obtained are listed in the Table

? ? , these are needed to plot the additive arrested development tendency against the existent traffic strength for each category type defined.

Figure 1.16: Linear arrested development Trend Graph for Class A – Actual traffic strength together with matching additive arrested development prognosis

Figure 1.17: Linear arrested development Trend Graph for Class B – Actual traffic strength together

with matching additive arrested development prognosis

Figure 1.18: Linear arrested development Trend Graph for Class C – Actual traffic strength together with matching additive arrested development prognosis

Figure 1.19: Linear arrested development Trend Graph for Class D – Actual traffic strength together with matching additive arrested development prognosis

Observation for additive arrested development: It can be seen that the prognosis is rather close to the existent values.

1.6.5 Mean Absolute Percentage Error ( MAPE )

Eleven a?’ Fi

Pn

This is M AP E =

i=1

Ten I

N

where Eleven

is the existent value at the blink of an eye I, Fi

is the

prognosis value for the blink of an eye I and N is the figure of sample. Mean absolute per centum

mistake is favoured compared to other methods since it measures the absolute mistake value and the way of the mistake is non required in this experiment. The MAPE for the experiments above is listed in the Table? ? . Linear arrested development has a better public presentation on the overall compared to other theoretical accounts.

Categorization

Lumen

Exponential

Second Order Polynomial

Lawrencium

Class A: 2 TRX 14 TCH

100 %

10 %

128 %

4 %

Class B: 2 TRX 24 TCH

100 %

18 %

122 %

6 %

Class C: 3 TRX 33 TCH

100 %

37 %

232 %

6 %

Class D: 4 TRX 53 TCH

99 %

30 %

41 %

3 %

Table 1.6: Mean Absolute Percentage Error for Each Forecasting Technique

Lumen: least average square method, LR: additive arrested development method, Exponential: exponential method and 2nd order multinomial: 2nd order multinomial method

1.7 Conclusion based on consequences

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

? ? for additive arrested development prediction technique is significantly less compared to the other techniques. However uniting several techniques utilizing weights will non needfully assist towards supplying a more accurate prognosis. The Methods do hold some Mistakes and com- bining them could minimise utmost mistakes but these will non be precise as they are merely cut downing absolute mistakes utilizing some weights. Even if utilizing the Linear Regression com- bined with some other techniques will merely deviate the prognosis. Therefore Linear Regres- Zion Forecast Technique is the best tantrum for the web under survey. ( ? ) investigates and indicates similar consequences for other state of affairs.