This section describes the design of experiment (DOE) methods that have been used to model the power consuming pattern of the Ethernet switch. Here statistical analysis methods have been applied to identify the most influential parameters affecting the power consumption of the Ethernet switch within the range and domain of these experiments. Two methods have been used to model the equation. One is full factorial method and another is linear regression analysis.
5.1 Full Factorial
A Full Factorial Design of Experiment provides responsive information about factor main effects and factor interactions. It also provides the process model‟s coefficients for all the factors and interaction. A full factorial DOE is a planned set of tests on the response variable or variables with one or more inputs factors with all possible combinations of levels. If we have n factors, with the i-th factor having ki levels, and if each experiment is repeated for r times, then
the total number of experiments: .
One of the most important parts of full factorial method is to design the experiment. Because to get an effective result, it is important to have the appropriate model. Key steps in designing an experiment include:
1) Identify factors of interest and a response variable: Before starting the experiment it is important to have the list of variables that will be varied and also the response variable or variables that will be measured. In other words these factors of interests are the key to run the experiment. By varying the values of these factors we will get different values of response variable. That will ultimately help us to analysis the behavior of the system.
2) Determine appropriate levels for each explanatory variable: The second step is to find out the appropriate level for the varying factors. If a factor is continuous then it is important to define some fixed level for the experiment. One factor can have several levels. Minimum
31 number of level is two. For example: it could be high and low. However this leveling is entirely depends on the designer of the experiment and the factors.
3) Determine a design structure: Next key step of the experiment is to decide the structure of the design. As discussed earlier each varying factors can have several levels. And depending on the design the total number of experiment run is determined. Therefore if the number of
variable is a big number and if every variable has several levels then the number of experiments might not be feasible. To keep the feasibility in mind, the design should be effective and at the same time within the limit.
4) Randomize the order: Randomization is another important feature of the full factorial experiment that should be kept in mind. There should not be any sort of pattern while varying the factors. Any pattern can have some hidden effect on the experiment that is unwanted.
5) Organize the results: The last step of the experiment is organize the results in order to draw appropriate conclusions. From a full factorial analysis several outcome can be achieved. The target is to find the right result that is required.
Full factorial experiments provide two types of conclusion. One is main effects another one is interaction effects. Main effects often mean the impact of the changing factors level. Main effect usually shows a single factors effect on the overall response variable. Main effect is calculated by finding out the difference between factor average and grand mean. Whereas grand mean is the overall average of the result. Main effects plots are a quick and efficient way to visualize effect size. In addition to determining the main effects for each factor, it is often significant to identify how multiple factors interact in effecting the results. An interaction occurs when one factor affects the results in a different way depending on a second factor.
Interaction plots are used to conclude the effect size of interactions.
The main objective of full factorial method is to learning the most from as few numbers of experiments as possible. It identifies the factors affect mean and variation which usually helps to identify if the parameter is necessary for the model or not. And then lastly it produces prediction equations which can be used for validation.
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5.2 Linear Regression Analysis
Regression analysis is the method of fitting straight lines to set of data. As discussed by Robert(2014) in a linear regression model, the variable of interest in this case is power
consumption which is predicted from k other variables using a linear equation. If Y denotes the dependent variable or the response, and X1, …,Xk, are the independent variables, then linear regression analysis would be look like:
Y= c0 + c1 X1 + c2X2 + … + ckXk
Where, c0, c1, c2, ck are constant, independent and identically distributed. C0 is the so-called catch of the model. In other words, the expected value of Y when all the X‟s are zero or has no value to mention. And ci is the coefficient (multiplier) of the variable Xi. All the c‟s and X‟s together is the overall model. Each X has its own coefficient and these coefficients are non-related with others. This is about one of the simplest possible model for predicting one variable or response from a group of other variables. The model is built on few strong assumptions.
1. The value of Y which is the response variable, is a linear function of the X variables. This means:
a. When there is a change in the value of Xi by an amount of ∆Xi, considering this change could be both positive and negative and holding all the other variables fixed, then the projected value of Y changes by a proportional amount ci∆Xi. This ci is a constant value that could be both positive and negative number.
b. The value of ci is always the same, regardless of values of the other X‟s.
c. If we calculate the separate effects of the X‟s on the response value Y, and sum these effects then it would be the same of the total effect of the X‟s on the response value of Y.
2. There could be some unexplained variations of Y which are independent random variables.
The equation for Y is modeled based on the provided relationship with Y and X‟s.
3. They are normally distributed.
33 Nevertheless these assumptions will never be exactly justified in the real world experiment.
The values from real world experiment are not ideal so there will be variation. The regression model constructs very well-built assumptions about the way in which Y depends on the X‟s. It can be said that the causal or predictive effects of the X‟s with respect to Y are linear and additive and non-interactive and that any variations in Y that are not explained by the X‟s are statistically independent of each other and identically normally distributed under all conditions.
However, here linear regression analysis with two way interactions has been used. It considers all the possible interactions between all the parameters. Stepwise regression method has been deployed which is used in the probing stages of model building to find out a useful subset of factors. The process step by step adds the most significant variable or the combination of the variables and removes the least significant variable or the combination of the variables.
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