Comparative evaluation of ARIMA and ANFIS for modeling of wireless network traffic time series

3.1 Autoregressive integrated moving average

In the framework of regression models, the computation of present output is done as a linear combination of some pre-specified number of past outputs and moving average of random white Gaussian noise [3].

Let us denote Ω as the lag operator such that ΩX(t) = X(t - 1). In general we write Ω ^τ X(t) = X(t - τ). Also let us denote Δ as the difference operator so that ΔX(t) = X(t) - X(t - 1). It can be observed that Δ ^τ X(t) = (1 - Ω) ^τ X(t). Let us also define two polynomial functions ϕ(Ω) = (1 - ϕ₁Ω-.......- ϕ _m Ω ^m ) and θ(Ω) = (1 - θ₁Ω-.......- θ _n Ω ⁿ ) where ϕ₁, ϕ₂,....ϕ _n and θ₁, θ₂,....θ _n are coefficients of the lag operator Ω; m and n are the degree of the polynomials, respectively.

Given these notations, the definition of regressive models follows next.

where ε(t) is random white Gaussian noise.

It can be seen that ARIMA is the most general of all the three regressive models discussed above. Although other more generalized regressive models are also available, ARIMA will be the focus of our study in this paper.

3.2 Adaptive neuro fuzzy inference system

A fuzzy inference system (FIS) is a framework for computation based on the concepts of fuzzy set theory, fuzzy if then rules and fuzzy reasoning [13, 19]. As shown in Figure 1, a FIS mainly has three conceptual components, viz., rule base, database, and reasoning mechanism. The rule base is a collection of fuzzy if-then rules which decide the system’s behavior and response under different possible situations. The database contains the information about the membership functions in terms of their type and shape. Finally, the reasoning or decision-making mechanism is used to infer and derive output from the system. It may be noted that a FIS may need a fuzzification interface to convert crisp input values to fuzzy values suitable for processing. However, when the inputs themselves are fuzzy then this may not be required. Similarly at the output side, a defuzzification interface is used because in almost all of the real-world application, we need a crisp output value.

A neural network following the above discussed framework of a FIS results in ANFIS. For introductory purpose, a first-order Sugeno-type FIS (see [19]) in Figure 2 and equivalent ANFIS architecture in Figure 3 has been shown next. The following common rule set for first-order Sugeno fuzzy model can easily be verified:

Rule 1. If x is P₁ and y is Q₁, then d₁ = a₁x + b₁y + c₁.

Rule 2. If x is P₂ and y is Q₂, then d₂ = a₂x + b₂y + c₂.

In the ANFIS architecture shown in Figure 3, each node in the same layer has the similar function. Here we denote output of the i th node in the layer l by $O_i^l$.

In layer 1, a linguistic label is associated with each input in terms of its membership grade. This membership grade can be defined by suitable membership functions ${{\mu }_P}_{{}_i}\left(x\right)$ and ${{\mu }_Q}_{{}_i}\left(x\right)$ with appropriate parameters. The parameters associated with these membership functions are called premise or nonlinear parameters.

The output of this layer is often called the firing strength of the corresponding rule.

Here, a _i , b _i , and c _i ; i = 1, 2 are called consequent or linear parameters of ANFIS. The total number of parameters of ANFIS is the sum of premise and consequent parameters.

It must be noted that the structure of ANFIS explained above is not unique, and, in fact, arbitrary but meaningful assignment of node functions and configurations is possible.

A Sugeno-type FIS as in MATLAB Fuzzy Logic Toolbox is shown in Figure 4.

4.1 Network traffic data collection

After completing a brief introduction of modeling approaches of ARIMA and ANFIS, we now proceed to implement these concepts to the real-world network traffic data.

To begin, real-time network traffic trace is the first thing required. In the networking research community, Wireshark [20] is the most popular and sophisticated network traffic monitoring tool. It can capture data packets from the network and provides important information like packet size, packet transfer rate, and packet capture time as well as data packet contents. We collected the packet statistics from an institutional wireless network, discarding the user data for this study. Matshark [21] was used to extract the data from Wireshark and export it into MATLAB memory space.

4.2 Data pre-processing

The collected network traffic data was at nonuniform time scale. To get a time series data, samples at a uniform time scale are required. Data samples were extracted from the traffic trace at intervals of 0.1 s in MATLAB resulting in a time series data.

This ensured that all samples remain within the range [0, 1] so that the RMSE values after applying different models can be effectively compared.

where y _m and ${\widehat{y}}_m$ denote m th actual and model trained data samples, respectively. N is the sample size.

5.1 ARIMA approach

5.2 ANFIS approach

MATLAB Fuzzy Logic Toolbox [23] was used for ANFIS modeling. The following discussion details the results obtained under each case.

An RMSE of 0.080 was obtained with ANFIS model under case 1. Four past values of data were used as input, and the present value was used as output with each input having two membership functions (2-2-2-2 architecture). The variation of error with increasing epoch numbers is shown in Figure 5. It is observed that error continues to decrease till about 140 epochs and then it becomes almost constant. ANFIS specification for this case has been presented in Table 2.

An RMSE of 0.087 was obtained with ANFIS model under case 2. Six past values of data were used as input, and the present value was used as output with each input having two membership functions (2-2-2-2-2-2 architecture). The variation of error with increasing epoch numbers is shown in Figure 6. It can be seen that error remains almost unchanged beyond seven epochs. ANFIS specification for this case has been presented in Table 3.

A careful reader might have observed that the error along the Y-axis in Figure 6 remains almost equal to 0.0871 (shown only until four decimal places). This means that the error does not decrease appreciably with increasing epochs.

RMSE of 0.081 was obtained with ANFIS model under case 3. Six past values of data were used as input and present value as output with two inputs having three and rest others having two membership functions (2-2-3-2-3-2 architecture). The variation of error with increasing epoch numbers is shown in Figure 7. It is observed that the error curve is smooth contrary to earlier two cases and also that the error does not decrease appreciably with increasing epochs similar to case 2 above. ANFIS specification for this case has been presented in Table 4.

From Table 5, we see that ANFIS model results in lower RMSE as compared to ARIMA in all the three cases considered here. At the same time, it can also be observed that the number of parameters in ANFIS is much larger than ARIMA in each of these cases. Computational complexity is empirically related to the number of parameters of a model which means that ANFIS is computationally more expensive and complex than ARIMA. The difference between the numbers of parameters becomes even larger when the number of inputs and the number of MFs of inputs of ANFIS are increased.

Hence, it is clear from the above results that although ANFIS performs better than ARIMA, this is achieved at the cost of complexity in computation which must be taken into consideration when ANFIS is used for network traffic modeling.