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(b) Fit the ARMA model with the order of (p, q) to the time series data or the transformed time series data if the transformation is performed in Step 1a, using Statistica [4] The Autocorrelation Function (ACF) plot and the Partial Autocorrelation Function (PACF) plot [2, 4] of the time series data can be used to gain insights into the order of the ARMA model that ts to the data The initial values of p and q can be determined accordingly (c) Use the ARMA model from Step 1b to predict the time series data over time (d) Compute the Mean Squared Error (MSE) of the predicted time series data from the original time series data as follows: MSE = 1 n 1
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where xt represents the data observation at time t, and xt represents the predicted data value at time t (e) Verify the ARMA model with its autoregressive and moving average parameters using the Autocorrelation Function (ACF) plot and the Partial Autocorrelation Function (PACF) plot of the time series data to see if the ACF and PACF plots agree with the ARMA model For example, the following is the ARMA (1, 2) model tted to the transformed attack data of Network Interface\Bytes Received/sec under the ARP Poison attack: yt = 01140yt 1 + et 07570et 1 06599et 2 Figures 161 and 162 show the ACF plot and the PACF plot of the attack data
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Autocorrelation Function \\ALPHA02-VICTIM\Network Interface(Intel[R] PRO_1000 MT Network Connection - Packet Scheduler Miniport)\Bytes Received/sec: D(-10) (Standard errors are white-noise estimates) Lag 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Corr SE +701 0371 +435 0371 +139 0371 +089 0371 +029 0370 -009 0370 -044 0370 -212 0370 -371 0369 -498 0369 -347 0369 -209 0369 -078 0368 -055 0368 -014 0368 0 10 05 00 05 Q p 3564 0000 4938 0000 5079 0000 5136 0000 5142 0000 5143 0000 5157 0000 5487 0000 6495 0000 8318 0000 9201 0000 9524 0000 9568 0000 9591 0000 9592 0000 0 10 Conf Limit
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Figure 161 The Autocorrelation Function (ACF) plot of the attack data for Network Interface\Bytes Received/sec under the ARP Poison attack
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Partial Autocorrelation Function \\ALPHA02-VICTIM\Network Interface(Intel[R]PRO_1000 MT Network Connection - Packet Scheduler Miniport)\Bytes Received/sec: D(-10) (Standard errors assume AR order of k-1) Lag 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Corr SE +701 0372 -111 0372 -243 0372 +255 0372 -092 0372 -124 0372 +094 0372 -403 0372 -191 0372 -013 0372 +196 0372 -071 0372 -046 0372 +070 0372 +008 0372 0 10 Conf Limit 05 00 05 10
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Figure 162 The Partial Autocorrelation Function (PACF) plot of the attack data for Network Interface\Bytes Received/sec under the ARP Poison attack
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The ACF plot in Figure 161 shows the exponential decay of positive spikes, while the PACF plot in Figure 162 shows the oscillating decay of positive and negative spikes This is agreeable to the ARMA model of order (1, 2) with one positive autoregressive (AR) coef cient and two negative moving average (MA) coef cients (f) Repeat Steps 1(b) 1(d) with different values of p and q which vary from their initial values until an ARMA model is found to produce a good t to the data with a small value of MSE (g) If the transformation is performed in Step 1(a), the data model for the original time series data is constructed using the ARMA model from Step 1(f) For example, the following data model is constructed for the attack data of Network Interface\Bytes Received/sec under the ARP Poison attack, using the ARMA model in Formula 169 and the differencing transformation in Formula 167, as follows: x t = xt 10 + yt yt = 01140yt 1 + et 07570et 1 06599et 2 (1610) (1611)
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2 Repeat Step 1 but replace the attack data sample with the normal use data sample for the text editing in the training data set to develop the text editing data model 3 Repeat Step 1 but replace the attack data sample with the normal use data sample for the web browsing in the training data set to develop the web browsing data model For example, Network Interface\Bytes Received/sec has the autocorrelation increase (A+) characteristic under the ARP Poison attack as shown in Table 102 and Table 131 Formulas 1610 and 1611 above de ne the attack model for this variable under the ARP Poison attack
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