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10.1 Auto-associative NN for nonlinear PCA using barcode generating for none control to generate, create none image in none applications.creating barcode is known to be robu none for none st to outliers in the data (Section 6.2). Figure 10.

6(c) is the solution selected based on minimum H with the MAE norm used. While wiggles are eliminated, the solution underestimates the curvature in the parabolic signal. The rest of this section uses the MSE norm.

In summary, with noisy data, not having plentiful observations could cause a exible nonlinear model to over t. In the limit of in nite number of observations, over tting cannot occur in nonlinear regression, but can still occur in NLPCA due to the geometric shape of the data distribution. The inconsistency index I for detecting the projection of neighbouring points to distant parts of the NLPCA curve has been introduced, and incorporated into a holistic IC H to select the model with the appropriate weight penalty parameter and the appropriate number of hidden neurons (Hsieh, 2007).

An alternative approach for model selection was proposed by Webb (1999), who applied a constraint on the Jacobian in the objective function. 10.1.

4 Closed curves While the NLPCA is capable of nding a continuous open curve solution, there are many phenomena involving waves or quasi-periodic uctuations, which call for a continuous closed curve solution. Kirby and Miranda (1996) introduced an NLPCA with a circular node at the network bottleneck (henceforth referred to as the NLPCA(cir)), so that the nonlinear principal component (NLPC) as represented by the circular node is an angular variable , and the NLPCA(cir) is capable of approximating the data by a closed continuous curve. Figure 10.

2(b) shows the NLPCA(cir) network, which is almost identical to the NLPCA of Fig. 10.2(a), except at the bottleneck, where there are now two neurons p and q constrained to lie on a unit circle in the p-q plane, so there is only one free angular variable , the NLPC.

At the bottleneck in Fig. 10.2(b), analogous to u in (10.

5), we calculate the pre-states po and qo by po = w(x) h(x) + b ,. Monarch and qo = w(x) h(x) + b(x) ,. (10.16). where w(x) , w(x) are weight parameter vectors, and b parameters. Let 2 2 r = ( po + qo )1/2 ,. and b(x) are offset (10.17). then the circular n none none ode is de ned with p = po /r, and q = qo /r, (10.18). satisfying the unit circle equation p 2 + q 2 = 1. Thus, even though there are two variables p and q at the bottleneck, there is only one angular degree of freedom. Nonlinear principal component analysis from (Fig. 10.2(b )), due to the circle constraint.

The mapping from the bottleneck to the output proceeds as before, with (10.3) replaced by h (u) = tanh((w(u) p + w(u) q + b(u) )k ). k (10.

19). When implementing N none none LPCA(cir), Hsieh (2001b) found that there are actually two possible con gurations: (i) a restricted con guration where the constraints p = 0 = q are applied; and (ii) a general con guration without the constraints. With (i), the constraints can be satis ed approximately by adding the extra terms p 2 and q 2 to the objective function. If a closed curve solution is sought, then (i) is better than (ii) as it has effectively two fewer parameters.

However, (ii), being more general than (i), can more readily model open curve solutions like a regular NLPCA. The reason is that if the input data mapped onto the p-q plane cover only a segment of the unit circle instead of the whole circle, then the inverse mapping from the p-q space to the output space will yield a solution resembling an open curve. Hence, given a dataset, (ii) may yield either a closed curve or an open curve solution.

It uses 2lm + 6m + l + 2 parameters. Hsieh (2007) found that the information criterion (IC) H (Section 10.1.

3) not only alleviates over tting in open curve solution, but also chooses between open and closed curve solutions when using NLPCA(cir) in con guration (ii). The inconsistency index I and the IC are now obtained from I =1 1 C( p, p) + C(q, q) , and H = MSE I, 2 (10.20).

where p and q are f rom the bottleneck (Fig. 10.2(b)), and p and q are the corresponding nearest neighbour values.

For a test problem, consider a Gaussian data cloud (with 500 observations) in 2-dimensional space, where the standard deviation along the x 1 axis was double that along the x2 axis. The dataset was analyzed by the NLPCA(cir) model with m = 2, . .

. , 5 and P = 10, 1, 10 1 , 10 2 , 10 3 , 10 4 , 10 5 , 0. From all the runs, the solution selected based on the minimum MSE has m = 5 (and P = 10 5 ) (Fig.

10.7(a)), while that selected based on minimum H has m = 3 (and P = 10 5 ) (Fig. 10.

7(b)). The minimum MSE solution has (normalized) MSE = 0.370, I = 9.

50 and H = 3.52, whereas the minimum H solution has the corresponding values of 0.994, 0.

839 and 0.833, respectively, where for easy comparison with the linear mode, these values for the nonlinear solutions have been normalized upon division by the corresponding values from the linear PCA mode 1. Thus the IC correctly selected a nonlinear solution (Fig.

10.7(b)) which is similar to the linear solution. It also rejected the closed curve solution of Fig.

10.7(a) in favour of the open curve solution of Fig. 10.

7(b), despite its much larger MSE..
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