unh_cmac.ps

一个老外写的CMAC Neural Network源代码和说明

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( positive. A similar)3209 2977 MS
(expression can be easily formulated, however, for )496 3083 MS
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(It is impossible to quantify the improvement in performance to be gained in the general case)646 3239 MS
(by using a relatively uniform lattice of receptive fields, rather than the )496 3345 MS
(diagonalized arrangement)3322 3345 MS
(used by )496 3451 MS
(Albus. In our experience, a more uniform arrangement of receptive fields typically)843 3451 MS
(provides learning system  performance equal to or better than that achieved in the same)496 3557 MS
(application using the )496 3663 MS
(Albus arrangement \(sometimes substantially better\), with relatively little)1360 3663 MS
(increase in computational effort. We typically select C to be the smallest power of 2 which is)496 3769 MS
(equal to or greater than 4*N, in which case a good displacement vector is simply the first N odd)496 3875 MS
(integers [12].)496 3981 MS
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(Receptive Field Sensitivity Functions)496 4190 MS
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(We have also investigated CMAC networks with graded, rather than all-or-none, receptive)646 4347 MS
(field sensitivity functions [12-14], as have others [16]. In this case, the CMAC output is)496 4453 MS
(influenced more by receptive fields for which the input vector is near the center of the active)496 4559 MS
(range, and is influenced less by receptive fields for which the input is near the limits of the)496 4665 MS
(active range. The CMAC output is then a weighted average of the C addressed adjustable)496 4771 MS
(parameters, rather than a simple average as in equation 5. This provides a continuous function)496 4877 MS
(approximation \(rather than the piece-wise constant function approximation of the conventional)496 4983 MS
(CMAC\). Any function which is maximum at the center and decreases smoothly to near 0 at the)496 5089 MS
(edges is satisfactory \(e.g., linear decrease\) for generating continuous outputs. Smooth outputs)496 5195 MS
(require that the slope of the function also approach 0 near the receptive field edges \(e.g., cubic)496 5301 MS
(spline, )496 5407 MS
(gaussian, etc.\). The critical issue is how to form the )787 5407 MS
(multi-dimensional receptive field)2904 5407 MS
(sensitivity function from the one-dimensional primitives.)496 5513 MS
(An obvious choice would be to simply base the receptive field sensitivity function on the)646 5669 MS
(radial distance from the center of the receptive field. While there is substantial evidence)496 5775 MS
(supporting the use of radial basis functions for general system approximation [17,18], the fixed,)496 5881 MS
(relatively sparse distribution of receptive fields inherent in CMAC family networks must be)496 5987 MS
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(6)2495 6197 MS
(considered. In the normalized input space defined in the previous section, each CMAC)496 603 MS
(receptive field spans the interior of a )496 709 MS
(hypercube of side C. The distance from the center to the)2002 709 MS
(nearest edge \(the center of a face of the )496 815 MS
(hypercube\) of the receptive field is C/2, while the)2167 815 MS
(distance from the center to the farthest edge \(a corner of the )496 946 MS
(hypercube\) is N)2988 946 MS
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( C/2. A radial)3708 946 MS
(basis function which tapers to a small value at the nearest edge of the receptive field will be)496 1052 MS
(very small in most of the corner region, confining the significant response to a limited region of)496 1158 MS
(the hypothetical receptive field. On the other hand, if the radial basis function tapers to a small)496 1264 MS
(value at the corner, it will have a significant output at the nearest edge of the receptive field,)496 1370 MS
(which is counter to the objective of a tapered receptive field.)496 1476 MS
(A second alternative is to use the distance from the input point to the nearest face of the)646 1632 MS
(receptive field as the single parameter in the sensitivity function \(rather than the radial distance)496 1738 MS
(from the center\). This provides a receptive field sensitivity function which is maximum at the)496 1844 MS
(center and which has the same value at all points on its boundary. The sensitivity function will)496 1950 MS
(be continuous throughout the receptive field, but will have discontinuous slopes along select)496 2056 MS
(hyperplanes. We have found this to be the preferred alternative for CMAC neural networks [12].)496 2162 MS
(The CMAC computations described in the previous sections can easily be modified to)646 2318 MS
(accommodate non-constant receptive field sensitivity functions. Let )496 2424 MS
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( represent the )3325 2424 MS
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(component of the receptive field virtual address )496 2530 MS
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( in the normalized input space, as given in)2525 2530 MS
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(equation 10. Let )496 2636 MS
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( represent the )1278 2636 MS
(jth component of the real-valued, normalized input vector )1873 2636 MS
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('':)4277 2636 MS
n
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( )3109 2856 MS ( )3134 2856 MS ( )3160 2856 MS ( )3185 2856 MS ( )3211 2856 MS ( )3236 2856 MS ( )3262 2856 MS ( )3287 2856 MS ( )3313 2856 MS ( )3338 2856 MS
( )3364 2856 MS ( )3389 2856 MS ( )3415 2856 MS ( )3440 2856 MS ( )3466 2856 MS ( )3491 2856 MS (\()3517 2856 MS (1)3547 2856 MS (1)3598 2856 MS (\))3650 2856 MS
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(The corresponding faces of the receptive field occur at )496 3092 MS
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( = )2837 3092 MS
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( - 0.5 and )3020 3092 MS
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( = )3531 3092 MS
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( + C - 0.5 in the)3714 3092 MS
(normalized space. For an arbitrary input point, the minimum distance )496 3205 MS
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(d)3319 3205 MS
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(i)3364 3218 MS
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( to any face of receptive)3377 3205 MS
(field )496 3318 MS
(i is then given by:)691 3318 MS
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(N)3213 3471 MS ( )3279 3471 MS ( )3305 3471 MS ( )3330 3471 MS ( )3356 3471 MS ( )3381 3471 MS ( )3407 3471 MS ( )3432 3471 MS ( )3458 3471 MS ( )3483 3471 MS
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(=)1135 3471 MS
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(m)1210 3471 MS (i)1286 3471 MS (n)1306 3471 MS (\()1358 3471 MS
(.)1840 3471 MS
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(.)2436 3471 MS
(\))2703 3471 MS
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( )3509 3471 MS ( )3534 3471 MS ( )3560 3471 MS ( )3585 3471 MS ( )3611 3471 MS ( )3636 3471 MS ( )3662 3471 MS (\()3687 3471 MS (1)3717 3471 MS (2)3769 3471 MS
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(In this case, )496 3664 MS
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( = 0 corresponds to any point on a face of the receptive field, while )1072 3664 MS
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( = C/2)3862 3664 MS
(corresponds to the single point at the center of the receptive field. )496 3784 MS
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( varies linearly along any)3250 3784 MS
(linear path from any point on a face of the receptive field to the point at the center.  Equation 5)496 3897 MS
(can then be modified to give the new CMAC output:)496 4003 MS
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[91.559 0 0 -91.559 0 0]/Helvetica MF
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[91.559 0 0 -91.559 0 0]/Symbol MF
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