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The nearest-neighbor approaches described in the previous section can be thought of as approximating the target function f ( x ) at the single query point x = x, Locally weighted regression is a generalization of this approach It constructs an explicit approximation to f over a local region surrounding x, Locally weighted regression uses nearby or distance-weighted training examples to form this local approximation to f For example, we might approximate the target function in the neighborhood surrounding x, using a linear function, a quadratic function, a multilayer neural network, or some other functional form The phrase "locally weighted regression" is called local because the function is approximated based only on data near the query point, weighted because the contribution of each training example is weighted by its distance from the query point, and regression because this is the term used widely in the statistical learning community for the problem of approximating real-valued functions Given a new query instance x,, the general approach in locally weighted regression is to construct an approximation f^ that fits the training examples in the neighborhood surrounding x, This approximation is then used to calculate the value f"(x,), which is output as the estimated target value for the query instance The description of f^ may then be deleted, because a different local approximation will be calculated for each distinct query instance

Figure 12-17

About two-thirds of widescreen movies are filmed at 185 (flat) aspect ratio When a 185 film is transferred directly to full-frame 4:3 by including the extra picture at the top and bottom, the actual size of the images on the TV are the same as for a letterbox version In other words, letterboxing only covers over the part of the picture that also was covered in the theater A pan and scan transfer to 4:3 makes the I didn t pay good money for my 30-inch TV just to watch black bars crowd happy But a letterbox transfer to anamorphic 16:9 is still needed to appease the videophiles who demand the theatrical aspect ratio and to keep the I paid good money for my widescreen TV crowd from revolting Ordinarily, this would mean two separate products, but not with DVD The producer of the disc can put the 4:3 version on one side (or one layer) and the letterboxed 16:9 version on the other This more or less doubles the premastering cost and slightly increases the mastering and replication costs, but with production runs of 100,000 copies or more, this adds less than a dollar to the unit cost On the other hand, the widescreen letterbox transfer is sometimes reserved for a special edition and sold as a separate product at a higher price As widescreen TVs and HDTVs slowly replace traditional TVs, 4:3 transfers will become less common, and even letterboxed 4:3 transfers will become more appreciated In Japan and Europe, where widescreen TVs already outsell standard TVs, letterboxed video is more popular

Let us consider the case of locally weighted regression in which the target function f is approximated near x, using a linear function of the form

(continued)

Summary

As before, a i ( x ) denotes the value of the ith attribute of the instance x Recall that in 4 we discussed methods such as gradient descent to find the coefficients wo w, to minimize the error in fitting such linear functions to a given set of training examples In that chapter we were interested in a global approximation to the target function Therefore, we derived methods to choose weights that minimize the squared error summed over the set D of training examples

Figure 12-18

Putting everything together (ignoring the option of cropping during transfer) gives the following variations to the four basic output formats: 1 4:3 full frame (4:3 original) a Direct transfer 2 4:3 pan and scan (wide original) a Pan and scan transfer b Automatic pan and scan done by player (1) On direct transfer (16:9 original) (2) On pan and scan transfer (not 16:9 original) (3) On letterbox transfer (not 16:9 original) 3 4:3 letterbox (wide original) a Letterbox transfer b Automatic letterbox done by player (1) On direct transfer (16:9 original)

Listing 12-3

(2) On pan and scan transfer (not 16:9 original) (3) On letterbox transfer (not 16:9 original) 4 4:3 anamorphic (wide original) a Direct transfer (16:9 original) b Pan and scan transfer (not 16:9 original) c Letterbox transfer (not 16:9 original)