**5.1 Example**

**Figure 8** illustrates an example of the classification of a query cell *xs* after considering both spaces: *xs*'s neighbors in the feature space, under the title "Feature space" (**Figure 8(left)**), and *xs*'s neighbors in the geographical space, under the title "Geographical space" (**Figure 8(right)**).

In this example, the class assignment is based on the two samples that are closest in the feature space, and there are four land-use classes. *xs*'s two nearest neighbors in the feature space are *xr* and *xq*, and their land-use arrays are *Ar* ¼ ð Þ 0, 0, 1, 0 and *Aq* ¼ 0, <sup>3</sup> <sup>4</sup> , 0, <sup>1</sup> 4 with computed weights *w*ð Þ*<sup>r</sup> <sup>s</sup>* <sup>¼</sup> 1 and *<sup>w</sup>*ð Þ*<sup>q</sup> <sup>s</sup>* ¼ 4, respectively. Notice that *w*ð Þ*<sup>r</sup> <sup>s</sup>* and *w*ð Þ*<sup>q</sup> <sup>s</sup>* are set, respectively, according to the *xr* and *xq*feature space distances from the query cell *xs*. In **Figure 8(left)**, under the title "Feature space," the two bar graphs represent the land-use arrays of *xr* and *xq*, which are *Ar* and *Aq*, respectively. For example, because *Ar* has 100% confidence of being attributed to class 3, the value

#### **Figure 8.**

*Land-use classification of a query cell based on (left) only the feature space, (right) only the geographical space, and (bottom) both using neighbor smoothing.*

of the bar of class 3 is 1, and the values of the other bars are 0. *xs* 0 *s* land-use array is computed as a weighted average of *Ar* and *Aq* (Eq. (2)): *As* <sup>¼</sup> *<sup>w</sup>*ð Þ*<sup>r</sup> <sup>s</sup> Ar*þ*w*ð Þ*<sup>q</sup> <sup>s</sup> Aq w*ð Þ*<sup>r</sup> <sup>s</sup>* <sup>þ</sup>*w*ð Þ*<sup>q</sup> s* ¼ ð Þ 0, 0*:*6, 0*:*2, 0*:*2 , as is demonstrated in **Figure 8(left)**, and it is the result of the weighted average of *Ar* and *Aq*. Without neighbor smoothing, assigning a class to *xs* would have been decided at this point, and *xs* would have been assigned to the class which is the highest in *As*, that is, class 2.

But here, we integrate the neighbors' land use in the classification decision. Let us assume *xs* has two geographical neighbors *xa* and *xb*, and their land-use arrays are *Aa* <sup>¼</sup> ð Þ 0, 0, 0, 1 and *Ab* <sup>¼</sup> ð Þ 0, 0*:*1, 0*:*1, 0*:*<sup>8</sup> , and their weights are *<sup>W</sup>*ð Þ *<sup>a</sup> <sup>s</sup>* ¼ 2 and *W*ð Þ *<sup>b</sup> <sup>s</sup>* <sup>¼</sup> 3, respectively. Notice that *<sup>W</sup>*ð Þ *<sup>a</sup> <sup>s</sup>* and *W*ð Þ *<sup>b</sup> <sup>s</sup>* are set according to the Euclidean geographic distance of *xa* and *xb* from the query cell *xs*. In **Figure 8(right)**, under the title "Geographic space," the two bar graphs represent the land-use arrays of *xa* and *xb*. *xs*'s neighbors' land-use array is computed by a weighted average of *Aa* and *Ab* (Eq. (7)): *NAs* <sup>¼</sup> *<sup>W</sup>*ð Þ*<sup>a</sup> <sup>s</sup> Aa*þ*W*ð Þ *<sup>b</sup> <sup>s</sup> Ab W*ð Þ*<sup>a</sup> <sup>s</sup>* <sup>þ</sup>*W*ð Þ *<sup>b</sup> s* ¼ ð Þ 0, 0*:*06, 0*:*06, 0*:*88 . *NAs* is demonstrated in **Figure 8 (right)** under the title "Geographical component." The maximal value of 0.88, based on the influential geographic neighbors of *xs* 0 *s*, challenge the cell's previous assignment of class 2 to that of class 4.

The final decision about assigning a class to *xs* is after combining the feature component *As* and the geographic component *NAs*. Let us set the smoothing parameter *σ* at 0.1, and thus the weight of the neighbors' component is (Eq. (9)): *nbrsq* <sup>¼</sup> 2, *<sup>σ</sup>* <sup>¼</sup> <sup>0</sup>*:*<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*11*:xs* 's integrated land-use array (Eq. (8)) is *IAs* ¼ 0*:*11∙*NAs* þ ð Þ 1 � 0*:*11 ∙*As* ¼ ð Þ 0, 0*:*54, 0*:*18, 0*:*28 , as is demonstrated in **Figure 8(bottom)** under the title "Land-use array *xs*." If we consider *IAs*'s 0.54 confidence high enough, then *xs* would be classified as class 2 and added to the training set *G* for the next iteration.
