2.3.2. Selection of Informative Pixels

The next step is to select candidate pixels to be used as the new training data from the initial SVM classification results. Pixels with higher uncertainty tend to be located near a hyperplane determined by the initial classification, and therefore are more likely to be mixed pixels [24,26]. If the class label of these pixels is correctly defined, then further classification with these pixels as new training data could properly modify the decision boundary, resulting in improved classification performance.

Among various approaches for the selection of pixels with high uncertainty [37–39], the breaking ties (BT) algorithm [40], which is simple to implement, was adopted in this study. The BT algorithm first computes the difference between the largest and the second largest *posteriori* probabilities (Δ*P*) as,

$$
\Delta P = P(\omega\_1 | D) \ - \ P(\omega\_2 | D) \tag{2}
$$

where *P*(*ωi*|*D*) is the *posteriori* probability of the *i*th class (*ωi*) computed using Equation (1). *ω*<sup>1</sup> and *ω*<sup>2</sup> are the classes with the largest and the second largest *posteriori* probabilities, respectively.

The larger the difference between these two *posteriori* probabilities meant the decision to select the pixel is less ambiguous. The pixels with smaller differences are extracted as the ones with the higher uncertainties. The pixels with high uncertainty selected through the BT algorithm are used as candidates in the initial training data.

#### 2.3.3. Generation of Rule Information and Prediction of Class Labels

Once informative pixels have been selected, the next critical step, which is the core part of the self-learning approach, is to define the class labels for the candidates. Rule information on sequential land-cover changes are first defined by comparing past land-cover maps (i.e., CDLs in this study), then the rule information is used to assign the class labels to the selected candidates. The rule information extracted in this study can be regarded as a form of temporal contextual information. Some land-cover classes, such as urban and water, tend to remain unchanged. Conversely, some crops are likely to change to others or become fallow by certain cropping systems in the study area. For example, crop rotations such as a corn-soybean rotation are very common in the USA. If such temporal change information is properly characterized, it could be used to predict the class labels of the candidates for new training data.

The main concept of defining rule information on sequential land-cover patterns is illustrated in Figure 3. Suppose that pixels #1 and 3 undergo bi-annual crop rotations in Figure 3. If these pixels are selected as candidates for new training data, their class labels can be predicted for *Tn* by considering the sequential rotation patterns. Conversely, pixels #2 and 4 remain unchanged during the same time period, and are predicted to be unchanged for *Tn*.

**Figure 3.** Illustrations for the process of generating rule-based class labels. Past land-cover maps are stacked and compared, where *Tn* is the reference year under consideration.

Our goal is to predict the class label for the considered time (*Tn*) using past land-cover maps, which will be used to define the class labels for the candidates of new training data. When past *k* years (*Tn-i*, *i* = 1, ···, *k*) are considered, the rule information on sequential land-cover patterns for *Tn* can be formulated as,

$$\left\{\omega^{T\_{n-k}}(\mathbf{x}), \,\omega^{T\_{n-k-1}}(\mathbf{x}), \,\cdots, \,\omega^{T\_{n-2}}(\mathbf{x}), \,\omega^{T\_{n-1}}(\mathbf{x})\right\} \to \,\omega^{T\_{\mathbf{n}}}(\mathbf{x})\tag{3}$$

where *ωTj*(**x**) denotes the class at a location (**x**) for the year *Tj*.

In this study, a simple but efficient heuristic approach is applied to predict the class labels from sequential land-cover patterns using past land-cover maps. Under the assumption that the typical sequential patterns of land-cover changes in the study area could be maintained, the class label can be predicted for the considered time. This assumption has been routinely adopted for classification processes using temporal contextual information. More specifically, by overlaying past time-series land-cover maps, sequential land-cover patterns are first identified at each pixel and unique sequence rules from the specific sequences of land-covers in successive years are then defined. Each unique sequence rule has its own non-overlapping combination of sequential land-cover patterns. Pixels with different colors in Figure 3 have their corresponding unique sequence rules. Based on unique sequence rules that provide useful information on the prediction of class labels, the class labels for the considered time can be predicted by adopting above assumption. For example, suppose that a certain pixel has a unique sequence such as corn-soybean-corn-soybean-corn, which reflects a corn-soybean rotation, like pixel #1 in Figure 3. For any pixel with this unique sequence rule, the class label for *Tn* is predicted as soybean to account for a corn-soybean rotation.
