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Intensity Analysis has become popular as a top-down hierarchical accounting framework to analyze differences among categories, such as changes in land categories over time. Some aspects of interpretation are straightforward, while other aspects require deeper thought. This article explains how to interpret Intensity Analysis with respect to four concepts. First, we illustrate how to analyze whether error could account for non-uniform changes. Second, we explore two types of the large dormant category phenomenon. Third, we show how results can be sensitive to the selection of the domain. Fourth, we explain how Intensity Analysis’ symmetric top-down hierarchy influences interpretation with respect to temporal processes, for which changes during a time interval influence the sizes of the categories at the final time, but not at the initial time. We illustrate these concepts by applying Intensity Analysis to changes during one time interval (2000–2004) in a part of Central Kalimantan for the land categories Forest, Bare and Grass.

Intensity Analysis is a quantitative framework to account for differences among categories as summarized by a square transition matrix for which the rows’ categories are identical to the columns’ categories [

Our article examines concepts that apply to a single time interval, while the concepts apply also to case studies that analyze each of several consecutive time intervals. The first concept addresses how to interpret results when researchers suspect that the data have error but do not know exactly how much error. The second concept concerns the large dormant category phenomenon, where the inclusion of a single large category influences the results for all other categories. The third concept relates to the fact that it is frequently not obvious how to select the domain for a case study, so we illustrate conceptually how Intensity Analysis can be sensitive to the selection of the domain, with particular attention to inclusion of various amounts of persistence. The fourth concept relates to the fact that Intensity Analysis uses a top-down hierarchy in which the sizes of the categories at the initial and final times help to set uniform baselines for comparison to the observed transitions. However, the amounts of change and persistence during a time interval influence the sizes of the categories at the final time, but do not influence the sizes of the categories at the initial time. Thus, it can be more intuitive to interpret change intensities that are conditional on the sizes of the categories at the initial time, rather than the final time, depending on whether one views the change processes as top-down or bottom-up.

This article illustrates the concepts with a case study that has three land categories, because three categories show the concepts as clearly as possible. Our case study is on the island of Borneo in Central Kalimantan, Indonesia.

Map in the upper left shows the location of the study site in the southern part of the island of Borneo, which is shared by Indonesia, Malaysia and Burnei. Kalimantan is the Indonesian portion of Borneo. The study site is in Central Kalimantan Province. The middle and lower maps on the left show land-cover categories at 2000 and 2004. Maps on the right show changes during 2000–2004.

The underlying data are two Landsat scenes from path 118 and row 62: one ETM+ scene from 16 July 2000 and one TM scene from 17 June 2004. The satellite images were geometrically corrected with ground control points that were evenly distributed in space, and the root mean square error was within one pixel. Atmospheric correction was performed using the FLAASH module, available in the ENVI 4.0 software. The atmospheric correction removed the influence of aerosols that affect reflectance values primarily in the short wavelength regions of the electromagnetic spectrum.

We applied unsupervised image classification using the ISODATA clustering algorithm with 100 initial spectral clusters. The spectral clusters were labeled using the information from color composites, NDVI images, land-cover spectral reflectance characteristics, and field knowledge. Three land-cover categories were assigned: Forest, Bare, and Grass. Forest is peat swamp forest. Bare is bare soil including burn scars that do not have vegetation. Grass is fern grass, sparse shrubs, and regenerating secondary forest. Then we applied a majority filter with a window size of 3 by 3 pixels to reduce isolated pixels in the land-cover maps. A mask eliminated pixels that are water or clouds at either time point. We do not have information concerning the accuracy of the maps because we do not have ground information for 2000 and 2004.

After we created the classified maps, we constructed a transition matrix by overlaying the two land-cover maps.

Transitions as a percent of the domain. Superscript

2004 | 2000 | Interval | ||||
---|---|---|---|---|---|---|

Forest | Bare | Grass | Total | Loss | ||

76.8 | 5.7^{α} |
4.0 | 86.5 | 9.7 | ||

4.0^{α} |
5.5 | 1.8^{τ} |
11.2 | 5.7 | ||

1.1 | 0.3^{τ} |
0.9 | 2.4 | 1.5 | ||

81.9 | 11.4 | 6.7 | 100.0 | |||

5.1 | 6.0 | 5.8 | 16.9 |

_{ij}_{ji}

Mathematical notation for Intensity Analysis.

Symbol | Meaning |
---|---|

number of categories, which equals 3 in our case study | |

index for a category at the interval’s initial time point | |

index for a category at the interval’s final time point | |

index for the losing category for the selected transition | |

index for the gaining category for the selected transition | |

_{ij} |
number of pixels that transition from category |

total change as percent of domain, which equals the uniform intensity for the category level | |

_{j} |
intensity of gain of category |

_{i} |
intensity of loss of category |

_{in} |
intensity of transition from category |

_{n} |
uniform intensity of transition from all non- |

_{mj} |
intensity of transition from category |

_{m} |
uniform intensity of transition from all non- |

^{G}_{j} |
hypothesized commission of category |

^{G}_{j} |
hypothesized omission of category |

^{L}_{i} |
hypothesized commission of category |

^{L}_{i} |
hypothesized omission of category |

Intensity Analysis is a mathematical framework that compares a uniform intensity to observed intensities of temporal changes among categories. We use Intensity Analysis at two levels: category and transition.

At the category level, we compare _{j}_{i}_{j}_{j}_{i}_{i}

Equations (6) and (7) concern Intensity Analysis for the transition from an arbitrary category _{in}_{n}_{in}_{n}_{in}_{n}

Equations (8) and (9) concern Intensity Analysis for the transition from a particular losing category _{mj}_{m}_{mj}_{m}_{mj}_{m}

The transition from category _{mn}_{n}_{mn}_{m}_{mn}_{n}_{mn}_{m}

We do not have reference information to measure errors in the maps from the two time points. Nevertheless, we suspect our maps have some errors and we wonder whether errors in the maps could account for the deviations from uniform intensities that Intensity Analysis reveals. Equations in this subsection allow us to compute the minimum hypothetical error in the data that could account for the deviation between the uniform change intensity

Equations (10–13) compute the hypothetical error in the map of the final time that could account for a deviation between _{j}_{j}_{j}

Equations (14–17) compute the hypothetical error in the map of the initial time that could account for a deviation between _{i}_{i}_{i}

Total change separated into quantity and allocation.

Gain, persistence and loss by category.

Intensity of gains and losses by category.

Intensity of transitions given Forest’s gain on the positive axis and given Forest’s loss on the negative axis.

Intensity of transitions given Bare’s gain on the positive axis and given Bare’s loss on the negative axis.

Intensity of transitions given Grass’ gain on the positive axis and given Grass’ loss on the negative axis.

Hypothetical error intensities within the change region that could account for deviations from uniform category level losses and gains. If the actual error intensities are less than the ones in the figure, then there is evidence that the category level gains and losses are non-uniform.

It is common that researchers do not know the sizes and types of errors in the data, especially because satellite images are widely available for historic dates but it is impossible to go back in time to collect ground reference information that would be necessary to assess accuracy. If accuracy information were to exist, then it would be desirable to compare the hypothetical error intensities to the actual error intensities, but ignorance of actual errors is not a sufficient reason to ignore possible errors and is not a sufficient reason to disregard historic satellite images. Researchers can draw on their experience with similar types of classifications as researchers interpret the hypothetical errors.

The Kalimantan case study illustrates one type of the large dormant category phenomenon, in which the presence of a large dormant category causes the intensities of other categories to be greater than they would be in the absence of the large dormant category [

Our case study illustrates one type of the large dormant category phenomenon that is different than a second type in which the large dormant category plays a small role in total change. Water is a typical example of the second type of large dormant category. Water might be necessary to include in some land change studies where humans convert water to land via infill or convert land to water via dams. It is not clear how much of the persistent water should be included in a study of change, especially for coastal studies where most of the water category persists as ocean. The size of any category affects the intensities of other categories; thus, researchers can be tempted to exclude water to eliminate this effect. In fact, we masked water from the Kalimantan case study because water is small and not particularly relevant to our research question. If water were excluded from an analysis, then the analysis might miss some important transitions that involve water, depending on the research question. The large dormant category phenomenon requires more research to determine general principles concerning how to select a study’s domain.

Selection of the domain is influential regardless of whether one uses Intensity Analysis or some other analytical technique. For example, Equation 1 computes the total change as a domain’s percent, which includes persistence in the denominator. Equation (1) is usually the first calculation in an investigation of land change. This subsection illustrates how Intensity Analysis is sensitive to the amount of persistence that the domain includes for each category, thus it is usually helpful to consider how the results of Intensity Analysis can be sensitive to the selection of the domain, especially concerning possible elimination of some persistence or entire categories, such as water.

Matrix that shows Forest’s gain targets Bare and Bare targets Forest’s loss.

Final Time | Initial | Interval | ||||
---|---|---|---|---|---|---|

Forest | Bare | Grass | Total | Loss | ||

1 | 2 | 1 | 4 | 3 | ||

2 | 1 | 0 | 3 | 2 | ||

1 | 0 | 2 | 3 | 1 | ||

4 | 3 | 3 | 10 | |||

3 | 2 | 1 | 6 |

Matrix that shows uniform transition intensities to Forest and from Forest for active Forest.

Final Time | Initial | Interval | ||||
---|---|---|---|---|---|---|

Forest | Bare | Grass | Total | Loss | ||

1 | 2 | 1 | 4 | 3 | ||

2 | 2 | 0 | 4 | 2 | ||

1 | 0 | 1 | 2 | 1 | ||

4 | 4 | 2 | 10 | |||

3 | 2 | 1 | 6 |

Matrix that shows Forest’s gain avoids Bare and Bare avoids Forest’s loss.

Final Time | Initial | Interval | ||||
---|---|---|---|---|---|---|

Forest | Bare | Grass | Total | Loss | ||

1 | 2 | 1 | 4 | 3 | ||

2 | 3 | 0 | 5 | 2 | ||

1 | 0 | 0 | 1 | 1 | ||

4 | 5 | 1 | 10 | |||

3 | 2 | 1 | 6 |

Matrix that shows uniform transition intensities to Forest and from Forest for dormant Forest.

Final Time | Initial | Interval | ||||
---|---|---|---|---|---|---|

Forest | Bare | Grass | Total | Loss | ||

4 | 2 | 1 | 7 | 3 | ||

2 | 0 | 0 | 2 | 2 | ||

1 | 0 | 0 | 1 | 1 | ||

7 | 2 | 1 | 10 | |||

3 | 2 | 1 | 6 |

We had considered excluding persistence from the equations as we developed Intensity Analysis. Specifically, we considered making the denominator of Equation (6) equal to the loss of category

Intensity Analysis has a top-down hierarchy in which broader information determines the context for more detailed information. Specifically, Intensity Analysis interprets category level intensities _{i}_{j}_{in}_{mj}_{n}_{m}

Furthermore, Intensity Analysis’ equations concerning the matrix’s rows are symmetric with its equations concerning the matrix’s columns. However, a temporal change process is not symmetric in time because the change during a time interval influences the sizes of the categories at the final time but not at the initial time. Therefore, it can be more intuitive to interpret intensities that are conditional on the initial time, than to interpret intensities that are conditional on the final time. Specifically, at the category level, it can be more intuitive to compare _{i}_{j}_{i}_{j}_{in}_{n}_{mj}_{m}

We designed symmetry into

Let us illustrate further with our case study by considering transitions to Forest, while focusing on transition intensities that are conditional on the initial time. We hypothesize that Forest would grow more from Grass than from Bare, because we hypothesize a natural process of recovery from Bare to Grass to Forest. However, the size of the transition from Bare to Forest is larger than the size of the transition from Grass to Forest (

Now let us consider the transitions from Forest, while focusing on transition intensities that are conditional on the final time. We hypothesize that the change processes of Forest loss in our study area are fire, agriculture, and logging [

If Intensity Analysis were to give information identical to the information that we could see easily by a direct comparison of the sizes of the transitions, then there would be no need for Intensity Analysis. Intensity Analysis probes the transition matrix to reveal the matrix’s detailed patterns. The transition matrix describes patterns of change, which are caused by processes of change. Researchers must use qualitative knowledge concerning processes of change in order to interpret Intensity Analysis in a manner that can help to develop a cause and effect understanding. Intensity Analysis can help to assess the evidence for a particular hypothesized process of change, and can help to develop new hypotheses concerning processes of change. For proper interpretation, researchers must consider whether the hypothesized processes of change match the hierarchical structure of Intensity Analysis.

We are beginning to develop a method to detect whether top-down processes can account for detailed transitions or whether various bottom-up transitions are required to account for a particular matrix, because usually there can be many possible combinations of transitions that are consistent with a set of marginal totals and persistence for each category. If the matrix’s marginal totals could explain all the transitions, then there would be evidence that top-down processes are operating. If the matrix’s marginal totals cannot explain the transitions, then there would be evidence that bottom-up processes are operating. Future research should examine this approach to link patterns with processes.

It is interesting to compare Intensity Analysis to the Markov approach, which is a popular method to analyze a transition matrix [

Some potential applications of Intensity Analysis are not temporal, and would therefore not have the above-mentioned complications concerning interpretation of temporal cause and effect relationships. For example, Intensity Analysis could compare two classifications of a single image, where the rows indicate the categories according to one method of classification and the columns indicate the categories according to an alternative method of classification. The research question would ask how the two classifications are associated. In this situation, there is not a cause and effect relationship among the rows and columns, because the process of one classification does not affect the process of the other classification. For such cases, the symmetrical architecture of Intensity Analysis matches the symmetry of the research question concerning the association between the two methods of classification.

This article examines the design of Intensity Analysis and offers guidance concerning its interpretation. We illustrate four important concepts using a matrix of transitions among the categories Forest, Bare, and Grass over one time interval in Central Kalimantan, Indonesia. These four concepts concern: error analysis, the large dormant category phenomenon, sensitivity to the selection of the domain, and the top-down hierarchical symmetric structure of Intensity Analysis. The results illustrate how Intensity Analysis gives information that is different than the information obtained from a direct comparison of the sizes of the entries in the transition matrix. In our case study, Forest is the only dormant category for both gains and losses, in spite of being involved in most of the changes. The transition from Forest to Bare is systematically avoiding, in spite of being the largest transition. These types of insights can help researchers test and develop hypotheses concerning processes of change. This article’s concepts are generally applicable, so we hope researchers of other case studies will benefit from these ideas during the application and interpretation of Intensity Analysis.

We thank the Japan Science Technology (JST), Japan International Cooperation Agency (JICA), and Japan Society for the Promotion of Science (JSPS) for their financial support. The United States National Science Foundation funded grant DEB-0620579 for Safaa Aldwaik to create a computer program that we used and that is available for free at the Intensity Analysis web site. Anonymous reviewers supplied comments that improved this article.

The authors declare no conflict of interest.