Methods processing 2 time markers (2 limits)

In many situations the time dimension is only describe through the status of observation properties at the beginning and the end of the considered time period (tmin, tmax). Assuming a context of a univariate description of multiple observations, information can be structured as a two dimensional table with rows corresponding to observations and columns expressing the two time limits.

Number of inhabitant and political majority in 1900 and 1990Number of inhabitant and political majority in 1900 and 1990

As stated previously, property change can be considered either individually or globally among all spatial features or individually but relatively to the global behaviour of features.

Individual property change (comparison of the two properties):

As stated previously, property change can be considered either individually or globally among all spatial features or individually but relatively to the global behaviour of features.

Property difference (quantitative)

D index is the difference between property value V at tmax and tmin for each i spatial feature:


The normalised difference ND expresses the change rate based on a reference time (often tmin) for each i spatial feature:


Property ratio (quantitative)

R index is the ratio between property value V at tmax and tmin for each i spatial feature:


Another property ratio can be produced that allows comparison between individual change rate and the global one (for the whole set of considered features), Rtot. This relative change rate index RRi (Relative Ratio) is the ratio between Ri and Rtot:

Change classification (qualitative)

C index expresses the type of change between property value V at tmax and tmin for each i spatial feature. C values can be simply binary (with 0 for no change and 1 for change) or multiple to describe the type of change between the two considered categories. C results from a classification process.
In the case of nominal level content, C value is either 0 or 1 expressing a change or not. However, for variables at ordinal or cardinal level, one might want to differentiate between three situations of change: a decrease, no change and an increase. The 3 possible values of C index could then be derived from the classification (recoding) of Di values, according to the following scheme:

Illustration

Let us apply this set of individual property change indices to the two variable sets listed in Table above. The change of individual features between the two intervals of time is described as follow.




In the example it shows that Ri amplifies growth changes compared to NDi, but the reference value for a decrease is now less than 1 (municipality I).

The spatial analysis of behaviour change can then be carried on as a further step by making use of index mapping.



Mapping of property change indices for the number of inhabitant (quantitative)Mapping of property change indices for the number of inhabitant (quantitative)

Mapping of property change index for the political majority (qualitative)Mapping of property change index for the political majority (qualitative)

Global property change:

One can be interested in evaluating changes among the whole set of spatial features.

Summary statistics (quantitative / qualitative)

Individual change indices can be summarised using appropriate central tendency and dispersion indicators. Of course only change indices relevant for comparison between features should be considered.

For quantitative change indices (ordinal and cardinal levels):


For qualitative change indices (nominal level):


These relevant central tendency and dispersion descriptors can then be used for a relative description of individual change behaviour. At ordinal and cardinal levels individual feature change can be compared to the global change by grouping its change value into classes around the central tendency:
- at ordinal level: interquartiles around the median;
- at cardinal level: standard deviation units around the mean value.

Scattergramme (quantitative)

Another efficient way for comparison of change between features is to plot them according to their individual properties for the two dates. This can be seen as a “time change map” with a pair of coordinates locating them into this two-dimensional space.

Such a diagramme allows different types of interpretation:


Next figure illustrates the use of a diagramme representation for the mapping of the 9 municipalities. Their change behavior can be compared to each other as well as to global references (mean and standard deviation) and types of change.

Graphical representation of property change for the number of inhabitant using a scattergramme as a “time change map”Graphical representation of property change for the number of inhabitant using a scattergramme as a “time change map”

EXERCISE

On the last figure:

Transition matrices (qualitative /quantitative)

Now our interest is in the nature of transitions from one state to another. We can use techniques that sacrifice all information about individual observation properties but provide in return information on the tendency of one state to follow another.

Due to the use of a two dimensional cross-table, the number of considered properties is limited. This approach is fully adapted for qualitative data (categories at nominal level), but also for classes at ordinal level and even at cardinal level, assuming original properties are grouped into a limited number of classes.

Let us illustrate the exploitation of termtransition matrices with our data set on Political majority of municipalities in 1900 and 1990 illustrated in previous table. We would like to summarise the change from the property in 1900 to another one in 1990. Furthermore we could identify the tendency of one property to follow another.

A 4 × 4 matrix (or cross-table) can be constructed showing the number of times a given property –political majority- is succeeded by another, a matrix of this type is called a transition frequency matrix and is shown in the following table. In order to avoid confusion between properties and frequencies of change patterns, let us recode property values with letters (L for Liberal, R for Republican, D for Democratic and S for Socialist).

The considered sample contains 9 observations, so there are 9 transitions. The matrix is read from rows to columns meaning, for example, that a transition from state L to state D is counted as an entry element a1,3 of the matrix. That is if we read from the row labelled L to the columns labelled D, we see that we move from state L into state D one time in the set of observations, but we can observe that there is no occurrence of move from state D into state L (entry element a3,1). The transition frequency matrix is asymmetric and in general ai,j ≠ aj,i. The transition frequency matrix is a concise way of expressing the incidence of one state or property following another, the transition pairs.

A transition frequency matrix showing property change patterns in political majority between 1900 and 1990A transition frequency matrix showing property change patterns in political majority between 1900 and 1990

The tendency for one state to succeed another can be emphasised in the matrix by converting the frequencies to decimal fractions or percentages. Different types of relative frequencies can be derived:

The transition relative frequency matrix showing all the possible types of transitions in political majority between 1900 and 1990The transition relative frequency matrix showing all the possible types of transitions in political majority between 1900 and 1990
The transition proportion matrix showing the tendency of one state of political majority in 1900 to follow another in 1990The transition proportion matrix showing the tendency of one state of political majority in 1900 to follow another in 1990

The transition relative frequency matrix shows that 44% of the municipalities had the property R in 1900 and this proportion has decreased to 22% in 1990. This 22% of loss compensates for the loss of state L during the same period. The property S has the highest proportion of resulting states with 44% of municipalities. On the other hand the indicates the same tendency for state L to move to states D and S (L/D and L/S = 0.5), as opposed to the state R where the tendency of unchanged is 0.5 (R/R).

Assuming a representative sample of features, relative frequencies can be interpreted as probabilities of occurrence. This extended approach can make use of Markov chains for estimating the probability of occurrence of a state based on the existence of a previous stage. This method will be discussed in the next section relative to time series analysis of a sequence of data. It can be used to describe individual transition pattern.

EXERCISE

Compare the two last tables and apply the same reasoning for the final state S.