Methods for time series

Evolution of thematic properties throughout time can be described in greater details by measuring the property of individual observation (each spatial feature) at numerous intervals of time during the considered period. Time intervals can be either regular or irregular. However many methods require a same regular interval that is common to all observations in order to allow comparison. For each observation this sequence of property values is called a time series. Time series collected for a set of observations can be structured as a two-dimensional data table as illustrated below.

Individual property change:

The objective is again to summarise the individual time change behaviour with a single index value. This can be done by the use of typical statistical descriptors (central tendency and dispersion) or by modelling the trend in change.

Summary statistics

With a simple statistical descriptor some aspects of the individual change behaviour can be described such as its central tendency, its variability or a combination of both. Depending on the nature of data analysed (qualitative or quantitative), the following statistical descriptors are applied to summarise the change:

• For qualitative data time series: the mode and the diversity
• For quantitative data time series: the mean or the median, the standard deviation or the interquartile, the coefficient of variation and the range.

Let us illustrate the use of statistical descriptors for summarising the time change behaviour of features described in the two tables above.

The time series describing the political majority during the period 1900-1990 for 9 municipalities are qualitative data measured at nominal level, therefore two indicators can be applied, the mode for deriving their central tendency and the diversity for their dispersion as expressed below

The time series describing the number of inhabitant during the period 1900-1990 for 9 municipalities are quantitative data measured at cardinal level, therefore six different indicators can be applied for deriving their central tendency and dispersion

Change trend modelling

The objective of such modelling is to summarise with a single index value the time change behaviour of each observation. Such a constraint limits the panel of functions to very simple ones. Two models expressing the trend of change could be retained: the linear regression and the allometric function. More precisely, a single coefficient for each model will be retained in order to express the change trend.

The trend of a linear model can be summarised as the slope coefficient b of the linear regression function Y = a + b X. In the case of time change modelling, the variable X is the time and Y is the considered phenomenon. In a growth process, as in many change processes, the property values of a phenomenon do not often increase in a linear manner with respect to time. Time series values can be transformed by the application of a logarithmic (log) or a square root (sqrt) function in order to compensate for this non linear increase. As the linear trend coefficient should describe best the change behaviour, the analyst should apply to the original time series values the most efficient transformation to compensate for this non linear behaviour.

This linear model can then take the following forms:

• For a linear increase in the original time series values: Y = a + b X
• For a non linear increase in the original time series values: logY = a + b X or sqrtY = a + b X
  with X as the time variable and Y the considered phenomenon

The principle of modelling the evolution trend is illustrated below. From the three time series illustrating property change of three hypothetical situations throughout time, one can observe that the slope coefficient b of a linear regression function is capable to summarise the evolution pattern of each series.

Assuming the adjustment of the linear trend is satisfying for each individual time series, then b coefficient can be interpreted as the growth rate. The coefficient value expresses the strength of change, a positive value indicates an increase as a negative value suggests a decrease throughout the period of time. However the comparison between coefficient values is limited due to the influence of the size of values on the trend coefficient b. There are several techniques proposed to standardise this coefficient for comparison purposes, but an efficient way to perform comparison is the use of an allometric function.
Allometry is a model developed in the context of biology. It attempts to describe the relative growth of a body part with respect to the growth of the whole body (organism). With the rise of the systemic approach in sciences, this concept of relative growth was then associated with principles of interactions between a set and its subsets (parts). Let us start with a definition: “Allometry: the relative growth of a part in relation to an entire organism or to a standard” (Anonymous). Thus one can summarise the relative growth of individual features with respect to the growth of the whole, in our case the set of considered features.

The allometric law is defined as follow:

• In original form:
  
  
• In linear form:
  
  In this linear form the allometric coefficient b -which is also the slope coefficient of the linear regression model- can be interpreted as follow:
  
  

The b change index provides information about the type and the rate of relative change of individual features. Comparison between bi index values is then made possible, as well as mapping their spatial distribution within the study area.

Global property change:

Another way to summarise the behaviour of individual features throughout time is to synthesise the time dimension variables into synthetic component. Finally index values will be attached to each feature (component scores) but they are derived from a global analysis of the whole set of features and time variables. The objective is again to summarise the individual time change behaviour with a couple index values. Such a transformation is called Principal component transformation or analysis (PCA), it produces principal component scores for each feature.

Standardised principal component scores

One can read the content of the following table as a set of time series describing the change in number of inhabitant during a period of time for the nine considered municipalities (row direction reading). This table structure can be approached from the column direction as well. This time census dates are interpreted as time variables. Each of them describes the number of inhabitant in the nine municipalities at a specific date. As properties change smoothly from a census date to another, one can expect a fairly high degree of correlation between time variables. Therefore it seems reasonable to synthesise this set of variables (here 10 dates or 10 variables) into a few number of relevant components.

This high degree of correlation is illustrated by the corresponding correlation matrix in the next table. One can observe the following:

• The strongest correlation takes place between a date and its preceding or its next one.
• The degree of correlation decreases with the distancing of census dates.
• The variable 1960 has the overall highest degree of correlation with all other variables.

Correlation matrix for the ten time variables. It shows the strong degree of correlation between variables.

In order to remove the undesirable scale effect, variables should be standardised prior to the component transformation.

The principal component transformation produces two significant components, summing up 99.5% of the total variance:
- component 1 (PC1): 85.3% of variance
- component 2 (PC2): 14.2% of variance

The contribution of original variables to the two retained principal components is expressed by their respective weights in the component matrix. The figure below illustrates this contribution. One can observe the following:

• Years 1900 to 1960 have a strong influence on the component 1 (1960 with the strongest), while 1970 to 1990 have a lower one (1990 with the weakest). All weights are positive.
• For the component 2 weights range from negative to positive. Once again the 2 groups of variables can be identified. The order of weights almost follows the sequence of years.

Weight of variables on the two principal components.

Now let’s consider the properties of the 9 municipalities for the two principal components. Their respective score values can be plotted for identification of groups with similar evolution behaviour. From the next figure three groups of features can be identified:

• Group 1: made of a single outsider B
• Group 2: made of municipalities F and H. Their two index values are positive
• Group 3: the largest group that includes the 6 remaining municipalities. Index values (scores) for both components are almost negative. However, municipality G is slightly away from other members of this group, with a positive index value for the first component.

Score values of the 9 municipalities for the two principal components. Interpretation of score values as a global behaviour index is made possible by comparing with the diagram of population change during this period of time.