Keeping these cribs aside, the book is useful as a quick reference to functions from survival , coxme , cmprsk and eha packages. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. This site uses Akismet to reduce spam. Learn how your comment data is processed.
The Multidimensional Scaling MDS tool is adopted in order to compare and to extract relationships among the data. Finally, we propose an amplitude-space embedding technique that produces a clear fire pattern classification. Ignitions might have different sources, as natural causes, human negligence or human intentionality, among others. The data analysed in this paper was retrieved in December, Each record contains information about the events date, time with one minute resolution , geographic location and size in terms of burnt area.
The burnt area is expressed in logarithmic units and is related to the color of the marks. We can note two annual cycles: the first is weaker and includes the months of February and March; the second is stronger and is due to the major incidence of fires during summer . In Fig. It is visible the increasing number of events as well as the strong activity verified around the middle of the decade — Nevertheless, the charts reveal a large volatility and pose difficulties to capture some trend.
We observe minimal values for years , , …, , and maximum values for and , but no straightforward method to correlate data points. The results shown above illustrate through simple statistics the increasing importance of understanding the behavior of forest fires and characterizing the spatiotemporal distributions unveiled by such a complex phenomenon.
For that purpose, in the next sections we adopt several complementary mathematical tools. In this section we adopt the mutual information to correlate forest fires annual patterns. First we compute the mutual information, based on events size i. Second, we use a hierarchical clustering algorithm to find relationships among the data. Visualization trees are used to highlight the interpretation of the results.
If X i and X j are two discrete random variables, then the mutual information, I X i , X j , is given by: 3 where p x i , x j is the joint probability distribution function of X i , X j , and p x i and p x j are the marginal probability distribution functions of X i and X j , respectively. The concept of mutual information comes from the information theory  and has been adopted in the study of complex systems from diverse fields, namely in experimental time series analysis, in DNA and symbol sequencing and in providing a theoretical basis for the notion of complexity  — .
In this section, instead of expression 3 , we use the normalized mutual information, I N X i , X j , given by  : 4 where H X i , X j represents the joint entropy between X i and X j : 5. Forest fires are analysed in an annual basis. This means that the events are modelled as Dirac impulses, where A k represents fire size i.
The mutual information is calculated to correlate events occurred in different years of the analysed time period. The probabilities for calculating the mutual information are estimated from the histograms of amplitudes A k , constructed considering bins, each one having width equal to 0. The map reveals strong correlations between certain years, corresponding to higher values of mutual information. Nevertheless, the analysis is not totally assertive and requires multiple comparisons. Having in mind an efficient method to visualize and to compare results, a hierarchical clustering algorithm is adopted, based on the mutual information, I N X i , X j , between pairs of objects.
The goal of hierarchical clustering is to build a hierarchy of clusters, in such a way that objects in the same cluster are, in some sense, similar to each other  ,  — . Based on a measure of dissimilarity between clusters, those are combined or, alternatively, split for agglomerative or, alternatively, divisive clustering. This is achieved by using an appropriate metric, quantifying the distance between pairs of objects, and a linkage criterion, defining the dissimilarity between clusters as a function of the pairwise distances between objects.
The results of hierarchical clustering are presented in a phylogenetic tree adopting the successive agglomerative clustering and average-linkage method Fig. Both representations of Fig. In this section we adopt the MDS tools to handle information and the relationships embedded into the data. MDS is a statistical technique for visualizing data that can reveal similarities between objects. MDS assigns a point to each object in a m -dimensional space and arranges the set in order to reproduce the observed similarities.
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A shorter larger distance between two points means that the corresponding objects are more similar distinct. The Shepard and the stress plots assess the quality of the MDS maps. The Shepard diagrams Fig. On the other hand, the stress plot reveals that a three dimensional space describes adequately the data Fig.
This can be concluded by observing the stress line, which diminishes strongly until the dimensionality is two, moderately towards dimensionality three and weakly from then on. Often, the maximum curvature point of the stress line is adopted as the criterion for deciding the dimensionality of the MDS map. The MDS maps of Fig.
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Comparing Fig. The MDS maps, in particular the 3D plot, are more intuitive than the phylogenetic tree. Moreover, most software for MDS analysis allows the user to rotate and visualise the maps from different perspectives, easing the identification of clusters. This is useful especially when dealing with large amounts of data. In this section we study forest fires in a complementary line of thought, namely by considering spatial information.
First, we divide the geographic territory under study i.
Second, for characterizing the histograms, we calculate the Shannon entropy, S i , given by: 8 where the probabilities p i m , n are approximated by the relative frequencies. The results are visualized in the phylogenetic tree of Fig.
The evolution of S i versus year is represented in Fig. In this chart is clear a large volatility and apparently some increase of entropy along time. In a more global perspective, we verify that amplitude and space data lead to distinct observations. In this line of though, we embed amplitude and space data into a single graph by adding to the bidimensional MDS plot of Fig.
We study two special cases of time-dependent covariates in more detail and show that if the time-dependent Cox model satisfies the proportional hazards assumption, there will be attenuation in the sense that the landmark regression coefficient is between the time-dependent Cox regression coefficient and 0. We show that the degree of attenuation depends on the rate of change of the time-dependent covariate. An illustration using the Stanford heart transplant data is provided.
Intuitively, the landmark model employs an old value, X s , instead of the current value X t to describe the hazard at time t. In what follows we consider two of such models. The first of these is the common situation where X t is dichotomous, the other is a specific case where X t is continuous. Let X t be a dichotomous time-dependent covariate.
An irreversible illness-death multi-state model, with response as the illness state. States 0 and 1 correspond to the values of the time-dependent covariate response being 0 and 1, respectively, and state 2 is the death state. The original landmarking paper did not consider a possible transition back from state 1 to 0, but we will allow this here. The attenuation as time between s and t increases can be clearly seen. The time-dependent landmark regression coefficients from cox.
The mean for the time-dependent Cox regression was 1. With censoring, the mean for the time-dependent Cox regression was 0. All curves in Fig.
Event History Analysis With R (chapman & Hall/crc The R Series) Download
Compared to a situation with no removal of subjects because of death, for a given X s , subjects with higher X t have a higher probability of being removed. As a result, subjects with lower X t remain in the population, which results in the dotted curves being pulled downwards. This behaviour is quite similar to the selective removal of subjects with high frailty values in frailty models, the effect of which is also stronger with increasing frailty variance.
Further simulation studies not shown here indicated that the effect of selective removal is stronger i. The event time is time from admittance to the waiting list until death; interest is in the effect of a heart transplant on survival. Of the patients, 69 received a heart transplant, and a total of 75 patients died, 45 with a heart transplant and 30 without a heart transplant. Median follow-up calculated by reverse Kaplan—Meier was 2.
An important covariate predictive of the effect of heart transplant is the mismatch score, measured for those patients with a heart transplant. Median mismatch score was 1. Because different effects of the heart transplant may be expected for patients with a high mismatch score and patients with a low mismatch score, we distinguish between patients with a mismatch score in the highest quartile and the rest. We define two time-dependent covariates of the type studied in Sect. It is clear that, compared to the hazard ratio of 1.
In this paper we derived relations between the regression coefficients obtained in a landmark analysis and those of a time-dependent Cox regression, when interest is in the effect of a time-dependent covariate on survival. In case the time-dependent covariate has no effect on survival at all, i.