Geophysical Research Letters

Changes in the predictability of the daily thermal structure in southern South America using information theory


  • Gustavo Naumann,

    1. National Scientific and Technological Research Council, Department of Atmospheric and Oceanic Sciences, Faculty of Sciences, University of Buenos Aires, Buenos Aires, Argentina
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  • Walter M. Vargas

    1. National Scientific and Technological Research Council, Department of Atmospheric and Oceanic Sciences, Faculty of Sciences, University of Buenos Aires, Buenos Aires, Argentina
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[1] Spatiotemporal behaviors of predictability of climate system were studied. These were analyzed as changes in persistence and system memory using information theory. This study was performed by coupling a cluster analysis algorithm and conditional entropy in southern South America. The spatial analysis of the entropy showed that a meridional gradient exists in the entire region, and its maximum is in the southern region. In this study, the gradients of this property in the northern regions yield predictabilities that are twice those in the southern part of South America. Temporal changes in conditional entropy were observed with quasi-cyclical variations. The low frequency variability estimate in the conditional entropy indicates that the dominant wave is approximately 18 years. The changes observed in the persistence and conditional entropy, especially in groups that representing the warm and cold days, suggest that changes in objective forecasting are necessary.

1. Introduction

[2] The character of the dynamics, linear or non-linear, and the precision of the measurement of the initial states of a system determine the horizon of predictability. For most complex systems like meteorological or financial processes, exists at best a few general ideas about their predictability [Feistel and Ebeling, 1989; Ebeling, 2002].

[3] The problem discussed here concerns spatial and temporal behaviors and the ability to make good predictions of future states when considering persistence, as well as cases where predictions of the different weather states are rather poor in the southern South America. A basic tool used to analyze these questions is information theory using conditional entropies introduced by Shannon [1948, 1950] and used by many researchers such as Leung and North [1990], Ebeling [1997], Werner et al. [1999], Molgedey and Ebeling [2000], Roulston and Smith [2002], Balling and Roy [2004], DelSole and Tippett [2007], and Tang et al. [2008].

[4] In order to differentiate between the processes involved, a non-hierarchical cluster analysis is introduced for the reference series of daily maximum and minimum temperature anomalies. With this algorithm, each day is represented by each group, which is characterized by the values of their center of mass, and the algorithm assigns each day to one of four groups: warm, wet, cold and dry [Vargas and Naumann, 2008]. This work also shows that the thermal properties of each group are associated with atmospheric processes of the synoptic-scale, with circulation patterns defined for each group. The wet days are primarily responsible for the changes observed in the series of maximum and minimum temperatures. These findings are consistent with trends in the series of precipitation in the region. In addition, a significant increase in the occurrence of warm days and a decrease in the frequency of cold days were detected.

[5] In this study, the conditional entropy of the spell length of n-states clustered temperature is analyzed by Shannon's information-theoretical methods. A spatial and temporal analysis of these parameters allows for inferences to be made about the regions or the periods in which there are more risks to the prediction accuracy due to the internal variability of the climatic system.

[6] Section 2 presents the daily data used for the study. Conditional entropy and its relationship with the spell length are presented in Section 3. The spatial distribution of the uncertainties and its climatological interpretation are discussed in Section 4. Section 5 shows the temporal behavior of the conditional entropy. Finally, Section 6 summarizes the main conclusions.

2. Data and Method

[7] The stations or reference series used in the analysis must have long records (ideally covering the instrumental period) with high quality measurements, and must represent different or specific climatic regions. A reference series of daily maximum and minimum temperatures provided by National Weather Service and Claris Project was selected in order to have a large quantity of data to produce a stable estimation of the clusters over the daily temperatures (in this case, more than 20000 values); a representative geographic distribution of stations was selected to include as many climatic regions of southern South America as possible, as well as to cover a wide latitudinal selection (covering S23°–S55°; see Table 1).

Table 1. Description of the Eight Reference Stations Analyzed
WMO IDStationLongitudeLatitudeStartEnd
87623Santa Rosa−64.26−36.5419372004
87925Río Gallegos−69.45−51.9918962004
87585Buenos Aires (O.C.B.A.)−58.42−34.5719062000

[8] In a cluster analysis, p-dimensional explicative variables X are used for N objects; the objective is to group them into K groups (K\N), so that the variables of items that belong to one group resemble each other as much as possible and differ as much as possible from those of other groups.

[9] After the cluster analysis, four groups were obtained and characterized by their four centroids. These provide a new series of discrete values in which each day is no longer represented by various temperature anomalies, but by a symbol indicating that it belongs to a particular group. For more details on the classification, see Vargas and Naumann [2008].

[10] To characterize climatological processes, it is important to have detailed knowledge of the temporal behavior of the parameters, like the memory of the process [Ebeling, 2002]. These specific properties of the system can be analyzed as changes in persistence and system memory represented by conditional entropy.

[11] According to the cluster analysis algorithm, each day is described by only one of the symbols wet, dry, cold or hot. Pi (X1; …; Xi) was defined as the probability of each group or symbol sequence of length i. This probability describes the persistence of each cluster over i time steps. The sequence of the probabilities pi(x) states that x will be maintained over i days before it switches over to another sequence. This quantity is called the “exit time distribution” [Nicolis et al., 1997].

[12] Figure 1 shows the probability that the weather corresponds to each cluster for Buenos Aires. For cold and warm days the distribution is similar to a Markov process, but for the wet and dry days, the persistence is higher than for a Markov processes.

Figure 1.

Calculated (solid line) and exponential fit (dashed line) exit time distribution for (a) wet, (b) dry, (c) warm and (d) cold days at Buenos Aires, and the attached values of the Markov process (squares).

[13] If this distribution changes with time, it may be associated with a climatic change. The exit time distribution can generally be fit to an analytical function. Several possible expressions were tested [Gabriel and Neumann, 1962; Nicolis et al., 1997]. In most cases, the best fit for each cluster corresponds to an exponential distribution:

equation image


equation image

where p1 is the transition probability for each group and p(i) is the probability of a sequence of I days in the same state.

3. Conditional Entropy

[14] The Shannon Entropy [Shannon, 1948, 1950] is computed as follows: Let x be a discrete variable that may take i=1…m possible values. In this work, x are the clustered temperatures that can take four possible states: wet, dry, cold and hot.

equation image

From equation (2) it can be seen that the highest amount of uncertainty from an information source is realized when the output symbols of the source are equally probable. The entropy varies from 0 to log(m), so the entropy may be standardized such that it would range from 0 to 1 by dividing it by its maximum.

[15] It may be easier to compare the amount of disorder of two systems if it is known that one system encountered more states than did the other. Conditional entropy relies on transitional probabilities for the conditional probability of state j of variable Y given state I of variable X, [p(Y=jX = i)]. The equation for computing the conditional entropy is

equation image

Conditional entropy (mutual information) measures the uncertainty of predicting a state one step into the future given a history consisting of n states; the present state and the previous states are known [Ebeling and Nicolis, 1992]. Predictability in this work is measured by differences in Shannon entropies, i.e., by conditional entropies.

[16] The existence of long correlations is expressed by long decreasing tails of the conditional entropies. In general, our expectation is that any long-range memory decreases the conditional entropies and improves the predictions.

[17] Let the n-tuple {A1….An} be the λ states of a given spell of length n. Further, let p(A1…An) be the probability to find a spell with the states A1…An, so the Block − entropy per spell of length n is defined as

equation image

From this entropy, the conditional entropy hn is derived as the differences

equation image

The maximum of the uncertainty (in units of log(λ)) is hn = 1. Hence, one can define the average predictability as the difference between the maximal and the actual uncertainty

equation image

This means that predictability is related to the certainty concerning the next state in the future in comparison to the available knowledge.

[18] In order to analyze these properties, Figure 2 shows conditional entropy as a function of spell length for four reference series. Here, the differences between the conditional entropies of the six n-tuple are larger between the third and fourth spell lengths than between the others. Thus, the growth in information is larger between this first and second day than between the following days. For the forecast of climatological parameters, the consideration of longer durations leads to an improvement in the predictability of the following day.

Figure 2.

Conditional entropy (uncertainty) as a function of n-tuple length at Campinas (grey), Tucumán (dotted), Buenos Aires (dashed) and Rio Gallegos (Black).

[19] Analysis of the spatial behavior of this parameter shows a latitudinal behavior with greater uncertainties in Río Gallegos (lat = S52°) and lesser uncertainties in the tropical regions (Tucumán and Campinas).

4. Spatial Variations in the Uncertainties

[20] Following this line of reasoning, the spatial behavior of the entropy was analyzed for all of the stations in the region that ensure collection of the daily maximum and minimum temperatures for at least 20000 days to guarantee a stable estimation of the clusters (taking into account the typical spell length in the atmosphere processes and the states defined).

[21] Figure 3 shows the field of entropies H for the spell length 2 calculated according equations (5). Figure 3 reinforces the idea of a meridional gradient, initially suggested by the results in Figure 2, and some inferences concerning the regional circulation and its predictability emerge.

Figure 3.

Spatial distribution of Block-entropy H for spell length 2 at 54 stations in the region.

[22] In the Patagonia region (southern than S40°), where the higher entropies were observed, the preferential direction of the synoptic system is highlighted (cold fronts, cyclonics and anticyclonic systems, etc.). Another aspect that should be mentioned is the role of the Andes in the synoptic system trajectories, as evidenced by a zonal gradient mainly in Patagonia, where the west winds are dominant.

[23] In central Argentina and the coastal regions of Brazil there is a transition region between the mid-latitudes and tropical regimes. This region is characterized by a greater predictability due to the persistence of the warm advections from the Amazonia.

[24] These results suggest that the risks associated with the same prediction model are different for each region. In other words, entropy measures the quantity of information that is needed to build an efficient model.

5. Temporal Variations of Conditional Entropy and Persistence

[25] For practical applications, there is little interest in an average uncertainty of predictions, but instead in a concrete prediction based on the observation of a concrete string of finite length n. Figure 4 shows the conditional entropies for warm and cold days and transition probabilities between cold to cold and hot to hot days and between cold to hot and hot to cold; these indicate the persistence and antipersistence, respectively.

Figure 4.

Conditional entropy (dashed line) (top) for hot days [H] and (bottom) for cold days [C] and the transition probability (solid line) between (a) H-H, (b) H-C, (c) C-C, and (d) C-H at Buenos Aires.

[26] Here, the conditional entropy shows significant variations with time. In both cases, this variation is ± 0.10, which represents changes in the predictability values between 15–25%. This means that there are special years where the predictability increases/decreases by a factor of two in comparison with the average.

[27] Similar to the way the nearly inverse course reveals that the conditional entropy of cold/warm days must decrease, in the case of increasing persistence the conditional entropy increases in the presence of transitions from warm to cold and cold to warm events.

[28] Some periodicities throughout the region were found in this variation. Figure 5 shows the power spectrum of the conditional entropy for the warm group at Buenos Aires.

Figure 5.

Spectral density of the conditional entropy for the warm group at Buenos Aires (solid line) and 95% significance level for red noise (dashed line).

[29] This shows a periodicity of about 18 years where the security of prediction based on persistence increased/decreased with time. An increase in conditional entropy is observed in the 1930s, 1950s and at the end of the 1970s; the last two maximums are coincident with two climatic jumps observed for many variables of the region [Minetti and Vargas, 1998]. This may be related because an increase in the disorder of the system could imply a change of state. After these events begin, there is a decrease in conditional entropy that reaches its minimum approximately ten years after.

6. Conclusions

[30] By coupling a cluster analysis algorithm with an analysis of system memory through conditional entropy and the exit time distribution complex, climatological structures can be more accurately described [Werner et al., 1999].

[31] A spatial analysis of entropy indicated that there was a meridional gradient throughout the entire region, with its maximum in the southern region of South America (westerlies region). Taking into account the reduced predictability of this system, small changes in conditional entropy imply significant variability. In this study, the gradient implies that the predictability in the northern regions is two times that of the westerly part of South America.

[32] Evidence of temporal changes in conditional entropy was observed with quasi-cyclical variations. The spectral estimate of the conditional entropy suggests that the dominant period is about 18 years. A nearly inverse course reveals that the conditional entropy of cold/warm days must decrease in the case of increasing persistence and the conditional entropy increase in the presence of more transitions of warm to cold and cold to warm events.

[33] The changes observed in persistence and conditional entropy, especially in groups that represent the warm and cold days, were taken into account. These results suggest that changes in objective forecasting models that involve time series are warranted.

[34] Finally, observed changes in the distribution of the circulation patterns [Vargas and Naumann, 2008] could directly affect the objective forecast. Moreover, the study of the long-term variability of the entropy would be a useful tool to diagnosis of climatic jumps.


[35] This research was sponsored by projects UBA X-228 and CONICET PIP 5139.