Background

An association between violence or conflict and temperature has been studied since late 60s. And a growing body of studies have tried to measure and prove quantitatively that relationship. There are two mechanism that could affect violence and those are:
* A temperature affect production of food and in consequence has an effect on distribution of resources and income.
* Psychical mechanisms could increase short term predisposition to violence.
The long term hypothesis has been tested but the results are not robust. Burke, et al (2009) started a debate using prio database and year data finding statistically significant association between temperature, rainfall and conflict. Not long after that Buhaug, et al (2010) proved that the result was not robust. In 2013, trying to reconcile the literature Hsiang et al compile most of the studies and unintentionally make it clear that the result was not robust but model dependent.
It was also clear from Hsiang that the short term hypothesis was not tested because of the lack of data. In this post we are going to go over a) why are the result non consistent, and b) testing the short term effect on political violence.

It’s not normal

While this studies have all been careful in the methodology they ignored for some reason an old empirical regularity, conflict is not normally distributed. In the mid 40’s the physicist Lewis Fry Richardson was the first one in proving that fatalities in conflict followed a power law. Power law are an easy kind of distribution, which makes them abound in nature. In our case \(p(fatalities)=C*fatalities^{-\alpha}\), where \(\alpha\) is an scaling factor that determine the proportion to which scale the density of conflict. Usually social scientist have relied in the central limit theorem but it assumes that the sample has a finite mean and variance. Power laws, when this scaling factor is less than 2 have none of those, and when it’s less than 3 they still have infinite variance. This can be seen graphically with fat tails.
Following Clauset, et al (2009) we tested statistically 101 cities in Africa. There were few of them that, for not having enough variation could not be tested. (In the next section I explain where those cities come from )

Power law with cutoff Exponential with cutoff Exponential Lognormal with cutoff Lognormal City Alpha
0.9900 0.8404 0 0.0000 0.0000 Algiers 2.338630
0.9808 0.9760 0 0.0000 0.0000 Omdurman 1.672696
0.9172 0.0036 0 0.0000 0.0000 Bujumbura 2.100756
0.9968 0.6412 0 0.0000 0.0000 Koulikoro 2.610468
0.9844 0.4320 0 0.0000 0.0000 Niamey 4.669515
1.0000 1.0000 0 0.0000 0.0000 Casablanca 3.885390
0.9940 0.9212 0 0.0000 0.0000 Brazzaville 3.211769
0.9708 0.2340 0 0.0000 0.0000 Durban 4.337487
0.9888 0.3556 0 0.0000 0.0000 Abidjan 2.724768
0.8416 0.4888 0 0.0000 0.0000 Lomé 1.937416
0.9976 0.9920 0 0.0000 0.0000 Monrovia 1.651437
0.9900 0.0192 0 0.0000 0.0000 Benin City 7.104060
0.9948 0.9608 0 0.0000 0.0000 Lubumbashi 1.988023
0.9968 0.6960 0 0.0000 0.0000 Ouagadougou 1.797253
0.9820 0.6744 0 0.0000 0.0000 Kigali 1.802619
0.9876 0.0076 0 0.0000 0.0000 Pretoria 5.217446
0.9924 0.9908 0 0.0000 0.0000 Marrakech 1.436810
0.9916 0.8156 0 0.0000 0.0000 Kano 1.927519
0.9988 0.3416 0 0.0000 0.0000 Cape Town 5.598933
0.9788 0.9304 0 0.0000 0.0000 Kinshasa 1.944211
0.9960 0.9912 0 0.0000 0.0000 Maseru 2.168013
0.9968 0.0032 0 0.0000 0.0000 Lagos 2.747822
1.0000 0.0752 0 0.0000 0.0000 Mafikeng 6.770780
0.9996 0.0840 0 0.0000 0.0000 Port Elizabeth 5.932607
0.9892 0.0008 0 0.0000 0.0000 Nairobi 2.756673
0.9952 0.8700 0 0.0000 0.0000 Juba 2.120821
0.9988 0.0052 0 0.0000 0.0000 Lusaka 4.818737
0.9928 0.5952 0 0.0000 0.0000 Kumasi 2.870616
0.9992 0.0012 0 0.0000 0.0000 Port Harcourt 3.626469
0.9932 0.8972 0 0.0000 0.0000 Cairo 1.903925
1.0000 0.0000 0 0.0000 0.0000 Thiès Inf
0.9424 0.8716 0 0.0000 0.0000 Addis Ababa 1.732384
0.9800 0.1700 0 0.0000 0.0000 Entebbe 2.688481
0.9992 0.9976 0 0.0000 0.0000 Luanda 1.650291
0.9956 0.9844 0 0.0000 0.0000 Tiaret 2.033822
0.9712 0.3352 0 0.0000 0.0000 Pietermaritzburg 3.648300
1.0000 0.4128 0 0.0000 0.0000 Harare 4.029948
0.9664 0.0000 0 0.0000 0.0000 Mogadishu 2.851409
1.0000 0.9476 0 0.0000 0.0000 Enugu 3.935612
0.9832 0.9828 0 0.0000 0.0000 Huambo 3.238733
1.0000 1.0000 0 0.0000 0.0000 Freetown 4.640957
0.0000 0.0244 0 0.0000 0.0000 Rabat 6.049433
0.9996 0.4832 0 0.0000 0.0000 Kismayo 3.697045
0.9816 0.1412 0 0.0000 0.0000 Ibadan 2.264199
0.9804 0.0104 0 0.0000 0.0000 Maputo 3.802871
0.9852 0.0000 0 0.0000 0.0000 Benghazi 3.126131
0.9988 0.9968 0 0.0000 0.0000 Brikama 1.847727
0.9992 0.3724 0 0.0000 0.0000 Antananarivo 3.824208
0.9996 1.0000 0 0.0000 0.0000 Banjul 1.810526
1.0000 0.9840 0 0.0000 0.0000 Bhisho 2.895543
0.9980 0.8168 0 0.0000 0.0000 Bloemfontein 2.674332
1.0000 0.0048 0 0.0000 0.0000 Mombasa 8.407462
0.9992 0.8684 0 0.0000 0.0000 N_Djamena 1.555654
0.0000 0.0004 0 0.0000 0.0000 Porto-Novo 8.213475
0.0000 0.0000 0 0.0000 0.0000 Nelspruit 9.656170
0.9920 0.9996 0 0.0000 0.0000 Moundou 1.567524
0.9992 0.9992 0 0.0000 0.0000 Accra 1.659544
0.9984 1.0000 0 0.0000 0.0000 Luxor 1.795299
0.9736 0.9732 0 0.0000 0.0000 Nzérékoré 1.917194
0.0000 0.0000 0 0.0000 0.0000 Tamale 9.656170
0.9912 0.4772 0 0.0000 0.0000 Tunis 2.303232
0.9996 0.7900 0 0.0000 0.0000 Douala 2.019887
0.0000 0.0000 0 0.0000 0.0000 Blantyre 6.289882
0.0000 0.0000 0 0.0000 0.0000 Bulawayo 25.525816
0.9884 0.9180 0 0.0000 0.0000 Abuja 1.936262
0.0000 0.0000 0 0.0000 0.0000 Mutare 12.541560
0.9756 0.9972 0 0.0000 0.0000 Conakry 2.137536
0.9972 0.5516 0 0.0000 0.0000 Asmara 1.872015
0.9984 0.3780 0 0.0000 0.0000 Ndola 3.019773
0.8084 0.7012 0 0.0000 0.0000 Bissau 1.538388
1.0000 1.0000 1 0.9976 0.9964 Malabo 2.116221
0.0000 0.0000 0 0.0000 0.0000 Lilongwe 9.175272
0.8856 0.0948 0 0.0000 0.0000 Libreville 3.066542
1.0000 1.0000 0 0.0000 0.0000 Bobo-Dioulasso 1.850566
1.0000 0.9244 0 0.0000 0.0000 Djibouti 2.194491
0.9748 0.0000 0 0.0000 0.0000 Tripoli 2.196964
0.0000 0.0000 0 0.0000 0.0000 Polokwane 21.197731
1.0000 0.9980 0 0.0000 0.0000 Port Said 1.830699
1.0000 1.0000 0 0.0000 0.0000 El Aaiún 7.952119
1.0000 1.0000 0 0.0000 0.0000 Zanzibar City 3.182713
0.9888 0.9848 0 0.0000 0.0000 Alexandria 2.168547
0.9872 0.5568 0 0.0000 0.0000 Hargeisa 3.234091
1.0000 1.0000 0 0.0000 0.0000 Yamoussoukro 3.885390
1.0000 0.7796 0 0.0000 0.0000 Mwanza 2.938273
0.9956 0.9880 0 0.0000 0.0000 Suez 1.904632
0.9860 0.9380 0 0.0000 0.0000 Dar es Salaam 2.221197
0.9028 0.0020 0 0.0000 0.0000 Timbuktu 2.164284
0.9992 0.8576 0 0.0000 0.0000 Bangui 2.251300
0.9924 0.5152 0 0.0000 0.0000 Sabha 3.050307

This table shows the proportion of samples that we fail to reject as each of the distributions for 2500 replication using the values of the original data. and the \(\alpha\) associated with the distribution. As we can see, of the 89 cases only 9 fail to reject as power law, in contrast, we fail to reject the exponential in 26 cases (including the 9 that were not classified as power law). We reject all the other distributions for every single city. The average alpha is \(2.93\) and the median alpha is \(2.30\). This is a quite strong evidence that the central limit theorem is not likely applicable here.

The Data

In order to test our hypothesis we collect the data from ACLED data for all Africa between 1997 and 2017. We subtracted the 100 most populated cities of the continent according to Wikipedia and use only the fatalities that fall between 50 miles of the city. Here we have a map with all those cities, the size of the circle is proportional to the total number of fatalities there. It is clear that some cities are way more violent than others. The only exclusion that we did is “remote violence” from ACLED database. and that is because temperature in a place x can only affect violence there, not in another remote place.

For the temperature data we collect rainfall and average temperature from NASA to the centroid that each buffer had. we are going to analyze individually each time series. By city we have more than 7000 days with the records of political violence. One of the problems of working with temperature is that it’s extremely seasonal. in order to know whether it was just the seasonal circle or the anomalies of the real data we extract the cycle and created 100 surrogates. We assigning equal anomalies that the real data has per city but randomly distributed trough the year.
## Empirical Dynamic Model

We are working on power law that behaves in a quite similar to chaotic. Based on Taken’s theorem when we have a chaotic system when we observe the lags of the system we can reconstruct the motion of those. When the lags tend to infinity we should be able to reconstruct the motion and as the lags tend to infinity it should become deterministic. This last part is not true for real data because we are adding also noise and measurement problems.
Surcharge, et al (1994) developed an empirical dynamic model based on the conclusions of the taken model and the idea of granger causality. It essentially predict where \(x_\tau\) is going to be using nearest neighborhood of k lags. The number k should ideally come from theory. But there is no study that I know on how long does the impact of a fatality least in time. The alternative is doing brute force an run the algorithm for the time series for lags between 1 and n. it is to say \(x_\tau~f(x_{\tau-1}), x_\tau~f(x_{\tau-1}, x_{\tau-2}),...,x_\tau~f(x_{\tau-1}, x_{\tau-2},...x_{\tau-n})\). given the computational demand I defined that the maximum lag allowed is 8, but there are no theoretical reasons to believe so. The fact that most of the cities used a lag between 7 and 8 make me believe that we should use more. But it is a computationally intensive work that already takes more than 48 hours in processing each loop.
Using this model we estimate what is the correlation between the forecast using real data and the real values. Then we estimate the correlation between the forecast and each of the surrogates, and we create a box plot on what is the range of precision of the surrogates. Those are, again, just the temperature cycle with random anomalies. If the forecast with real data is better than the forecast with surrogates that would and it is different to zero it would imply that there is causation.
This table is sorted by increasing distance from equator. It is to say, the first cities in the table are the closest to the equator and the latest one are the farthest. we can notice that as in general the prediction of the real data is better. And not only that, as we move away from the equator, where there is more variability in the temperature, the difference between the cycle and the anomalies becomes more important.

Meassuring the Direction of That Relationship

Unlike a linear regression that produce a single coefficient and it’s standard deviation that let’s us know how x impacts on y EDM is mostly a forecast tool. In order to measure the direction of the relationship we generated an small perturbed state of equal to +/- .05 of the standard deviation of the sample, about 0.16°. We estimated the result of what would be the change in forecast of a positive and a negative increase in temperature and then we spited that by the delta in temperature that we introduced. To make it comparable across the different cities, we normalized the fatalities.
As we can see the effect is close to zero. Here the cities are again ordered by distance to equator. and it is more noticeable that the impact when we are far from equator are actually higher than those close.

Conclussion

It’s a work in progress and I still want to run the same model with maximum daily temperature to check robustness. But based on this analysis there are a few important conclusions to take:
* Conflict is not normal and we can not use the central limit theorem to justify regressions
* Using this methodology we can settle the non robust part of the discussion
* We found that temperature does explain conflict,
* We also found that the relationship is less dependent of the seasonal cycle as we move away from the equator which implies that the consequences of global warming are going to be worse there.

Bibliography

Buhaug, Halvard , (2010), Climate not to blame for African civil wars, Proceedings of the National Academy of Sciences, 107 (38) 16477-16482.
Burke; Marshall B. , et al, (2009), Warming increases the risk of civil war in Africa Proceedings of the National Academy of Sciences, 106 (49) 20670-20674 Clauset; A., et al, (2009), “Power-law distributions in empirical data” SIAM Review 51(4), 661-703
Hsiang, Solomon, Marshall Burke, and Edward Miguel, (2013), Quantifying the Inuence of Climate Change on Human Conict, Science, 341.
* Sugihara G, et al. (2012) Detecting causality in complex ecosystems. Science 338(6106): 496-500
* Ye H, et al. (2015) Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling. Proc Natl Acad Sci USA 112(13):E1569-E1576.