A devastating 7.0magnitude earthquake occurred on January 12, 2010 in Haiti (Figure 1) and killed over 316,000. In October 2010 Haitians were faced an outbreak of Cholera. Haiti recorded cases of cholera in its Centre and Artibonite Departments in October 2010. Following the earthquake and several months prior to the outbreak, spokespersons from the United States (USA) Centers for Disease Control and Prevention (CDC) said, “An outbreak of cholera was very unlikely”. The CDC’s assessment was reasonable, as Haiti had not experienced a cholera outbreak in over a century. However, several weeks after the earthquake Cholera rapidly spread, initially lamed by the CDC largely on “Haiti’s uniformly poor water and sanitation infrastructure”. In truth, even before the earthquake, the nation lacked adequate washing facilities. The lack of sanitation facilities was particularly grave; as of 2008, only 17 percent of the population had access to improved sanitation facilities. In the study reported on below, several mathematical methodologies, one empirical and one statistical, are employed to decompose the Cholera and other medical condition time series. From the analyses, predictive capabilities are created to forecast future outbreaks and related correlated medical conditions. The public health center data sets showed twenty additional medical conditions of patients tested, and many were found to be statistically correlated with cholera, with some not. As such, the implications of an atrisk population are quite challenging to the medical community.
Figure 1: Topographical map of Haiti.
Cholera; Haiti
In October 2010, researchers at North Carolina State University (NCSU) were approached by representatives of Doctors without Borders (DWB) and asked if they could look at the Haiti Cholera data, from 7 Public Health Centers in Haiti (Figure 2) to determine its initial spread and to see if a prediction could be made regarding its possible future reappearance. The NCSU Team reported to the DWB that the outbreaks occurred at 6 of the 7 stations as delta function outbreaks, so it was as if the disease was carried to those locales. With these results, researchers at the CDC determined that the strain of cholera responsible for the outbreak was consistent with strains found in South Asia and may have been introduced into Haiti by peacekeepers from Nepal, part of the United Nations Stabilization Mission in Haiti. Furthermore, “patient zero” was identified by the DWB as a 28yearold Haitian who was exposed to cholera while bathing in, and drinking from, a river near the peacekeepers’ camp. Ultimately, by the end of 2011, the outbreak had resulted in over 500,000 infections and 7,000 deaths. Cholera had also spread to the Dominican Republic; which, as of the end of 2011, had recorded over 21,000 cholera cases and 363 resulting deaths. Cholera rapidly spread, largely because of Haiti’s “uniformly poor water and sanitation infrastructure”.
Figure 2: CDC Cholera outbreak map with case numbers.
Figure 2 was provided by the US CDC and concisely documents cholera data as a function of Public Health center locations. It is of note that in the figures to follow, the NCSU SAS Inc. software and the EEMD algorithm package [1], were utilized to create the data depictions. These software packages cannot be altered.
The Haiti cholera outbreak was determined by the CDC to be the worst epidemic in recent history and as of April 2013, had killed at least 7,000 Haitians and sickened a few hundred thousand more while spreading to neighboring countries including the Dominican Republic and Cuba. Since the outbreak began in October 2010, more than 6% of Haitians have had the disease. While there had been an apparent lull in cases in 2014, by August 2015, after the rainy season that year brought a spike in cases, more than 700,000 Haitians had become ill with cholera and the death toll had climbed to 9,000.
In figure 3, the NCSU time series of the data of the 7 time series of documented cases of Cholera in Haiti provided by the DWB to NCSU are presented. In figure 4 the official Cholera cases plots are provided to show that the DWB and NCSU statistical data sets are identical.
Figure 3
Figure 4: The DWB time series of Haiti Cholera time series.
As the cholera and other medical condition data sets required additional medical and mathematical expertise, researchers at Coastal Carolina University (CCU) contributed to this study.
The Hilbert Transform (HT), the Empirical Mode (EMD) and Ensemble Empirical Mode Decomposition (EEMD)
The Hilbert Transform [2], has been applied to data time series that are both nonperiodic and nonlinear. The Hilbert Transform (HT) is thus more versatile than the Fourier Transform (FT), which can only transform periodic and linear data sets [3,4]. The Haiti Cholera data are both nonperiodic and nonlinear. Due to the HT being a global domain integral, the instantaneous amplitude and instantaneous frequency obtained using the HT is neither temporally local nor instantaneous, the direct quadrature algorithm is implemented to obtain the instantaneous amplitude and instantaneous frequency. In the process of employing the HT, one obtains a series of frequency and amplitude modulated internal modes of variability buried within a time series, which Huang et al. [3], deemed the Empirical Mode Decomposition (EMD). To eliminate spurious noise within the EMD Wu and Huang [57], created the Ensemble EMD, the EEMD.
In the EMMD, data time series x(t) are decomposed in terms of “Intrinsic Mode Functions” (IMFs), c_{j}, i.e.
and r_{n} is the residual of the data x(t), after n Intrinsic Mode Functions (IMFs) are extracted from the instantaneous frequency, omega from high frequency to low frequency intrinsic modes from j=1 to the finite number “j=n”, determined via the “sifting” process and which constitute the limits of the integral (i.e. 1 to finite number “j=n”). Here “instantaneous frequency” is defined in context, and the integral can be considered as the local mean for IMF c_{n}. Clearly, The IMFs expressed in Equation (1b) are simple oscillatory functions with relatively slowly varying and nonnegative amplitude and relatively fast changing and nonnegative frequency at any temporal point.
In practice, the EEMD is implemented through a sifting process that uses only local extrema. From any data set, x(t)=r_{j1}, say, the procedure is as follows: 1) Identify all the local extrema (The combination of both maxima and minima) and connect all these local maxima (Minima) with a cubic spline as the upper (Lower) envelope; 2) Obtain the first component h by taking the difference between the data and the local mean of the two envelopes; and 3) Treat h as the data and repeat steps 1 and 2 as many times as is required until the envelopes are symmetric about zero within a certain tolerance. The final h is designated as c_{j}. A complete sifting process stops when the residue, r_{n}, becomes a monotonic function or a function only containing one internal extremum from which no more IMFs can be extracted. In this method, multiple noise realizations, such as 100, are added to one time series of observations to mimic a scenario of multiple realizations from which an ensemble average approach for corresponding IMFs can be used to extract scaleconsistent signals. The major steps of the EEMD method are as the following:
After a time series is decomposed into IMFs, natural amplitudefrequency modulated oscillatory functions, various methods can be applied to obtain instantaneous frequencies for each IMF that lead to timefrequencyenergy representation of data.
In figures 5ah, the EEMD IMF’s are shown for all of the Public Health Locales in Haiti, along with the overall total.
Figure 5: The IMF decompositions of the 7 data sets of Cholera cases all display 6 modes of higher to lower frequency modes. Mode 6, the red line in the top panels, is the trends.
In table 1, the IMFs as a function of station location are provided. The IMFs are all temporally consistent with each other, showing that the cases of Cholera are consistent across Haiti.
Station=> Mode # Is Down 
St Michel 
Grosmorne 
Marmelade 
Ennery 
Govaives 
Terr Neuve 
AnseRouge 
Mode 6: The “Trend”. This is also the Base to which all other modes Either add to Or subtract From, for a specific EpiWeek. 
Starts at~5 on S42, then rises to a peak at ~20 on S27 and then drops to 0, on S7, S8 
Starts at~31 on S42, then goes down to 0, on S7, S8 
Starts at~12 on S42, then drops to~3 on S15 
Starts at~2 on S42 then rises to a peak of ~ 20 on S27, then falls to 0 on S13 
Starts at~19 on S42, then drops to 0 on S15 
Starts at~3 On S42, Then drops to 0 on S15 
Starts at 6 on S42, then drops to 1 on S15 
Mode 5: Period of~71 to 74 Weeks; an Oscillation 
Has a recordlength period with a+7 or4 cases amplitude; starts positive then goes negative 
Has a recordlength period with a+3 or 4 cases amplitude; starts negative then goes positive 
Has a recordlength period with a+1 or 1 cases amplitude; starts negative then goes positive 
Has a record length period with a+3 or 3 cases amplitude; starts positive then goes negative 
Has a record Length period with a+2 or 2 cases amplitude; starts negative then goes positive 
Has a record length period with a+1 or 1 cases amplitude; starts negative then goes positive 
Has a record length period with a +1 or 1 cases amplitude; starts negative then goes positive 
Mode 4: Period Of ~39 to 50 weeks 
Amplitude of 10 to +20 cases 
Amplitude of 8 to +9 cases 
Amplitude of 7 to +7 cases 
Amplitude of 7 to +14 cases 
Amplitude of 6 to +6 cases 
Amplitude of 1 to +1 cases 
Amplitude of 4 to +4 cases 
Mode 3: Period of 1520 weeks 
Amplitude of 15 to +15 cases 
Amplitude of 5 to +7 cases 
Amplitude of 10 to +10 cases 
Amplitude of 20 to +25 cases 
Amplitude of 5 to +9 cases 
Amplitude of 2 to +2 cases 
Amplitude of 4 to +4 cases 
Mode 2: Period of ~78 weeks 
Amplitude of 15 to +15 cases; mostly from S19S29 
Amplitude of 10 to +10 cases mostly S42S50 
Amplitude of 3 to +7 cases 
Amplitude of 15 to +15 cases 
Amplitude of 10 to +10 cases 
Amplitude of 3 to +4 cases 
Amplitude of 5 to +5 cases 
Mode 1: Period of ~3 weeks 
Amplitude of 15 to +15 cases; all be for S39 
Amplitude of 10 to +17 cases; all before S50 
Amplitude of 7 to +7 cases 
Amplitude of 10 to +15 cases 
Amplitude of 10 to +30 cases; all before S50 
Amplitude of 3 to +6 cases 
Amplitude of 10 to +10 cases 
Time Series dominated by: 
Modes 1,2,3,4,5 all important 
Mode 1 before S50, then Modes 2,3,4 dominate 
Modes 1,2,3,4,5 all important 
Modes 1,2,3,4 dominate 
Mode 1 before S50, then Modes 2,3,4 dominate 
Modes 1,2,3,4 dominate 
Modes 1,2,3,4,5 all important 
Summation Plot 
Modes 1, 2 only important from S42 S52 at disease onset 
Mode 3 important up to S35 
Mode 4 Important up to S39 
Mode 5 important throughout 
Trend= Mode 6 30 to 0 by S39 


Table 1: Summary Matrix of EEMD IMF Plots.
From a onetoone comparison of the IMFs presented in figure 5, we find visually that for IMF C5 (InterAnnual, about 1.5 years, variability), C4 (~ Annual or between 4050 week variability), C3 (~ seasonal or 34 month variability), C2 (~ 2 months or 78 weeks) and C1 (~ 3 weeks): a) C5 shows that Cazale, Ennery and St. Michel are InPhase; b) C5 shows that Anse Rouge, Govaives, GrosMorne, Terr Neuve and Marmelade are InPhase; c) C5 that CazaleEnnerySt. Michel are OutofPhase with Anse Rouge, Govaides, Grosmorn, Terrneuve and Marmelade; d) C4 that Cazale, Ennery and St. Michel are InPhase; e) C4 that Govaides and GrosMorne are InPhase; f) C4 that AnseRouge, Marmalade and Terr Neuve are InPhase; g) C3 that Cazale, Ennery, Govades, GrosMorne and St. Michel are In Phase; h) C3 that AnseRouge and Marmalade are InPhase; i) That TerrNeuve had very few cases in the C3 frequency mode; j) C2 that Anse Rouge, Ennery, Grosmorne, Govades and St. Michel are InPhase; k) C2 that Cazale, Marmalade and Terr Neuve go InPhase and OutofPhase with each other and the other stations as well; and l) C1 is very high frequency with mixed results.
Unfortunately, this visual correlation is not highly conclusive. Therefore, attempting to go beyond “visual correlations”, we take a different approach, mixing our empirically derived results with a statistical analysis. This approach has been reported in the literature.
Statistical analyses of the empirically derived IMFs resulting in highly correlated modes, well correlated modes and uncorrelated modes
We believe that the above EEMD IMF results can be compared directly using the Statistical Correlation Matrix approach. We now present statistical EEMD IMF correlations between IMF Modes on a Mode by Mode basis. To our knowledge this type of analysis has never previously been done or certainly has not been reported upon in the literature. Tables 27 display those crosscorrelations. We will employ Pearson Correlation Coefficients [8], (Sample N = 78), where Probability > r under H0: Rho=0.
If r < 0.22 this is considered to be not statistically significant,
If r >/= 0.23 this is considered to be statistically significant,
If r > 0.363 this is considered to be very statistically significant and
If r >/= 0.364 this is considered to be extremely statistically significant.
Mode 1 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.1903 
0.0438 
0.0711 
0.0924 
0.0092 
0.3251 
Ennery 
0.1903 
1.0000 
0.1430 
0.2283 
0.3725 
0.2761 
0.1821 
Govaives 
0.0438 
0.1430 
1.0000 
0.9483 
0.5451 
0.1990 
0.4604 
Gros_morne 
0.0711 
0.2283 
0.9483 
1.0000 
0.6060 
0.0938 
0.5087 
Marmelade 
0.0924 
0.3725 
0.5451 
0.6060 
1.0000 
0.0195 
0.1929 
Stmichel 
0.0092 
0.2761 
0.1990 
0.0938 
0.0195 
1.0000 
0.0420 
Terrneuv 
0.3251 
0.1821 
0.4604 
0.5087 
0.1929 
0.0420 
1.0000 
Table 2: Mode1 is the mode that represents the highest frequency of variability.
Mode1 is the mode that represents the highest frequency of variability, ~ 23 weeks, of the time series at each station. For Mode 1, Govaives has the highest positive correlation with, followed with a second strong negative correlation (0.6060) between stations Gros Morne, Marmalade and Terreuve. The above correlations of the first EEMD mode indicates that Govaives, Gros Morne, Marmalade and Terrnruve are most strongly connected, while station St Michel is only statistically correlated with station Ennery and is independent of other stations. Anse rouge has a very statistically significant correlation with Terrneuve, but has no significant correlation with other stations.
Although Ennery, Marmelade and St Michel are connected, their correlations are not extremely significant.
Mode 2 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.7238 
0.3480 
0.3573 
0.1710 
0.3522 
0.7048 
Ennery 
0.7238 
1.0000 
0.1155 
0.1290 
0.4661 
0.3609 
0.5394 
Govaives 
0.3480 
0.1155 
1.0000 
0.9219 
0.2109 
0.2964 
0.4038 
Gros_morne 
0.3573 
0.1290 
0.9219 
1.0000 
0.2928 
0.3617 
0.2698 
Marmelade 
0.1710 
0.4661 
0.2109 
0.2928 
1.0000 
0.0433 
0.2402 
Stmichel 
0.3522 
0.3609 
0.2964 
0.3617 
0.0433 
1.0000 
0.0548 
Terrneuv 
0.7048 
0.5394 
0.4038 
0.2698 
0.2402 
0.0548 
1.0000 
Table 3: Mode 2 depicts about 7  8 weeks in variability.
Mode 2 depicts about 7  8 weeks in variability. Govaives has a very strong correlation with Gros Morne and to a less or degree, Terrnruve, however it has no significant correlation with Marmalade. Govaives is significantly correlated to Anserouge and St Michel. Anse rouge and Ennery are also significantly correlated. Terrneuve is significantly correlated with all other stations except for St Michel, which itself is well correlated with all stations but Marmalade and Terreneuve.
Mode 3 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.1573 
0.0071 
0.2266 
0.6207 
0.1480 
0.3454 
Ennery 
0.1573 
1.0000 
0.7859 
0.4387 
0.3791 
0.5267 
0.3329 
Govaives 
0.0071 
0.7859 
1.0000 
0.6888 
0.1683 
0.4503 
0.4117 
Gros_morne 
0.2266 
0.4387 
0.6888 
1.0000 
0.2876 
0.4666 
0.5137 
Marmelade 
0.6207 
0.3791 
0.1683 
0.2876 
1.0000 
0.1781 
0.3223 
Stmichel 
0.1480 
0.5267 
0.4503 
0.4666 
0.1781 
1.0000 
0.1110 
Terrneuv 
0.3454 
0.3329 
0.4117 
0.5137 
0.3223 
0.1110 
1.0000 
Table 4: Mode 3 shows 1520 weeks in variability.
Mode 3 shows 1520 weeks in variability. The strongest correlation occurs between Govaives and Ennery and Gros Morne. Ennery and Govaives are significantly correlated to all other stations, except to Anse rouge, which is correlated with Marmalade and Terrneuve.
Mode 4 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.2983 
0.4917 
0.5053 
0.8123 
0.2750 
0.7139 
Ennery 
0.2983 
1.0000 
0.4666 
0.4066 
0.4701 
0.9702 
0.3231 
Govaives 
0.4917 
0.4666 
1.0000 
0.9760 
0.0797 
0.5591 
0.5988 
Gros_morne 
0.5053 
0.4066 
0.9760 
1.0000 
0.0578 
0.4713 
0.6092 
Marmelade 
0.8123 
0.4701 
0.0797 
0.0578 
1.0000 
0.4788 
0.4295 
Stmichel 
0.2750 
0.9702 
0.5591 
0.4713 
0.4788 
1.0000 
0.2739 
Terrneuv 
0.7139 
0.3231 
0.5988 
0.6092 
0.4295 
0.2739 
1.0000 
Table 5: Mode 4 shows 4050 weeks of variability.
Mode 4 shows 4050 weeks of variability. Ennery, GrosMorne and Terrneuve are all significantly correlated to other stations. The highest correlation is between GrosMorne and Govaives (0.9760) with the second highest between Ennery and St Michel. Notably, the overall strong positive correlation between stations implies that Mode 4 of the time series in those stations is inphase and has similar variation patterns.
Mode 5 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.5104 
0.8939 
0.9375 
0.6336 
0.1807 
0.9444 
Ennery 
0.5104 
1.0000 
0.7878 
0.7238 
0.9681 
0.9150 
0.7318 
Govaives 
0.8939 
0.7878 
1.0000 
0.9908 
0.8866 
0.4973 
0.9541 
Gros_morne 
0.9375 
0.7238 
0.9908 
1.0000 
0.8253 
0.4034 
0.9762 
Marmelade 
0.6336 
0.9681 
0.8866 
0.8253 
1.0000 
0.8314 
0.8017 
Stmichel 
0.1807 
0.9150 
0.4973 
0.4034 
0.8314 
1.0000 
0.4239 
Terrneuv 
0.9444 
0.7318 
0.9541 
0.9762 
0.8017 
0.4239 
1.0000 
Table 6: Mode 5 displays 1.5 year variability and all stations.
Mode 5 displays 1.5 year variability and all stations, except for Anse rouge and Terrneuve, are strongly correlated. The highest correlation occurs between Gros Morne and Govaives, second highest correlation between Marmalade and Ennery. We note that, excepting for St. Michel and Anse rouge all stations are either strongly positively or negatively correlated over the year and a half time scale.
Mode 6 
Anse_rouge 
Ennery 
Govaives 
Gros_morn 
Marmelade 
Stmichel 
Terrneuve 
Anse_rouge 
1.0000 
0.4259 
0.8390 
0.7979 
0.8877 
0.6721 
0.8925 
Ennery 
0.4259 
1.0000 
0.1349 
0.2055 
0.0385 
0.9562 
0.0279 
Govaives 
0.8390 
0.1349 
1.0000 
0.9974 
0.9953 
0.1610 
0.9942 
Gros_morne 
0.7979 
0.2055 
0.9974 
1.0000 
0.9858 
0.0899 
0.9840 
Marmelade 
0.8877 
0.0385 
0.9953 
0.9858 
1.0000 
0.2556 
0.9999 
Stmichel 
0.6721 
0.9562 
0.1610 
0.0899 
0.2556 
1.0000 
0.2658 
Terrneuv 
0.8925 
0.0279 
0.9942 
0.9840 
0.9999 
0.2658 
1.0000 
Table 7: Mode 6 represents the longterm trends of the various time series.
Mode 6 represents the longterm trends of the various time series. All trends, if well correlated, are positive, meaning that time series varied either in an ascending or decreasing sense as time progressed. The maximum correlation coefficient is between Govaives and GrosMorne, with the second largest between Govaives and Marmalade. AnseRouge displays an extremely significant correlation with all other stations, while Ennery is only significantly connected to AnseRouge and St Michel, and is independent of the other stations.
Statistical analyses of the original 7 data sets and the development of “patterns”
To begin the Statistical analyses of the original 7 stations, we plot the Data on the original scale with the conventions that circles are data, and lines are smoothing splines running through the data. The results are shown in figure 6.
Figure 6: Observations (from figure 2) indicate that some coherency is apparent in a few of the series.
We then regressed the data on 7 sinecosine pairs and via employing a linear trend, we seem to capture the patterns nicely, as shown in figure 7.
Figure 7: Disease data.
From figure 7, we see that when there are high peaks in the time series, we see much more variability than when the levels are low, a typical symptom of the statistical need to log transform the data.
Here in figures 8a & b, are the same graphs as in figure 7, shown on the log scale: Y→log(Y+1). Note the more uniform variations throughout the time sequences, as well as the tighter coherency toward the end of Year 2 and into Year 3.
Figures 8: a) Data on a Log scale; b) Without the actual data points.
So, from figures 68, the diagnostic statistical analyses and the sinecosine pairs with the linear trend produced remarkably good agreement. This suggests that a reasonable prognostic capability has been established, and that the future can be predicted from the past. Thus the time series could be run out for the next year in a predictive mode. However, before attempting to do so, we should test whether or not the strength of the trends are significant.
We can test, on a case by case, whether the trends are significant. A “pvalue” (Pr>F) that is less than 0.05 indicates significance:
Test trend results
Dependent Variable 
Mean 

Source 
DF 
Square 
F Value 
Pr>F 

St Michel 
Numerator 
1 
25.08248 
363.71 
<.0001 
Grosmorne 
Numerator 
1 64 
21.82081 
123.62 
<.0001 
0.17651 

Terr Neuve 
Numerator 
1 
0.05312 
0.18 
0.6755 
0.30041 

Ennery 
Numerator 
1 
14.76168 
35.46 
<.0001 
0.41628 

Anse Rouge 
Numerator 
1 
3.94366 
5.65 
0.0205 
0.6986 

Govaives 
Numerator 
1 
15.70193 
52.37 
<.0001 
0.29983 

Marmelade 
Numerator 
1 
2.83516 
6.27 
0.0148 
0.45227 
We can test case by case for seasonality as measured by the test for the seven sinecosine pairs (fundamental period 52 plus 5 harmonics) used in the model. A pvalue (Pr>F) that is less than 0.05 indicates significance:
Test spectral results
Dependent Variable 
Mean 

Source 
DF 
Square 
F Value 
Pr>F 

StMichel 
Numerator 
12 
2.65954 
38.56 
<.0001 
Gros morne 
Numerator 
12 
1.65998 
9.4 
<.0001 
0.17651 

Terr Neuve 
Numerator 
12 
0.98276 
3.27 
0.001 
0.30041 

Ennery 
Numerator 
12 
6.49655 
15.61 
<.0001 
0.41628 

Anse Rouge 
Numerator 
12 
2.20902 
3.16 
0.0014 
0.6986 

Govaives 
Numerator 
12 
1.25012 
4.17 
<.0001 
0.29983 

Marmelade 
Numerator 
12 
4.71071 
10.42 
<.0001 
0.45227 
All series have a trend, which is not to say they have the same slope and all have seasonal pattern, which is not to say they are coherent, but that they just share the same frequencies. Just a plain correlation between pairs of series is an indication of coherency that does not involve models for the data. We have noted those greater than 0.50.
Pearson Correlation Coefficients, N = 78, Prob > r under H0: Rho=0
StMichel 
Gros morne 
Terr Neuve 
Ennery 
Anse Rouge 
Govaives 
Marmelade 

StMichel 
1.00000 
0.54773 
0.03195 
0.69814 
0.16504 
0.60621 
0.11285 
<.0001 
0.7812 
<.0001 
0.1487 
<.0001 
0.3252 

Gros morne 
0.54773 
1.00000 
0.20647 
0.42553 
0.40479 
0.82625 
0.21543 
<.0001 
0.0697 
0.0001 
0.0002 
<.0001 
0.0582 

Terr Neuve 
0.03195 
0.20647 
1.00000 
0.22192 
0.59794 
0.18275 
0.24927 
0.7812 
0.0697 
0.0508 
<.0001 
0.1093 
0.0278 

Ennery 
0.69814 
0.42553 
0.22192 
1.00000 
0.49289 
0.61795 
0.34792 
<.0001 
0.0001 
0.0508 
<.0001 
<.0001 
0.0018 

Anse Rouge 
0.16504 
0.40479 
0.59794 
0.49289 
1.00000 
0.47641 
0.52272 
0.1487 
0.0002 
<.0001 
<.0001 
<.0001 
<.0001 

Govaives 
0.60621 
0.82625 
0.18275 
0.61795 
0.47641 
1.00000 
0.32042 
<.0001 
<.0001 
0.1093 
<.0001 
<.0001 
0.0042 

Marmelade 
0.11285 
0.21543 
0.24927 
0.34792 
0.52272 
0.32042 
1.00000 
0.3252 
0.0582 
0.0278 
0.0018 
<.0001 
0.0042 
We can then visualize the above correlations in a “scatter plot matrix” (Figure 9) in which a grid of bivariate plots is presented. For a graph in one of the cells, the label for the vertical axis is the one in that same row of cells and the label for the horizontal axis is the one in that column of cells. The labels run down the diagonal of the matrix. The Scatter plot overlays on the data to which the function presented above was fitted.
Figure 9: ScatterPlot Matrix of the Cholera data.
Figure 9 is a straightforward scatter plot while the scatter plot shown in figure 10 has 50% confidence ellipses drawn in using a multivariate normal assumption which would be incorrect when there are a lot of 0’s as in Terre Neuve (Right column and bottom row, which is why we moved it to last). Still, it does enhance the visuals somewhat for example Govaives and Ennery show a narrow ellipse tilted at about 45 degrees indicating a high degree of correlation; as do other stations with other stations.
In figure 10, we present the Scatter Plot Matrix of the Cholera data but with 50% Confidence Ellipses drawn as well. Basically, the higher the positive correlation between stations, the closer the major axis of the Ellipses lie to a 45 degree line, or an ellipse particularly one that is elongated, from lower left to upper right, in each of the Scatter Plot Squares; such as St. Michel with Ennery and GrosMorne.
Figure 10: Same as figure 9 with 50% Confidence Ellipses.
Next, suppose we compute, on the log scale, which seems appropriate from the graphs previously submitted, the mean of the 7 sites for each week then regress on the 7 sinecosine pairs (fundamental and 5 harmonics) and time t to get the linear downward trend. This is presented in figure 11. Here are the predicted values and 95% prediction limits for individual weekly means. We see a peak at t=6 repeating 52 weeks later at week t=46. So, we have seen it at week 46 of 2 years. Between these markers is a larger peak at t=27. There is no week 27 value in either of the other years. Note that t=1 is the first week of the middle year with negative t’s for the previous year.
Figure 11: A Prediction of the cumulative Cholera data set.
Next we convert back to the actual number scale. In figure 12, that plot is shown.
Figure 12: The conversion form the log scale (Figure 11) to the actual number scale.
We note that in figure 12, a statistically significant downward trend is found. The trend shows about a 3% drop in rate per week (0.02926). From the statistical perspective, claims of significant seasonality based on less than 2 years of data is on very shaky ground. If forced to do so, it would appear that there is a bimodal pattern peaking around weeks 27 and 46 of the year but notice that the only case where we saw more than one instance of the peak was the leftmost and rightmost lines (at week 46). The week 27 peak is based on only one of the observed years. Note that on the original scale our effects are multiplicative so when the level is lower, like in that second instance, the multiplicative increase will produce a less impressive peak than that at the higher level. A 5% increase when the rate is 100 cases per million people is just 5 people whereas it is 50 people when the rate is 1000 cases per million people. That is why the second instance is a lower peak than the first. It is a percentage increase from a lower point on that linear trend.
By taking a mean over the 7 cases, we are ignoring any phase alignment issues but we think there was reasonable coherency in the previous graphs. Also, there was no accounting for autocorrelation when computing the prediction bands or t tests below. Autocorrelation could render bands a little wider and tests less significant (more uncertainty). On the positive side we are explaining about 90% of the variation on the log scale with this model as the R^{2} below shows.
Analysis of Variance
Analysis of Variance 

Source 
DF 
Sum of Squares 
Mean Square 
F Value 
Pr > F 
Model 
13 
66.78874 
5.1376 
39.62 
<.0001 
Error 
64 
8.29866 
0.12967 

Corrected Total 
77 
75.08739 
Root MSE 
0.36009 
RSquare 
0.8895 
Dependent Mean 
1.46301 
Adj RSq 
0.867 
Coeff Var 
24.61306 

Parameter Estimates 

Variable 
DF 
Parameter Estimate 
Standard Error 
t Value 
Pr > t 
Type I SS 
Intercept 
1 
2.39648 
0.06925 
34.6 
<.0001 
166.95182 
t 
1 
0.02926 
0.0019 
15.38 
<.0001 
40.94076 
s1 
1 
0.58774 
0.0608 
9.67 
<.0001 
14.17099 
c1 
1 
0.36806 
0.06112 
6.02 
<.0001 
3.41315 
s2 
1 
0.03822 
0.06235 
0.61 
0.5421 
0.01769 
c2 
1 
0.3796 
0.06148 
6.17 
<.0001 
4.42778 
s3 
1 
0.15804 
0.06097 
2.59 
0.0118 
1.1067 
c3 
1 
0.12265 
0.06087 
2.01 
0.0481 
0.51172 
s4 
1 
0.07161 
0.06063 
1.18 
0.242 
0.17631 
c4 
1 
0.10346 
0.06141 
1.68 
0.0969 
0.16741 
s5 
1 
0.03663 
0.06102 
0.6 
0.5505 
0.02031 
c5 
1 
0.19436 
0.06051 
3.21 
0.0021 
0.98003 
s6 
1 
0.10149 
0.05897 
1.72 
0.09 
0.37707 
c6 
1 
0.11512 
0.05991 
1.92 
0.0591 
0.4788 
There does seem to be autocorrelation around 0.30 in the residuals which we can take into account. There is no change in the overall conclusions. The graphs are jagged as the predictions are now one step ahead predictions using the previous residual and autocorrelation to adjust the forecast:
Statistically speaking, the patterns are intriguing, but we have so few years to go on that we are hesitant to think our analyses pinned down the pattern. We do get a nice fit, explaining 90% of the variation in the historic weekly mean incidence with our model. The question is whether that will hold up in future years. Unfortunately, we have not seen enough years to feel confident about repetitive behavior. The rate does seem to be trending downward in reports from Haiti.
Cazale data set and the EEMD IMF decomposition
The Cazale data set was provided well into the study of the original data sets. We cannot include them into what has already been done above. However, the great value of the Cazale data is that it is daily confirmed cases. This is in contrast to the previous data sets presented above which were compiled over a week’s period of time for each locale. Cazale’s location is shown in figure 13. Figure 13: Cazale’s location in Haiti.
The EEMD IMF decomposition of the Cazale daily time series (Figure 14a) indicates that the increase in the frequency of sampling has added three higher frequency IMFs to the suite of modes. This offers us the opportunity of doing weekly averages over the data set, which can then be compared and contrasted with the Original Time Series that we presented above. We present the weekly averaged time series decomposition in the figure 14b. Sure enough, the three very high frequency modes disappear.
Figure 14: a) IMF Modes 19 of the daily Cazale time series of Cholera cases; b) IMF modes 16 of the weekly averaged Cazale time series.
Modes 1,2,3 in figure 14a above indicate 12, 23 and 35 day variability in the data. Modes 4 through 8 in figure 14a and Modes 1 through 5 in figure 14b can be compared directly and are the same modes of internal variability in the data.
Stacked EEMD intrinsic mode functions of All data sets including original 7 plus Cazale (CZ) in search of phase relationships
Below, in table 8, we present all of the cholera data IMF modes compared against each other, figures 5 ah, a virtual stacking of the EEMD IMF modes. We assess the phase relationships in each in an effort to determine the temporal and spatial propagation of the outbreak of cholera. We note that this is an experimental product and has never been previously tried to our knowledge. We see which Station leads another Station by Mode and by sequence; so 1 is 1^{st}, 2 is 2^{nd} , and so on, 3, 4, 5, 6, 7, finally the last station to be hit, 8. We note that some sets of stations are hit at nearly the same time so they receive multiple #s.
Modes (Down) 
Stations => 
CZ 
AR 
EM 
GV 
GM 
MA 
SM 
TN 
1 

8 
7 
1  4 
5  6 
5  6 
1  4 
1  4 
1  4 
2 

4  5 
4  5 
6  7 
2  3 
1 
8 
2  3 
6  7 
3 

1  2 
8 
5  6 
3  4 
3  4 
7 
1  2 
5  6 
4 

8 
1  3 
7 
1  3 
1  3 
6 
4  5 
4  5 
5 

1 
4 
2 
5  6 
5  6 
8 
3 
7 
Total Time Series 

8 
4, 5, 6 
4, 5, 6 
3 
2 
4, 5, 6 
7 
1 
Table 8: Total time series.
The conclusion reached from table 8 is that no definitively clear pattern of the temporalspatial spread of the cholera epidemic is evident over the various time scales of variability (weekly (Mode 1) to biweekly (Mode 2) to monthly (Mode 3) to seasonally (Mode 4) to annually) (Mode 5) from station to station.
The Total Time Series (TTS) suggests that the spatial movement was from TerrNeuve to Grosmorne to Govaives to Ennery/Marmelade/AnseRouge to St Michele and then to Cazale. However, this analysis was not nearly as revealing as the Matrix Analyses that we presented previously. Clearly the TTS supports the movement of the UNESCO Nepalese Task Force that introduced and carried and spread Cholera across Haiti.
Statistical analyses of the cazale daily data set
We chose to analyze weekly composites rather than the daily data, so as to conform to the earlier data sets. We note, however, that we could have done a statistical analysis on the daily data if necessary. First, to get 52 weeks per year, one needs to deal with the extra days (52*7=364, not 365 or 366) so we started week 1 on Jan. 1 each year then at week 52 we incorporated the leftover day or two of the year and converted that sum to a 7 day basis by taking the mean of those 8 (for example) numbers and multiplying by 7. A log transformation seemed appropriate and figure 15 is a plot of the results.
Figure 15: Log Transform of the Cazale Data Set.
When a model with a fundamental frequency of 1 cycle per year and 5 harmonics plus a trend is fit (as per the EEMD decomposition), the picture in figure 16 below emerges where the thick red line is the trend plus a sinusoids part and the black line includes the adjustment for autocorrelation (1 lagged residual is helpful in predicting the next residual). Interestingly only the fundamental cycle is statistically significant.
Figure 16: The Log Scale decomposition of the Cazale Data Set, converted back (Figure 15), and with the 5 EEMD IMFs included in blue.
If we then refit the Cazale Data with only the fundamental frequency of 1 cycle per year we get the simpler picture shown in figure 17. The equation of the red curve is the weekly case count = exp (4.58640.0158t0.1174S0.7522C) where S and C are the sine and cosine of 2πt/52, t is time counted in weeks through the data set and exp is exponentiation. The two years end at weeks t=10 and t=62 as marked by the vertical lines. As in our last statistical report above, when the trend line is lower (the right side) the effect of a multiplicative sinusoid will be less than when it is higher (the left side). This is reflected in the amplitude of the waves when transformed back to the original scale. The linear decrease 0.0158 per week or approximately a 1.58% decrease week to week is statistically significant even after accounting for autocorrelation. You can take the derivative of 0.1174S0.7522C with respect to t and find out the points t where the local maxima are.
Figure 17: Cholera.
The implications of the above analysis for a projected “Outlook” are provided below.
Projected outlook for future cholera outbreaks at the original 7 stations
Using the log scale (log(Y+1)) model with 1 fundamental and 5 harmonics plus a linear trend, we generated predictions and prediction intervals on the log scale. These can be converted back to the original scale by exponentiation. In the graphs below, in figures 18af) predictions were extended 52 weeks past the end of the data (red line) and it is fairly obvious where the peaks are but we are also providing the 52 predictions for each site of the exact week as might be required. A 95% confidence band is shown around each prediction. This is a band for individual observations and thus takes into account both the inaccuracy in estimating the curve and the variation of data around that curve. Black lines delineating the years are included in each graph.
Figure 18: Statistically predicted cases of Cholera by Station Location
Predicted Cases:
Obs 
t 
year 
week 
lead 
pSt Michel 
PGros morne 
PTerr Neuve 
pEnnery 
PAnse Rouge 
P Govaives 
p Marmelade 
79 
68 
3 
16 
1 
2.3853 
0.04114 
0.05317 
1.2811 
0.1799 
0.24273 
0.5534 
80 
69 
3 
17 
2 
2.9606 
0.12607 
0.08452 
1.8709 
0.10291 
0.30884 
0.61572 
81 
70 
3 
18 
3 
3.5791 
0.28437 
0.29493 
2.8259 
0.0259 
0.42123 
0.68138 
82 
71 
3 
19 
4 
4.2203 
0.40575 
0.47395 
4.1327 
0.04901 
0.54712 
0.73885 
83 
72 
3 
20 
5 
4.959 
0.46575 
0.48175 
5.5643 
0.11122 
0.66683 
0.75013 
84 
73 
3 
21 
6 
5.9611 
0.44219 
0.28915 
6.7784 
0.13675 
0.79895 
0.66805 
85 
74 
3 
22 
7 
7.4246 
0.29399 
0.02467 
7.6675 
0.08944 
1.00565 
0.47435 
86 
75 
3 
23 
8 
9.4625 
0.07032 
0.16907 
8.5593 
0.08242 
1.37782 
0.20533 
87 
76 
3 
24 
9 
11.8905 
0.80273 
0.22653 
10.0603 
0.45346 
2.00735 
0.06913 
88 
77 
3 
25 
10 
14.0119 
1.98905 
0.11534 
12.8015 
1.09313 
2.91285 
0.29197 
89 
78 
3 
26 
11 
14.7951 
3.29512 
0.21895 
17.0577 
1.91657 
3.88501 
0.44115 
90 
79 
3 
27 
12 
13.6615 
3.8965 
0.81961 
21.8322 
2.47764 
4.41883 
0.51853 
91 
80 
3 
28 
13 
11.0985 
3.35628 
1.5336 
24.1879 
2.23405 
4.0852 
0.52941 
92 
81 
3 
29 
14 
8.2077 
2.21383 
1.89703 
21.5247 
1.31202 
3.06112 
0.46703 
93 
82 
3 
30 
15 
5.8072 
1.20025 
1.59811 
15.2878 
0.38579 
1.94669 
0.2991 
94 
83 
3 
31 
16 
4.1272 
0.57444 
0.93931 
9.3303 
0.1744 
1.15122 
0.05262 
95 
84 
3 
32 
17 
3.0435 
0.27486 
0.3878 
5.6011 
0.4046 
0.74071 
0.74262 
96 
85 
3 
33 
18 
2.3422 
0.18354 
0.11972 
3.884 
0.39685 
0.63974 
2.01221 
97 
86 
3 
34 
19 
1.8398 
0.19889 
0.13846 
3.5175 
0.10931 
0.75984 
4.06315 
98 
87 
3 
35 
20 
1.4176 
0.22214 
0.47488 
4.0947 
0.73836 
0.97913 
6.637 
Obs 
t 
year 
week 
lead 
pSt Michel 
PGros morne 
PTerr Neuve 
pEnnery 
PAnse Rouge 
P Govaives 
p Marmelade 
99 
88 
3 
36 
21 
1.0286 
0.17051 
1.21692 
5.198 
2.65714 
1.09312 
8.5957 
100 
89 
3 
37 
22 
0.6812 
0.03139 
2.30739 
5.8552 
5.54355 
0.91776 
8.57181 
101 
90 
3 
38 
23 
0.4028 
0.13189 
3.21756 
4.9593 
7.29712 
0.49919 
6.56189 
102 
91 
3 
39 
24 
0.2103 
0.24553 
3.25057 
2.8609 
6.01788 
0.06511 
3.98989 
103 
92 
3 
40 
25 
0.102 
0.26666 
2.46524 
0.9981 
3.29385 
0.22641 
2.04714 
104 
93 
3 
41 
26 
0.0634 
0.16878 
1.54764 
0.0202 
1.29133 
0.35055 
0.95704 
105 
94 
3 
42 
27 
0.072 
0.07764 
0.94423 
0.4397 
0.32622 
0.32086 
0.48719 
106 
95 
3 
43 
28 
0.0998 
0.48115 
0.7152 
0.5595 
0.01692 
0.12142 
0.4369 
107 
96 
3 
44 
29 
0.1185 
0.94716 
0.80008 
0.4954 
0.01159 
0.28871 
0.77102 
108 
97 
3 
45 
30 
0.1094 
1.24023 
1.12252 
0.2121 
0.26818 
0.87756 
1.60687 
109 
98 
3 
46 
31 
0.0718 
1.16529 
1.51933 
0.427 
0.80244 
1.39962 
3.09397 
110 
99 
3 
47 
32 
0.0188 
0.79263 
1.70405 
1.4746 
1.42672 
1.51323 
5.07224 
111 
100 
3 
48 
33 
0.0341 
0.35859 
1.48522 
2.5332 
1.77763 
1.17968 
6.75592 
112 
101 
3 
49 
34 
0.079 
0.02367 
0.99515 
2.8961 
1.63583 
0.69136 
7.22045 
113 
102 
3 
50 
35 
0.1173 
0.18167 
0.51457 
2.4177 
1.18612 
0.30574 
6.38979 
114 
103 
3 
51 
36 
0.156 
0.28761 
0.19794 
1.6133 
0.73908 
0.09557 
4.99985 
115 
104 
3 
52 
37 
0.1998 
0.33399 
0.05701 
0.9306 
0.44316 
0.03658 
3.73372 
116 
105 
4 
1 
38 
0.2468 
0.35574 
0.05255 
0.4776 
0.29781 
0.08169 
2.82929 
117 
106 
4 
2 
39 
0.2876 
0.37805 
0.13327 
0.1924 
0.24376 
0.17 
2.23115 
118 
107 
4 
3 
40 
0.3107 
0.41064 
0.23219 
0.0097 
0.20555 
0.22589 
1.79419 
119 
108 
4 
4 
41 
0.3076 
0.44746 
0.27993 
0.1834 
0.12062 
0.19356 
1.3907 
120 
109 
4 
5 
42 
0.2768 
0.47582 
0.25061 
0.3425 
0.02406 
0.07908 
0.9652 
121 
110 
4 
6 
43 
0.2263 
0.48669 
0.18075 
0.4741 
0.18888 
0.06366 
0.54364 
122 
111 
4 
7 
44 
0.1761 
0.4794 
0.12975 
0.5639 
0.3243 
0.18501 
0.18599 
123 
112 
4 
8 
45 
0.1526 
0.46221 
0.13585 
0.6081 
0.40493 
0.26635 
0.07013 
124 
113 
4 
9 
46 
0.1726 
0.4498 
0.2013 
0.6107 
0.42856 
0.31481 
0.22341 
125 
114 
4 
10 
47 
0.2301 
0.45607 
0.28957 
0.5791 
0.40449 
0.3498 
0.29146 
126 
115 
4 
11 
48 
0.299 
0.48523 
0.33492 
0.5257 
0.3498 
0.39052 
0.29484 
127 
116 
4 
12 
49 
0.351 
0.52978 
0.28737 
0.4677 
0.28991 
0.44443 
0.25336 
128 
117 
4 
13 
50 
0.3671 
0.57831 
0.16415 
0.4193 
0.25162 
0.50357 
0.18826 
129 
118 
4 
14 
51 
0.33823 
0.62405 
0.03663 
0.37719 
0.24858 
0.55275 
0.12086 
130 
119 
4 
15 
52 
.26237 
0.66724 
0.02757 
0.31804 
0.27511 
0.58019 
0.06513 
From the above we reach the following summary of the prognostic forecast for Station and Numbers of Cases by Week(s) and Number (od cases).
Here we conclude this prognosticforecast discussion by pointing out that: We need to be sure to put in some heavy caveats about predictingforecasting a repeating pattern having only seen it once (more or less). Also we note, for your reading pleasure, that by using six frequencies in this Fourier analysis for all sites, we are leaving in all frequencies that are significant in at least one of the sites but if we went site by site we almost certainly would not need them all. This way we have some consistency but risk that some of the wiggling we see is just noise rather than true signal. There was at least one site where the fifth harmonic was statistically significant, in other words. For some of these it is easily possible to run a horizontal line completely through the confidence bands so from a practical point of view the seasonality and trend are not dramatic at all. And finally, because this is a periodic model, the peaks are forced to occur in the same week every year, as the model has no ability to do anything else.
Statistical analyses of multidisease data sets.
In the Boxes shown in figure 19, we present the various disease and or medical condition data sets provided to us for possible analyses. The data were collected at the 7 Public Health facilities in Haiti. While we are not in a position to develop causal relations, thus attribution of a medical condition to the outbreak and spread of cholera.
Figure 19: Documented numbers of diseases and medical conditions at the 7 Public Health Stations in Haiti.
Next we present our Statistical Analyses of the above diseases and medical conditions. First, we eliminate all diseases with 0 occurrences. These are: Leprosy, Diphtheria, Dengue, Leishmaniasis, Chagas Disease, Schistosomiasis (Bilharzia), and Onchocerciasis.
Next we label the remaining diseases. They are:
String Label
Next we ran a correlation matrix on these showing only those with correlations greater than 0.75. Note there is no a priori reason to choose the 75% of probability but this is a very strong %. In any case, omitting correlations less than 0.75, we get:
Variable 
D1 
D2 
D3 
D4 
D5 
D6 
D8 
D9 
D10 
D11 
D13 
D15 
D16 
D17 
D18 
D20 
D21 
D1 
. 
99 
. 
98 
79 
. 
98 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D2 
99 
. 
. 
96 
76 
. 
97 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D3 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D4 
98 
96 
. 
. 
82 
. 
97 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D5 
79 
76 
. 
82 
. 
. 
77 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D6 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D7 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D8 
98 
97 
. 
97 
77 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D9 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D10 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
D11 
. 
. 
. 
. 
. 
. 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
. 
D12 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D13 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D14 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D15 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
75 
89 
85 
. 
D16 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D17 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
75 
. 
. 
. 
91 
. 
D18 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
89 
. 
. 
. 
81 
. 
D19 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D20 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
85 
. 
91 
81 
. 
. 
D21 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
Now here we note that there are 99% correlation relationships which are highly unlikely. So revisiting the data sets we note some apparent singularities at the onset of the data time series. So we will eliminate those and redo the Correlation Matrix. We next take away the first observations and a very different story emerges:
Variable 
D1 
D2 
D3 
D4 
D5 
D6 
D8 
D9 
D10 
D11 
D13 
D15 
D16 
D17 
D18 
D20 
D21 
D1 
. 
. 
. 
83 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D2 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
84 
D3 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D4 
83 
. 
. 
. 
. 
. 
78 
. 
. 
. 
. 
. 
. 
8 
. 
9 
. 
D5 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D6 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D8 
. 
. 
. 
78 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D9 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D10 
. 
. 
. 
. 
. 
. 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
. 
D11 
. 
. 
. 
. 
. 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
. 
. 
D12 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D13 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D14 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D15 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
89 
84 
. 
D16 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D17 
. 
. 
. 
8 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
91 
. 
D18 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
89 
. 
. 
. 
81 
. 
D20 
. 
. 
. 
9 
. 
. 
. 
. 
. 
. 
. 
84 
. 
91 
81 
. 
. 
D21 
. 
84 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
Next, eliminating variables that have no correlations exceeding 0.75 with others we get:
Variable 
(D1 
D4 
D8) 
(D10 
D11) 
(D17 
D20 
D15 
D18) 
(D2 
D21) 
D1 
. 
83 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D4 
83 
. 
78 
. 
. 
8 
9 
. 
. 
. 
. 
D8 
. 
78 
. 
. 
. 
. 
. 
. 
. 
. 
. 
D10 
. 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
D11 
. 
. 
. 
76 
. 
. 
. 
. 
. 
. 
. 
D17 
. 
8 
. 
. 
. 
. 
91 
. 
. 
. 
. 
D20 
. 
9 
. 
. 
. 
91 
. 
84 
81 
. 
. 
D15 
. 
. 
. 
. 
. 
. 
84 
. 
89 
. 
. 
D18 
. 
. 
. 
. 
. 
. 
81 
89 
. 
. 
. 
D2 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
84 
D21 
. 
. 
. 

.. 
. 
. 
. 
. 
84 
. 
These are remarkable groupings. They do not indicate “causality” but they do indicate very, very highly correlated relationships which are either medical in nature or socioeconomic or other.
Here we see the clustering of:
Basically strong Positive Correlations imply that: if you know one disease or medical condition is present and increasing (or decreasing) then the other disease or medical condition will also be present and be increasing (or decreasing). Alternatively, strong Negative Correlations imply that: if you know one disease or medical condition is present and increasing (or decreasing) then the other disease or medical condition will also be present and be decreasing (or increasing). Again, this does not prove “causality” but it does show powerful correlative connectivity. To wit, if you know one, then you know the other.
Basically the empirical and statistical analyses show highly complementary and revealing results. Moreover, both suggest that the downward trends indicate that these two principal peaks were related but singular events; likely related to the circumstances following the Earthquake and a combination of environmental and socioeconomic factors. Without further evidence we see no compelling evidence for a cholera breakout of any significant magnitude this coming year. However, in a section above (and summarized below), we have extended our diagnostic analysis into the next period and presented our prognostic analysis; with appropriate caveats.
More advanced analyses would incorporate a consideration of environmental factors such as precipitation, water and air temperatures, locale, etc., and oceanic, hydrologic and atmospheric long term factors such as atmospheric temperatures, precipitation, water quality, various climate factors, along with public health variables such as hospital size, road access, dedicated medical teams, especially potable water, infrastructure, etc. The environmental factors could be coupled to socioeconomic factors and then decomposed both statistically and empirically, with the cross correlations occurring utilizing intrinsic mode functions. A more revealing retrospective, diagnostic analysis would result and it is possible and highly likely that a prognostic capability could be created; one more sophisticated than that which we presented above and summarized below.
If we had summed over all 7 stations and then smoothed the data, followed by an empirical decomposition we find that: There are 4 principal peaks:
The Cazale Data Set: The Cazale data was compiled daily. This is in contrast to the previous data sets presented above which were compiled over a week’s period of time for each locale. The empirical decomposition indicates that the increase in the frequency of sampling added three higher frequencies Intrinsic Mode Functions, to the suite of modes; so 9 versus 6 of the original 7 stations. This offered us the opportunity of doing weekly averages over the data set, which we then compared and contrasted with the original 7 station time series already discussed above. We also presented the weekly averaged time series decomposition and sure enough the three very high frequency modes disappeared. Basically, in the 9 mode decomposition Modes 1 and 2 indicated “noise” in the data. Mode 3 was 23 day variability which could not be revealed in the weekly sampling of the original 7 stations. Mode 5 was biweekly to monthly variability. Mode 6 shows monthly to seasonal variability. Mode 7 is nearly annual variability. Mode 8 is interannual variability. Mode 9 is the record length trend.
Citation: Pietrafesa LJ, Dickey DA, Henwood TM, Gayes PT, Bao S, et al. (2019) Statistical and Empirical Analyses of Haiti Cholera Post Mortems. J Environ Sci Curr Res 2: 012.
Copyright: © 2019 Pietrafesa LJ, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.