Figure 1.
Satellite image of West Africa.
Panel A: the Sahara desert and the savannah occupy the northern and southern West African landscapes, respectively, while the Sahel spans the intermediate fringe zone—Mali is transected by all three landscapes. Panel B corresponds approximately to an enlargement of the red demarcation in Panel A. The black line on the top of this panel delineates the southeastern Mauritanian border; the depicted segment of the Niger River flows in the southwest-northeast direction; the district of Niono, which is located 330 km northwest of Bamako and 100 km north of the Niger River along the Canal du Sahel (Segou Region), is situated within the red rectangle. This satellite image places the district of Niono in the Sahelian zone: poverty is extensive in the northern (semi-desert) and central (irrigated) regions; contrarily, poverty diminishes southward (near savannah areas) where mixed crops prevail. Image source: adapted with permission from Globalis, http://globalis.gvu.unu.edu (08/2007) [11].
Figure 2.
Irrigation system and stagnant water reservoirs in the district of Niono, Mali.
This composite panel depicts irrigation canals (which support mainly rice monoculture) and stagnant water reservoirs where Schistosoma haematobium transmission may occur. District communities not only ingest water from the irrigation system but also wash their belongings, bathe, excrete, and amuse themselves in the canals, considerably increasing exposure to S. haematobium. Furthermore, rainfall precipitation fluctuations prompt the local authority (Office du Niger) to adjust irrigation management accordingly; for example, the Office du Niger may relax water control amid increased precipitation to better irrigate drier areas whilst collaterally enhancing water-flow through typically well-served agricultural fields—S. haematobium transmission suitability might then simultaneously increase and decrease in the former and latter scenarios, respectively.
Figure 3.
(1) Prior time-series (TS) observations initialize (2) the program that selects the best-performing exponential smoothing (ES) method within the state-space forecasting (ETS) framework, according to Equations 2 & 3 (Methods) as well as the Akaike's Information Criterion (AIC). Then, (3) Equations 2 & 3 simulate h-month horizon forecast path distributions with the best-performing ES method via B = 1000 ordinary residual bootstraps. (4) Mean forecast and 95% prediction interval (PI) values obtain as described in the Methods section. Subsequently, (5) the 1-month horizon forecast plus (6) the available TS (including the most contemporaneous observation) is supplied to (2, 3) the execution program to (4) revise forecasts and their 95% PI values. The automatic supply of contemporaneous TS observations into (2–6) yields revised on-line forecasts, i.e. external predictions. Basically, contemporaneous forecasts obtain via TS extrapolation whereby previous deviations between forecasts and their corresponding observations are exponentially adjusted with smoothing control values. For example, (1) the Schistosoma haematobium TS observations from January 1996 to December 1998 for the district of Niono, Mali, initialize (2–4) the ETS execution program that predicts consultation rates for January 1999 to May 1999 (assuming a 5-month horizon forecast). Once (5) the January 1999 forecast plus (6) the available TS (including the most contemporaneous observation of January 1999) become available to the on-line system, (2–4) the execution program cycles again and optimizes all considered ES methods, selecting the best-performing one (which may or may not be the same one employed prior to the arrival of this new observation). As a result, revised consultation rate predictions for February 1999 to June 1999 become available. This process repeats ceaselessly. This diagram was adapted from Medina et al. [11].
Table 1.
Demographic and consultation record descriptions for the district of Niono, Mali.
Table 2.
Selected exponential smoothing methods within the state-space forecasting framework.
Figure 4.
State-space forecasts of Schistosoma haematobium consultation rate time-series for the district of Niono, Mali.
Observed Schistosoma haematobium consultation rate time-series (TS) in the district of Niono, Mali, are depicted as black lines in this composite panel while red traces correspond to contemporaneous h-month horizon forecasts; 95% prediction interval (PI) bounds are symbolized by red dots of the same color. Abscissa projections span 102 months (01/1996–06/2004) while ordinate scales represent the number of newly diagnosed (or forecasted) S. haematobium–induced terminal hematuria cases per 1000 individuals. Forecasts were generated with exponential smoothing (ES) methods, which are encapsulated within the state-space forecasting (ETS) framework (Methods). Panels A, B, C, and D correspond to 2-, 3-, 4-, and 5-month horizon forecasts, respectively. These forecasts are, by definition, external predictions. Predictions were superimposed onto the original TS to allow visual prediction accuracy evaluation. This figure should be considered dynamically. As observations and forecasts became available to and from the on-line execution program, the actual graphing of forecasts (red traces) preceded that of observations (black lines) by exactly h-month horizon.
Table 3.
ETS framework smoothing controls.
Figure 5.
Schistosoma haematobium consultation rate time-series forecasting accuracy and dispersion for the district of Niono, Mali.
Panel A: Mean absolute percentage error (MAPE) values between Schistosoma haematobium time-series (TS) observations for the district of Niono, Mali, and their corresponding h-month horizon forecasts measure external accuracy. The average coefficient of variance () for h-month horizon forecast probability density functions reflect prediction dispersion. MAPE and
values are displayed as a function of h-month horizon forecasts. MAPE and
values for 1–5 month horizon forecasts were circa 25 and 45%, respectively. Therefore, panels A and B demonstrate that forecast accuracy and dispersion are reasonable for short horizons. Of note, MAPE, unlike
, values assess the skill of h-month horizon forecasts.
and PI values are rarely reported outside the econometric literature; yet, they have paramount importance for calculating, e.g., the probability that a future observation will be smaller or greater than the expected forecast distribution mean by a certain margin. Alternatively, the number of individuals at risk may be calculated for a specified probability.