<i>Schistosoma haematobium</i> consultation rate time-series forecasting accuracy and dispersion for the district of Niono, Mali.

<p>Panel A: Mean absolute percentage error (<i>MAPE</i>) values between <i>Schistosoma haematobium</i> time-series (TS) observations for the district of Niono, Mali, and their corresponding <i>h</i>-month horizon forecasts measure external accuracy. The average coefficient of variance () for <i>h</i>-month horizon forecast probability density functions reflect prediction dispersion. <i>MAPE</i> and values are displayed as a function of <i>h</i>-month horizon forecasts. <i>MAPE</i> and values for 1–5 month horizon forecasts were <i>circa</i> 25 and 45%, respectively. Therefore, panels A and B demonstrate that forecast accuracy and dispersion are reasonable for short horizons. Of note, <i>MAPE</i>, unlike , values assess the skill of <i>h</i>-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.</p

Irrigation system and stagnant water reservoirs in the district of Niono, Mali.

<p>This composite panel depicts irrigation canals (which support mainly rice monoculture) and stagnant water reservoirs where <i>Schistosoma haematobium</i> 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 <i>S. haematobium</i>. Furthermore, rainfall precipitation fluctuations prompt the local authority (<i>Office du Niger</i>) to adjust irrigation management accordingly; for example, the <i>Office du Niger</i> may relax water control amid increased precipitation to better irrigate drier areas whilst collaterally enhancing water-flow through typically well-served agricultural fields—<i>S. haematobium</i> transmission suitability might then simultaneously increase and decrease in the former and latter scenarios, respectively.</p

<i>On-line</i> forecast flow-chart.

<p>(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 (<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#s2" target="_blank"><i>Methods</i></a>) as well as the Akaike's Information Criterion (<i>AIC</i>). Then, (3) Equations 2 & 3 simulate <i>h</i>-month horizon forecast path distributions with the best-performing ES method <i>via B</i> = 1000 ordinary residual bootstraps. (4) Mean forecast and 95% prediction interval (PI) values obtain as described in the <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#s2" target="_blank"><i>Methods</i></a> 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 <i>on-line</i> forecasts, i.e. external predictions. Basically, contemporaneous forecasts obtain <i>via</i> TS extrapolation whereby previous deviations between forecasts and their corresponding observations are exponentially adjusted with smoothing control values. For example, (1) the <i>Schistosoma haematobium</i> 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 <i>on-line</i> 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 <i>et al. </i><a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Medina1" target="_blank">[11]</a>.</p

Demographic and consultation record descriptions for the district of Niono, Mali.

<p>The total projected (2004) population in the district of Niono, Mali, is 278 741 individuals, inhabiting approximately 20 000 km². The projected number of individuals served by each community health center (CSCOM) service area within this district is tabulated under the <i>Population</i> heading. The population from each CSCOM service area was adjusted with the national annual population growth rate (3.2%) before the <i>Schistosoma haematobium</i> consultation rate time-series (TS) was calculated with Equation 1 (<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#s2" target="_blank"><i>Methods</i></a>) <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-USAID1" target="_blank">[19]</a>,<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Division1" target="_blank">[20]</a>. Potential records are listed under <i>Time-series period</i>. Unavailable CSCOM service area records appear under <i>Missing dates</i>—the number of missing monthly records for each year is listed in parenthesis otherwise records for the whole year are missing. These are totaled under <i>Missing months</i> and expressed as percentages from the total number of possible records (across all CSCOM service areas and years) under the <i>% missing</i> heading. Of note, the Niono CSCOM service area, which includes the district center and immediate periphery, is one of the 17 CSCOM service areas within the district of Niono, Mali. This table was adapted from Medina <i>et al.</i><a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Medina1" target="_blank">[11]</a>.</p

Selected exponential smoothing methods within the state-space forecasting framework.

<p>All exponential smoothing (ES) methods within the state-space forecasting (ETS) framework (Equations 2 & 3) were optimized with a likelihood function analog as new <i>Schistosoma haematobium</i> time-series (TS) observations for the district of Niono, Mali, became available; the best-performing method was continuously re-selected with the Akaike's Information Criterion (<i>AIC</i>) to generate optimum forecasts (<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#s2" target="_blank"><i>Methods</i></a>). Throughout the investigational period, only 3 from a total of 15 ES methods considered within the ETS framework were re-selected; they are: the multiplicative error/ trendless/ aseasonal (MNN); multiplicative error/ damped additive trend/ aseasonal (MA_dN); and, multiplicative error/ damped multiplicative trend/ aseasonal (MM_dN) ES methods. Notice that none of them are seasonal. Although a full portrayal of the ETS state-space framework (Equations 2 & 3) encapsulating all 30 ES methods <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Medina1" target="_blank">[11]</a>, <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Holt1" target="_blank">[21]</a>–<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Hyndman3" target="_blank">[31]</a> is beyond the scope of this investigation, those ES methods which have been selected at least once during the TS analysis are described herein in terms of <i>E</i>[<i>F</i>(<i>y_t</i>|<i>I_t</i>_-1)] and <i>x_t</i> recursions—<i>α</i>, <i>β</i> , and <i>φ</i> control smoothing of level (<i>l_t</i>), trend (<i>r_t</i>), and <i>r_t</i>-dampening, respectively. Large <i>α</i>, <i>β</i>, and <i>φ</i> values confer greater weights to recent information and effectively shorten the smoothing “memory”, i.e. the recent-past has a more pronounced influence on estimated components than does the distant-past <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Medina1" target="_blank">[11]</a>, <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Holt1" target="_blank">[21]</a>–<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Hyndman3" target="_blank">[31]</a>. For example, MA_dN state-space Eqs. 2 & 3 may be written in explicit matrix form as: <i>F</i>(<i>y_t</i>|<i>I_t</i>_-1) = A•<i>x_t</i>_-1•(1+<i>ε_t</i>) & <i>x_t</i> = B•<i>x_t</i>_-1+A•<i>x_t</i>•C•<i>ε_t</i> where A = (1, <i>φ</i>)′, <i>x_t</i>_-1 = (<i>l_t</i>_-1, <i>r_t</i>_-1), C = (<i>α</i>, <i>β</i>), and B is a 2×2 matrix whose entries b_1,1, b_1,2, b_2,1, b_2,2 are 1, <i>φ</i>, 0, <i>φ</i>, respectively.</p

Satellite image of West Africa.

<p>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 <i>Canal du Sahel</i> (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. <i>Image source</i>: adapted with permission from Globalis, <a href="http://globalis.gvu.unu.edu" target="_blank">http://globalis.gvu.unu.edu</a> (08/2007) <a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#pntd.0000276-Medina1" target="_blank">[11]</a>.</p

State-space forecasts of <i>Schistosoma haematobium</i> consultation rate time-series for the district of Niono, Mali.

<p>Observed <i>Schistosoma haematobium</i> 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 <i>h</i>-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) <i>S. haematobium</i>–induced terminal hematuria cases <i>per</i> 1000 individuals. Forecasts were generated with exponential smoothing (ES) methods, which are encapsulated within the state-space forecasting (ETS) framework (<a href="http://www.plosntds.org/article/info:doi/10.1371/journal.pntd.0000276#s2" target="_blank"><i>Methods</i></a>). 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 <i>on-line</i> execution program, the actual graphing of forecasts (red traces) preceded that of observations (black lines) by exactly <i>h</i>-month horizon.</p

Forecasts for acute respiratory infection (ARI) consultation rate time-series.

<p>Observed ARI consultation rate time-series are depicted as black lines while red and blue traces correspond to contemporaneous 2- and 3-month horizon forecasts, respectively; their prediction interval bounds are symbolized by dots of the same colors. Forecasts and prediction interval bounds are calculated with a bootstrap-coupled seasonal multiplicative Holt-Winters method (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0001181#s4" target="_blank"><i>Methods</i></a>). In each panel, the abscissa spans 102 months (01/1996–06/2004) while the ordinate represents the number of newly diagnosed cases <i>per</i> 1000 age category-specific individuals. Panel A: 0–11 months; Panel B: 1–4 years; Panel C: 5–15 years; and, Panel D: >15 years. Forecasts deteriorate slightly towards older age categories owing to seasonality attenuation. Of note, age category-specific 2- and 3-month horizon ARI consultation rate forecasts roughly overlap because of slowly shifting level and negligible trend time-series components. Consequently, 2- and 3-month horizon 95% prediction interval bounds also overlap.</p

<i>On-line</i> forecast flow-chart.

<p><i>On-line</i> forecasts imply that historical records are automatically and continuously supplied to the program, which revises forecasts. (1) Prior time-series observations and pseudo-parameter initialization are inputted into (2) the program that executes the multiplicative Holt-Winters (MHW) forecasting method, according to Equations 1–4 in the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0001181#s4" target="_blank"><i>Methods</i></a> section. (3) The program cycles (red arrow) through hundreds of residual bootstraps (Equation 5 in the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0001181#s4" target="_blank"><i>Methods</i></a> section) to produce (4) median forecasts and their 95% prediction interval bounds. Subsequently, (5) these forecasts plus (6) contemporaneous time-series observations are supplied to (2, 3) the MHW execution program, which revises (4) median forecasts and their prediction interval bounds. The automatic supply of contemporaneous time-series observations into (2–6) yields <i>on-line</i> forecasts.</p

Multiplicative Holt-Winters method: pseudo-parameter and mean absolute percentage error (MAPE) values.

<p>Median (and inter-quartile range) pseudo-parameter <i>α</i>, <i>β</i>, and <i>γ</i> values—which smooth control level, trend, and seasonal time-series components, respectively—reflect fitting of <i>B</i> = 500 bootstrap-generated full-length pseudo-time-series with the seasonal multiplicative Holt-Winters method. The greater the pseudo-parameter value, the shorter the smoothing memory, i.e., information from the recent-past have more pronounced effects on estimates than those from the distant-past. Generally, the strength of pseudo-parameters follows <i>γ</i>≫<i>α</i>≥<i>β</i>, which is expected for time-series with highly seasonal and negligible trend components. Furthermore, large mean absolute percentage error (MAPE) values between observed monthly consultation rates and their median forecasts imply low accuracy and <i>vice-versa</i>. Thus, 92% of the 24 time-series (TS) forecasts generated here (2 forecast horizons, 3 diseases, and 4 age categories = 24 TS forecasts) are reasonably accurate, i.e., their MAPE values are <i>circa</i> 25%. MAPE values from a seasonal adjustment (SA₃) forecasting method <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0001181#pone.0001181-Abeku1" target="_blank">[32]</a> are also listed for benchmark comparison. The MHW performance is equal or superior to that of the SA₃forecasting benchmark in 87.5% of the 24 TS forecasts generated here, as implied by equal or smaller MAPE values.</p