5 An application: the Italian Farm Structure Survey (FSS)
Marco Ballin and Giulio Barcaroli
The sampling frame used for the selection of 2003 Italian Farm Structure Survey (FSS) sample contains 2,153,710 farms. For the purposes of FSS sample design, the auxiliary variables considered are the following:
- regions (21 different values);
- provinces (103 different values);
- legal status (2 classes);
- sector of economic activity (9 classes);
- economic size unit (3 classes);
- agricultural area utilized (3 classes);
- livestock unit (3 classes);
- altimetry of the headquarter of the holding (5 classes).
Fourteen different target variables have been considered as the main target of FSS, on which the required precision levels (in terms of maximum coefficient of variations) have been fixed at regional level (domains of interest). The list of variables and related precision constraints are reported in table 5.1.
Both the 8 auxiliary and the 14 target variables have been observed during the previous 2000 Agricultural Census, so their values are available for each unit in the frame. This gives the possibility to calculate means and standard deviations related to whichever defined stratum.
Firstly, the current "manual� procedure followed in 2003 to choose the most suitable stratification for sample selection is described.
2003 manual configuration of strata to select the FSS sample
In the first step, a take-all stratum was defined in each region on the basis of local characteristics. The thresholds for defining the take-all strata were determined using the Hidiroglou method (1986).
In the second step, a choice between a stratification based on provinces or on the region as a whole, was chosen region by region, on the basis of local organizational considerations.
In the third step, the other six variables were alternatively used in each region or provinces (depending on the result obtained in the second step) as stratification variables. For each of such alternative stratifications, the optimal sample size was computed (the minimum sample size in each stratum had been fixed to 50) (in the cost function, fixed cost has been set to zero, and variable costs were set equal to 1 in each atomic stratum: so the cost function coincides with the total sample size). The stratification supporting the overall minimum sample size in each region (usually defined on different variables) was considered as the output of this step.
In the fourth step, the remaining five variables were used separately to refine the stratification previously obtained. For each of these refined stratifications the optimal sample size was computed considering the same constraints used in the third step.
This stepwise procedure was repeated on a regional basis, by refining the best stratification obtained in each step, using the remaining available variables until the obtained stratification revealed to be less efficient than the stratification in the previous step.
By so doing, the total amount of planned sample size was fixed to 42,465 units (actually, the sample size used for 2003 FSS was increased to 52,713 in order to obtain better estimates at national level. Here we consider the number of 42,465 to correctly compare the results obtained with the genetic algorithm).
Use of the genetic algorithm to identify optimal strata and best allocation
The most detailed available stratification of the frame, obtained as a Cartesian product of all the auxiliary variables, consists of 24,454 different strata, 1,787 of which have been defined as take-all strata. So, the atomic strata are given by the 22,667 sampling strata obtained by subtracting the 1,787 take-all strata. The latter are collapsed in only one stratum, whose 6,971 units will always be selected for whatever sample.
Actually, one of the auxiliary variables, region, is considered as the domain variable. So, our task consists in optimising the frame stratification and the sample allocation distinctly for each one of the different 21 Italian regions. For instance, the first region (Piemonte) is characterised by 105,074 units in 1,646 sampling strata, and 597 units in 129 take-all strata.
Precision constraints (once again expressed in terms of upper limits on coefficients of variation) have been set, for each one of the 14 different target variables, at the same values chosen on the occasion of manual configuration of strata carried out for the 2003 survey: they are 5%, 6% or 10% for the most important variables in each region. Table 5.1 reports the complete set of the coefficient of variations used in planning the 2003 FSS.
| Region | Cereals | Industrial crops | Fresh vegetables | Flowers | Vineyards | Olives | Citrus fruit | Fruits | Bovines | Pigs | Sheep | Economic size units | Utilized agricultural surface | Livestock unit |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Piemonte | 5.0 | 10.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||||
| Val d'Aosta | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Lombardia | 5.0 | 10.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||||
| Bolzano | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Trento | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Veneto | 5.0 | 10.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||||
| Friuli V.G. | 5.0 | 10.0 | 5.0 | 6.0 | 6.0 | |||||||||
| Liguria | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Emilia R. | 5.0 | 10.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||
| Toscana | 5.0 | 10.0 | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||
| Umbria | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Marche | 5.0 | 6.0 | 6.0 | |||||||||||
| Lazio | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||||
| Abruzzi | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Molise | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Campania | 5.0 | 10.0 | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | ||||||
| Puglia | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||||
| Basilicata | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||||
| Calabria | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | ||||||||
| Sicilia | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 | |||||
| Sardegna | 5.0 | 5.0 | 5.0 | 6.0 | 6.0 |
Table 5.2 reports the results of the two solutions in terms of required sample size: the one planned in 2003 by the expert sample designer of the FSS (column 6), and the one obtained by applying the genetic algorithm (column 7).
| (1) Domain (region) | (2) Total number of units in the frame | (3) Number of atomic sampling strata in the frame | (4) Number of units in the sampling strata | (5) Number of units in take-all strata | (6) Sample size by 2003 stratification | (7) Sample size by Genetic Algorithm solution | (8) Number of strata in GA solution | (9) % relative difference (7) vs (6) |
|---|---|---|---|---|---|---|---|---|
| Piemonte | 105,671 | 1,646 | 105,074 | 597 | 2,687 | 1,497 | 9 | -44.29 |
| Valle d'Aosta | 6,125 | 65 | 6,074 | 51 | 408 | 317 | 7 | -22.30 |
| Lombardia | 71,257 | 1,902 | 69,495 | 1,762 | 3,428 | 2,151 | 7 | -37.25 |
| Bolzano | 23,362 | 127 | 23,202 | 160 | 692 | 430 | 7 | -37.,86 |
| Trento | 30,021 | 124 | 29,908 | 113 | 676 | 523 | 7 | -22.63 |
| Veneto | 176,999 | 1,450 | 176,064 | 935 | 3,531 | 1,868 | 11 | -47.10 |
| Friuli | 32,981 | 638 | 32,805 | 176 | 807 | 498 | 6 | -38.29 |
| Liguria | 29,992 | 584 | 29,967 | 25 | 766 | 485 | 7 | -36.68 |
| Emilia R. | 103,702 | 2,157 | 102,922 | 780 | 2,584 | 2,022 | 11 | -21.75 |
| Toscana | 107,288 | 1,959 | 106,964 | 324 | 2,099 | 1,337 | 16 | -36.30 |
| Umbria | 46,074 | 435 | 45,897 | 177 | 1,354 | 751 | 7 | -44.53 |
| Marche | 60,439 | 1,005 | 60,271 | 168 | 918 | 488 | 8 | -46.84 |
| Lazio | 162,109 | 1,304 | 161,801 | 308 | 3,233 | 2,216 | 14 | -31.46 |
| Abruzzi | 67,117 | 888 | 66,941 | 176 | 1,035 | 743 | 10 | -28.21 |
| Molise | 28,890 | 375 | 28,834 | 56 | 1,190 | 630 | 6 | -47.06 |
| Campania | 212,145 | 1,271 | 211,833 | 312 | 2,559 | 1,883 | 13 | -26.42 |
| Puglia | 288,087 | 1,026 | 287,877 | 210 | 4,712 | 2,009 | 14 | -57.36 |
| Basilicata | 68,470 | 504 | 68,355 | 115 | 703 | 493 | 7 | -29.87 |
| Calabria | 145,812 | 1,624 | 145,654 | 158 | 2,798 | 1,792 | 17 | -35.95 |
| Sicilia | 295,637 | 2,345 | 295,472 | 165 | 3,955 | 3,140 | 22 | -20.61 |
| Sardegna | 91,532 | 1,238 | 91,329 | 203 | 2,330 | 982 | 7 | -57.85 |
| Italia | 2,153,710 | 22,667 | 2,146,739 | 6,971 | 42,465 | 26,255 | 213 | -38.17 |
As the determination of the best stratification has been carried out separately for each region, 21 independent results can certify the great convenience of the algorithm in most domains. A dramatic decrease of the required overall sample size can be observed, as shown by a 38.17 % saving on the previous total. This result is differentiated region by region, with a maximum decrease for Sardegna (-57.85%) and a minimum for Sicilia (-20.61%). Also in terms of strata, from the initial number of atomic strata (22,667), a huge reduction occurs to the final stratification, characterised by only 213 different strata (ranging from a minimum of 6 strata in region Friuli, up to 22 strata in Sicilia).
As for the setting of the parameters used to obtain the above result, the most important revealed to be the following:
- number of iterations (or generations);
- generation size (number of individuals, or solutions, evaluated at each iteration);
- mutation chance;
- initial number of strata;
- factor for increasing the initial number of strata.
Their final values have been determined, after numerous trials, on the basis of the analysis of the runs for each region.
In particular, by inspecting the convergence plot, it is possible to understand if the number of iterations is sufficient to ensure that the final solution is definitely the best obtainable, or if otherwise a higher number of iterations is needed. This can be done by analysing the behaviour of the two curves in the plot: the lower one reports the best evaluation value, while the upper one refers to the mean evaluation value. When the mean evaluation value is still decreasing, together with the best evaluation value, it is worthwhile to go on iterating. When the best value line becomes stably constant (and typically the mean value line begins to fluctuate up and down), no further gain can be expected by new iterations. This is the case, for instance, of the convergence plot for Trento region, shown in figure 5.1.
A convenient value for iterations parameter was found to be 5,000. As for the mutation chance, a suitable value was found to be 0.001: this means that, for any chromosome in the genome (any value in vector ), a mutation occurred on average only once out of a thousand. A critical point is in fixing the initial number of strata. Since the final solution is very sensitive on the number of strata, we decided to let the algorithm to choose it. This can be done, as already said in section 4, by assigning a low value to initialStrata, and by giving a value greater than zero to addStrataFactor: this enables the algorithm to explore solutions characterized by a wide range of number of strata. In our experiment, we set the initial number of strata to the value 5, while assigning a value 0.01 to the factor for increasing the initial number of strata (this means that, any time a mutation occurs, there is a probability of 1% to increase by 1 the current number of strata).
Figure 5.1 Best and mean evaluation value for the Trento region
From a computational point of view, the overall task required an elapsed time of 641,820 seconds (more than 178 hours, nearly one week) (the job was run on a desktop AMD Athlon (2.90 Ghz, 3 GB RAM)).
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