Crop Allocation Model

Fig 1: Overview of Spatial Production Allocation Model (SPAM)
Source: You, Wood-Sichra and Wood (2009)

Our spatial allocation model uses the concept of information entropy (Jaynes, 1979; Golan, Judge and Miller, 1996). We will start with the concept of information entropy.

Cross Entropy Approach

The origin of entropy goes back to Boltzmann’s distribution law in thermodynamics (Jaynes 1979). It is a measure of “disorder” of molecules in a system. One of the fundamental laws of nature is the second law of thermodynamics, which says the entropy of a closed system never decreases and increases whenever possible.  Shannon (1948) introduced information entropy to measure the uncertainty (state of knowledge) of the expected information, which gives birth to information theory. According to Shannon’s definition, information is statistical property of a message. Any probability distribution pi, i= 1, 2, …, n, of a random variable provides some information about that variable. Shannon defined entropy H(p) as a weighted sum of the information –lnpi, i= 1,2,…,nwith respective probabilities as weights:

(Eq.1)

with convention that 0ln0=0. E(lnp) is expected value of lnp.

Jaynes (1957) proposed the maximum entropy principle in statistical inference: the least informative probability distribution pi can be found by maximizing the entropy H(p). In (Eq.1), the solutions are: pi = 1/n, i= 1, 2, …, n, H(p) = ln n. Put the principle in other words: in the absence of information to the contrary, all possible states of system are equally likely. Generalized maximum entropy (GME) approach is based upon this principle.

Following (Eq.1), the cross-entropy of one probability distribution p={p1, p2 , …, pn} with respect to another probability distribution q={q1, q2 , …, qn} can be defined

(Eq.2)               

This is actually a measure of distance between two probability distributions p and q. If we choose the non-informative q, i.e, q = {1/n, 1/n,…, 1/n}, then CE(p, q) becomes:

(Eq.3)                

Therefore maximizing entropy is in fact a special case of minimizing cross-entropy with respect to a uniform distribution. The cross entropy (CE) approach can be stated as a minimization problem where the cross entropy (objective function) is minimized subject to applicable constraints and the prior knowledge.

A comprehensive book by Golan, Judge and Miller (1996) has brought much interest in applying the entropy approach (Lencer and Miller 1998; Paris and Howlitt 1998; Robinson, Cattaneo and El-Said 2000; Zhang and Fan 2001). The unique feature of the entropy approach is to overcome two empirical problems that hamper traditional econometrics: multi-collinearity and ill-posed problems (particularly due to underdetermined or incomplete data) (Paris and Caputo 2001; Golan, Judge and Miller 1996). The idea is to remove irrelevant information at the beginning of a problem rather than taking pains to make dubious assumptions. Preckel (2001) compares least squares and entropy methods from a penalty function perspective, and concludes that the differences between these two approaches boil down to how the supports for errors and coefficients are defined in a generalized cross-entropy approach. When the supports are specified to be symmetric, wide, and centered on zero for the residual errors, the coefficient estimates are essentially indistinguishable (Preckel 2001). Shen and Perloff (2001) estimate a ratio of parameters using different methods and concludes that GME (and Bayesian method of moments) estimator has a much smaller mean square errors and average biases than do ordinary least squares (OLS). Bera and Bilias (2002) does an excellent synthesis on different estimation approaches such as method of moments, maximum entropy, maximum likelihood, empirical likelihood, estimating function and generalized methods of moments. The paper compares many of these estimation techniques with a unified framework and puts these techniques in an interesting historical perspective. Our production allocation problem is underdetermined with quite incomplete data, and the entropy approach is ideally suited for our spatial allocation model.

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Spatial production allocation model

This section describes the spatial allocation model using the entropy approach with partial sub-national data and new irrigation maps. The overall model procedure could be simplified as the following: we first break down the crop areas by production systems (4 different management intensities) at the national level. We then generate the initial crop area distribution (pi_ijl in Eq.13) using the subnational census data and crop suitability surfaces. Finally, we use cross-entropy method to optimize the initial distribution according to minimum cross-entropy distance measure under different spatial constraints (e.g., crop land area per pixel, irrigated areas). The following will describe the detailed equations.

Let s_ijl be the area share allocated to pixel i and crop j at input level l with a certain country (say X) in the world. CropArea_jl is the total physical area for crop jat input level l for a certain spatial allocation unit, and Aijl  the area allocated to pixel i for crop j at input level l in country X. Therefore:

(Eq.4)                                        

Let pijl be the prior area shares (see below) for pixel i and crop j at input level l in country X. The modified spatial allocation model can be written as follows:

(Eq.5)                

subject to:

(Eq.6)                

(Eq.7)                

(Eq.8)                

(Eq.9)                

(Eq.10)              

(Eq.11)              

where:

i : i = 1, 2, 3, …, pixel identifier within the allocation unit, and

j: j = 1, 2, 3, …, crop identifier (such as maize, cassava, rice) within the allocation unit, and

l: l = irrigated, rainfed-high input, rainfed-low input, subsistence, management and input levels for crops

k: k = 1, 2, 3, …, identifiers for sub-national geopolitical units

J:  a set of those commodities which sub-national production statistics exist

L:  a set of those commodities which are irrigated within pixel i.

Avail_i: total agricultural land (ha) in pixel i, which is equal to total agricultural area estimated from land cover satellite image as described in the previous section.

Suitable_ijl : the suitable area(ha) for crop j at input level l in pixel i, which comes form FAO/IIASA suitability surfaces as introduced in the previous section.

SubCropArea_jk: Available sub-national crop area (ha) for crop j and sub-national geopolitical unit k.

IRRArea_i : the irrigation area (ha) in pixel i from global map of irrigation (Siebert et al 2001).

Comparing to the original spatial allocation model (You and Wood 2006), there are two new constraints: equations (3.9) and (3.10).  Constraint (3.9) sets the sum of all allocated areas within those sub-national units that have existing statistical data to be equal those corresponding sub-national statistics. Constraint (3.10) includes the irrigation information: the sum of all allocated irrigated crop areas in any pixel must not exceed the total area identified by Siebert et al. (2001) as being equipped for irrigation within that pixel. The objective function and all other equations are similar to the original model. The modified model is capable of handling mixed scales of production data disaggregation, thus both broadening it’s range of practical application, and increase its reliability since it can use the highest level of production data disaggregation available for each crop,

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Prior

Obviously, an informed prior (pijl) is very important for the success of the model. We create the prior based upon a range of available evidence. First for each pixel, we calculate the potential revenue per pixel as

(Eq.12)              

where Pricej are the price for crop j in the country X. SuitYieldijl is the agro-climatically suitable yield for crop j at input level l and pixel i, Accessi  is the proxy of physical market accessibility for pixel i, as previously described.  In the case of subsistence production, however, we replace the revenue measure by population density. For subsistence production, we assume that crops are grown primarily to meet for food security reasons and as an expression of preference for certain food staples, even at relatively low levels of biophysical suitability. We then pre-allocate the available statistical crop areas (at various geopolitical scales) into pixel-level areas by simple weighting:

(Eq.13)                         

where Areaijl is the area pre-allocated to pixel i for crop j at level l, Percentjl is the area percentage of crop j at input level l . Those geopolitical units without crop area statistics are combined and a total crop area for that merged unit is derived by subtracting the sum of available sub-national areas from the national total. After this pre-allocation, we calculate the prior by normalizing the allocated areas over the whole country.

(Eq.14)

Figure 1(above) shows the overview of spatial allocation model to clarify the flow of information in the spatial allocation model. We started with the administrative (geopolitical) units for which we have been able to obtain production statistics. These may typically be national or sub-national administrative regions such as states, districts, or counties (Figure 1a). We reinterpret the already classified land cover imagery (Figure 1b) into crop land and non crop land (Figure 1c). This crop land surface provides valuable information on where and how much agricultural land is at the pixel level. Figure 1d shows the crop-specific (maize in the figure) suitability judging from local climate and soil conditions. The spatial allocation model utilizes all these input data, and applies a cross entropy approach to obtain the final estimation of crop distribution. Comparing Figure 1e, the model results, with Figure 1a, the starting point, we could see the tremendous improvement in revealing spatial heterogeneity within the administrative units.

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Production and yield

To convert the allocated crop areas into production, we need consider both the broader production systems and the spatial variation within the systems. We first calculate an average potential yield within a SRU, , for crop j in production system l using the allocated areas (A_ijl) as weight:

(Eq.14)                                                           

Then estimate the actual crop yield of crop j in production system l and pixel i  (Y_ijl) as

(Eq.15)                                                                      

where Yield_jl is the statistical yield (from census data) for crop j in production system l. The production of crop j in production system l, and pixel i, (Y_ijl) , could be calculated as the following:

(Eq.16)

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References

Bera, A. K. and Y. Bilias.   2002. The MM, ME, ML, El, EF and GMM approaches to estimations: A synthesis. Journal of Econometrics 107:51-86

Golan, A. G. Judge and D. Miller. 1996. Maximum entropy econometrics: Robust estimation with limited data.  New York: John Wiley & Sons.

Jaynes, E. T. 1957. Information theory and statistical methods I, Physical Review 106: 620-630

Jaynes, E. T. 1979. Where do we stand on maximum entropy?  In The maximum entropy formalism, ed., Levine, R.D. and M. Tribus.   Cambridge, MA: MIT Press.

Lence, S. H. and D. J. Miller. 1998. Estimation of multi-output production functions with incomplete data: a generalised maximum entropy approach. European Review of Agricultural Economics 25: 188-209

Paris, Q., R.E. Howitt. 1998. Analysis of ill-posed production problems using maximum entropy. American Journal of Agricultural Economics 80:124-138

Preckel, Paul V.  2001. Least squares and entropy: a penalty function perspective. American Journal of Agricultural Economics 83 (2): 366-377.

Robinson, Sherman, A. Cattaneo, and M. El Said. 2000. Updating and estimating a social accounting matrix using cross entropy methods. Trade and Macroeconomics Division Dicussion Paper No.58. Washington D.C.: International Food Policy Research Institute.

Shen, E. and J. M. Perloff. 2001. Maximum entropy and Bayesian approaches to the ratio problem. Working paper January 2001, Department of Agricultural and Resource Economics.  Berkeley, Calif.: University of California Berkeley.

Siebert, Stefan, P. Döll and J. Hoogeveen. 2001. Global map of irrigated areas version 2.0. Center for Environmental Systems Research, Kassel, Germany: Univerity of Kassel, and Rome, Italy: Food and Agriculture Organization of the United Nations.

You, Liangzhi and S. Wood, 2003. Spatial allocation of agricultural production using a cross-entropy approach.  Environment and Production Technology Division Discussion Paper No. 126. Washington D.C.: International Food Policy Research Institute.

You, L. and S. Wood. 2006. An entropy approach to spatial disaggregation of agricultural production. Agricultural Systems Vol.90, Issues1-3 p.329-347.