Statistical Inferences of Soil Properties and Crop Yields as Spatial Random Functions ESHEL BRESLER* AND ASHER LAUFER ABSTRACT Crop yields are often highly variable due (partly) to spatial het- erogeneity of soil properties. The conditional multivariate normal (CMVN) method is systematically described and applied to 24 spa- tial random functions. These include: 12 soil properties, four soil variables, and two yield components of each of four field crops [wheat (Triticum aestivum L.), vetch (Vicia sativum L.), corn (Zea mays L.), and peanut (Arachis hypogaea L.)]. The statistical parameters char- acterizing the joint probability density function in a two-dimensional field were estimated, from 20 to 60 measurements, by the maximum likelihood procedure. An estimation procedure for three parameters of four covariance models (piecewise linear, exponential, spherical, and Gaussian), constant mean, and linear drift, is described. It was found that the covariance function of all the 24 spatial functions can be represented by the four models and that there is a significant two-dimensional linear drift of the mean. The parameter that rep- resents the nugget effect (i.e. the variance of uncorrelated values of the spatial function) is estimated to be either zero or nonsignificant for the soil properties but has a small significant value for most crop yield components. Soil properties as well as crop yield components are characterized by the existence of a significant correlation scale on the order of 10 to 30 m. Estimations of conditional expectation, conditional covariance, the variance, and the variance of the esti- mated variance were made and demonstrate that the CMVN method can be applied to real field data. The specific covariance model used generally has a small effect on the estimated results. Comparisons with kriged values and kriging variance demonstrate the similarity and differences in the results. P ELD MEASUREMENTS have indicated consistently that soil properties as well as crop yields vary throughout the field space in an irregular manner (e.g., Biggar and Nielsen, 1976; Bresler et al, 1981, 1982; Nielsen et al., 1973; Russo and Bresler, 1981; Trang- mar et al., 1986). Values of these variables at different points in the field are subjected to uncertainty due to a limited number of observation points and because the measurements themselves are error prone. To deal with uncertainty and spatial variability, soil properties and various components of crop yield are regarded as random functions of the space coordinates, or as re- gionalized variables in the geostatistical terminology (Delhomme, 1978; Journel and Huijbregts, 1978; Matheron, 1971, 1973). The actual field is viewed as a given realization of an ensemble of all possible re- alizations, of which only one realization (comprised of the actual measurements) is available for the sta- tistical analysis. Hence, the statistical structure of any soil property or a crop yield component can be inferred only if some kind of statistical assumptions are made. Field mea- surements of a soil property or a crop yield compo- nent can be used for the inference of the probability density function (pdf) and its parameters. The con- ditional geostatistical approach uses the observations themselves to reduce the uncertainty in the values of the random function at any point in the field other than the observation points. The approach investi- gates the stochastic interpolation problem by using the kriging method (Delhomme, 1978; Journel and Huijbregts, 1978), which predicts the best unbiased estimates of the expectations and covariances condi- tion on the observations. The statistical structure (i.e. Dep. of Soil Physics, Inst. of Soils and Water, A.R.O., The Volcani Ctr., P.O. Box. 6, Bet Dagan 50-250, Israel. Contribution 2096-E, 1987 series, from the Inst. of Soils and Water A.R.O., Israel. Re- ceived 27 July 1987. "Corresponding author. Published in Soil Sci. Soc. Am. J. 52:1234-1244 (1988).
Transcript
Statistical Inferences of Soil Properties and Crop Yields as Spatial Random FunctionsESHEL BRESLER* AND ASHER LAUFER
ABSTRACTCrop yields are often highly variable due (partly) to spatial het-
erogeneity of soil properties. The conditional multivariate normal(CMVN) method is systematically described and applied to 24 spa-tial random functions. These include: 12 soil properties, four soilvariables, and two yield components of each of four field crops [wheat(Triticum aestivum L.), vetch (Vicia sativum L.), corn (Zea mays L.),and peanut (Arachis hypogaea L.)]. The statistical parameters char-acterizing the joint probability density function in a two-dimensionalfield were estimated, from 20 to 60 measurements, by the maximumlikelihood procedure. An estimation procedure for three parametersof four covariance models (piecewise linear, exponential, spherical,and Gaussian), constant mean, and linear drift, is described. It wasfound that the covariance function of all the 24 spatial functions canbe represented by the four models and that there is a significanttwo-dimensional linear drift of the mean. The parameter that rep-resents the nugget effect (i.e. the variance of uncorrelated values ofthe spatial function) is estimated to be either zero or nonsignificantfor the soil properties but has a small significant value for most cropyield components. Soil properties as well as crop yield componentsare characterized by the existence of a significant correlation scaleon the order of 10 to 30 m. Estimations of conditional expectation,conditional covariance, the variance, and the variance of the esti-mated variance were made and demonstrate that the CMVN methodcan be applied to real field data. The specific covariance model usedgenerally has a small effect on the estimated results. Comparisonswith kriged values and kriging variance demonstrate the similarityand differences in the results.
PELD MEASUREMENTS have indicated consistentlythat soil properties as well as crop yields vary
throughout the field space in an irregular manner (e.g.,Biggar and Nielsen, 1976; Bresler et al, 1981, 1982;Nielsen et al., 1973; Russo and Bresler, 1981; Trang-
mar et al., 1986). Values of these variables at differentpoints in the field are subjected to uncertainty due toa limited number of observation points and becausethe measurements themselves are error prone. To dealwith uncertainty and spatial variability, soil propertiesand various components of crop yield are regarded asrandom functions of the space coordinates, or as re-gionalized variables in the geostatistical terminology(Delhomme, 1978; Journel and Huijbregts, 1978;Matheron, 1971, 1973). The actual field is viewed asa given realization of an ensemble of all possible re-alizations, of which only one realization (comprisedof the actual measurements) is available for the sta-tistical analysis.
Hence, the statistical structure of any soil propertyor a crop yield component can be inferred only if somekind of statistical assumptions are made. Field mea-surements of a soil property or a crop yield compo-nent can be used for the inference of the probabilitydensity function (pdf) and its parameters. The con-ditional geostatistical approach uses the observationsthemselves to reduce the uncertainty in the values ofthe random function at any point in the field otherthan the observation points. The approach investi-gates the stochastic interpolation problem by using thekriging method (Delhomme, 1978; Journel andHuijbregts, 1978), which predicts the best unbiasedestimates of the expectations and covariances condi-tion on the observations. The statistical structure (i.e.Dep. of Soil Physics, Inst. of Soils and Water, A.R.O., The VolcaniCtr., P.O. Box. 6, Bet Dagan 50-250, Israel. Contribution 2096-E,1987 series, from the Inst. of Soils and Water A.R.O., Israel. Re-ceived 27 July 1987. "Corresponding author.
Published in Soil Sci. Soc. Am. J. 52:1234-1244 (1988).
BRESLER & LAUFER: STATISTICAL INFERENCES OF SOIL PROPERTIES AND CROP YIELDS 1235
the variogram) which has been inferred by some ar-bitrary averaging procedure, is assumed to be knownwith certainty. The fact that the parameters charac-terizing the pdf are also random variables has beengenerally disregarded. An exception is the work of Ki-tanidis and Vomvoris (1983). Dagan (1982, 1985) sug-gested an alternative approach based on the classicalstatistics of conditional probability assuming multi-variate normal (MVN) pdf (Mood et al., 1974).
Unfortunately the estimated parameters character-izing the MVN pdf are random variables because theymust be inferred from observations of a finite sample,and they are thus subjected to uncertainty. Recogniz-ing this additional source of uncertainty, Feinermanet al. (1986) developed a method to infer the spatialdistribution of the statistical moments of a stationarynormal spatial function, based on a finite set of mea-sured values. Estimates of the expectations, covari-ances, and variances at any arbitrary point in the fieldwere demonstrated. The estimates were conditionedon simulated observations points, however, rather thanon real measurements.
In this paper the new conditional multivariate nor-mal (CMVN) approach (Feinerman et al., 1986) willbe applied to a large set of real field data comprisedof soil properties, soil variables, and various crop yieldcomponents. These observed data were inferred to bedrawn from a population with normal or lognormaldistribution (e.g., Bresler et al., 1981; Russo and Bres-ler, 1981; Stern and Bresler, 1983). The methodologyfor CMVN pdf is systematically described. Subse-quently, the set of statistical parameters characterizingthis pdf is estimated for 24 soil properties and cropyield components. Estimations are carried out withthe aid of the maximum likelihood procedure (Theil,1971) and applied, using four covariance models, forthe measured data representing two-dimensional ran-dom functions. The results are then analyzed and pa-rameter estimations of different covariance models arecompared. Results of estimated conditional MVN ex-pectation, covariance and variance, and the varianceof estimated variance, are illustrated. Numerical val-ues of the inferred expected values and variances, fordifferent covariance models, are compared. Resultsbased on kriging are also compared with CMVN re-sults.
STATISTICAL PRELIMINARIESConsider a soil property or any component of crop yield
as a generic two-dimensional random function, or a region-alized variable. It has a value z(x) with x being the coordi-nate vector of a point in an agricultural field. As in Feiner-man et al. (1986), it is assumed that z = (z,, z2, . . ., ZK) isa vector or random variables, the statistical structure of whichis denned by the joint pdf/(z,, z2, . . . ZK) at an arbitrary setofK points (x,, x2, . . ., \K) within the field domain. It is alsoassumed that z, or its residual vector defined by z' = z —E(z), is stationary in the ordinary sense (i.e., that fa) is in-variant under translation in the field space of the points x,)and that fa) is a nonsingular multivariate (A"-variate) nor-mal pdf/z) - (27r)-^2|fl|-"2exp{-0.5[z -
X [z - E(z)]} . [I]Here fl > 0 is the covariance matrix, |fl| its determinant, Q~'
its inverse, superscript T denotes transpose, and E(z) is theexpected value of z defined by
£(z) = Jz./(z)rfz. [2]The covariance fi is defined for two arbitrary points (x,,
XH), by, x,,) =
= ff2p(xi,xn) [3]where a2 is the variance, p is the autocorrelation of z'(x,)with z'(x,,), and the covariance Q(x,, xn) depends only onthe lag r, r = |xn — x,|. In particular, for r = 0 the variance
= fl(x,, x,) = Q(xn, xn) = «2 [4]is constant. It should be emphasized that the definition ofstationarity can be generalized to consider, in addition tothe case of constant mean (i.e., E(z) = a,), the case whena, is not constant and z has a trend with mean E(z) = «i(x).Thus, when z has a drift its residual z' is stationary in anordinary sense.
Let us subdivide the vector of z of size .AT into z = (Y, y)where Y and y are subvectors of z; Y = (Yt, Y2, . . . , YN)represents the random sample of size N, and y = (y,, y2, . . .. WA/) of size M with M = K — N. The conditional proba-bility density function of y given Y is obtained from
= Ay, Y)/A(Y) = [5]where Xy»Y) is the joint pdf of y and Y, and /z(Y) is themarginal pdf of Y. Since z has a multivariate normal (MVN)pdf (Eq. [1]), the marginal pdf of its subvectors are alsoMVN, characterized by their expectations £(y) and E(Y) andby their covariances J2y and QY (e.g., Press, 1982, p. 71-72).The conditional pdf of y given Y = Y* (Eq. [5]) is alsonormal with the mean vector a linear function of Y* andthe covariance matrix independent of Y*, where Y* is a vec-tor of observed values of the random variable. That is (e.g.Press, 1982, p. 73)£[y(x)|Y = Y*]
+ nyY.Q[y(x)|Y = Y*]
where QyY is defined by
[Y* - EV.(Y*)] [6]flyY. (QY.)-' 0Y.y [7]
[8]INFERRING THE STRUCTURE OF A SPATIAL
RANDOM FUNCTIONThe uncertainty of the random function z(x) can be re-
duced by making the inference of the statistical structureconditional on a subset of given realizations y(x,), F(x2),. . ., Y(xN), i.e. on a vector of measured values of the soilproperty, or crop yield, Y(y,, Y2, . . . , YN). For each arbitrarypoint in x, the expected value of y(\), conditional on themeasurement vector Y, can be inferred in the assumed sta-tionary system from Eq. [6] and [7], using
£[y(x)|Y] = «,(Y)|(x) + £ X/x)7,7-1
where X/x) and £(x) are defined by[9]
[10]
[11]Note that here and hereafter the hatted variables denote es-timates which are therefore random variables. For two ar-bitrary points x,, x,,, the conditional MVN pdf, j(y\,yn\Y), is
1236 SOIL SCI. SOC. AM. J., VOL. 52, 1988
bivariate and is characterized, in addition to the conditionalexpectation, Eq. [9], by the two-points covariance (C) con-ditional on the measurement vector Y, and is estimated (seeAppendix I for details) from:C(x,, x,,|Y) = [12]
Here, the estimate ffoLh xlt) is the two-point equivalent ofthe expression
(see Eq. [7], [8], [3], and [4]) and is denned by
x,,) = p(x,, x,,) - . [13]
Hence, for any arbitrary point x in the field, the N valuesof X,(x) (Eq. [10]) to be used in Eq. [9], [11], and [13] areobtained from the solutions of the linear systemN
Tx/x) C(x,, x,) = C(x, x,) [14]
or (using EQ. [3]) from the solution of
i/x) p(x7, x*) = p(x, XA.) [15]j-i
k=l,2,...,Nwhere x, and XA are measurement points.
The best attainable estimate of the estimated conditionalcovariance C is given by the expected value [E(Q] of C inEq. [12], considering the estimators a, and «2 to be deter-ministic (Feinerman et al., 1986), and calculated from
; x,,)] = C(xI)XlI|y)
= a2(Y)7?(x,) . [16]
Equation [16] is a general formula for conditional MVNcovariance by inclusion of the last term, which incorporatesthe uncertainty of the estimate 5,, i.e., a|,. Since the estimateC is a random variable, its variance can be estimated bytaking the variance over C in Eq. [16] as
Var (Q ,, x,,) . [17]
For x, — xn = x, Eq. [16] and [17] predict the estimatedvariance of the error and the estimate of its variance, re-spectively, as
t, x) = «2(Y)n(x, x) +
x).
(x, x) [18]
[19]
Given a set of measured values ¥(7,, Y2, ... , YN) takenfrom field locations x,, x2, ... \N, the aim is to predict, orto estimate at any arbitrary point (x) in the field, the ex-pected value of y, the best attainable estimate of the covar-iance, and also the variance of its variance. The predictionproblem is thus to find
E[y(x)], C = E[C(x)], and <r£<x)conditional on the measured values Y(Ylt ... , YN). Sincethe true values are not known, and the only available in-formation is the set of W measured values Y, this aim isobtained from the estimations given in Eq. [9], [16], or [18],and [17] or [19], based on the measurements vector y.
ESTIMATIONS OF FIELD PARAMETERS BYTHE MAXIMUM LIKELIHOOD METHOD
To infer the conditional moments Eq. [9] and [12], theuncertainty involved in the estimated parameters a (i.e., «i,a2, p) must be taken into account. As these parameters arenot known, their values must be estimated from the onlyavailable information about y, i.e., from the measurementsvector Y = (7,,..., YN). By using the maximum likelihood(ML) method applied tofy), Eq. [1], a set of estimates a(Y)are obtained for the vector of parameters a. For a sufficientlylarge number of observation points and under quite generalconditions it is generally assumed that the estimates a areasymptotically normal, unbiased, and efficient estimates i.e.,they are of minimum variance equal to the lower bound ofthe Cramer-Rao inequality (Theil, 1971, p. 392-395). More-over, the maximum likelihood (ML) procedure provides anestimate of the variance-covariance matrix of the ML esti-mates of a which can serve for statistical inference of theestimated values of a. Hence, parameter estimations madewith the aid of the maximum likelihood (ML) procedure areconsidered to be a very powerful statistical method (Kitan-idis and Vomvoris, 1983). To simplify matters further, theML variance-covariance matrix will be taken here as deter-ministic, although strictly speaking, it is also random, be-cause it is an estimator.
Models for the Statistical StructureMany models may be considered to describe the statistical
structure (first two moments) of soil properties and the re-sultant crop yield. Two models were selected in this studyfor the mean, and four models were studied for the covar-iance of each soil property or crop yield component. For themean, constant mean and linear drift have been examined.For the covariance (Eq. [3]), the four models are: (i) linearcovariance (C); (ii) exponential C; (iii) spherical C, and (iv)Gaussian model for C.
These models are formulated as follows(a) Constant mean a, = 00 [20](b) Linear drift «j = /?„ + /3, x, + 02 x2 [21]
where x, and x2 are the horizontal, two-dimensional spatialcoordinates.(i) Linear covariance: C(r) = a2[l — (a4/a2)][l —
(r/a3)] for r < a3
C(r) = 0. for r > a3 [22](ii) Exponential covariance:
C(r) = 0. for r > a3 [24](iv) Gaussian covariance: C(r) = a2{exp[ — (r/a3)2]
[1 - («4/«2)]}. [25]Here again, r is the separation distance, al is the expected
value, «2 is the variance, and a3 is the correlation rangerelated to the integral scale. Specifically, «3 is equivalent tothe linear integral scale of model (ii) in which estimatedvalues of «3 represent the spatial correlation (i.e., the dis-tance over which a soil property or a crop yield componenttends to be uncorrelated); while «4 is the value of the whitenoise or the nugget, and is a measure of the nonstructuralvariability, i.e., small scale variance. The parameters ft, i =0, ... , 2 are the linear drift constants.
The ML method has been used to estimate «,-, / = 1, . . . ,4, and ftj, j = 0, ... , 2 (depending on the existence or
BRESLER & LAUFER: STATISTICAL INFERENCES OF SOIL PROPERTIES AND CROP YIELDS 1237
nonexistence of a linear trend or drift) for each data set ofany soil property, or its log transformation, and any cropyield component. To derive the maximum likelihood esti-mates of a, and 0,-, the method ofSchweppe (1973) has beenused (similar method was adopted by Kitanidis and Vom-voris, 1983). The procedure involves the minimizing of thenegative log of the MVN (Eq. [1]) for the measured valuesYI, Y2,..., YN. The solution (see Appendix II) was obtainedwith the aid of the MINOS computer program (Saunders,1980), which had been designed to solve general nonlinearminimizing problems. The search algorithm was inaugu-rated as follows: for the parameters /30, 0i, and 02 by a linearregression analysis of the Y data as dependent variables andthe spatial coordinates x{ and x2 as independent variables;for the parameters a2, «3, and «4 from a preliminary vario-gram calculation with a constant number of points to av-erage a given lag r.
DATA SETS OF TWO-DIMENSIONALRANDOM FUNCTIONS
The data to be used for the statistical inference are fieldmeasurements of certain soil properties and a few crop yieldcomponents along with spatial coordinates of the locationsof the observation points in the field from which the mea-surements were taken. Russo and Bresler (1981) provided aset of field-measured data on six different hydraulic prop-erties of a Rhodoxeralf soil at Bet Dagan, Israel. These prop-erties are saturated hydraulic conductivity (Ks), air entry val-ues (hw), sorptivity (5), saturated and residual water contents(6S, and 8n respectively), the constant /J characterizing thesoil water characteristic curve, and the constant f, the powerin the unsaturated conductivity function (both are constantsin the Brooks-Corey equations). Additional soil data whichhave been obtained from the samples taken from the samelocation and at the same time are: soil solution concentra-tion of Na and Ca given as sodium adsorption ratio (SAR),electrical conductivity of the soil solution (EC), and soil tex-ture data (percent clay, percent sand, and percent silt).
Crop yield data were collected from the same sites in thesame Bet Dagan field on two components of peanut yield(Bresler et al., 1981) and sweet corn yield (Stern and Bresler,1983). Additional sets of yield data (Y) and seasonal root-zone averages of water contents (6,, and 8W) were measuredduring the winters of 1980-1981 and 1981-1982 on vetchhay and wheat (Triticum durum Dest). A summary list ofdata sets of 24 spatial random functions is given in Table1. The sample sizes of the data sets are also given in Table1. Note that a sample size smaller than N = 20 does notcontain sufficient information for reasonably accurate esti-mates of the six parameters (if trend is assumed) and evennot for the four parameters (in the constant mean models).On the other hand, data sets sizes > 60 cause computationaldifficulties associated with transforming the covariance mat-rices and inverting them in the ML procedure.
RESULTS OF ESTIMATED PARAMETERSTable 2 summarizes the ML analysis which was per-
formed on each data set (Table 1) using the four Co-variance models with a linear drift (Eq. [21]-[25]). Thistable gives the best estimated values of the vector a,i.e., 0o, £i, 02, «2> «3) and «4, and also the significancelevels of the estimated parameters. For this purposethe Students' t parameters were calculated from /„ =a/a;, and Mests were performed to test the hypothesisof zero a values, i.e., H0:a = 0. The NS superscriptsadjacent to the numbers in Table 2 indicate that theestimated values of these parameters are nonsignifi-cant at the 0.05 level. One asterisk denotes the 0.05level of significance, and two asterisks denote the 0.01
Table 1. A list of variable names, sample sizes and data sourcesof 12 soil properties, four soil variables, and yield componentsof four field crops.
VariableNo. Type Name
Sample———— size SourcesSymbol (JV) of data
1. Soil2.3. Properties4.5.6.78.9.
10.11.12.
13. Soil14.15. Variables16.
17. Crop18. Yield19. Components20.21.22.
23.24.
Saturated conductivityLog scaling parameterPower in unsaturated KPower in retention curveLog in retention curveAir entry valueResidual water contentSaturated water contentSorptivity% clay% silt% sand
Electrical conductivitySodium adsorption ratioAverage 9 for vetchAverage 8 for wheat
New dataNew dataNew dataNew dataNew dataNew dataStern and
Bresler,1983
Bresleret al., 1982
t T = logtlffs/X*)1'2], and e and 0 are Brooks-Corey parameters (for details,see Dagan and Bresler, 1979).
level of significance; all the other nonsuperscripted es-timated parameters in Table 2 are significant at levels> 0.01.
An examination of Table 2 suggests several obser-vations. First, all soil properties and yield componentsare characterized by a nonconstant mean and the ex-istence of a significant linear trend in the data. Thissuggests that the residuals Y' should be considered forthe analysis, and that Eq. [21], in combination withEq. [22] to [25] (with or without a4 = 0), gives muchbetter estimates than Eq. [20] with [22] to [25], al-though the latter may be preferred because it containsless parameters than the former (i.e., 3 or 4, as com-pared with 5 or 6). Second, in general, soil propertiesand yield components are characterized by the exist-ence of a significant (a3 > 0) correlation scale on theorder of 10 to 30 m. Out of the 14 soil parameters andfield properties, a statistically significant nonzero cor-relation scale was calculated in 12 cases, and in onlytwo cases the «3 estimates were not statistically sig-nificant. This does not necessarily indicate a lack ofspatial correlation in these two parameters because thevalues of the nonsignificant correlation lengths («3)were generally comparable with the correlation scaleof the remaining 12 significant cases (excluding theEC case). The latter exceptional case may be indica-tive of the difficulties in determining the range in whichthe autocorrelation exists. All yield components arealso characterized by a significant correlation scale,the value of which is generally smaller than the esti-mated equivalent value for the soil properties (a3 is,in general, between 10 and 30 m for soil variables andbetween 2 and 10 m for various crop yield compo-nents).
1238 SOIL SCI. SOC. AM. J., VOL. 52, 1988
Table 2. The ML values of the estimated parameters,
The third general observation is related to the nug-get effect as represented in Table 2 by the estimatedvalue of a4. Generally speaking, most soil propertiesare characterized by the lack of a nugget effect (esti-mated values of a4 are either zero or nonsignificant).Again, out of the 14 soil properties and field variables,
only five soil parameters show nonzero «4 estimatesbut just one or two exhibit nugget effects with signif-icant estimates of «4 values. These cases were gener-ally associated with soil parameters which are difficultto determine and may therefore show a relatively highmeasurement error. Unlike the soil observations, most
BRESLER & LAUFER: STATISTICAL INFERENCES OF SOIL PROPERTIES AND CROP YIELDS 1239
10o, 0i, and fa (Eq. [21]) and «2, 03, and 04 (Eq. [22]-model I, Eq. [23]-model II, Eq. [24]-model III, and Eq. [25]-model IV), and ML estimates of 0o (Eq.[20]) 02, a3, and at (Eq. [23]-model VI).
t The numbers without superscripts designate the significant level of the estimated parameters greater than 0.001; * and ** designate significance at thelevels of 0.05 and 0.01, respectively; and NS means nonsignificant estimates.
yield components have a nonzero significant nuggeteffect or small scale variability. This probably resultsfrom genetic or other plant variations rather than frommeasurement errors or other small scale effects of soilproperties on crop yields.
A summary of the data relative to the estimatedparameters of the trend, variance, correlation scale,and the nugget, for each soil property and crop yieldcomponent (Table 2), indicates that the covariancesof the random functions in Bet Dagan are describedin a similar manner by all four models. In general, theestimated parameters (except «4 and a few «3 cases)are highly significant in all the models.
CRITERIA FOR SELECTING THE "BEST"COVARIANCE MODEL
The maximum likelihood (ML) method leads to op-timum parameter estimates for one of the models givenby Eq. [20] to [25], with no indication which of themodels provides the best estimate of the real system.For those cases in which all parameters in all fourmodels are statistically significant at the same level ofsignificance, several objective criteria, based on thelikelihood concept, can be used to rank the alternativemodels. Carrera and Neuman (1986) suggested fourdiscrimination models by which the most appropriateML model can be identified,
1240 SOIL SCI. SOC. AM. J., VOL. 52, 1988
Table 3. Model structure discrimination for the data sets of Table Table 3. Continued.1 and for tl
(iii) *(«) - 2L(a) + CMln[ln(7V)] [28](iv) d(a) = 2L(a) + In (N/2*) + In \F\ [29]
Here, a is the parameter vector as estimated by theML method, L is the minimum value of the negativelog-likelihood for the fitted model, N is the number of
observations points, Mis the dimension of significanta, i.e., the number of independently adjusted signifi-cant (suitable) parameters of the model, C ^ 2 is aconstant (normally C = 2), and \F\ is the determinantof the Fisher information matrix (see Theil,, 1971).The "best" selected model is the one with a minimumcriterion (AIC, BIC, $, or d) value. Obviously, wheneverything but M is the same, the model with thesmallest number of estimated parameters (M) will have
BRESLER & LAUFER: STATISTICAL INFERENCES OF SOIL PROPERTIES AND CROP YIELDS 1241
the lowest criterion value and therefore be selectedwith its estimated values of a being used as the bestchosen model. This criterion must be accompanied bythe requirement that these values of a are statisticallysignificant at any desired level.
MODEL STRUCTURE DISCRIMINATIONThe eight possible models given by Eq. [20] to [25]
are denoted as follows: (i) For linear drift, I-Linearcovariance, II-Exponential covariance, Hi-Sphericalcovariance, and IV-Gaussian covariance, (ii) For con-stant mean, V-Linear, VI-Exponential, Vll-Spherical,and VIII-Gaussian. To identify the best model struc-ture among the eight options given by these combi-nations, the criteria in Eq. [26] to [29] have been ap-plied to each model using the data sets of Table 1.Results of structure discrimination of five differentmodels are summarized in Table 3. The table lists, foreach set of data of soil properties or crop yield com-ponents, the minus log-likelihood L of Eq. [A5], thefour discrimination criteria (Eq. [26]-[29]), and theoptional models designated by the Roman numbers.All four options of the linear drift model (I-IV) arelisted in the table but only one—the best constant meanmodel—is given for illustration purposes. It should beemphasized that since, generally, the estimated nuggetparameters («4) are zero or statistically insignificant(see Table 2), the number of independent and signif-icant estimated parameters for the data given in Table2 is either M = 5 (/30, ft, ft,, a2, and «3, for the lineardrift models) or M = 3 (/30, a2, and «3). The preferredmodel among the five listed in Table 3 for each soilproperty or crop yield component, is the one with thelowest discrimination criterion. This model is markedby the asterisks attached to the appropriate numberin the table.
An examination of Table 3 shows that, generally,the constant mean model is not necessarily the pre-ferred, although it contains two less significant param-eters. It is also important to note that generally onlysmall differences exist between the various values ofthe calculated criteria, for the same number of param-eters and the same sample sizes. This may indicatethat one can use either one of models I to IV with thesame level of confidence. A similar conclusion mayalso be drawn for models V to VIII because the sametendency was observed. The fact that the five-param-eter models, although containing two more significantparameters, generally have smaller discrimination cri-teria than the three-parameters models (after elimi-nating the trend from the original data), suggests thesuperiority of the former over the latter, at least forthe Bet Dagan field.
INFERRED EXPECTATIONS, KRIGEDVALUES, VARIANCES, AND VARIANCES OF
ESTIMATED VARIANCESGiven the ML estimates of /? and a based on the
measured values vector Y (Table 2), we can now pre-dict (see Appendix III) the value ofE(y), its variance,and the variance of its variance at any arbitrary pointx in the field. It should be remembered that X, andhence £(x) and »j(x, x) depend on a3 in a nonlinearmanner. It is assumed, for simplicity that «3 is deter-
ministic so that X/ are also deterministic variables.Hence, only a{ (including ft) and a2=a2
y are random.The variance-co variance matrix of a and /8 needed forthe inference procedure (Eq. [All] and [A 12] is takenfrom the inverse of the Fisher information matrix (seeTheil, 1971, Chapter 8), which gives values of al, Sjand the associated covariances.
In the estimation procedure we divided the entirefield domain into a rectangular grid containing K =441 nodal points distributed regularly. Subsequently,at each of the TV locations of coordinate x,, the valuesof X/x) are determined by solving the linear system ofEq. [14] with C(x}, xk) taken from Eq. [22] to [25].Then the predictor E(y(x)^), the estimated varianceof the predicted error a2
E, and the estimate of its var-iance, were computed by Eq. [Al 1], [A12], and [A13],respectively, at each of the A!" nodal points of the grid.Results, using all covariance models, were obtainedfor all the 24 random functions (see Bresler and Lau-fer, 1986), but to economize on space only a few rep-resentative numerical values are presented in Tables4 and 5.
Effect of the Covariance ModelThe contour lines' maps of the estimates of the pre-
dictor of E[y(x)], its variance or error, and the vari-ances of the predictor error, using all the covariancemodels and for the 24 random functions were givenelsewhere (Bresler and Laufer, 1986). Small part of thenumerical values that were used to construct thesecontour maps are demonstrated in Table 4 for onesoil property (Ks) and one crop component (dry matterof wheat). It can be seen in these examples that thespecific covariance model used has a negligible smalleffect on the estimated results of both variances andexpectation. In addition, these models do not affectthe quality of the expectation estimates to serve asinterpolators of the observation points and not theuncertainty (or error) involved in the interpolation andits uncertainty which is related to the conditional sec-ond-order statistics.
Comparison with Kriging MethodThe analogy between the kriging method (Del-
homme, 1 978) and the estimated CMVN method (firstintroduced by Feinerman et al., 1986) may be attainedfrom the following. Since /(x) = y(x) — a,(x) hasstationary increments, one may use the variogramwhich in cases of bounded variance (a2) is given by
r(x,, XH) = ff2 - C(x,, xn) . [30]
Substituting Eq. [30] into [14] yields for X,
= T(x, k = 1, 2, . . . , N [31]For a finite value of [a2[\ — 2jLi X/x)]} and for a verylarge variance of y (i.e., a2
y —> °°), then
H - ;1 -» 0. or, x,. = 1 . [32]j-\ j-\Hence, the conditional expected value E\y(x) Y] as ob-
1242 SOIL SCI. SOC. AM. J., VOL. 52, 1988
Table 4. Numerical examples of the estimated expectation (Eq.[All]), estimated variance (Eq. [A12]), and variance of estimatedvariance (Eq. [A13J of two random functions (Ks and dry mat-ter of wheat) in nine locations of coordinate (xj, X2) as inferredusing the four covariance models and kriging.
tained by kriging to spatial random functions of un-bounded variance (a2
y —> °°), with a well denned var-iogram, and of stationary increments, is identical tothat of Eq. [9], whereas the conditional covariance inEq. [16] is replaced in kriging by
C(Xl,
x,,) . [33]
Note that in the case of a bounded variance, the re-sults based on the CMVN probability may differsomewhat from those which are based on kriging (Ta-ble 5). Furthermore, since the parameters ab a2, and«3 are estimates and are not known with certainty,then the kriging approach may give different estimatesof the conditional moments Eq. [9] and [33] than thosebased on the CMVN probability approach Eq. [All]and [16]. Note that for better accounting of the un-certainty in the inference of the conditional moments,the estimated parameter «3 should have been consid-ered as random rather than deterministic. This wouldhave resulted in larger differences between the krigingand the MVNC approaches, but would largely com-plicate the calculations of the stochastic, rather thanthe deterministic vector X.
Table 5. Numerical examples of kriged values, estimated expec-tation (Eq. [All]), kriging variance, estimated variance (Eq.[A12]), of wheat grain in 48 spatial points of coordinates (Xi,X2). The exponential covariance model (II) was used for thecalculations.
For the comparison between the CMVN method ofstochastic interpolation and the traditional krigingmethod (Journel and Huijbregts, 1978), the same K= 441 grid points were used. The variograms wereassumed to have the analytical exponential form ofEq. [23] substituted into Eq. [30] with the unknownparameters a being estimated by the same ML pro-cedure (Table 2). Results of the contour maps of krigedvalues and the kriging variances of all the 24 data sets(Table 1) were superimposed on the CMVN resultsand were presented in Fig. 6 and 7 of the BARD report(Bresler and Laufer, 1986). The maps showed that most(18 out of 24) of the estimated predictors obtained bythe kriging method were identical to the results of theCMVN procedure. In six cases the results differed butwere close, as is seen in the case demonstrated in Ta-ble 5, in which the expectation contours deviate themost from kriged values. The two methods yield alsoclose, but not necessarily identical, predication vari-ances (see Tables 4 and 5). Generally, the estimatedvariances of the two methods differ in areas more re-mote from the measured points, and also the krigingvariances are smaller than those based on the CMVNdistribution.
It should be noted that the estimated variance ofestimation of the variance (Eq. [A 13], Table 4), cannotbe obtained by the classical kriging approach, becausekriging variance is considered to be deterministic. Thedata illustrated in the last three columns of Table 4may serve as a measure of reliability of the data ofestimated variance results. The variance data may alsoserve as a guide for selecting additional observationpoints in future experiments (see Russo and Bresler,
BRESLER & LAUFER: STATISTICAL INFERENCES OF SOIL PROPERTIES AND CROP YIELDS 1243
1982). The estimates a2 and a\2 appear to be a signif-icant factor in economic optimization of the com-mercial yield F(Feinerman et al., 1985), which in turnmay depend, in a linear or nonlinear manner, on thesoil properties, and both are spatial random functions.
SUMMARYIn the present paper the multivariate normal meth-
odology for statistical inference of spatial randomfunctions has been applied to inferred expectations,variances and variances of the inferred variances ofcertain soil properties and crop yield components ina Bet Dagan field. Inferences of the spatial distributionof the statistical moments of the properties, as normalrandom spatial functions, are based on a finite set ofmeasured values. In this inference method the uncer-tainty of estimation of the parameters characterizingthe multivariate normal pdf is incorporated in a sys-tematic manner using classical statistics theory in theprocess of inference, prediction, and conditioning onthe actual measured data. The method has been ap-plied to 12 soil properties (Ks, e, /3, In 0, T, hm 8S, Bnpercent clay, percent sand, percent silt, and 5), foursoil variables (SAR, EC, average 6n and 6W\ and twocomponents of four crops (vetch for hay, wheat, corn,and peanuts). The procedure has been applied to casesin which a linear stochastic trend is presented.
ACKNOWLEDGMENTThis study was supported by Grant US-420-81 from
BARD, the United States-Israel Binational Agricultural Re-search and Development Fund.
APPENDIX/. Estimation of the Conditional Covariance (Eg. [12])
The estimate of the conditional covariance (Eq. [12]) isobtained as follows. Add to and subtract from the estimateC(x,, x,,|Y) = E[[yi - E(yM][yu - E(yn^)\]\Y\ [Al]the expectations E(y{Y) and E(yn\Y) and obtain:C(x,, x,,|Y) = E{[y{ - E(y,\Y)
[A2]Rearrange [A2]C(x,, x,,(Y) = E{[(yt -
E{(A E{(H D + F
» - E(yu |Y]; .
[A3]where
BH = \y, -D = [£(y,,|Y) -£(y,,|Y)] = K = CONSTANTF = [£(y,|Y) - £(j>,|Y)] = K = CONSTANT
Using [9], [1 1] and [12] and because K E[H] + KE[G] =0 + 0 = 0, then,C(x,, = E{(A E{(H D + F G)|Y}
[A4]
II. Estimations of the Parameters a and ft (Eg. [20] to [25])by the Maximum Likelihood (ML) Method
The problem is to select the parameter vectors a and ffsuch that the negative log of the MVN Eq. [1] is minimized.Hence, the problem is to minimize L with the best fitted aand /?, where
L = A/2 ln(2ir) + 0.5 [In \Q + b Y'] [A5]C is the (symmetric) covariance matrix C(r,7) of one of themodels (Eq. [20]-[25]) with r,y = |x, — xj, and Y' the resid-uals and b are vectors defined by
[A6][A7]
Y' = Y - £(Y) = Y - a,b = Y' rC-'.
The minimizing procedure is carried out with the aid ofthe computer code MINOS (Saunders, 1980) as follows:
1. Input: (i) the first guess of the parameters a and /? andtheir uppermost and lowest possible values (first guesses ofP0, 0!, and /32 are obtained from a linear regression analysisof Eq. [21]; first guesses of «2, a}, and «4 are obtained froma preliminary calculation of the variograms); (ii) an indexfor the selected covariance (variogram) model from Eq. [20]to [25]; (iii) number of the data points (the sample size N);(iv) N values of the vector Y and its respective coordinates
2. Calculate: (i)r/, = |x, — xj; (ii) C from the selected model(one from I through VIII); (iii) the determinant |C|; (iv) thevector Y' from [A6] using a, of Eq. [20] for the nondriftcases or of [21] for the drifted cases; (v) the vector b aftercalculating the transpose T of Y' and the inverse of C.
3. Calculate the partial derivatives: (i) for each element ofthe vector /8
[A8]
[A9]
= band (ii) for each element of the vector adL/da = 0.5 Tr [C~l (dC/det)]
- 0.5 b(dC/<3a)(C-')Y'where Tr is the trace (i.e., sum of diagonal terms). Equations[A8] and [A9] transfer scalar values to MINOS in which thebest fitted a and /? are estimated.
4. Using the estimated values of a and ft calculate the Mjkelements of the Fisher Information matrix (Theil, 1971) fromMJk = 0.5 Tr[C-l(dC/dOj)C-l(dC/dOk)
+ (dcuWpC-^daiWk)where 0,, 6k are the best fitted (estimated) parameters a and0.
5. Calculate the estimated variances of the estimated ex-pectations of the parameters a and 0 (i.e., al), and the as-sociated covariances, from the inverse of the Fisher infor-mation matrix.
6. Calculate the Students' t statistics from t, — 6/0,--.1. Print all 6 (i.e., «,, a3, a4, j30, 181, 0->, L, Mu, Q), and all
, andIII. CalculationsThe computation procedure to predict the expectation,
the variance, and the variance of the predicted variance (ofeach soil property and crop component and at any point xin the field domain) involves several steps, as follows:
1. Select the "best" covariance (variogram) model fromone of the models of Table 2 (Eq. [20] through [25]).
2. Input measurements values Y = y,-, j = 1 , 2, . . . , N,the respective space coordinates x,, values of a = J30, /?,, |92,
1244 SOIL SCI. SOC. AM. J., VOL. 52, 1988
a2, a3, a4 from Table 2 and values of£«„, */>„ • • - , ̂ , C(P0, &), C(p0, /?,) C(fr, P2) from the var-iance-cpvariance matrix of the ML output (see Appendix II).
3. Divide the field into Nx X Ny = K grid points, whereNx and Ny are number of grid points in the x = xt and y= x2 directions, respectively.
4. Calculate the lag \tj = |x, — xj, ij = 1, 2 , . . . , N, andthe lag rkm = \xk - \m\, k, m = 1, 2, . . . , K, and computethe covariance matrix C(r) from the model selected fromTable 3 and Eq. [22] to [25]), using the best-fitted and sig-nificant parameters a (Table 2).
5. Calculate X,- from Eq. [14] or [15], |(x) from Eq. [11]and Ti(xh xn) from Eq. [13]. Note that since, in general, asignificant linear drift has been inferred (Table 2), then«i(Y)£(x) in Eq. [9] is given by
«,(Y)|(x) = £ ft £,(x); [A10]i-O
where,N
£o(x) = 1 - I X,-;1=1
N N
£,(x) = x, - £x,- 4 £2(x) - x2 - X Xj x{7=1 7=1
where xb x2 are the horizontal space coordinates, and /3, arethe ML estimates for a,(Y) (Eq. [21]).
6. Calculate%(x)|Y] = 3o£(x) + 3,l,(x)
A1
+ tex) + S\<x)^ [AH]7-1
from Eq. [9] and [A 10].7. Calculate
*KX) = 527,(X) + ^2(x) + £2,£2(x) + J2^2(x)
+ 2Ca^^(x)^(x) + IQ^Wtfx) + 2Q2a,|,(x)^(x)
[A12]from Eq. [18] but after replacing the nondrifted term<rl^2(x) by the relevant drifted one.
The variance of estimated covariance, or variance (Eq. [A13])is related exclusively to the uncertainty of the estimation of
2Note'that in Eq. [All], [A12], and [A13] x, = x,, = x,and that the estimated parameters a3 and a4 are assumed tobe deterministic (nonstochastic) parameters, so that the aboveare appropriate versions. If the significant nugget is sto-chastic such that
C(x,, x,,) = 0:4 5UI + a2 P(\I,XU) [A 14](where {M, is the Kronecker delta) then a2 jj(x, x) in Eq. [12]and [16] are replaced by
N
a2 + «4 - «2 Z X7 P(X> x>) • [A1517-1
Note also that in case of a significant nugget, the values ofoi4 and of C,iatri(x, x) should be added to Eq. [A 12]. It is
assumed, however, that all aji,, Q,.̂ ., /', j = 0, . . . , 2,and a«3 (a} being the scale) are deterministic.
9. Output: K values of Efy(\)\\)], alfx), and CT|X,X)(X)and the associated field coordinates.
