Estimation of the Parameters of Isotropic Semivariogram Model ...

Applied Mathematical Sciences, Vol. 9, 2015, no. 103, 5123 - 5137

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.54293

Estimation of the Parameters of Isotropic

Semivariogram Model through Bootstrap

K. N. Sari

Department of Mathematics and Natural Sciences

Bandung Institute of Technology, Indonesia

U. S. Pasaribu



O. Neswan



A. K. Permadi

Department of Petroleum, Bandung Institute of Technology

Indonesia

Copyright © 2015 K. N. Sari et al. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

In isotropic semivariograms, ordinary least squares can estimate nugget effect and

sill by partitioning its range. By conducting simulation, a semivariogram model

with previously given parameters will be estimated through bootstrap method. Least

square-bootstrap (LS-Bootstrap) will be applied to estimate the parameters of the

model after resampling the errors of the model. The selection of the resulting

semivariogram model from bootstrap method will be affected by the number of

distance lags, the precision level of the range partitions, the number of bootstrap

iterations, and the given reference model. The exponential and Gaussian models are

sufficiently good in the estimation for the models with the same references. Meanwhile, the estimation yielded from spherical model is quite far from the reference

5124 K. N. Sari et al.

exponential and Gaussian models, with the mean square error value reaching 713.

The estimation with bootstrap method which is the same as the reference model will

be faster to converge with the maximum iteration of 50. Besides, bootstrap method

enables to obtain the point estimates and interval estimates of the nugget effect, sill,

and range parameters.

Keywords: Bootstrap, confidence interval, isotropy, least square, semivariogram

1. Introduction

There are some well-known resampling methods such as randomization test,

cross-validation, bootstrap, and jackknife. These methods are often applied in

regression, time series, and principle component analysis. Randomization test has

been used to test linear independence between random variables in wavelength

selection in near-infrared spectral analysis [1]. Cross-validation has been applied to

estimate unbiased prediction error [2]. Both of bootstrap and jackknife have been

to estimate residuals of a predictor of the random variables and apply in model

autoregressive (AR)-sieve bootstrap model in time series [3]. Both also have been

used to give stability index of principle component analysis result [4].

Resampling methods above are also widely used in geostatistics especially

bootstrap and jackknife. Those methods often applied in kriging. Kriging is a

method of calculating estimates of regionalized variable at a point, over an area, or

within a volume, and uses as a criterion the minimization of an estimation variance

[5]. Until now, kriging used jackknife to obtain accurate estimate for variable values

observed by removing one by one observation values in a certain location, and then

estimating the parameters with the rule that the square error has to be as small as

possible. While, Den Hertog et al. [2006, 6] developed bootstrapped kriging to

estimate the predictor variance as a function for an unobservable location. In

application, Waterman [7] used weighted jackknife-ordinary kriging in estimation

of gold and copper ore deposits at Grasberg, Papua. That method was better than

jackknife ordinary kriging method because it was used for data that had outliers and

not symmetrical.

In kriging, one method used to estimate the value at an unobservable location

is determining semivariogram model. Semivariogram is a measure of variance from

difference between two spatial locations that separated by certain distance.

Semivariogram is divided based on the presence or absence of the influence of the

angle between pair of locations respectively called anisotropic and isotropic

semivariogram. That model has 3 parameters such as nugget effect, sill, and range.

Nugget effect is initial semivariance when autocorrelation is highest; or just the

uncertainty where distance (d) is close to 0, sill is the value that semivariance flat,

and range is lag distance where the sill is reached. There are 7 semivariogram

models such as: nugget effect, linear, spherical, exponential, power functions,

Estimation of the parameters of isotropic semivariogram model… 5125

Gaussian, and hole effect. Mostly, the model commonly used are exponential,

Gaussian, and spherical. The three parameters and semivariogram models are

illustrated in Figure 1.

Figure 1. Plot of semivariogram models (γ(d)) and 3 parameters model (nugget

effect, sill, and range)

The main problem in semivariogram modeling is to estimate parameters model.

Till now, those parameters are estimated by applying numerical methods because

the nonlinearity of the semivariogram function involved. Several methods have

been proposed to estimate parameters of the semivariogram models, such as: least

squares [8], generalized least squares [8], maximum likelihood [9], restricted

maximum likelihood [9], and weighted least squares. Zimmerman and Zimmerman

[10] found that weighted least squares is sometimes the best procedure and never

does badly, whereas the others are subject to erratic behavior in some requirements

has made it the primary choice among semivariogram model estimation methods.

Some background for weighted least squares method is the ordinary least squares

and generalized least squares.

In that estimation, the number of lag distance as sample points only a few

frequently, so it needs the nonparametric techiques such as jackknife and bootstrap

to add sample point. Along with the development of computation field, the

nonparametric technique are developed rapidly. Moreover, using of resampling

methods in the estimation semivariogram model is still limited. Therefore, this

paper will applied bootstrap method and the least square method to estimate the

vector of parameters isotropic semivariogram model. It’s expected that the

parameter estimation results obtained from applying least square-bootstrap (LS-

Bootstrap) method will be close to the given semivariogram model within a certain

mean square error.


2. Estimation of The Parameters of Isotropic Semivariogram

Model

2.1 Notation

The following notations will be used to formulate mathematical symbols that

needed to study the estimation of parameters semivariogram model:

d - Distance between pair of locations

0θ , 0θ̂ - Nugget effect and its estimator

1θ , 1θ̂ - Sill and its estimator

2θ , 2θ̂ - Range and its estimator

0 1 2, , 'θ θ θθ - Parameters vector

0 1 2ˆ ˆ ˆˆ , , 'θ θ θθ - Estimated model-parameters vector

0 1 2ˆ , ,B θ θ θθ

- Estimation of parameters vector by bootstrap

1ˆ( )

ˆ( )

ˆ( )n

γ d

γ d

γ d

- Empirical semivariogram

1( , )

( , )

( , )n

γ d

γ d

γ d

θ

θ

θ

- Semivariogram model

2

ˆ( ) ( , )d

γ d γ d θ

- Sum of the squared residuals

1

( )n

i

i

d

- Sum of the experimental semivariogram where n is

the number of distance lag

ˆ

1

1 i

nd

i

e

- Sum of the exponential function

ˆ

1

( ) 1 i

nd

i

i

d e

- Sum of the multiplication between the experimental

semivariogram and the exponential function

2ˆ

1

1 i

nd

i

e

- Sum of the squared the exponential function

2

ˆ

1

1 i

nd

i

e

- Sum of the Gaussian function

2

ˆ

1

( ) 1 i

nd

i

i

d e


semivariogram and the Gaussian function

2

2

ˆ

1

1 i

nd

i

e

- Sum of the squared the Gaussian function


3

1

ˆ ˆ1.5 0.5n

i i

i

d d

- Sum of the spherical function

3

1

ˆ ˆ( ) 1.5 0.5n

i i i

i

d d d


semivariogram and the spherical function

2

3

1

ˆ ˆ1.5 0.5n

i i

i

d d

- Sum of the squared the spherical function

p - Number of lag distance

B - Number of bootstrap iteration

2.2 Estimation of parameters

This research applies ordinary least squares to find the estimator of θ by

minimizing 2

ˆ( ) ( , )d

γ d γ d θ . In ordinary least squares, all the empirical

semivariogram are assumed to be uncorrelated with each other and to have the same

variance. The objective function to be minimized in ordinary least squares is can be

written in matrix notation as ˆ ˆ( ) ( , ) ' ( ) ( , ) 'γ d γ d γ d γ d ε ε θ θ .

Through ordinary least squares method, obtained θ̂ . Range will be determined

first by partitioning it using systematic random sampling by taking minimum

distance interval until maximum of lag distance. Then, nugget effect and sill will

be estimated using ordinary least squares method, as in the estimation of the

parameters in simple linear regression model. In this paper, parameter estimations

are conducted for 3 semivariograms, i.e. exponential (exp), Gaussian (gauss), and

spherical (sph). These models are respectively formulated as follow:

exp 0 1

2

ˆ ˆˆ ( ) 1 expˆ

dγ d θ θ

θ

,

2

gauss 0 1

2

ˆ ˆˆ ( ) 1 expˆ

dγ d θ θ

θ

, and

3

sph 0 1

2 2

ˆ ˆˆ ( ) 1.5 0.5ˆ ˆ

d dγ d θ θ

θ θ

.

2.3 Parameters estimation for exponential model

Let 2

1ˆ

ˆ

,

1

( )n

i

i

d a

, ˆ

1

1

1 i

nd

i

e b

, ˆ

1

1

( ) 1 i

nd

i

i

d e c

and

2ˆ

1

1

1 i

nd

i

e d

then the estimators are 1

0 1ˆ ˆ ba

n n

and

11

1 2

1

1

ˆ

abc

n

bd

n

.


To obtain the estimation of nugget effect, the sill will be estimated first. Meanwhile,

the range will always appear in the two parameters. By considering exponential

model form, the range can be written as

1

0 1

ˆlnˆ

ˆ ˆ ˆln ( )i id d

.

2.3 Parameters estimation for gauss model

Let 2

1ˆ

ˆ

,

1

( )n

i

i

d a

, 2

ˆ

2

1

1 i

nd

i

e b

,

2ˆ

2

1

( ) 1 i

nd

i

i

d e c

and

2

2

ˆ

2

1

1 i

nd

i

e d

, then the estimators are 2

0 1ˆ ˆ ba

n n

and

22

1 2

2

2

ˆ

abc

n

bd

n

.

If range is also estimated by applying ordinary least square, the formula will be

obtained as follows:

2 2 2

ˆ ˆ ˆ2 2 2

0 1

1 1 1

ˆ ˆ( ) i i i

n n nd d d

i i i i

i i i

d d e d e d e

2 2ˆ2

1

1

ˆ 0i

nd

i

i

d e

To estimate range, the solution for 2

ˆ1 id

iu e

should be computed, also the

solution of 2 2 2 2 2

0 1 1

1 1 1 1

ˆ ˆ ˆ( ) 0n n n n

i i i i i i i i i

i i i i

d d u d u d u d u

.

2.4 Parameters estimation for spherical model

Let 2

1ˆ

ˆ

,

1

( )n

i

i

d a

, 3

3

1

ˆ ˆ1.5 0.5n

i i

i

d d b

, 3

3

1

ˆ ˆ( ) 1.5 0.5n

i i i

i

d d d c

and 2

3

3

1

ˆ ˆ1.5 0.5n

i i

i

d d d

, then the estimators are 3

0 1ˆ ˆ ba

n n and

33

1 2

3

3

ˆ

abc

n

bd

n

.

If the range is also estimated using ordinary least square, then the formula in the

form of polynomial will be obtained: 2 3 5ˆ ˆ ˆ ˆ 0 p q r s t where

2 3 3 4

1 0 1

1 1 1 1

ˆ ˆ ˆ1.5 , ( ) , 2 ,n n n n

i i i i i

i i i i

p d q d d d r d

6

1 0

1 1 1

ˆ ˆ0.5 , ( )n n n

i i i i

i i i

s h t d d d

.

Each of the parameters estimation above contains another parameter such as

range and sill. The range always appears in nugget effect and sill formula,

meanwhile the sill always appears in nugget effect formula. With this reason, the

process of estimating parameter can only be executed numerically, by computing

the range first as written above. Along with the rapid development in computation

methods, the parameter estimation methods are also developing, i.e the ones using

nonparametric methods.


2.5 Bootstrap method

Bootstrap method, initially introduced by Efron in 1979. This is a procedure

for resampling data with replacement, expecting the samples can represent the

population. Bootstrap enables to do statistical inference without making any

assumptions of strong distributions for any estimator’s sampling distribution or

without any available analytical formulation. The data used doesn’t need to be

assumed strictly in its distribution, it is small-sized and original, not resulting from

any simulation [11].

In the estimation of the semivariogram parameters, if real data is unavailable,

then it is a must to do simulation to generate normally distributed errors with 0 as

the mean and 1 as the variance which will be added to semivariogram model to

obtain the semivariogram values considered as experimental semivariogram.

Isotropic semivariogram model can take one of the models: exponential, Gaussian,

and spherical. Then bootstrap method will be applied to resample the errors of

semivariogram model to obtain the estimations of model parameters as many as the

bootstrap iterations. Semivariogram model resulting from bootstrap will be

obtained by taking average of those three parameter estimations of the model.

Bootstrap procedure for estimation will be described in Chapter 3.

3. Results and Discussions

3.1 LS-Bootstrap Procedure

Firstly, least square is used to estimate effect nugget and sill by partition its

range, the following steps are conducted:

i. Suppose one of the reference semivariogram model: (1) exp, (2) gauss, or (3)

sph with respectively parameters vector 0 1 2, ,θ θ θref ref ref

θ .

ii. Select p for forming semivariogram function.

iii. Select d1, d2, ... , dp with d1 < d2 < ... < dp. di for i = 1,2,...,p is choosen

systematically with divided distance interval 2 times range with p dan d1 will

be fixed randomly.

iv. Compute the expected semivariogram ˆ( ), 1,2,...,iγ d i p with di on iii.

v. Take iε is normal distribution with mean 0 and variance 1.

vi. Add , 1,2,...,iε i p to the ˆ( )iγ d .

vii. Estimate parameter vectors, i.e. 0 1 2ˆ ˆ ˆˆ , ,θ θ θθ by least square for respectively

the semivariogram model: (1) exp, (2) Gauss, or (3) sph as estimation model.

viii. Make 3 semivariogram model *ˆ ( )iγ d (exp, Gauss, and sph) with input the θ̂

on vii.

ix. Compute the mean square error (MSE), formulated as 2

*

1

1ˆ ˆ( ) ( )

n

i i

i

γ d γ dp

.

x. Repeat step i.


From the 1st procedure, the least square estimator estimator θ̂ is close to θ , in other

word called unbiased estimator.

Secondly, the bootstrap is used to resample the error of semivariogram model

from the first procedure above. The following steps are conducted:

i. Suppose one of the reference semivariogram models: (1) exp, (2) gauss, or

(3) sph with respectively parameters vector 0 1 2, ,θ θ θref ref ref

θ .

ii. Compute the errors *ˆ ˆ( ) ( ), 1,2,...,i i iε γ d γ d i p .

iii. Define p for each semivariogram model.

iv. Take iε as the bootstrap sample with p as the size and B as the number of

bootstrap repetitions.

v. Add , 1,2,...,iε i p to the ˆ( )iγ d .

vi. Estimate B parameter vectors, i.e. 0 1 2ˆ ˆ ˆˆ , ,j j j jθ θ θθ , 1,2,...,j B by least square

for respectively the semivariogram model: (1) exp, (2) gauss, or (3) sph as

estimation model.

vii. Compute the average of the estimated parameter vectors 0 1 2ˆ , ,B θ θ θθ where

0 0

1

1 ˆ ,B

i

i

θ θB

1 1

1

1 ˆB

i

i

θ θB

and 2 2

1

1 ˆB

i

i

θ θB

.

viii. Make 3 semivariogram models *ˆ ( )iγ d (exp, Gauss, and sph) with input the

parameter estimation on vii.

ix. Compute the MSE, formulated as 2

*

1

1ˆ ˆ( ) ( )

n

i i

i

γ d γ dp

.

x. Repeat step i.

The best estimated model can be determined by finding the one with the least MSE.

Besides, by using bootstrap, the (1-α)% confidence interval can be estimated for the

parameter vector, where α is the significance level.

3.2 Simulation of the number of sample point and Isotropic Semivariogram

Model

The data used will be the permeability of a reservoir at Jatibarang field,

Indonesia. Jatibarang reservoir is one of the famous reservoirs because of its special

characteristics. There are volcanic stones with fractures and low sulfur content. The

volcanic layer is the largest oil producer among the Jatibarang reservoirs. This

reservoir is located in the north of West Java and the oil field area has an elongated

position +10 kilometers north-south and +16 kilometers west-east. Since 1969,

there have been +200 opened and in 1998 the production reached a cumulative

production of nearly 13 million m3 [12].

Out of the existing 132 wells, 12 wells have been selected through

systematic random sampling to be analyzed with spatial analysis. The data consists

of the coordinates of the selected 12 wells and k-fracture values. k-fracture values

show the oil permeability near a well in milli-Darcy (mD) field unit. By applying

Surfer software, obtained 3 semivariogram models with the parameter estimates as


follows: expˆ ( ) 171.2 245 1 exp

1.900

dγ d

,

2

gaussˆ ( ) 125.4 185 1 exp

0.552

dγ d

and 3

sphˆ ( ) 29.32 268.1 1.5 0.5

0.881 0.881

d dγ d

.

The selection of p was simulated to form three isotropic semivariogram

models. p was taken systematically but the first distance interval will be fixed

randomly, later will be called as sample point. p is selected 6 sample points with

consideration of many lag distance may be formed for a Gaussian model that has a

curve inflection. Furthermore, by doing 1st procedure, the result of estimation of

the parameter vector is presented in Table 1 and Figure 1.

Table 1 The parameter vector and MSE for three semivariogram models (exp, gauss, and sph).

The reference model The parameter vector and MSE

Exp Gauss Sph

Exp

(171.2, 245, 1.900)

(155.39, 264.83, 1.98)

81.6

(209.75, 183.97, 1.98)

115.5

(189.26, 157.50, 1.98)

519.4

Gauss

(125.4, 185, 0.552)

(69.87, 265.13, 0.53)

174.5

(116.56, 197.79, 0.53)

36.4

(111.41, 148.72, 0.53)

1,156.5

Sph

(29.32, 268.1, 0.881)

(77.95, 292.26, 0.95)

802.2

(138.89, 196.62, 0.95)

1,170.6

(103.42, 188.86, 0.95)

303.7

.(a) (b)

(c)

Figure 2 Semivariogram model plots for the three reference semivariogram (a) the exponential

reference model, (b) the Gaussian reference model, and (c) the spherical reference model.


Figure 3a MSE plots for the three semivariogram models with p sample points and B number of

bootstrap iterations (25, 50, 75, and 100) for the exponential model as the reference.

Figure 3b MSE plots for the three semivariogram models with p sample points and B number of

bootstrap iterations (25, 50, 75, and 100) for the Gaussian model as the reference.


Figure 3c MSE plots for the three semivariogram models with p sample points and B number of

bootstrap iterations (25, 50, 75, and 100) for the spherical model as the reference.

From Figure 2a, exponential model is the best model that estimated its

reference model with point estimator for each model parameter is closed and the

MSE value is the smallest reaching 81.6. The spherical model has the highest MSE

to estimate the exponential as the reference model. From Figure 2b, Gaussian model

is the best estimator for Gaussian model as the reference model with MSE reaching

36.4. That estimation is followed by exponential model that estimate better than

spherical model because exponential model is base form from Gaussian model.

While from Figure 2c, the spherical model can estimate better its reference model

than exponential and Gaussian model. The conclusion, estimation of parameter

vector using ordinary least square with certain reference model could be estimated

by same model with MSE value less than 100 for exponential and Gaussian model,

and MSE value less than 500 for spherical model.

Figure 3 show that exponential model always close to Gaussian model. From

Figure 3a, if the reference model is exponential, it is clear that exponential model

could estimate better than Gaussian model, while spherical model gave a far model

estimates with MSE reaching 713 in the 50th and 75th bootstrap iteration. By

selecting 4 sample points and 50 bootstrap iterations, exponential model can

estimate the reference model very well with the MSE is 56. Then, for each bootstrap

iteration, some repetitions are applied to see the convergence of MSE values. If less

than 10 number of sample points is selected, for the estimations using exponential,

Gaussian, and spherical models, the MSE will converge respectively at the 50th,

50th, and 75th iteration. It can be concluded that if the exponential model as the


reference is estimated using Gaussian and spherical models for small-sized samples

(not more than 8), then the MSE values for each models are, respectively, 70 – 348

and 290 – 662. While for big-sized samples (more than 8), the MSE values for each

models are 134 – 230 and 577 – 713. So, for those three models, it is sufficient to

take 4-6 sample points.

If the reference model is Gaussian in Figure 3b, it is clear that Gaussian model

will be the best estimates, followed by exponential model which has slightly

different model form. By selecting 6 sample points and 50 bootstrap iterations,

Gaussian model can estimate the reference model very well with the MSE is 39. If

less than 10 number of sample points is selected, for the estimations using

exponential, Gaussian, and spherical models, the MSE will converge at the 50th

iteration for all models. It can be concluded that if the Gaussian model as the

reference is estimated using exponential and spherical models for small-sized

samples (not more than 8), then the MSE values for each models are, respectively,

118 – 279 and 681 – 1,357. While for big-sized samples (more than 8), the MSE

values for each models are 156 – 287 and 940 – 1,339. So, for those three models,

it is sufficient to take 4-6 sample points.

If the model is estimated by the spherical in Figure 3c, then the MSE will

converge at the 25th iteration. If the spherical model as the reference, by selecting 4

sample points and 25 bootstrap iteration, the spherical model can estimate the

reference model very well with the MSE of equal to 90. If less than 10 number of

sample points is selected, for the estimations using exponential, Gaussian, and

spherical models, the MSE will converge respectively at the 75th, 50th, and 25th

iteration. It can be concluded that if the spherical model as the reference is estimated

using exponential and Gaussian models for small-sized samples (not more than 8),

then the MSE values for each models are, respectively, 526 – 1,395 and 615 – 1,753.

While for big-sized samples (more than 8), the MSE values for each models are 797

– 1,834 and 1,210 – 1,651. So, for those three models, it is sufficient to take 4-6

sample points.

From the result, the parameter vector can be estimated by selecting the number

of sample points and bootstrap iterations. For the exponential, Gaussian, and

spherical model as the reference, number of sample point and number of iteration

bootstrap are selected respectively 4 and 50, 6 and 50, 4 and 50. The results of that

estimation are presented in Table 2 and Figure 4.

Table 2 Estimations of parameter vector for 3 semivariogram models. The reference

semivariogram model and the number of bootstrap iteration (B) are given.

The reference model Point Estimator, Confidence interval of parameter vector, and MSE value

Exp Gauss Sph

Exp

(171.2, 245, 1.900)

(148.61, 277.81, 1.95)

29.99, 183.82

252.10, 326.68

0.59, 3.30

83.0

(245.39, 118.40, 1.95)

173.26, 265.43

102.50, 157.22

0.59, 3.30

273.4

(142.57, 161.86, 1.95)

0, 314.67

3.13, 360.39

0.59, 3.30

402.9


Table 2. (Continued): Estimations of parameter vector for 3 semivariogram models. The reference

semivariogram model and the number of bootstrap iteration (B) are given.

Gauss

(125.4, 185, 0.552)

(36.29, 317.48, 0.56)

0, 80.78

281.28, 399.29

0.11, 0.99

349

(147.65, 148.57, 0.56)

63.45, 173.15

127.89, 200.45

0.11, 0.99

166

(59.86, 158.27, 0.56)

0, 233.55

0.61, 230.29

0.11, 0.99

1,032

Sph

(29.32, 268.1, 0.881)

(21.94, 332.88, 0.81)

0, 61.91

293.95, 396.36

0.18, 1.44

1,521

(143.36, 205.04, 0.81)

0, 177.42

170.57, 302.10

0.18, 1.44

1,474

(78.90, 138.34, 0.81)

0, 237.77

1.09, 381.17

0.18, 1.44

215

. . (a) (b)

(c)

Figure 4 Plots of reference semivariogram model and three semivariogram models (exponential,

Gaussian, and spherical) from bootstrap result for (a) the exponential model as the reference with p

= 4, B = 50, (b) the Gaussian model as the reference with p = 6, B = 50 and (c) the spherical

model as the reference with p = 4, B = 50.

4. Conclusion

By applying bootstrap method, the estimations of the semivariogram

parameters such as nugget effect and sill with partitioned range in a certain

precision will give point and interval estimations with a certain confidence level.

The number of the range partitions affects the MSE value between the experimental semivariogram and its reference model. The selection of the reference semivariogram


model will also affect the appropriate semivariogram model. With the exponential

as reference model, the estimation of exponential and Gaussian model will be the

best estimation of the reference model. Meanwhile, the spherical model will be the

poorest estimation of the model with the MSE reaching 713. The model estimations

using bootstrap method will converge faster to the reference semivariogram model

with the maximum number of iterations of 50. The computation time spherical

model as the reference model, the estimation was obtained within the fastest

computation time with the number of bootstrap iterations of 25. Meanwhile, the

spherical model is the slowest computation time for estimating the exponential and

Gaussian model with the number of bootstrap iterations of 75.

References

[1] H. Xu, Z. Liu, W. Cai and X. Shao, A wavelength selection method based on

randomization test for near-infrared spectral analysis, Chemometrics and

Intelligent Laboratory System, Elsevier, 97 (2009), 189 - 193.

http://dx.doi.org/10.1016/j.chemolab.2009.04.006

[2] Y. Bengio and Y. Grandvalet, No Unbiased Estimator of the Variance of K-

Fold Cross-Validation, Journal of Machine Learning Research, 5 (2004),

1089 - 1105.

[3] J. P. Kreiss and S. N. Lahiri, Bootstrap Methods for Time Series, Time Series

Analysis: Methods and Applications, Elsevier, 30 (2012), 3 – 26.

http://dx.doi.org/10.1016/b978-0-444-53858-1.00001-6

[4] P. Besse, PCA stability and choice of dimensionality, Elsevier, 13 (1992), 405

– 410. http://dx.doi.org/10.1016/0167-7152(92)90115-l

[5] M. E. Hohn, Geostatistics and Petroleum Geology, 2nd, Kluwer Academic

Publishers, London, 1999. http://dx.doi.org/10.1007/978-94-011-4425-4

[6] S. Waterman, Weighted Jackknife-OK Dalam Penaksiran Sumber Daya

Mineral, Dissertasion, Institut Teknologi Bandung, 2003.

[7] J. P. C. Kleijnen and E. Mehdad, Classic Kriging versus Kriging with

bootstrapping or conditional simulation: classic Kriging’s robust confidence

interval and optimization, CentER Discussion Paper, 2014-076 (2014), ISSN

0924-7815.

[8] S.N. Lahiri, Y. Lee and N. Cressie, On asymtotic distribution and asymtotic

efficiency of least squares estimators of spatial variogram parameters,

Journal of Statistical Planning and Inference, Elsevier, 103 (2002), 65 - 85.

http://dx.doi.org/10.1016/s0378-3758(01)00198-7

http://dx.doi.org/10.1016/j.chemolab.2009.04.006

http://dx.doi.org/10.1016/b978-0-444-53858-1.00001-6

http://dx.doi.org/10.1016/0167-7152%2892%2990115-l

http://dx.doi.org/10.1007/978-94-011-4425-4

http://dx.doi.org/10.1016/s0378-3758%2801%2900198-7


[9] N. Cressie and S.N. Lahiri, The asymtotic distribution of REML estimators,

Journal of Multivariate Analysis, 45 (1993), 217 - 233.

http://dx.doi.org/10.1006/jmva.1993.1034

[10] D. Zimmerman and M. Zimmerman, A comparison of spatial semivariogram

estimators and corresponding ordinary Kriging predictors, Technometrics,

33 (1991), 77 - 91. http://dx.doi.org/10.1080/00401706.1991.10484771

[11] B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman and

Hall, New York, 1993.

[12] Damayanti, Spatial Point Process Untuk Prediksi Distribusi Peluang

Permeabilitas Reservoir Jatibarang, Final project, Bandung Institute of

Technology, 2003.

Received: April 17, 2015; Published: August 3, 2015

http://dx.doi.org/10.1006/jmva.1993.1034

http://dx.doi.org/10.1080/00401706.1991.10484771

Estimation of the Parameters of Isotropic Semivariogram Model ...

Documents

Transcript of Estimation of the Parameters of Isotropic Semivariogram Model ...