By George A. Milliken, KSU Jeff Willers, USDA, ARS Chuck O’Hara, MSU

Using Spatial Information in the Using Spatial Information in the Design and Analysis of Design and Analysis of

Experiments Used to Evaluate Experiments Used to Evaluate the Effectiveness of Precision the Effectiveness of Precision

Agriculture Management Agriculture Management SystemsSystems

By By George A. Milliken, KSUGeorge A. Milliken, KSUJeff Willers, USDA, ARSJeff Willers, USDA, ARS

Chuck O’Hara, MSUChuck O’Hara, MSU

22 Department of Statistics Department of Statistics Kansas State UniversityKansas State University

Team ApproachTeam ApproachTeam: Subject Matter Specialists needed to identify

the problems and provide the resources for carrying out the study—GLUE THAT HOLDS THE TEAM TOGETHER

GPS and Computing Artist gather all of the layers of data and construct a data set

Statistician to identify the physical structures and processes (with their help) of carrying out the experiment and use the data types to formulate a model to extract useful information from the available data


Objectives of ExperimentObjectives of Experiment

Determine the effect of a plant growth Determine the effect of a plant growth regulator rates on the production of cotton regulator rates on the production of cotton for several varietiesfor several varieties

Use landscape and plant characteristics to Use landscape and plant characteristics to determine a “Prescription” for applying the determine a “Prescription” for applying the plant growth regulator to the fieldplant growth regulator to the field


Experiment ProcessExperiment Process

Collect all of the site characteristic dataCollect all of the site characteristic data

Use site characteristic data to form Use site characteristic data to form prescription for plant growth regulatorprescription for plant growth regulator

Apply the Plant Growth RegulatorApply the Plant Growth Regulator

Plant the Cotton VarietiesPlant the Cotton Varieties

Harvest the CottonHarvest the Cotton

These two may be reversed


Distance

Hillshade Slope

Aspect


Concavity

Section of field where experiment was conducted


Process for Plant Growth Regulator Process for Plant Growth Regulator

Collect all of the site characteristic data and use expert opinion to determine areas at which one applies the various rates of PGR

These areas form a prescription that can be applied to the field with equipment using GPS and computers


Example Field with Prescription for 3 rates of PGR—A, B, C

B

A

A

C

B




Sprayer Path covers 24 row at a time B

A

A

C

B


Process for Plant Growth RegulatorProcess for Plant Growth Regulator

Currently have to spray entire width with same rate. Form rectangles within each sprayer path to finish prescription

B

A

A

C

B

A

A

AA A

AA

AA

B

B

B

B

B B BB

B

BB

B B

B B

CC C

CCC

B

B

B

B

A

A

B


Planting The VarietiesPlanting The Varieties

Plant 12 rows in the Planting Path—2 Planting Paths/Sprayer Path

Randomly Assign Varieties to Sprayer Path

B

A

A

C

B

V1 V1 V2 V2V1V2V3 V3 V3


Harvesting ProcessHarvesting Process

B

A

A

C

B

V1 V1 V2 V2V1V2V3 V3 V3

Harvest 6 rows at a time, yield measurement each 2 seconds – 36 Harvester passes


ModelingModeling

B

A

A

C

B

V1 V1 V2 V2V1V2V3 V3 V3 Sprayer Strip Path

Planting Path

Harvester Pass

Yield Monitor Area

EXPERIMENTAL UNITS


ModelingModeling

Have data at each yield monitor site

Yield (y)

PGR rate (r)

Site Characteristics(x1,x2,…,xp)

Model Yield of Each Variety as function of PGR rate and site characteristics


Modeling—Error TermsModeling—Error Terms

B

A

A

C

B

V1 V1 V2 V2V1V2V3 V3 V3

εijklmn Yield Monitor Error

Similar sizes

Eijk Experimental Units

Very Different Sizes


Modeling—Error TermsModeling—Error Terms

B

A

A

C

B

V1 V1 V2 V2V1V2V3 V3 V3

Sprayer Path within Variety Sij

Planter Path within Variety pijkl

Harvester Pass within Planter path of a Variety hijklm


ModelModel

1

p

ijklmn i i ijklmn ri rijklmn ij ijk ijkl ijklm ijklmnr

y r X s E p h

where yijklmn is the yield recorded on the nth yield monitor reading in the mth harvester pass of the lth planter path of the kth experimental unit or replication within the jth spraying path of the ith variety,

Rijklmn is the PGR rate,

Xrjklmn is the value of the rth site characteristic from the yield monitor site,


ModelModel

1

p


y r X s E p h

φi is the slope corresponding to the rate,

βri is the regression coefficient corresponding to Xrijklmn for the ith variety,

sij is the effect of the jth sprayer path of the ith variety,

Eijk is the effect of the kth experimental of the ith variety within the jth sprayer path


ModelModel

1

p


y r X s E p h

pijkl is the planter path,

hijklm is the harvester pass effect, and

εijklmn is the effect of the yield monitor area.

Note: Planter Path, Harvester Pass and Yield Monitor Area are ALL nested within the Experimental Unit.


ModelModel

1

p


y r X s E p h

ij

ijklm

11 12 sv 1

ij 1 ij2 ijt 2

ijk ijk1 ijk2 3

ijkl ijkl1 ijkl2 4

ijklm ijklm1 ijklm2 ijklmn

s=[s ,s , , s ]' ~ N(0, Q ),

=[ ,E , ,E ]' ~ N(0, Q ),

p =[p , p ]' ~ N(0, Q ),

=[h , h ]' ~ N(0, Q ),

and [ , , , ] ~ (0, )ijklm

ij

n

E E

h

N R


ModelModel

1

p


y r X s E p h

Q1 is the correlation structure among the sprayer paths, which can be described by a one dimensional spatial covariance matrix using the path centroids as the centers and distance between centroids to describe the covariances,

Q2 is the correlation structure among the experimental units within a sprayer path, which can be described by a one dimensional spatial covariance matrix using the experimental units’ centroids as the centers and distance between centroids to describe the covariances,


ModelModel

1

p


y r X s E p h

Q3 is the correlation structure among the planter paths within and an experimental unit, which can be described by a one dimensional spatial covariance matrix using the planter paths centroids as the centers and distance between centroids to describe the covariances,Q4 is the correlation structure among the harvester passes within a planter path and an experimental unit, which can be described by a one dimensional spatial covariance matrix using the harvester pass’s centroids as the centers and distance between centroids to describe the covariances,


ModelModel

1

p


y r X s E p h

Rijklm is the correlation structure among the yield monitor values within a harvester pass within and experimental unit, which can be described by a one dimensional spatial covariance matrix using the yield monitor areas’ centroids of the yield monitor areas as the centers and distance between centroids to describe the covariances.

The site characteristic values can include transformations (squares, reciprocals, logarithms, etc.) of the actual site characteristics and their cross products or ratios.

jknR


PROC MIXED CODEPROC MIXED CODEPROC MIXED DATA=ONE;

CLASS VARIETY SPRAY EU PLANT HARV;

MODEL YIELD=RATE VARIETY*RATE X1 X1*VARIETY … XR XR*VARIETY / DDFM=KR;

RANDOM SPRAY*VARIETY/SUBJECT=INT TYPE=SP(GAU)(DIST_SPRAY);

RANDOM EU/SUBJECT=SPRAY*VARIETY TYPE=SP(GAU)(DIST_EU);

RANDOM PLANT/SUBJECT=EU(SPRAY*VARIETY) TYPE=SP(GAU)(DIST_P);

RANDOM HARV/SUBJECT=PLANT(EU SPRAY VARIETY) TYPE=SP(GAU)(DIST_H);

REPEATED / SUBJECT=HARV(PLANT EU SPRAY VARIETY) TYPE=SP(GAU)(DIST_YM);


PROC MIXED CODEPROC MIXED CODELSMEANS VARIETY/DIFF;

Provides a comparison of the varieties at the mean values of the rates and site characteristics.

This is like predicting each variety’s yield at each yield monitor site over the complete field and then comparing those means -- how would the varieties compare if the whole field was planted to each variety.


PROC MIXED CODEPROC MIXED CODE

LSMEANS VARIETY/ AT RATE=2 DIFF;

LSMEANS VARIETY/ AT RATE=6 DIFF;

These statements provide means at the mean values of the site characteristics using the specified rate – plant the complete field with each variety and use a blanket rate of PGR (same rate over the complete field.


ExampleExampleTREATMENT STRUCTRUE

17 Varieties

4 levels of PIX

DESIGN STRUCTURE

2 ROWS OF EACH VARIETY

PIX APPLIED TO MANAGEMENT ZONES


Pairs of Row for Each Variety


MANAGEMENT ZONES FOR PGR APPLICATION


Example’s ModelExample’s Model

1

p


y r X s E p h

1

p

ikmn i i ikmn ri rikmn ik ikm ikmnr

y r X E h

Original Model

New Model—only one spray path/ variety, so no S…

Only one planter path per variety, so no P…

Lose two subscripts!!!


Example’s ModelExample’s ModelUsing Rate as a continuous variable, so the E terms are absorbed in the regression part

Only have two error terms

Did some initial screening of the data,

then used LOESS smoother down a row,

deleted extreme residuals

All before fitting final mixed model


Examples’ ModelExamples’ ModelVariables Used – All scaled to Yield Monitor Areas

NDVI – normalized vegetation index

NDVI_diff – difference between two dates

Slo—Slope of land

FAC—square meters flowing into current area

DSM – Altitude

CVX- convexity of area--hold water or shed water

Euc – Euclidean distance from stream network


Examples’ ModelExamples’ Model

Rate

Rate2

Interactions of all variables with variety, rate and variety by rate

Backward Deletion to select final model


proc mixed data=predres1; where -800<res1<800;title 'analysis with site characteristics';class pass variety load_id;model say = variety rate rate*rate variety*rate variety*rate*rate Slo LogFac Cvx logEuc Dsm Ndvi_dif index one_cvx ndvi_717 logfac*cvx Dsm*variety Dsm*rate Cvx*variety*rate logEuc*variety*rate ndvi_717*variety*rate LogFac*variety*rate*rate logEuc*variety*rate*rate Dsm*variety*rate*rate ndvi_717*variety*rate*rate LogFac*rate cvx*variety*rate*rate Ndvi_dif*variety*rate*rate Slo*variety*rate*rate index*variety*rate Ndvi_dif*variety*rate Slo*rate logfac*dsm slo*dsm logfac*ndvi_dif one_ndvi dsm*dsm*variety /outp=preds solution;**say is smoothed yield;random load_id/subject=variety;repeated /type=sp(GAU)(new_y) subject=load_id(rateclass*variety) local;lsmeans variety/at means diff;lsmeans variety/at rate=0 diff;lsmeans variety/at rate=2 diff;lsmeans variety/at rate=4 diff;lsmeans variety/at rate=6 diff;lsmeans variety/at rate=8 diff;


Two passes of harvester – data are quite variable- similar side by side patterns, but lot of variability within a harvester pass


ExampleExample

Have only one strip per varietyHave only one strip per variety

Replication is obtained by “crossing” PGR Replication is obtained by “crossing” PGR rates with the varietiesrates with the varieties

Fit regression model using quadratic Fit regression model using quadratic function of rate, the site characteristics, function of rate, the site characteristics, and interactions with varietiesand interactions with varieties

Provided a model for each varietyProvided a model for each variety


ExampleExample

Carried out a backward deletion process Carried out a backward deletion process to simplify the modelto simplify the model

Following is AOV of variables in modelFollowing is AOV of variables in model


Type 3 Tests of Fixed Effects

Effect

Num

DF Den DF F Value Pr > F

variety 16 17 10.28 <.0001

rate 1 5979 2.82 0.0933

rate*rate 1 5979 4.45 0.0349

rate*variety 16 5979 6.18 <.0001

rate*rate*variety 16 5979 7.41 <.0001

Slo 1 5979 3.00 0.0834

logfac 1 5979 21.60 <.0001

Cvx 1 5979 544.65 <.0001

logeuc 1 5979 3.92 0.0477

Dsm 1 5979 527.19 <.0001

Ndvi_dif 1 5979 3.09 0.0788

index 1 5979 0.05 0.8207

one_cvx 1 5979 678.02 <.0001

Ndvi_717 1 5979 2.54 0.1112

logfac*Cvx 1 5979 290.96 <.0001

Dsm*variety 16 5979 10.35 <.0001

rate*Dsm 1 5979 2.80 0.0944

rate*Cvx*variety 17 5979 9.77 <.0001

Type 3 Tests of Fixed Effects in Final Model


rate*logeuc*variety 17 5979 1.91 0.0135

rate*Ndvi_71*variety 17 5979 5.94 <.0001

rate*rate*logf*varie 17 5979 8.29 <.0001

rate*rate*loge*varie 17 5979 1.89 0.0145

rate*rate*Dsm*variet 17 5979 7.40 <.0001

rate*rate*Ndvi*varie 17 5979 4.85 <.0001

rate*logfac 1 5979 1.52 0.2176

rate*rate*Cvx*variet 17 5979 9.62 <.0001

rate*rate*Ndvi*varie 17 5979 3.18 <.0001

rate*rate*Slo*variet 17 5979 64.09 <.0001

rate*index*variety 17 5979 8.76 <.0001

rate*Ndvi_di*variety 17 5979 3.85 <.0001

rate*Slo 1 5979 0.79 0.3748

logfac*Dsm 1 5979 22.91 <.0001

Slo*Dsm 1 5979 2.89 0.0894

logfac*Ndvi_dif 1 5979 51.67 <.0001

one_ndvi 1 5979 2.79 0.0947

Dsm*Dsm*variety 17 5979 55.82 <.0001

Type 3 Tests of Fixed Effects in Final Model(continued)


Predicted values from model, original data and smoothed data

One Harvester Pass


ExampleExample

Used LSMEANS to obtain predictions of Used LSMEANS to obtain predictions of each variety at rates of 0, 2, 4, 6,8 and each variety at rates of 0, 2, 4, 6,8 and mean (6.17). mean (6.17).

2 was not in the data set, but model 2 was not in the data set, but model provided predictionsprovided predictions

Following are the means and rank of the Following are the means and rank of the varieties within a rate.varieties within a rate.


Variety rate_0 rank_0 rate_2 rank_2 rate_4 rank_4 rate_6 rank_6 rate_8 rank_8

501 2715 1 2604 3 2542 4 2530 1 2569 1

991 2652 3 2666 1 2603 1 2463 2 2246 6

960 2488 5 2554 4 2543 3 2454 3 2287 4

5599 2295 14 2434 5 2484 5 2445 4 2317 3

215 2702 2 2660 2 2571 2 2436 5 2253 5

655 2302 13 2434 6 2461 6 2384 6 2203 9

4892 2236 16 2383 9 2430 7 2377 7 2224 8

451 2570 4 2397 8 2331 10 2372 8 2519 2

989B 2433 7 2383 10 2335 9 2289 9 2244 7

5415 2471 6 2432 7 2336 8 2183 10 1973 13

989R 2384 9 2281 13 2194 13 2123 11 2066 11

5303 2416 8 2310 11 2204 11 2098 12 1992 12

5690 2357 10 2226 15 2139 15 2096 13 2096 10

436 2292 15 2263 14 2194 14 2084 14 1933 15

4793 2352 11 2294 12 2201 12 2072 15 1908 16

521 2306 12 2212 16 2125 16 2045 16 1972 14

494 2162 17 2102 17 2008 17 1879 17 1715 17

Predictions from Model with Site Characteristics


ExampleExample

Computed the LSMEAN using mean, Computed the LSMEAN using mean, minimum and maximum standard minimum and maximum standard deviations from comparisons within a ratedeviations from comparisons within a rate


Table 4. Standard deviation information for pairwise comparisons among the varieties and 0.05 LSD values computed using the mean standard deviation (mn_std), the minimum standard deviation (min_std) and the maximum standard deviation (max_std) for each of the rates of PGR from a regression model using site characteristics.

ratete mn_std LSD_mn min_std LSD_min max_std LSD_max

0.00 92 194 86 180 104 218

2.00 64 134 59 125 72 152

4.00 55 115 46 98 66 138

6.00 38 80 34 72 43 90

6.17 37 77 34 71 41 85

8.00 58 122 39 81 98 206


Predicted Means from Model using Site Characteristics


ExampleExample

LSMEANS computed from mean, LSMEANS computed from mean, maximum and minimum standard error for maximum and minimum standard error for comparisons made from model without the comparisons made from model without the site characteristics.site characteristics.


Table 7. Standard deviation information for pairwise comparisons among the varieties and 0.05 LSD values computed using the mean standard deviation (mn_std), the minimum standard deviation (min_std) and the maximum standard deviation (max_std) for each of the rates of PGR from model without using site characteristics.

rate mn_std LSD_mn min_std LSD_min max_std LSD_max

0.00 216 455 214 449 232 487

2.00 190 398 163 342 223 468

4.00 203 427 168 353 248 520

6.00 144 303 125 263 169 354

6.17 138 290 122 256 159 335

8.00 190 399 178 373 208 436


Predicted Means from Model without Site Characteristics


SummarySummary

Team approach was required to assemble Team approach was required to assemble all necessary knowledge about the all necessary knowledge about the processes and toolsprocesses and tools

Built statistical model to describe data Built statistical model to describe data obtained from experiments ran on farms obtained from experiments ran on farms where site characteristics and where site characteristics and management procedures are taken into management procedures are taken into accountaccount


SummarySummary

Successfully put together a process that Successfully put together a process that can be used to answer many of the can be used to answer many of the questions being asked by researchers questions being asked by researchers needing to address research problems needing to address research problems associated with precision agriculturalassociated with precision agricultural

By George A. Milliken, KSU Jeff Willers, USDA, ARS Chuck O’Hara, MSU

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