Table 2 Cross-validation statistics from all interpolation methods and the appropriateness index (AI). AI-3 represents the first step of evaluation whereas AI-6 represents the second step of evaluation of the optimal interpolation technique. The values of cross-validation parameters are normalized using the minimum–maximum normalization method as discussed in section “Deterministic methods”.

From: Spatial–temporal characterization of rainfall in Pakistan during the past half-century (1961–2020)

Method

ME

RMSE

R2

MSE

RMSSE

ASE

AI-3

Rank AI-3

AI-6

Rank AI-6

IDW

0.9428

0.9351

0.9494

–

–

–

0.9430

4

–

–

GPI

0.9464

0.0000

0.0000

–

–

–

0.0000

26

–

–

RBF

0.7973

1.0000

1.0000

–

–

–

0.9280

7

–

–

LPI

0.4499

0.7589

0.8121

0.3564

0.9890

0.5263

0.6549

22

0.5524

15

EBK

0.9280

0.9527

0.9628

1.0000

1.0000

0.8530

0.9482

3

0.9379

2

OK-c

0.8314

0.9305

0.9456

0.8510

0.4728

0.6274

0.9020

10

0.7133

6

OK-s

0.8157

0.9432

0.9561

0.8261

0.4652

0.6252

0.9036

9

0.7070

7

OK-e

0.8716

0.9604

0.9691

0.8973

0.3678

0.5549

0.9333

6

0.6829

8

OK-g

0.8789

0.9202

0.9375

0.8855

0.4717

0.6004

0.9127

8

0.7174

5

OK-h

0.4922

0.9007

0.9253

0.4863

0.1819

0.3240

0.7452

17

0.4112

22

OK-k

0.9204

0.9338

0.9476

0.9476

0.4845

0.6328

0.9345

5

0.7495

3

OK-j

0.8387

0.9153

0.9343

0.8433

0.6094

0.6058

0.8961

11

0.7409

4

SK-c

0.3122

0.8316

0.8649

0.3494

0.6906

0.6766

0.6108

23

0.5162

18

SK-s

0.5312

0.8343

0.8661

0.5807

0.4865

0.5290

0.7291

18

0.5646

12

SK-e

0.5135

0.8825

0.9066

0.5601

0.6091

0.6571

0.7456

16

0.6206

9

SK-g

0.3140

0.8200

0.8552

0.3478

0.7249

0.6734

0.6069

24

0.5182

17

SK-h

0.2878

0.7573

0.8004

0.3486

0.5651

0.4512

0.5620

25

0.4345

20

SK-k

0.6010

0.8766

0.9013

0.6439

0.4888

0.5864

0.7821

14

0.6145

10

SK-j

0.0000

0.7974

0.8376

0.0000

0.7438

0.6565

0.0000

26

0.0000

23

UK-c

0.4652

0.8786

0.9071

0.5163

0.2183

0.3245

0.7208

19

0.4230

21

UK-s

0.4502

0.8889

0.9136

0.5110

0.3055

0.4062

0.7174

20

0.4710

19

UK-e

0.5655

0.9206

0.9379

0.6694

0.3469

0.4569

0.7893

13

0.5531

14

UK-g

0.4380

0.8543

0.8832

0.5552

0.4937

0.4571

0.6939

21

0.5289

16

UK-h

0.6553

0.8679

0.8943

0.9088

0.0000

0.0000

0.8000

12

0.0000

23

UK-k

1.0000

0.9219

0.9429

0.9932

0.1674

0.3701

0.9548

2

0.5567

13

UK-j

0.5679

0.8788

0.9032

0.6843

0.4948

0.4965

0.7688

15

0.5969

11

EBKRP

0.9629

0.9558

0.9653

0.9150

0.9364

0.9774

0.9617

1

0.9425

1

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