a design-based approach to k-nn technique in forest inventories 27 marzo 2008 graspa conference 2008...

A DESIGN-BASED APPROACH TO K-NN TECHNIQUE IN

FOREST INVENTORIES

27 Marzo 2008

GRASPA Conference 2008

F. Baffetta, L. Fattorini and S. Franceschi

INTRODUCTION

Several techniques for assessing natural resources employ information from remotely sensed imagery and ground data.

INTRODUCTION

Between these methodologies, the k-Nearest Neighbours (KNN)

technique is becoming increasingly popular

INTRODUCTION

Main application: forest attribute mapping

Data: auxiliary variables for all population units (the digital numbers from multispectral

remotely sensed imagery )

interest variable values for sampled units

Between these methodologies, the k-Nearest Neighbours (KNN)

technique is becoming increasingly popular

OBJECTIVE

The statistical properties of the k-NN method are still not clearly delineated.

The only advanced investigations lie in a complete model-based setting (Kim and Tomppo (2006) and McRoberts et al. (2007) )

OBJECTIVE



•Assumptions on superpopulation models frequently unrealistic

•Neglect the fact that data arise from a pre-determined scheme of sampling

THE GOAL:

Derive the statistical properties of the k-NN estimators in a completely design-based framework

no assumption about populations

OBJECTIVE



•Assumptions on superpopulation models frequently unrealistic

•Neglect the fact that data arise from a pre-determined scheme of sampling

Study region

POPULATION

Study region

POPULATION

usually a set of pixels

constituting a digital image

N,...,2,1U

Study region

POPULATION

usually a set of pixels

constituting a digital image

N,...,2,1U

Y the interest variable

Y1, Y2,..., YN population values

PARAMETERS

Njy j ,,2,1

PARAMETERS

Njy j ,,2,1

Aj

jA

A yN

Y1

PARAMETERS

Njy j ,,2,1

Aj

jA

A yN

Y1

N

jjy

NY

1

1

SATELLITE INFORMATION

q auxiliary variables

known for all population units

(the values of the spectral bands )

qXX ,,1

X-spaceUjjx

FOREST INVENTORY DATA

Sjp j

CONVERTING INVENTORY DATA TO PIXEL DATA

S jpy jj

CONVERTING INVENTORY DATA TO PIXEL DATA

reference set (sample)US

Sjy j

SU target set

K-NN ESTIMATION

Y

X-space

Y space

K=4

some notations

H(j,k) = label of the unit of S whose distance from unit j in U–S has rank k (k = 1,2,…,n)

unit j in U – S

units in S

K-NN ESTIMATION

straightforward choice K

wk1

SU

jywyK

kkjHkj

1),(

~

K-NN ESTIMATION

Aj

jA

NNkA yN

Y ~1~

N

jjNNk y

NY

1

~1~

(mean estimate in the subarea of interest)

(overall mean estimate)


wk1

SU

jywyK

kkjHkj

1),(

~

K-NN ESTIMATION

BIAS? ACCURACY?

Aj

jA

NNkA yN

Y ~1~

N

jjNNk y

NY

1

~1~

(mean estimate in the subarea of interest)

(overall mean estimate)


wk1

SU

jywyK

kkjHkj

1),(

~

DESIGN-BASED INFERENCE

Design-based inference is solely determinated by the characteristic of the sampling scheme and, in contrast with model-based inference, it avoids any assumption about the

population

see Gregoire (1998) for a revision of the approaches in forestry

y1, y2, …, yN fixed

S sample of pixels selected from U by a probabilistic sampling scheme

pixels inclusion probabilitiesN ,...,, 21


h(j,r) = label of the unit whose distance from unit j has rank r (r = 1,2,…,N-1)

h(j,r) are known

X-space

Dx


unit in S

S random H(j,k) are random variables

X-space X-space

K=4


• Model-based framework UNBIASED

• Design-based framework BIASEDjy~

1

1,

~N

rrjhjrj ywyE

jrw expected weight assigned to the pixel whose distance from j has rank r in the sequence of the N-1 distances from pixel j (completely known from the sampling scheme)

can be obtained in a very similar way but it cannot be estimated on the basis of the sample data.

jyMSE ~

jy~

DESIGN-BASED ESTIMATION OF MEAN

Owing to the design-based bias induced by each the estimator may provide a highly-biased estimator for the mean over the whole study area.

jy~

NNkY

~


Owing to the design-based bias induced by each the estimator may provide a highly-biased estimator for the mean over the whole study area.

jy~

NNkY

~

A more suitable estimator of turns out to beY

SU j j

jj

jjNNk

yy

Ny

NEYY

~1~1~~~

where denotes the Horvitz-Thompson (HT) estimator ofE~

UU j

jj

jj eN

yyN

E1~1


Bias of closed expression

Variance of cumbersome

Y~

Y~


Y~

Y~

•

• 00

2

1~~hj

j h hj

hjjhlin ee

NYVYV

U U

00jjj yye

K

kkjhkj ywy

1,

0

YYE

zyy

NYY

lin

jjj

j

jjlin

~

~1~~ 0

U

Bias of closed expression

Variance of cumbersome


Whenever an estimator of turns out to be

hjj h jhhj

hjjh eeN

YV

S S

2

1~ˆ

U jjh 0 YV~


Whenever an estimator of turns out to be

hjj h jhhj

hjjh eeN

YV

S S

2

1~ˆ

U jjh 0 YV~

otherwise, whenever some , conservative estimator of

turns out to be the Hunsen-Hurvitz variance estimator

Sj j

j EN

ne

nnYV

2~

1

1~~

0jh

YV~

ASYMPTOTIC RESULTS

To the derivation of the asymptotical properties of it is necessary to think of a sequence of populations with corresponding population sizes that increase along with sample sizes

Y~

lU lN

ln

Suppose that: , , , and any l lN ln by j lj U

ASYMPTOTIC RESULTS


Y~

lU lN

ln


• sufficient conditions for asymptotic unbiasedness are given

ASYMPTOTIC RESULTS


Y~

lU lN

ln



• sufficient conditions for consistency are given

• it is proved that under SRSWOR, is asymptotically unbiased

ASYMPTOTIC RESULTS


Y~

lU lN

ln



lY~

• sufficient conditions for consistency are given

ESTIMATION FOR SUB-POPULATIONS

Study region

N,...,2,1U


Study region

N,...,2,1U

Sub-region A

AA N,...,2,1


Study region

N,...,2,1U

Sub-region A

AA N,...,2,1

ASSA

• the auxiliary information is restricted to the ’s for each

• the sample information is restricted to the ’s for each

• the inclusion probabilities remain the same

jx

jy

Aj

ASj

RANDOM WEIGHTS

Very common choice

Kk,jkjHjd

kjHjdw K

k

jk ,,1',,

,,

1'

1

1

S

RANDOM WEIGHTS

Very common choice

Kk,jkjHjd

kjHjdw K

k

jk ,,1',,

,,

1'

1

1

S

Analogous results obtained for fixed weights

SIMULATION STUDY

Study region: Trentino (North Italy)

SIMULATION STUDY


Population of interest: N = 312 units (forest compartments)

SIMULATION STUDY



Interest variable Y : timber volume m3×ha (completely known)

SIMULATION STUDY



Interest variable Y : timber volume m3×ha (completely known)

Auxiliary variables: 6 spectral bands (obtained from the optimal bands of ETM+ acquisitions)

WHEN DOES K-NN WORK WELL?

Y

X-space

neighbouring structure on X-space similar to neighbouring structure on Y-space

the neighbouring units in the covariate space should be neighbour (i.e. they should have similar values) in the Y space

MULTIDIMENSIONAL SCALING (MDS)

X-space

Z0

MDS Dx = Dz

MULTIDIMENSIONAL SCALING (MDS)

NjzSYy jYj ,...,2,1,

X-space

Z0

MDS

YY

YS YSDY = const DZ

Dx = Dz

MULTIDIMENSIONAL SCALING

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60Z

ZSY Y

This should be the best situation for K-NN

12 YZR

MULTIDIMENSIONAL SCALING

jjYj ezSYy

Perturbing this situation should deteriorate

the performance of K-NN

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60Z

ZSY Y

This should be the best situation for K-NN

12 YZR

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60

PSEUDO-POPULATIONS

eSZY

Z

8.02 YZR

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60

PSEUDO-POPULATIONS

eSZY

Z

6.02 YZR

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60

PSEUDO-POPULATIONS

eSZY

Z

4.02 YZR

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60

PSEUDO-POPULATIONS

eSZY

Z

2.02 YZR

0

100

200

300

400

500

600

700

800

-40 -20 0 20 40 60

REAL POPULATION

Z

Y

3.02 YZR

MONTE CARLO EXPERIMENT

10,000 simulations

Sampling scheme adopted: SRSWORN =312, n = 16 (5%), 31 (10%), 62 (20%), 94 (30%)

K = 1,2,,…,12

MONTE CARLO EXPERIMENT

10,000 simulations

Sampling scheme adopted: SRSWORN =312, n = 16 (5%), 31 (10%), 62 (20%), 94 (30%)

K = 1,2,,…,12

YRB~

YRME~

YVRB~~

SIMULATION RESULTS YRME~

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

1 2 3 4 5 6 7 8 9 10 11 12

0%

1%

2%

3%

4%

5%

1 2 3 4 5 6 7 8 9 10 11 12

94n

3.02 YZR2.02 YZR 4.02 YZR 6.02 YZR 8.02 YZR 12 YZR

K

0%

2%

4%

6%

8%

10%

12%

14%

1 2 3 4 5 6 7 8 9 10 11 12

16n

K

31n

K

0%

1%

2%

3%

4%

5%

6%

7%

1 2 3 4 5 6 7 8 9 10 11 12

62n

K

SIMULATION RESULTS YVRB~~

3.02 YZR2.02 YZR 4.02 YZR 6.02 YZR 8.02 YZR 12 YZR

-21%

-16%

-11%

-6%

-1%

4%

9%

14%

19%

24%

1 2 3 4 5 6 7 8 9 10 11 12

-21%

-16%

-11%

-6%

-1%

4%

9%

14%

19%

24%

1 2 3 4 5 6 7 8 9 10 11 12

16n

-21%

-16%

-11%

-6%

-1%

4%

9%

14%

19%

24%

1 2 3 4 5 6 7 8 9 10 11 12

-21%

-16%

-11%

-6%

-1%

4%

9%

14%

19%

24%

1 2 3 4 5 6 7 8 9 10 11 12

62n

K

K

K

K

94n

31n

To compare k-NN with an alternative common way to handle auxiliary information

Generalized Regression Estimator (GREG)

(model assisted)

bxX ˆˆ TGREG yY

Sj

jyn

y1

Uj

jNxX

1

Sj

jnxx

1

SS jjj

j

Tjj y xxxb

1

ˆ

K-NN AND GREG

To compare k-NN with an alternative common way to handle auxiliary information

Generalized Regression Estimator (GREG)

(model assisted)

bxX ˆˆ TGREG yY

Sj

jyn

y1

Uj

jNxX

1

Sj

jnxx

1

SS jjj

j

Tjj y xxxb

1

ˆ

The performance of GREG is always superior than k-NN performance except in extrimely rare cases

K-NN AND GREG

PRACTICAL APPLICATIONS

Test area: a quadrat of 900 km2 in the north-eastern part of Tuscany.

Remotely sensed imagery:Landsat 7 ETM+ imagery acquired on June, 7h 2000

Ground data:Forest inventory (1998)

PRACTICAL APPLICATIONS

Population U: the imagery pixels N = 1,000,000

Pixel size: 30m x 30m

Auxiliary variables: digital numbers of the spectral bands 1,2,3,4,5,7

Target population F: forest pixel NF = 620,495

Target variable: timber volume (m3/ha)

FF

Fj

jyN

Y1

PRACTICAL APPLICATIONSForest inventory, two-phase sampling

1° phase: a grid of quadrates of side 400 m was randomly superimposed onto the study area and the grid dots were taken as the first-phase sample of points (equivalent to the aligned systematic scheme )

2° phase: a sample of points was selected from the first-phase sample by means of SRSWOR.

For each second-phase point in the forest part of the study area (nF = 229), a plot of size 600 m2 was centred at the point and the timber volume was recorded within.

610370/ Nnj

PRACTICAL APPLICATIONSk-NN estimation:

where

represents the label of the sampled pixel whose distance from j has rank k in the sequence of the distances with and

where

),( hjdX

K

kkjHkj ywy

1),(

~F

kjH ,F

FShF jh

FF

FFFF e

n

n

N

Ny

NY

jj

~1~

FSF

FF

F

F jj ee

nnn

n

N

NYV 2

2

)1(

1~~

FSF

Fj

jen

e1

KkK

wk ,..,11

PRACTICAL APPLICATIONSChoice of K:

020406080

100120140160180200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

K

YV

~~

ham.Y 317163~

F

111.36~~

FYV

%4.6~

EMR

13K

PRACTICAL APPLICATION

Is the k-NN estimation suitable for this case study?

Check the relationship between Z and Y values




Z

Y

Sj




Z

Y

PRACTICAL APPLICATIONPoor relationship between the Y- and Z-values ( R2 = 0.016)

serious doubts on the effectiveness of k-NN

estimation in the case study 0

100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z0

100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z


GREG estimation gave rise to no better results (R2 = 0.32 )

ham.Y 3Greg 829135ˆ F

8980ˆˆ .YV Grec F

Poor relationship between the Y- and Z-values ( R2 = 0.016)



100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z0

100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z


the spectral bands adopted as auxiliary variables do not constitute adequate proxies for timber volume.

GREG estimation gave rise to no better results (R2 = 0.32 )

ham.Y 3Greg 829135ˆ F

8980ˆˆ .YV Grec F

Poor relationship between the Y- and Z-values ( R2 = 0.016)



100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z0

100

200

300

400

500

600

700

800

900

1000

-80 -60 -40 -20 0 20 40 60 80

Y

Z

FINAL REMARKS

• k-NN estimators at the pixel level are generally biased with

mean squared errors which cannot be estimated from the

sample data

FINAL REMARKS



sample data

• the linear approximation of is unbiasedY~

FINAL REMARKS



sample data

• the linear approximation of is unbiased

• variance estimators are proposed

Y~

FINAL REMARKS



sample data



• a diagnostic tool is proposed

Y~

FINAL REMARKS



sample data



• a diagnostic tool is proposed

• it is stressed to use k-NN method when the neighbouring

structure of pixels in the space of the interest variable is similar

to the neighbouring structure in the space of auxiliary variables

Y~

ReferencesFattorini, L. (2006). Applying the Horvitz-Thompson criterion in complex designs: a computer-intensive perspective for estimating inclusion probabilities. Biometrika, 93, 269-278.

Fattorini, L., & Ridolfi, G. (1997). A sampling design for areal units based on spatial variability. Metron, 55, 59-72.

Fazakas, Z., & Nilsson, M. (1996). Volume of forest cover estimation over southern Sweden using AVHRR data calibrated with TM data. International Journal of Remote Sensing, 17, 1701-1709.

Franco-Lopez, H., Ek, A.R., & Bauer, M. E. (2001). Estimation and mapping of forest stand density, volume, and cover type using the k-nearest Neighbours method. Remote Sensing of Environment, 77, 251-1709.

Haapanen, R., Ek, A. R., Bauer, M. E., & Finley, A. O. (2004). Delineation of forest/nonforest land use classes using nearest neighbours methods. Remote Sensing of Environment, 89, 265-271.

Holmström H., & Fransson, J.E.S. (2003). Combining remotely sensed optical and radar data in k-NN estimation of forest variables. Forest Science, 4, 409-418.

Katila, M., & Tomppo, E. (2001). Selecting estimation parameters for the Finnish multisource National Forest Inventory. Remote Sensing of Environment, 76, 16-32.

Mäkelä, H., & Pekkarinen, A. (2001). Estimation of timber volume at the sample plot level by means of image segmentation and Landsat TM imagery. Remote Sensing of Environment, 77, 66-77.

Mäkelä, H., & Pekkarinen, A. (2004). Estimation of forest stand volume by Landsat TM imagery and stand-level field-inventory data. Forest Ecosystem and Management, 196, 245-255.

McRoberts, R.E., Nelson, M.D., & Wendt, D.G. (2002). Stratified estimation of forest area using satellite imagery, inventory data, and the k-Nearest Neighbours technique. Remote Sensing of Environment, 82, 457-468.

Tokola, T., Pitkänen, J., Partinen, S., & Muinonen, E. (1996). Point accuracy of non-parametic method in estimation of forest characteristics with different satellite materials. International Journal of Remote Sensing, 17, 2333-2351.

Tomppo, E. (1991). Satellite image-based national forest inventory of Finland. International Archives of Photogrammetry and Remote Sensing, 28, 419-424.

Tomppo, E., & Halme, M. (2004). Using course scale forest variables as ancillary information and weighting of variables in k-NN estimation: a genetic algorithm approach. Remote Sensing of Environment, In Press.

Tomppo, E., Nilsson, M., Rosengren, M., Aalto, P., & Kennedy, P. (2002). Simultaneous use of Landsat-TM and IRS-1C WiFS data in estimating large area tree stem volume and aboveground biomass. Remote Sensing of Environment, 82, 156-171.

MODEL - BASED INFERENCE

S fixed (reference and target set are fixed)

H(j,k) fixed neighbours are fixed

Example: S = {1,5,9}, U – S = {2,3,4,6,7,8}, K = 2

X-space

12

5

6

47

3

8

9

j H(j,1) H(j,2)

2 1 53 9 54 5 96 9 5 7 5 98 9 5

1 2 3

4 5 6

7 8 9


S fixed (reference and target set are fixed)

H(j,k) fixed neighbours are fixed

Example:

X-space

12

5

6

47

3

8

9

j H(j,1) H(j,2)

2 1 53 9 54 5 96 9 5 7 5 98 9 5

1 2 3

4 5 6

7 8 9

S = {1,5,9}, U – S = {2,3,4,6,7,8}, K = 2

MODEL - BASED INFERENCEMcRoberts – Tomppo – Finley – Heikkinen (2007)

jjjy

0jE 2jjV hjjhhjCov ,

MODEL - BASED INFERENCEMcRoberts – Tomppo – Finley – Heikkinen (2007)

jjjy

0jE 2jjV hjjhhjCov ,

BASIC ASSUMPTION

KkyEyE kjHj ,...,2,1,

neighbouring units have similar expectations

K-NN is a model-unbiased prediction

0|~ SyyE jjY

1.


STTETMSE

STTETMSE

SUjSyyEyMSE

Y

AAYA

jjYj

|~~

|~~

|~~

2

2

2

MODEL - BASED INFERENCEList of assumptions:

2. A parametric structure of covariance function (variogram)

3. See Appendix 1 (McRoberts, R. E. et al., 2007)

3a)

3b)

3c)

3d)

2

1

2),(

1j

K

kkjHk

2

1 1')',(),(2

1j

K

k

K

kkjHkjHk

2

1

2),(

1j

K

kkjHk

KkjkjHkjH ,...,2,12)',(),(

MODEL - BASED INFERENCE4. See Appendix 3 (McRoberts, R. E. et al., 2007)

KkjkjH ,...,2,1),(


KkjkjH ,...,2,1),(

assumption 1 and 4 involve 3

neighbouring units have similar mean and variances



ARE THESE ASSUMPTIONS REALISTIC?

KkjkjH ,...,2,1),(

assumption 1 and 4 involve 3

neighbouring units have similar mean and variances


Example

j H(j,1) H(j,2) ASSUMPTION 1

2 1 53 9 54 5 96 9 5 7 5 98 9 5

X space

12

5

6

47

3

8

9

5212 ,

5393 ,

9454 , 5696 , 9757 , 5898 ,

9,...,2,1 jj

homogeneity of expectation (and

variances)

ROBUSTNESS STUDIES UNDER PRESENCE OF SPATIAL TREND AND/OR HOTSPOT

a design-based approach to k-nn technique in forest inventories 27 marzo 2008 graspa conference 2008...

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