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Linear Mixture Modeling with
Autocorrelated Errors
Jayantha Ediriwickremal, Siamak Khorram2, Marcia
Gumpertz3 and John Brockhaus4
Abstract.-The
linear mixture model assumes that model errors are
spatially uncorrelated. Spatial continuities exist in most geographical
data. The data values close to each other are likely to have similar
spectral characteristics. Especially in remotely sensed data, spatial
autocorrelations are present among pixels. Residuals of the Advance
Very High Resolution Radiometer data are examined for any
deviation from the random assumption. The correlation among the
residuals of the neighboring pixels and Moran's I statistic
demonstrate the significance of non-random distribution of residuals.
This study develops an autocorrelated error model to calibrate the
linear mixture model. Seven different autocorrelation patterns are
considered, and for each pattern, the linear mixture model is
calibrated and land uselland cover class fractions are estimated from
the Advance Very High Resolution Radiometer data.
INTRODUCTION
The effects of spatial autocorrelation are not clearly understood in unmixing
mixed pixels by the linear mixture model (LMM). Autocorrelated errors are
common in remotely sensed data (Campbell and Kiiveri 1988). Campbell and
Kiiveri (1988) described point spread function of the sensor and atmospheric
effects as causes for the spatial autocorrelation among neighboring pixels. The
radiance values of the neighboring pixels may show similar spectral characteristics
due to scattering and similar land useAand cover (LUILC) class patterns. These
effects become apparent with increasing coarseness of the spatial resolution. The
autocorrelated structure of remotely sensed data is widely used as additional
knowledge in image classification. Belward (1992) used spatial attributes:
contextual information and autocorrelated errors to evaluate the Advance Very High
Resolution Radiometer (AVHRR) data in environmental monitoring. Campbell and
Kiiveri (1988) discussed the advantages of using the spatial relations of the
neighboring pixels in image classification, but they did not observe a significant
advantage due to the spatial autocorrelation when the spectral class separations were
high. Furthermore, no significant improvement was noticed in overall accuracy
when spatial autocorrelations were included in the classification process.
In the empirical method of unmixing mixed pixels by LMM, calibration
coefficients are estimated by multivariate regression analysis. The calibration
coefficients are usually estimated assuming the model errors are spatially
uncorrelated with constant variance. Iverson et al. (1989) used multivariate
regression analysis to develop an empirical relationship between the AVHRR data
1 Graduate Assistant, Computer Graphics Center, NCSU, Raleigh, NC.
2 Professor and Director, Computer Graphics Center, NCSU, Raleigh, NC, and Dean of Internationul Space
University, Parc d'lnnovation, Communaute Urbaine de Strasbourg, Blvd. Gonthier d' Andernach, 67400 Ilkirch, France
3 Associate Professor, Department of Statistics, NCSU, Raleigh, NC.
4 Assistunt Professor, Mapping, Charting and Geodesy, Department of Geography and Environmental Engineering,
United States Military Academy, West Point, NY.
and the forest cover. Pech et al. (1986) examined multivariate calibration methods
to estimate vegetation cover. In all these studies, the model errors were assumed to
be spatially uncorrelated with constant variance.
Multivariate regression models with autocorrelated errors are common in
statistics. Nevertheless, the spatial autocorrelation effects are usually assumed
insignificant in unmixing mixed AVHRR pixels. Autocorrelated error models are
widely used for spatially correlated data accounting for the interaction between
neighboring locations. Upton and Fingleton (1985) described a generalized least
squares (GLS) regression method for spatially autocorrelated models. Basically,
models with normally distributed spatially autoregressive errors fall in two
schemes: simultaneous and conditional autorenressive schemes. The autocorrelation
parameters in this study were estimated by a maximum likelihood method described
by Upton and Fingleton (1985) based on a simultaneous autoregressive scheme.
Seven spatial autocorrelation patterns were compared with the model that assumed
errors were spatially uncorrelated.
METHOD
The study area, North Carolina Piedmont showed a wide variation of LUlLC
patterns. The diversity of the LULC class patterns ranged from developed urban to
various distributions of residential, agriculture, and forests. The frequency of the
local variation was high within this area. A significant amount of autocorrelation
was anticipated among the AVHRR pixels especially due to mixed pixel effects.
Data
Classified Landsat Thematic Mapper (TM) data and NOAA AVHRR data
corresponding to the North Carolina Piedmont data set used by Khorram et al.
(1994) were used in this study. Two sets of ASCII data files were developed in this
process. One set was developed from the classified Landsat TM data and the other
set was derived from the AVHRR data. The file created from the Landsat TM data
described LU/LC class fractions within each one square lun area corresponding to
the AVHRR pixels. The ASCII files created from the AVHRR data contained digital
number values in 14 spectral bands (Ediriwickrema 1995).
The LMM with Autocorrelated Errors
The I M M assumed that there were no interactions between the six LULC
classes. It also assumed that the resultant radiation of each pixel could vary only
due to random noise. Correlation between AVHRR pixel values and atmospheric
errors were also considered as insignificant. Under these assumptions, composite
radiation of mixed pixels was related to the component LULC class fractions and to
the spectral values of the L U M class pure pixels by Eq. 1. A pure pixel was
defined as a pixel that was totally covered by only one defined LULC class.
where A,,= pixel value of the j" AVHRR pixel in band k, Fji = fraction of the ith
LULC class within the jth AVHRR pixel, Ri, = calibration coefficients of the ith
LULC class in band k, a,,= random noise of the jth AVHRR pixel in the band k,
and c = the number of LULC classes. The error term "q;'was assumed to be
spatially correlated. The Gauss-Markov condition in the OLS regression is not
satisfied when the residuals are spatially correlated. One model for spatially
correlated errors is the simultaneous autoregressive error model. In this model the
error for one pixel depends upon the errors of neighboring pixels. The error vector
a for one band having "n" pixels is a = pWa + u (Upton and Fingleton 198S),
which gives unx,= (I?,, -pWnx,)anx, and p is a constant -- autocorrelation
parameter. W is a proximity matnx in which each row indicates which pixels are
neighbors in the autocorrelation scheme, u = error vector that satisfies the GaussMarkov conditions, and I = identity matrix. Since the elements of "u" are
uncorrelated, the simultaneous autoregressive model can be fitted by OLS after pre
multiplying both sides by (I - pW).
In other words, A *,,, = F *,,, Rcxb+ unxb,where A *,, = (Inxn- pWnxn
)Anxb
and F *,, = (Inxn- pWnxn
)Fnxc
. Then the estimates of the calibration coefficients
( R,,,) are written as follows.
A reasonable p value for each spectral band was required to estimate Rcxb.In the
simultaneous autoregression, p was chosen such that it maximized the log
likelihood function "ln(L)" in Eq. 4.
)A
where 62= iiTii/ n and ii = (Inxn- pWnxn
)A ,,, - (Inxn- pWnxn ,,,(upton and
Fingleton 1985). In Eq. 4 the determinant I1 - pWI is not easy to compute for large
data sets. Upton and Fingleton (1985) describe formulas to avoid computational
complications as well as a way to compute I1 - pWI for large data sets. These
formulae are applicable for only one type of proximity matrix, the Rook's case
(Upton and Fingleton 1985). Therefore, a 20 column and 20 row subset was
selected from the upper half by considering available system resources to reduce
computational burden. This subset was used to calibrate the LMM. A similar size
subset was also selected from the lower half of the image to estimate LU/LC class
fractions, and thereby examine the effects of autocorrelated errors in the LMM.
Weightings
The proximity matrix (W) is constructed according to the autocorrelation pattern.
Upton and Fingleton (1985) described three weighting named Rook, Bishop and
Queen. Rook weighting considers only adjacent pixels within a row or column to
be neighbors. The Bishop weighting considered only the cells that touch at the
corners to be neighbors, while the Queen model considers all surrounding cells,
i.e., touching comers and sides to be neighbors. In this study, seven different
weighting structures were considered (Ediriwickrema 1995).
Proximity matrix
In the proximity matrix, the cells that account for contiguity are assigned one, and
the remaining cells are assigned zero. Each row in the proximity is divided by its
total so that row totals are all equal to one. For example, a lattice of 3 x 3 pixels
results in a 9 x 9 proximity matrix. The proximity matrix for the weighting Rook's
case will be similar to the one that is shown in (Ediriwickrema 1995). In that
proximity matrix, the boundary pixels are weighted differently than the other pixels.
Only the available surrounding pixels are accounted for; there are many other
possibilities for weighting pixels on the edges. When all pixels are not weighted
with the same number of pixels, the scaling results in an asymmetric " W matrix.
Analysis of Spatial Continuity in the Model Residuals
The spatial autocorrelation structure of the model residuals are commonly
examined by semivariogram plots of the data, h-scatter plots (Isaaks and Srivastava
1989) of the errors, Moran's I statistic (Upton and Fingleton 1985). The residuals
from the regression of AVHRR data on the LUlLC class fractions using ordinary
least squares (OLS) were used to analyze the spatial continuity in the model.
Ediriwickrema (1995) describes seven different spatial autocorrelation patterns. The
spatial autocorrelation coefficient of the pixel residuals from an OLS fit of the LMM
to their neighboring (as defined for each weighting) pixel residuals was calculated
for each spectral band. The significance of spatial autocorrelation of the OLS
residuals was tested using Moran's I statistic.
Autocorrelation parameters
The magnitude and the sign of the autocorrelation parameters measure the scale
and the direction of spatial continuity. The proximity matrices are commonly
designed with row totals equal to one, which restrains p to be between -1 and 1.
Haining (1990) pointed out that p is an eigenvalue of w ' ~ , so p must lie between
where Am is the largest eigenvalue of W and h,, is the smallest
I/&,, and l/&~,,,
eigenvalue of W. As described in section 'The LMM with Autocorrelated Errors,"
the p value for each band was calculated such that it maximized "ln(L)". In this
study, the log likelihood function "ln(L)" values were evaluated for different p
values ranging from - 1 to 1 with an increment of 0.0 1. The p value corresponding
to the maximum "ln(L)", was selected as the autocorrelation parameter estimate of
the respective spectral band.
Calibration of the LMM
Calibration coefficients were estimated by Eq. 5 based on spectral bands. For
each spectral band, six coefficients were estimated representing all LULC classes.
Each time when the calibration coefficients were estimated for a band, residual
errors (G,,) were also calculated from the same data. Finally, the overall calibration
coefficients matrix (kcx,)and the residual error matrix (ii,,) were developed. The
variance-covariance matrix of the residual errors was calculated by
T
*bxb= uAbxn
* 6nxb I n . In order to compare the autocorrelated models to the model
that assumed errors are spatially uncorrelated, calibration coefficients were
separately estimated assuming the model errors were spatially uncorrelated.
Estimation of the LULC Class Fractions from the AVHRR Data
The North Carolina Piedmont data set was divided into two portions. From the
upper portion, the subset (A,,,) was selected to calibrate the LMM model. Subset
(B,,) was selected from the second portion to estimate the LULC fractions (G,,)
and assess the autocorrelated errors in the LMM. B, = G,,QCxb + w , where
B,, = digital values of "m" AVHRR pixels in "b" spectral bands, Q,,,= calibration
coefficients of "c" LULC classes in "b" spectral bands, and w,, = residual error
matrix. Let Q,, = Rcx, and assuming the measurement errors associated with R , ~ ,
are insignificant, the regression model can be written as B, = G,,Rcxb + w,,
and can also be re-written by taking transpose as B:~, = R;~,G:~, + wzx,.
We propose a simple model incorporatin s atial correlation within and between
spectral bands. Let b = vec(BT) = [blT b2% b3T....b14T]T,where b, is an m x 1
vector of AVHRR values for the k" spectral band. Let
,
,
A
A
denote the covariance matrix among bands within a pixel. Let C , = (I -p,W)"
denote the square root matrix of the spatial autocorrelation for band k. Then let the
covariance matrix for all bands and all pixels be
This covariance model says that the covariance among two bands in two different
pixels is just the product of the components of spatial variance for each of the two
bands and the covariance among bands "k" and "1" within a pixel o,.The estimated
GLS estimator of the LULC class fractions for the j" pixel is then
where
$ is a 14 x 14 matrix of estimates of Var(b) corresponding to the jthpixel
with the elements of C estimated by *,,, = 51xn* fin,, 1n , and bj is a 14 x 1 vector
of AVHRR values for the j" pixel.
The LULC class fractions were found to be best estimated by the constraint least
square regression method (Ediriwickrema 1995). The same method was used in
this study to estimate the LU/LC class fractions from the AVHRR data. The LULC
class fractions were estimated separately considering each autocorrelated pattern,
and assuming errors were spatially uncorrelated (Ediriwickrema 1995).
Validation
To examine the developments in the autocorrelated error model to the OLS model,
the residual sum of squares was calculated for four sample areas. From the upper
left of each quadrant, a 20 x 20 sample area was selected.
RESULTS
Correlation of Model Residuals
The correlation of the model residuals among the neighboring pixels for each
weighting is described in Figure 1. Except for the bands 11 and 12 generally all
other bands show significant correlation among the neighboring pixels in all seven
spatial autocorrelation patterns. Horizontal and vertical spatial autocorrelation
patterns resulted in high correlation related to the diagonal spatial patterns.
B
Figure 1.-Correlation
C
D
Weighting
E
F
G
of model residuals among neighboring pixels in each
weighting and each spectral band.
Significance of Autocorrelated Errors
Except for the z-statistics in bands 11 and 12 in correlation patterns "b", "d", "e"
and "f' (Ediriwickrema 1995)' all other z-statistics are greater than the upper onetail critical value at the 95% confidence level (1.645) (see Figure 2). The z-statistic
values in all other spectral bands in all autocorrelation patterns are clearly greater
than 1.645.
B 1 B2
B3
B4 B5
B6
B7 B8
B9 B 1 0 B 1 1 B 1 2 B l 3 B l 4
Band
Figure 2.-z-statistics
of each AVHRR band in each autocorrelated pattern.
Validation of the Autocorrelated Error Model
The difference of the sum of squares of the residuals between the OLS and the
autocorrelated error model are shown in Figure 3. The residual sum of squares was
reduced in the autocorrelated error model for most of the bands in all four
quadrants. The spectral bands acquired in late June and in mid July show
remarkable improvement. Except for bands five and six in the upper right quadrant,
all other bands in all quadrants show an improvement with the autocorrelated error
model. The magnitude of the improvement is dependent on the spectral bands and
the distribution of the LULC class patterns. The data acquired in the infrared band
showed better improvement over the visible band.
Figure 3.-Difference
of sum of squares of the residuals between the OLS and
the autocorrelated error model.
CONCLUSION
The AVHRR data have autocorrelated errors. Moran's I statistics and correlation
coefficients among neighboring pixels in different spatial patter clearly
demonstrated the spatial association of the AVHRR data. The spatial autocorrelation
parameters were evaluated, and were corrected separately for each spectral band.
The magnitude and the sign of the spatial autocorrelation parameters were
dependent on many factors: size of the pixel, the spectral band of the image,
direction of the autocorrelation pattern and the shape and size of the region. Wilson
(1992) observed lower autocorrelation with the adjacent diagonal pixels than with
the adjacent vertical and horizontal pixels. In this study also, models using diagonal
pixels for the autocorrelation pattems showed the least autocorrelation. The Rook
and Bishop models further confirmed this observation. The decrease of the sum of
squares of residuals in the autocorrelated error model demonstrated the advantage in
accounting for the autocorrelated errors. The magnitude of the improvement
certainly depends on the spectral band, acquisition period, and the distribution of
the L U / ' class pattems. This study identified the advantage of analyzing the
effects of autocorrelated errors in unmixing mixed pixels using simulated data with
adequate background knowledge. The background knowledge is important to
understanding and interpreting the effects of autocorrelated errors in unmixing
mixed pixels.
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