here’s a data analysis problem: for the 2002 forest...
TRANSCRIPT
Here’s a data analysis problem:
For the 2002 forest inventory data (Finley et al 2008; BEF.dat, spBayes).
Problem: Replace a laborious outcome measurement with a function ofpredictors measured by satellites.
I Outcome: red maple total basal area (metric tons biomass).
I Predictors: Elevation, slope, brightness (TC1), greenness (TC2),wetness (TC3).
I 437 observations.
This problem could have been on the MS exam I wrote in 1982 . . .
EXCEPT these observations are spatially referenced
Red maple basal area Elevation
Here’s a standard model for analyzing data like this
At spatial locations indexed by s, model outcome y(s) as
y(s) = x(s)� + w(s) + ✏(s)
Ix(s) are covariates, including an intercept
Iw(s) is a stationary GP, mean 0, covariance function �2
s K (⇢)
I ✏(s) is iid N with mean 0, variance �2e
I Unknown parameters: �, �2s, ⇢, and �2
e .
. . . giving this mixed linear model and restricted likelihood
For observations at {s1, . . . , sn}, the mixed linear model is
y = X� + Inw+ ✏
IX ’s rows are the x(si )
Iw = (w(s1), . . . ,w(sn))0 ⇠ N(0,G ) for G = �2
s K ({si}; ⇢)I ✏ ⇠ N(0,R) for R = �2
e I
�2s, ⇢, �
2e can be estimated by maximizing the log restricted likelihood
� log |V |� log |X 0V
�1X |� y
0[V�1 � V
�1X (X 0
V
�1X )�1
X
0V
�1]y
where V = G + R , a dense matrix.
For models like this, we don’t have LM-quality tools
The RL doesn’t have a closed form; e↵ects of data features are obscure.
Variograms are useful but
I don’t give specific info about how the data determine �2s, ⇢, �
2e
I aren’t much help in assessing non-stationarity.
The usual residuals are problematic:
I This model can fit any dataset arbitrarily well.
I If the model smooths much, residuals are biased.
I Residuals don’t tell us how the data determine �2s, ⇢, �
2e .
Bose, Hodges, Banerjee Biometrics 2018
BHB Biometrics (2018) is Step 1 (maybe) in filling this gap.
This talk emphasizes ideas using a simplified problem:
data collected on a 1-D regular grid with no fixed e↵ects.
I’ll mention how we’ve addressed these simplifications.
The three ideas that make this tractable
1. Approximate the GP; transform the data.
2. The resulting (approximate) restricted likelihood has a simple form;use that form to understand how the data determine the fit.
3. Extend tools from linear models.
Idea #1: Spectral approximation for a stationary GP
Data taken at locations sj 2 {0, 1, ...,M � 1}, M a multiple of 2.
Frequencies !m 2 {0, 1M , ..., 1
2 ,�12 + 1
M , ...,� 1M }, m = 0, 1, ...,M � 1.
Approximate the GP w(sj) by
g(sj) = a0 +2
M2 �1X
m=1
(am cos(2⇡!msj)� bm sin(2⇡!msj)) + aM2cos(2⇡!M
2sj)
am, bm have independent mean zero Gaussian priors with variances
proportional to �2s �(!m; ⇢), the spectral density of the GP covariance.
Based on Paciorek (2007), Wikle (2002).
With the approximation g(sj
), the model becomes
y = X� + Zu + ✏
I Assuming no fixed e↵ects: X is a vector of 1’s.
I Omit a0 to avoid an identification problem.
IZ ’s columns are sin/cos functions and do not depend on unknowns.
IZ
0Z = Diag(1/2M, 1/2M, ..., 1/M); Z
01 = 0.
Iu ⇠ N(0,G ), G = �2
s Diag( 12M�(!m(j); ⇢),
1M�(!M/2; ⇢) )
I ✏ ⇠ N(0,R), R = �2e I
Columns 1 to 4 of the random e↵ects design matrix Z
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0 50 100 150 200
−2−1
01
2
Index
t(z)[1
, ]
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0 50 100 150 200
−2−1
01
2
Index
t(z)[2
, ]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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0 50 100 150 200
−2−1
01
2
Index
t(z)[3
, ]
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0 50 100 150 200
−2−1
01
2
Index
t(z)[4
, ]
Idea #1: Transform the data so the log RL is simple
Using the spectral approximation to the GP, the model is
y = X� + Zu + ✏
Pre-multiply this equation by (Z 0Z )�0.5
Z
0 to give:
v = (Z 0Z )
�0.5Z
0y = (Z 0
Z )0.5
u + (Z 0Z )
�0.5Z
0✏
Then
E(v) = 0
Cov(v) = �2sDiag(aj(⇢)) + �2
e I
aj(⇢) = �(!m(j); ⇢)
Idea #2: The (approximate) RL has a simple form.
v ⇠ N( 0, �2sDiag(aj(⇢)) + �2
e I )
v = (Z 0Z )
�0.5Z
0y
The (approximate) log RL for (�2s, ⇢,�
2e ) is the likelihood arising from v :
�1
2
M�1X
j=1
[ log(�2saj(⇢) + �2
e ) + v
2j / (�2
saj(⇢) + �2e ) ].
The keys to understanding this (approximate) RL as a function of
the data are the transformed data vj and the aj(⇢).
Given ⇢, the v
2j
follow a gamma-errors GLM
The log RL has the form of the likelihood from a gamma-errors GLMwith the identity link:
�1
2
M�1X
j=1
[ log(�2saj(⇢) + �2
e ) + v
2j / (�2
saj(⇢) + �2e ) ]
I The v
2j are the gamma-distributed “data”.
I 1/2 is the gamma’s shape parameter.
IE (v2
j ) = �2saj(⇢) + �2
e
I Var(v2j ) = 2 (�2
saj(⇢) + �2e )
2
How does aj
(⇢) change with ⇢?
Exponential covariance function K : aj(⇢) for ⇢ = 5 and 16.
The horizontal axis is j .
0 50 100 150 200
05
1015
aj165
For larger ⇢, the aj(⇢) start higher and decline to zero more sharply.
Idea #2: Use this (approximate) RL to understand the fit
The approximate RL uses the data only through the vj ,
projections of y onto sin/cos functions of di↵erent frequencies.
The projections vj don’t depend on any unknowns.
The projections vj are the same for all GP covariance functions.
Using di↵erent GP models for the random e↵ect, fitting di↵erent gamma regressions to the same transformed data.
The model for the v
2j is a GLM with 3 parameters ) no overfitting.
How do the data determine parameter estimates?
The “observations” are the v
2j ; parameters are fit such that:
E( v
2j | �2
s, ⇢,�2e ) = �2
s aj(⇢) + �2e .
The aj(⇢) are non-increasing in j and for large j ,
E( v
2j | �2
s, ⇢,�2e ) ⇡ �2
e .
Loosely,
I �2e is “in the middle” of the v
2j for large j .
I ⇢ fits the rate at which the v
2j decline for “small” j .
I �2s makes �2
saj(⇢)+ �2e go through the middle of the v2
j for “small” j .
Examples of v 2j
vs. j , with fits
Simulated: �2s = 2, ⇢ = 5,�2
e = 5 Red maple basal area
�2s smaller than �2
e Whopping non-stationarity
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0 50 100 150 200
010
2030
4050
Index
v^2
0 100 200 300 400 500
01000
2000
3000
4000
5000
6000 1
2
3
45678
91011121314
15
16171819
20
2122232425262728293031323334353637383940414243444546474849505152535455565758596061626364
6566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208
209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268269270271272273274275276277278279280281282283284285286287288
289290291292293294295296297298299300301302303304305306307308309310311312313314315316317318319320321322323324325326327328329330331332333334335336337338339340341342343344
345346347348349350351352353354355356357358359360361362363364365366367368369370371372373374375376377378379380381382383384385386387388389390391392393394395396397398399400401402403404405406407408409410411412413414415416417418419420421422423424425426427428429430431432433434435436437438439440
441442443444445446447448449450451452453454455456457458459460461462463464465466467468469470471472473474475476477478479480481482483484485486487488489490491492493494495496497498499500501502503504505506507508509510511512513514515516517518519520521522523524525526527528529530531532533534535536537538539540541542543544545546547548549550551552553554555556557558559
Some conjectures about how the data determine estimates
An outlier inflates the v
2j for large j ’s (high frequencies) ) inflated �2
e .
Little e↵ect on v
2j for “small” j (low frequencies) and thus on �2
s and ⇢.
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0 50 100 150 200
−50
510
15
Index
y
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Data contaminated with shift
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0 50 100 150 200
−50
510
Index
y
(a) data simulated from GP with�2s=2, �2
e
=5 and ⇢=5 with meanshift from 0 to 5 midway
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0 50 100 150 200
−2−1
01
2
Index
t(z)[1
, ]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0 50 100 150 200
−2−1
01
2
Index
t(z)[2
, ]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
0 50 100 150 200
−2−1
01
2
Indext(z
)[3, ]
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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0 50 100 150 200
−2−1
01
2
Index
t(z)[4
, ]
(b) first four columns of Z , thespectral basis matrix, on thedomain [1,2,...,199,200]
22 / 44
Hypothesize: how does this shift in mean a↵ect theestimates of the GP parameters ?
1v
22 will be inflated, this will lead to an inflated value of the estimateof �2
s .
2 So to capture the sharp decline in v
2j
’s the estimate of ⇢ will beinflated too.
3v
2j
’s for larger j ’s broadly una↵ected, hence the estimate of �2
e
willremain almost the same.
23 / 44
Parameter estimates (SE): data with mean shift
exact RL�2s �2
e
⇢actual values 2 5 5
uncontaminated 2.29 (0.11) 4.75 (0.11) 6.89 (0.82)contaminated 11.50 (0.36) 5.61 (0.06) 106.48 (6.06)actual values 10 0.1 16.67
uncontaminated 9.99 (0.34) 0.11 (0.01) 17.29 (0.67)contaminated 17.96 (0.77) 0.16 (0.01) 29.99 (1.32)
apprx RL�2s �2
e
⇢actual values 2 5 5
uncontaminated 2.39 (0.15) 4.90 (0.09) 13.51 (1.74)contaminated 11.60 (0.41) 5.33 (0.07) 122.0 (6.49)actual values 10 0.1 16.67
uncontaminated 10.02 (0.37) 0.32 (0.01) 32.70 (1.37)contaminated 19.70 (1.13) 0.37 (0.01) 61.49 (5.08)
25 / 44
The data contain little information about ⌫
0 10 20 30 40 50 60
0.000
0.010
0.020
0.030
Index
a
nu=inf,rho=0.1nu=inf,rho=0.025nu=0.5,rho=0.1nu=0.5,rho=0.025
Figure: aj
’s for Matern (⌫=0.5) and Matern (⌫=1) for the spectralapproximation on the domain [0,1,...,62,63] on the horizontal axis.
30 / 46
Idea #3: Extend tools from linear models and GLMs
Plot the v
2j and fitted values �2
saj(⇢) + �2e vs. j .
This shows the data and model fit corresponding to the RL.
It’s a direct look at the “signal” for non-stationarity.
Added variable plots show how the data produce a fixed e↵ect’s estimate.
An AVP can be done in both the
Observation domain (y) and
Spectral domain (the vj)
Aim andModel
Spectral-transformingthe data
Fit of thetransformeddata
Identifyingnonstationarity
Investigatingmissingpredictors
Summary
How to applyto real data
Real dataanalysis
Conclusion18/42
Added variable plot in observation domain
Adding predictor C to the model y = X� + C↵+ u + ✏,X contains predictor already in the model.
Multiply both sides of the model equation by V
�0.5, ˆ denotesestimates from fitting a model with X .
Then multiply both sides by
P = I � V
�0.5X (X 0
V
�1X )
�1X
0V
�0.5
• Plot PV�0.5y vs PV�0.5
C .
Aim andModel
Spectral-transformingthe data
Fit of thetransformeddata
Identifyingnonstationarity
Investigatingmissingpredictors
Summary
How to applyto real data
Real dataanalysis
Conclusion16/42
Added variable plot in the spectral domain
Adding predictor C to the model y = X� + C↵+ u + ✏,X contains predictors already in the model.
Multiply both sides by (I � PX ),then multiply by (Z 0
Z )�0.5Z
0, to get
v
⇤ = (Z 0Z )�0.5
Z
0(I � PX )y : the vj from the residual y and
v
⇤C = (Z 0
Z )�0.5Z
0(I � PX )C : the vj from the residual C .
• Plot Dv
⇤ vs Dv
⇤C ,
where D = Diag(1/p�2s a(⇢) + �2
e ), ˆ denotes estimatesobtained from fitting a model with only X , no C .
Aim andModel
Spectral-transformingthe data
Fit of thetransformeddata
Identifyingnonstationarity
Investigatingmissingpredictors
Summary
How to applyto real data
Real dataanalysis
Conclusion22/42
Added variable plots (AVPs) in the two domains
The AVPs in the spectral domain and in the observationdomain estimate the same coe�cient for a particularpredictor.
The spectral domain AVP highlights particular large scaletrends in the data. The observation domain AVPhighlights particular localized details.
The spectral domain AVP involves the spectralapproximation which the observation domain AVP doesnot.
Back to the forest inventory data
Outcome: Red maple basal area Predictor: Elevation
Undoing the simplifying assumptions
The paper and supplements discuss these at length.
• Spectral approximation in 2-D: Each design-matrix column correspondsto a pair of frequencies, one in each dimension (Paciorek 2007).
• Data not on a grid: Pre-smooth to a grid. (9 a better way?)
• Fixed e↵ects: Regress them out and apply the spectral approximationto the residuals.
Plots of v 2j
vs. j for outcome and predictor
Outcome: Red maple basal area Predictor: Elevation
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The horizonal axis is j ) low frequencies at left, high frequencies at right
AVPs for elevation
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24
Info from the AVPs vs. the actual fits
Elevation From the RLEstimate SE P-value �2
s ⇢ �2e
Intercept-only – – – 29.6 6.0 16.2AVP, Spectral -3.17 0.51 10�10 – – –
AVP, Observation -2.02 1.09 0.07 – – –Real fit -2.52 0.29 tiny 22.0 2.9 13.8
Focus on the ideas, not on our specific choices
The important thing is the 3 ideas:
1. Approximate the GP; transform the data.
2. ) simpler forms, which makes the fit understandable.
3. Extend tools from linear models and GLMs.
All the specific choices we’ve made could be replaced (I think).