Local Prediction of a Spatio-Temporal Process with Application to Wet Sulfate Deposition. Presented by Isin OZAKSOY. Benefits of Spatial-Temporal over Spatial :. Use larger sample size to support model estimation Spatio-temporal drift estimates Location- specific forecasts
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Goal of article :
(percip. mean weighted average of weekly obs.) * (total percip. of season)
(x,y) : spatial coordinates of locations in spatio-temporal space
t : temporal coordinates
X=(x,y,t)’ : spatio-temporal location
n : total number of spatio-temporal observations
fc : fraction of n used for prediction
nc =n* fc : number of observations used to predict at xo
Prediction Cylinder : Spatio-temporal space that holds the nc observations used to predict the process at xo .
NOTE : Spatial and temporal dimensions of cylinder are defined separately.
Step 1 :
mT≤ tlatest - tearliest
mT= tU - tL
tL = max ( tearliest , tU - mT )
tU = min ( tO , tO + (mT /2) )
nI observations within temp. interval
mTlarge enough so nC < nI
Step 2 :
Sort nI observations according to ||(xO,yO)’ – (x,y)’||
Sort nI observations according to | tO – t |
Step 3 :
Cylinder`s obs. are first nC of sorted nI observations.
Cylinder`s radious : Spatial distance between the nCth observation and (xO,yO)’Determining Cylinder`s nC Observations
tearliest tL tO tU tlatest
VC : Variance- covariance matrix between residuals at xO and the residuals at the observation locations.
E(RC(x)) is constant.
Cov(RC(x1), RC(x2))=CS,T(g((x1,y1)’, (x2,y2)’), h(t1,t2)’) where;
CS,T(. , .)’ : Spatio-temporal covariance function,
g((x1,y1)’, (x2,y2)’) = ||(x1,y1)’ – (x2,y2)’|| : spatial lag
h(t1,t2)’ = | t1 – t2 | : temporal lag
STEP 1 :
STEP 2 :
STEP 3 :
w : kriging weights
: lagrange multiplier
: estimate of UC from covariance function in Step 2
IMPORTANT : Errors must NOT be dependent !
STEP 1 : Find nS observation locations within the cylinder from the same seasonality level and temporally closest to the prediction time.
STEP 2 : Sort nS locations by .
STEP 3 : Let be sample variance of the second-stage residuals computed
at first nn of the sorted locations. Let be sample variance of the second-stage residuals computed at all nS locations.
is sample variance of the nn closest residuals to the prediction location in terms of seasonality level, time, and estimated drift.
STEP 4 : Calculate the heteroscedasticity function at xO by :
BAIS ASSESSMENT :
MCSTK predictor and it's estimated standard error is bias for E(YC(xO)) and se(xO) respectively.
Case : If VC is known and RC(x) is homoscedastic, GLS drift estimate is found by finding that minimizes .
Define predicted residuals to be . If is known, then the kriging and variance from the residual kriging equations is equal to :
PREDICTION BIAS :
For i=1,…,nCV define cross-validation residuals and the
standardized cross-validation residuals be
PREDICTION BIAS :
PREDICTION STANDARD ERROR ESTIMATE BIAS :
REASON FOR BIAS IN STANDARD ERROR
NOTE : Reliability of MCSTK estimated standard errors increases as reliability of semivariogram estimates increases.
Cross-validation is performed over a set of fC values and the smallest value of fC values selected for the prediction and standard error estimate bias are as small as possible.