Distributions, spatial statistics and a Bayesian perspective
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1 Distributins, spatial statistics and a Bayesian perspective Dug Nychka Natinal Center fr Atmspheric Research Distributins and densities Cnditinal distributins and Bayes Thm Bivariate nrmal Spatial statistics and the data prduct Supprted by the Natinal Science Fundatin DMS NCAR/IMAGe July 2007
2 Overview As a specific example we will use average July maximum temperatures fr an area arund Bulder ver the perid
3 Use the spatial predictin prblem t illustrate the cncepts f cnditinal distributins and Bayes therem. L F B
4 Densities A prbability density functin (pdf) is an idealized histgram. It is used t describe prbabilities fr a randm quantity. X = average July temperature fr Bulder f(x) pdf: Prbability that X is in the small interval [x, x + ] is apprximately f(x)
5 Bulder July temps with a nrmal distributin superimpsed: (µ = 65.4, σ = 1.6) Frequency Degrees F
6 Yu can see alt just by lking... (Ygi Berra) years degrees F Bulder I am ging t ignre any time trends!
7 Mre ntes There are many extic distributins, gamma, t, nnparametric, etc. Gaussian: f(x) e (x µ)2 2σ 2 the classic bell-curve shape density, µ and σ are parameters that cntrl the spread and lcatin. Discrete distributin A finite set f pints that are each assigned a prbability. Drawing a randm sample frm a pdf is ften a gd apprximatin t the cntinuus theretical distributin. Here the randm sample defines a discrete distributin.
8 degrees F Bulder data ( n=103) each pint is assigned prbability 1/103.
9 Discrete verses cntinuus distributins The cntinuus nrmal distributin, a randm sample (n=100) drawn frm it and the histgram summary. Frequency Degrees F
10 Statisticians have their mments! A distributin and a sample bth have a mean and a variance. But they appear t be defined differently and have different interpretatins! Sample mean and variance: : ˆµ = 1 n n j=1 X j = ˆσ 2 = 1 n 1 n j=1 n j=1 Mean and variance fr a pdf σ 2 = µ = X j (1/n) (X j ˆµ) 2 xf(x)dx (x µ) 2 f(x)dx
11 The cnnectin: If the sample is thught as a discrete distributin where the prbability f taking n each data is 1/n then the tw definitins agree. The Ensemble Kalman filter uses a discrete distributin at the heart f its statistical algrithm.
12 Sampling variability Same thing several times t shw the sampling distributin f the histgram and sample mean
13 Sme ther simple remarks: Mean versus a realizatin The mean describes the center f the distributin. If X is nt knwn the mean is the best predictin f X in terms f making the errr small. Hwever, the mean nt lk like a real X value! e.g. the mean f Bulder July temps ( 65.38) is nt equal t any year s value. Transfrming a distributin If X has sme pdf and we cnsider a functin f it say g(x) what is the distributin f g(x)? e.g. if X is nrmal then X 2 is χ 2 with 1 degree f freedm.
14 If X 1, X 2,..., X n is a randm sample frm the distributin then g(x 1 ), g(x 2 ),..., g(x n ) is a randm sample frm the transfrmed distributin. This is a very useful way t apprximate distributins when yu need t d a cmplicated transfrmatin. Fr the ensemble Kalman filter g is the frward step f the mdel, a nnlinear functin with n clsed frm.
15 Multivariate distributins OK this is really where things get interesting. A scatterplt f Bulder and Fraser mean July temps Bulder Fraser
16 Multivariate distributins f(x, y) The jint pdf, f(x, y), is defined s that prbability f bth X and Y being in a small bx with sides [x, x + ] and [y, y + ] is apprximately f(x, y)/ 2. Bivariate nrmal distributin: Cmpletely described by five parameters: mean(x), mean(y ), VAR( X), VAR( Y ) and COV( X, Y ) COV(X, Y ) = Cvariance matrix: (x µ X )(y µ Y )f(x, y)dxdy The VARs and COVs are rganized in a matrix: ( ) VAR(X) COV(X, Y ) Σ = COV(X, Y ) VAR(Y )
17 Multivariate nrmal density fit t the Bulder/Fraser data Bulder Bulder Fraser Bulder Fraser Z
18 Cnditinal distributins A key step in DA is t determine the distributin f the state f the system given the bserved data. The term given signals a cnditinal distributin. What is the distributin f Fraser temps given that the Bulder temp is 64.5 r say 67.5? This distributin is different frm: the jint distributin f bth Bulder and Fraser the climatlgical distributin f Fraser (if Fraser and Bulder are nt independent).
19 Mtivatin using the bserved data Take slices at 64.5 and 67.5, nly cnsider the data in a neighbrhd arund each value Bulder Fraser X X X X X X X X XX X X
20 A mre frmal definitin f Cnditinal Prbability A and B tw events e.g. A X 65, B Y 60 P (A), P (B) dente their prbabilities and P (AB) is the prbability f bth events happening tgether
21 A B Shaded area is P (AB) the cnditinal prbability f B ccurring given A ccurs is P (B A) = P (AB) P (A) The vertical bar is read as given.
22 Cnditinal densities f(x, y) the jint pdf fr (X, Y )and suppse that g(x) is the pdf just fr X. f(y x) = f(x, y) g(x) Here X is bserved ( fixed) and we have a distributin fr Y. A useful prperty f Multivariate nrmals is that the cnditinal distributins are als nrmal.
23 Sme useful ntatin fr pdfs: [Y ] the pdf fr the randm variable Y (Fraser temp in this case) [X, Y ] pdf fr jint distributin f X and Y [Y X] cnditinal pdf fr Y given X S the frmula fr the cnditinal is: [Y X] = [X, Y ]/[X] Als nte that [X, Y ] = [Y X][X]
24 Bayes Therem Bayes Therem gives a way f inverting the cnditinal infrmatin. In bracket ntatin it is just [Y X] = [X Y ][Y ] [X] The prf fllws by definitins: [Y X] = [X, Y ] [X] = [X Y ][Y ] [X] Nte that [Y X] is simply prprtinal t the jint density where the nrmalizatin depends n the values f X. (But in many cases the nrmalizatin is difficult t find.)
25 Cnditinal densities fr the Bulder/Fraser jint pdf Slicing the surface Fraser Bulder Z Bulder Fraser
26 Cnditinal densities fr the Bulder/Fraser jint pdf (Y is Fraser temps and X is Bulder) Frequency [Y X=64.5] [Y X=67.5] [Y] Degrees F
27 Ntes n this example Cnnectin with Least Squares (LS) If we use the sample statistics the cnditinal mean fr Frasier is identical t Fitting a linear regressin t the bserved data. Using the LS line t predict a new temperature. Cnnectin with frecast skill The variance f the distributin gives a measure f the uncertainty in the predictin. Analysis is nly as gd as the statistical assumptins!
28 Infilled Fraser means based n Bulder degrees F years
29 Three members f an ensemble fr Fraser Mean, ensemble member degrees F degrees F degrees F
30 Sme cmments All infills have the same cnditinal mean and the variability will reprduce the climatlgy.
31 Spatial Statistics The ntrius data prduct What des the temperature field lk like n a grid based n the bserved data? The mdel T are the field values (e.g. temperatures) n a large, regular 2-d grid (and stacked as a vectr). This is ur universe. T is multivariate nrmal with mean µ and cvariance matrix: Σ = COV (T ) usually Σ is related t the distance between lcatins
32 The data {Y 1,..., Y n } are the statin data at irregular lcatins with sme measuremnet errr. Y j = T (x j ) + e j e j is measurement errr,
33 Kriging slutin Find the cnditinal distributin f the gridded temperatures given the statin values! ˆT = µ T + COV (T, Y )COV (Y ) 1 (Y µ Y ) and the cvariance f the estimate is P = COV (T ) COV (T, Y )COV (Y ) 1 COV (Y, T ) In the next few lectures: This frmula in matrix frm is als the Kalman filter. This is als a Bayesian slutin.
34 Temperature fields fr the Frnt Range Estimating the means, variances and and crrelatins µ and Σ fr T are estimated frm what data we have FRASER July Means LONGMONT BOULDER July Standard deviatins FRASER LONGMONT BOULDER
35 Spatial crrelatin f temperature degrees F Lngmnt Bulder Fraser years
36 Dependence f crrelatin n distance Distance in Miles Crrelatin L F Distance in Miles Crrelatin Nte that the crrelatin is nt zer clse t zer distance! This may be due t measurement errr.
37 Example f the cnditinal mean Reprting statins Psterir mean surface Mst flks wuld stp here and call this their gridded data prduct!
38 Ensemble f fields fr July 1993 Mean and 5 draws frm the cnditinal distributin
39 Better ways t d this! Use elevatin as an explanatry variable. Perfrm each years predictin in a climate space say based n mean July temperatures and elevatins.
40 Summary pdf can be apprximated by samples cnditinal distributins are nt the same as the uncnditinal distributin. spatial predictin is an applicatin f cnditinal distributins.
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