Spatial methods and their applications to environmental and climate data

Spatial metho ds and their applications to en vironmen tal and climate data S p a t i a l F i e l d Behnaz Pirzaman bin Departmen t of Mathematical Statistics Lund Univ ersit y , Sw eden 2013 Abstract En vironmental and climate pro cesses are often distributed ov er large space-time domains. Their complexit y and the amoun t of av ailable data mak e mo delling and analysis a challenging task. Statistical mo delling of environmen t and climate data can ha v e sev eral different motiv ations including interpretation or characterisation of the data. Results from statistical analysis are often used as a integral part of larger en vironmental studies. Spatial statistics is an active and modern statistical field, concerned with the quan titative analysis of spatial data; their dep endencies and uncertainties. Spatio- temp oral statistics extends spatial statistics through the addition of time to the, t wo or three, spatial dimensions. The fo cus of this in tro ductory paper is to pro vide an o v erview of spatial metho ds and their application to environmen tal and climate data. This paper also gives an o verview of sev eral imp ortant topics including large data sets and non-stationary co v ariance structures. F urther, it is discussed ho w Ba y esian hierarc hical mo dels can pro vide a flexible wa y of constructing mo dels. Hierarchical models ma y seem to b e a go od solution, but they ha ve challenges of their own such as, parameter estimation. Finally , the application of spatio-temp oral mo dels to the LANDCLIM data (LAND cov er - CLIMate interactions in NW Europ e during the Holo cene) will b e discussed. CONTENTS 1 INTR ODUCTION 2 1.1 Basic concepts 3 1.1.1 Prop erties of random v ariable 3 1.1.2 Linear regression 4 1.1.3 P arameter estimation 5 2 SP A TIAL ST A TISTICS 6 2.1 Theory of Gaussian fields 6 2.2 The basic mo del 7 2.3 Prediction 8 2.4 P arameter estimation 9 2.5 Hierarc hical models 11 3 SP A TIO-TEMPORAL ST A TISTICS 13 4 NON-ST A TIONAR Y COV ARIANCE STRUCTURES 14 4.1 Pro cess conv olution 14 4.2 Deformation approac h 15 4.3 Allo wing for cov ariates in the co v ariance structure 16 5 HANDLING LAR GE SP A TIAL DA T A SETS 17 5.1 Lo w rank approximation 18 5.1.1 Fixed rank kriging 18 5.1.2 Predictiv e process 19 5.2 Co v ariance tap ering 19 5.3 Appro ximating Gaussian Random Fields with Gaussian Marko v Ran- dom Fields 20 6 APPLICA TION TO THE LANDCLIM D A T A 23 6.1 In tro ductory 23 6.2 Data 23 i 1 6.2.1 REVEALS, pollen-based v egetation reconstruction 24 6.2.2 A dditional data 25 6.3 Mo del 25 6.4 Preliminary results 26 6.5 F uture work 28 6.5.1 Iden tifying and ev aluating cov ariates 28 6.5.2 Assessing spatial structure 28 6.5.3 Iden tifying an indep endent human land-use model 28 6.5.4 Incorp orating the uncertain ty 29 REFERENCES 30 Chapter 1 INTRODUCTION Spatial statistics is an activ e and mo dern statistical field, concerned with the quan titative analysis of spatial data, their dep endencies and uncertainties. One of the main prop erties of spatial statistics is the handling of correlated data; i.e allowing for observ ations that are close to eac h other, to b e more similar than observ ations that are far apart. One of the earliest w ork including spatial considerations w as R.A Fisher’s dev el- opmen t of design-based inference for agricultural field exp eriments ( 1919 to 1933 , published in 1966 ). Fisher noted that field plots, rectangular units in the field, close to eac h other were more similar than the plots further apart, violating the assumption of indep enden t data. T o accoun t for the increasing dep endence, Fisher suggested the use of blo cking of the plots, a form of co v ariate adjustmen t. Hence, the larger blo cks of plots where appro ximately independent, and spatial v ariation, if it exists, w as constant within blo c ks. An alternativ e strategy suggested b y P apadakis [1] is to adjust the plot yields to take account of the av erage yield in neighbouring plots instead of the ov erall yield. This strategy has a close relation to Marko v random fields as p oin ted out b y Cox et al. [2] and Bartlett [3] (see further [4]). F urther developmen t of this concept lead to the general case of Gaussian Marko v Random Fields which are discussed in Section 5.3. Other imp ortan t w ork in spatial statistics w as done b y Krige [5] and Mathéron [6]. Their work fo cused on the c haracterization and prediction of spatial data, leading to Geostatistics . Another statistician who has done influen tial work in mo delling spatial dep endencies is Bertil Matérn, a Swedish forestry statistician. Matérn’s do ctoral dissertation is a remark able work in spatial statistics, ha ving one of the highest num b er of citations in the field [7]. He introduced the Matérn family of co v ariance functions whic h is one of the most popular mo dels for many geostatistical applications [4]. Spatial statistics can b e applied to data from man y differen t fields, including climate and en vironmental data. The developmen t of spatio-temp oral statistics during the last cen tury has aided scien tists in solving n umerous en vironmen tal and climate problems. The collab oration b etw een statistician and en vironmen tal scien tists has also led to imp ortan t developmen ts in spatial statistics, from the first work of Fisher [8], handling data from agricultural experiments, to modern da y methods, 2 Section 1.1. Basic concepts 3 handling the v ery large data sets that arise in climatological applications. F or example, en vironmen tal questions regarding aerosol forcing—the global distribution of aerosols, the transport of aerosols, and differences betw een satellite observ ations and global-climate-model outputs—were a motiv ation for equipping satellites to collect global data. These observ ations are noisy and contain missing v alues, Shi and Cressie [9] used spatial statistics to reconstruct the missing v alues and denoise the data. In P aleo ecology , scientists hav e questions ab out past forest composition: how relativ e v egetation abundances c hange ov er time, if forest comp ositions constantly shifting, how forest changed in response to past climate shift and how h umans affect forests in comparison to natural forest change. A recent study by Paciorek and McLac hlan [10] on US data used a multiv ariate spatio-temporal process to mo del forest comp osition for past time p erio ds, based on observ ations from the colonial era (1635-1800) and 20th centuries. The mo del of [10] estimates the spatial distribution of the relative vegetation abundances and gives uncertainties for these estimates. Throughout this pap er more en vironmental problems are given as examples in o ccurrence with the statistical metho ds that used to solve them. This pap er will provide an o verview of spatial statistics and its applications. The first section provides some basic concepts in probabilit y . Section 2 gives an in tro duction to spatial statistical mo delling, co vering parameter estimation and prediction. In Section 3 spatio-temp oral mo del are introduced. Section 4 introduces one of the challenges in spatial statistics, non-stationary cov ariance structures; some common solutions to the problem are also discussed. Section 5 discusses the issue of large spatial data sets. Finally , Section 6 discusses ho w spatio-temporal mo dels can be applied to the LANDCLIM data (LAND cov er - CLIMate interactions in NW Europe during Holocene), one of the pro jects in strategy research areas MER GE (Mo dElling the Regional and Global Earth system) with the focus on land-co v er/v egetation . 1.1 Basic concepts 1.1.1 Prop erties of random va riable Giv en a contin uous random v ariable with densit y f ( x ) the exp ectation is defined as E ( x ) = R xf ( x ) dx ( or E ( x ) = P x p ( x ) in discrete form) and the v ariance is V ( x ) = E ( [ x − E ( x )] 2 ) = E ( x 2 ) − E ( x ) 2 . The cov ariance of a bi-v ariate random v ariable is defined as C ( x, y ) = E ([ x − E ( x )] [ y − E ( y )]) = E ( xy ) − E ( x ) E ( y ) = C ( y , x ) . (1.1.1) And for multiv ariate random v ariables the cov ariance matrix is constructed as Σ ij = C ( x i , x j ) , note that Σ ii = C ( x i , x i ) = V ( x i ) . The cov ariance matrix has three prop erties; it is i) Square ii) Symmetric, Σ = Σ T and iii) Positiv e definite, a T Σ a > 0 if a 6 = 0 . The first tw o prop erties follo w trivially from (1.1.1) , and are easy to verify . The follo wing pro of sho ws why the co v ariance matrix ne eds to b e p ositive definite. 4 Introduction Chapter 1 Pro of (iii) Consider the cov ariance in matrix form, a T Σ a = E [ a T ( X − µ )( X − µ ) T a ] = E [  a T ( X − µ )  2 ] = V  a T X  > 0 if a 6 = 0 and V ( X )  0 . (1.1.2) F rom the pro of ab ov e one can see that the v ariance of linear com binations(e.g. a mean) of random v ariables is giv en b y a quadratic-form inv olving the co v ariance matrix. Additionally , v erifying that a matrix is p ositive definite, i.e. a v alid co v ariance, is hard since a T Σ a > 0 must hold for al l a 6 = 0 ; this will presen t a c hallenge when constructing co v ariance matrices. 1.1.2 Linea r regression A statistical mo del is a family of probability distributions, p , which one assumes that a particular data set, y , is sampled from. F or a parametric model, there are unknown parameters, θ , which control the distribution, p . Linear regression [11] is an example of a simple statistical mo del. Definition 1.1.1 (Linear regression) . Let Y = ( y 1 , y 2 , . . . , y n ) T b e a v ector of observ ations and X = ( X 1 , X 2 , . . . , X p ) be explanatory v ariables that are fixed and kno wn, then the linear mo del is constructed as y i = X p X i,p β p + e i e i ∈ N (0 , σ 2 ) or in matrix form, Y = X β + e e ∈ N (0 , I σ 2 ) . In linear regression one assumes that the observ ations Y follo w a Gaussian distri- bution with a mean that depends line arly on a set of kno wn explanatory v ariables X . The unkno wn parameters are: ho w muc h of eac h explanatory v ariable should b e used in the mean β ,and how large the v ariabilit y , σ , around the mean is. An estimate of β is ˆ β = ( X T X ) − 1 X T Y . Moreo ver, under the mo del assumptions, the residuals ˆ e = Y − X ˆ β , ha ve the follo wing prop erties, 1) They are indep endent, 2) Normally distributed, 3) Ha v e equal v ariance. In order to c hoose a better regression model, typically a model with optimal n umber of explanatory v ariables, one can p erform different statistical tests. One w ay of doing this is to define a criteria for the best mo del. T w o common criteria are Ak aike’s information criterion (AIC) [12], AI C ( p + 1) = − 2 log L ( ˆ β , ˆ σ ) + 2( p + 1) , Section 1.1. Basic concepts 5 and Sc h w artz’s Bay esian information criterion (BIC) [13], B I C ( p + 1) = − 2 log L ( ˆ β , ˆ σ ) + log n ( p + 1) . In general the second term in both AIC and BIC increases with p , how ev er larger p giv es smaller L ( ˆ θ ) , i.e. likelihoo d at ˆ θ , thus AIC and BIC attempts to balance b et w een model size and residual error. F or AIC the mo del size p enalty 2( p + 1) do es not dep end on num b er of observ ation, n , which, esp ecially for large n , can be a problem. The mo del size p enalty is adjusted in BIC with log n , hence BIC typically giv e smaller mo dels. 1.1.3 P a rameter estimation A common problem in statistics is to estimate parameters of a mo del. Giv en a mo del p ( y ; θ ) with parameter(s) θ , t w o common metho ds for estimation of the parameters are [14]: 1. Least squares (LS) 2. Maxim um lik elihoo d (ML) In least squares one tries to find the parameter(s) that minimizes the sum of square errors b et ween the observ ation and exp ected v alue giv en parameter(s), ˆ θ = argmin θ n X i =1 ( y i − E ( y i ; θ )) 2 and in the case of maximum lik el ihoo d, assuming that observ ations are indep endent the lik elihoo d is L ( θ | y 1 , . . . , y n ) = p ( y 1 , y 2 , . . . ; θ ) = indp n Y i =1 p ( y i ; θ ) hence, one tries to find parameter(s) that maximize the likelihoo d function, ˆ θ = argmax θ L ( θ | y 1 , . . . , y n ) . Chapter 2 SP A TIAL ST A TISTICS 2.1 Theo ry of Gaussian fields A statistical mo del describes and analyses randomness in a set of data. F or spatial data, one p ossible model is a random, or sto c hastic, field. Definition 2.1.1 (Sto c hastic field) . A random or sto chastic field, X ( u ) , u ∈ D ⊆ R d is a random function sp ecified b y its finite-dimensional join t distributions F u 1 ,...,u n ( x 1 , . . . , x n ) = P ( X ( u 1 ) ≤ x 1 , . . . , X ( u n ) ≤ x n ) for all finite n and all collection u 1 , . . . , u n of locations in D . Consider a field x defined in t w o dimensions, the field is called a Gaussian random field if any subset of points in the field, x ( u 1 ) , · · · , x ( u n ) , are jointly m ultiv ariate Gaussian, i.e. x ∈ N ( µ, Σ) , with density given by p ( x ) = 1 (2 π ) N/ 2 | Σ | 1 / 2 exp  − 1 2 ( x − µ ) T Σ − 1 ( x − µ )  . (2.1.1) Moreo ver, the field has a exp ectation function µ x ( u ) = E ( x ( u )) and cov ariance function r x ( u , v ) = C ( x ( u ) , x ( v )) . The stochastic field can further be stationary and/or isotropic. Definition 2.1.2 ( 2 nd order/w eak stationary) . A field is said to b e 2 nd order stationary if exp ectation and cov ariance are unchanged under translation. • µ x ( u ) = µ x ( u + h ) = constant. • r x ( u , v ) = r x ( u + h, v + h ) = ⇒ r x (0 , v − u ) = r x ( h ) . A stationary field is sometimes said to b e homogeneous. Definition 2.1.3 (Isotropic) . If a stationary co v ariance depends only on the distance b et w een the p oin ts and not on the direction, r ( h ) = r ( k h k ) , the field is said to be isotropic; otherwise it is anisotropic. 6 Section 2.2. The basic mo del 7 An alternative to the co v ariance function is the semi-v ariogram. Semi-v ariograms can b e defined even if the co v ariance function do es not exit; e.g. if the field lac ks finite expectation E ( x ) ≮ ∞ . Definition 2.1.4 (Semi-v ariogram) . A semi-v ariogram for a stationary and isotropic field is defined as γ ( k h k ) = 1 2 V ( x ( u + h ) − x ( u )) = r (0) − r ( k h k ) (2.1.2) Estimation of semi-v ariograms is more robust to miss-sp ecification of the mean than estimation of co v ariance function. This, since the mean cancels in the subtraction of x ( u + h ) and x ( u ) . If the field has constant mean then (2.1.2) b ecomes γ ( k h k ) = 1 2 E  ( x ( u + h ) − x ( u )) 2  In T able 2.1 some common co v ariance functions and corresp onding semi-v ariogram are sho wn. Name Co v ariance, r ( h ) Semi-v ariogram, γ ( h ) Matérn σ 2 ( h / ρ ) ν K ν ( h / ρ ) Γ( ν )2 ν − 1 σ 2  1 − ( h / ρ ) ν K ν ( h / ρ ) Γ( ν )2 ν − 1  Exp onen tial 1 σ 2 exp ( − h / ρ ) σ 2  1 − exp ( − h / ρ )  Gaussian 2 σ 2 exp ( − ( h / ρ ) 2 ) σ 2  1 − exp ( − ( h / ρ ) 2 )  Spherical ( σ 2  1 − 1 . 5( h ρ ) + 0 . 5( h ρ ) 3  0 ( σ 2  1 . 5( h ρ ) − 0 . 5( h ρ ) 3  h < ρ σ 2 o.w T able 2.1: Different co v ariance functions and semi-v ariograms, in general σ 2 denotes the v ariance of the field and ρ is the range parameter. 2.2 The basic mo del Geostatistical measurements are often made with some noise or error in measure- men ts, therefore any mo del describing the data has to acc oun t for the noise. Generally , the simplest mo del consists of a latent Gaussian field X ∈ N ( µ, Σ) at some locations { u i } n i =1 observ ed with additive noise, y i = x ( u i ) + ε i , ε i ∈ N (0 , σ 2 ε ) , 1 Matérn with ν = 1 / 2 . 2 Matérn with ν → ∞ . 8 Spatial Statistics Chapter 2 or in matrix form Y = A X + ε , ε ∈ N (0 , I σ 2 ε ) (2.2.1) where the matrix A is a sparse observ ation matrix that extracts the appropriate elemen ts from X . The join t distribution of X and Y is Z =  X Y  ∈ N  µ x µ y  ,  Σ xx Σ xy Σ yx Σ yy  =       µ Aµ  | {z } µ z ,  Σ Σ A T A Σ A Σ A T + I σ 2 ε  | {z } Σ z      (2.2.2) or Z ∈ N ( µ z ( θ ) , Σ z ( θ )) . T ypically the latent field is decomposed in to a deterministic mean and a stationary , mean zero field, X = µ + η , , η ∈ N (0 , Σ( θ )) . (2.2.3) The mean is often mo delled as a regression, µ = B β . A standard problem in spatial statistics is to reconstruct the latent field, X giv en observ ations Y . F or known parameters this is describ ed in Section 2.3, ho w ev er parameters are often unknown and need to be estimated, see Section 2.4 for more details. 2.3 Prediction The most famous metho d for reconstructing the field is kriging. The metho d was dev elop ed by Mathéron [6] from Krige [5] master’s thesis in geostatistics. It was mostly fo cus on spatial dependency and predicting v alues of field o v er a spatial region. Assuming a kno wn cov ariance , Σ , a regression formulation of the exp ectation, µ = B β , and given the observ ation at some lo cation, y ( u i ) , i = 1 · · · n , the aim is to predict v alues, i.e. reconstruct the field, at unobserv ed location, x ( u s ) . The reconstructions should b e: 1) Linear in the observ ations, ˆ x s = P k λ k y ( u k ) ; 2) Unbi- ased, E ( ˆ x s ) = E ( x s ) ; and ha ve 3) Minimum prediction v ariance; min λ V ( ˆ x s − x s ) . T raditionally Kriging has been divided in to three different cases: 1) Simple Kriging: µ is kno wn, 2) Ordinary Kriging: µ is unkno wn, but constant, 3) Universal Kriging: µ = B β , with β unkno wn. (Possibly , estimation of unkno wn Σ .) Note that µ = I β giv es an unknown constant making 2) a special case of 3) . F or a Gaussian pro cess the b est linear unbiased predictor is the conditional exp ectation [15], E ( X | Y ) = B x ˆ β + Σ xy Σ − 1 y y ( Y − B y ˆ β ) =  B x − Σ xy Σ − 1 y y B y  ˆ β + Σ xy Σ − 1 y y Y (2.3.1) Section 2.4. P a rameter estimation 9 where ˆ β =  B T y Σ − 1 y y B y  − 1 B T y Σ − 1 y y Y and B x is the co v ariate of the regression model used to estimate µ and B y = A ∗ B x . The prediction uncertaint y for simple kriging is V ( X | Y ) = Σ xx − Σ xy Σ − 1 y y Σ T y x (2.3.2) and for the ordinary and univ ersal kriging the β estimator’s uncertain t y is added to (2.3.2), V ( X | Y ) = Σ xx − Σ xy Σ − 1 y y Σ y x +  B T x − B T y Σ − 1 y y Σ y x  T  B T y Σ − 1 y y B y  | {z } V ( ˆ β | Y )  B T x − B T y Σ − 1 y y Σ y x  . Example 2.3.1 (T emperature map using Universal Kriging) . This example shows the estimate d temp er atur e for winter in U.S using 250 lo c ations. The estimate d values ar e pr e dicte d using universal kriging. In Figur e 2.1 , observe d lo c ations, pr e diction and standar d err or (SE) ar e shown. Observed temperature −10 −5 0 5 10 15 20 25 30 35 Predicted temperature −10 −5 0 5 10 15 20 25 30 35 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Average temperature using universal kriging SE of temperature Figure 2.1: Estimated temperature for winter in U.S using univ ersal kriging from 250 locations. 2.4 P a rameter estimation The reconstruction assumes a known cov ariance matrix Σ . Ho wev er, in practice Σ often has to b e estimated from data. Prop erties of Σ , suc h as symetry and p ositive definite, put constrains on the estimation. T o solv e this problem one often assumes that the cov ariance matrix is defined b y a parametric family of cov ariance functions. This assumption reduces the problem to estimate the parameters of the co v araiance function. Uncertainties in the estimated parameters can be accounted for b y using Mark ov Chain Mon te Carlo (MCMC) metho ds [16] or numerical integration (INLA) [17]. 10 Spatial Statistics Chapter 2 In order to estimate the cov ariance function using a non-parametric approac h, one migh t use the semi-v ariogram defined in Definition 2.1.4. One can estimate parameters of co v ariance function using the tw o common metho ds explained in Section 1.1.3; LS or ML. In LS, an estimated mean is subtracted from the observ ation, w = Y − B ˆ β , and the semi-v ariogram, ˆ γ ( k u i − u j k ) = 1 2 E ( w ( u i ) − w ( u j )) 2 , giv es one estimate for ev ery pair of observ ations. The estimates, ˆ γ , are often binned, i.e. group ed into blocks with similar distances ( u i − u j ) . The estimates are then a veraged within each blo ck. In order to handle the semi-definite restriction one ma y c hoose a parametric semi-v ariogram, e.g. one of those in T able 2.1, and pic k parameters, θ suc h that ˆ γ and γ ( h ; θ ) are as close as possible, that is ˆ θ = argmin θ X  γ ( h ; θ ) − ˆ γ  2 . (2.4.1) P arameter estimation in LS can b e sensitive to the n umber of bins. In ML one assumes that the observ ations, Y , form a Gaussian field with mean giv en b y a regression mo del, µ = B β and co v ariance giv en b y a cov ariance function, r ( h ; θ ) with unknown parameters, θ , Y ∈ N  B β , Σ( θ )  . The log-lik elihoo d of Y is ` ( θ , β | Y ) = constan t − 1 2 log | Σ( θ ) | − 1 2 ( Y − B β ) T Σ − 1 ( θ )( Y − B β ) , and ML-parameter estimation is given by maximizing ` . ( ˆ θ , ˆ β ) = argmax θ,β ` ( θ , β | Y ) , The maximization can b e done in tw o steps. First, for an y fixed v alue of parameters θ 0 there is a unique v alue of β that maximize ` . Second, for a given β ( θ 0 ) there is a ˆ θ that maximize ` ( θ ; Y ) . Eliminating β from the log-likelihoo d function in these t wo steps is called profiling, and ` ( θ ; Y ) is called the profile log-likelihoo d, ` profile ( θ ; Y ) = − 1 2 log | Σ ( θ ) | − 1 2 Y T P ( θ ) Y , (2.4.2) and ˆ β ( θ ) =  B T Σ − 1 ( θ ) B  − 1 B T Σ − 1 ( θ ) Y , (2.4.3) where P ( θ ) = Σ − 1 ( θ ) − Σ − 1 ( θ ) B  B T Σ − 1 ( θ ) B  − 1 X T Σ − 1 ( θ ) . Section 2.5. Hiera rchical mo dels 11 Ev en though ML is the most common wa y of estimating parameters, ML estima- tors are biased due to the reduction in degrees of freedom caused b y the estimation of β [18]. Instead one can use the restricted maximum lik elihoo d (REML) whic h reduces or even eliminates the bias in ˆ θ . REML estimates the parameter by maximizing the log-lik elihoo d function associ- ated with error con trasts. The error contrasts are the ( n − p ) linearly indep enden t com bination of observ ations, a T Y that ha v e exp ectation zeros for all β and θ . This assumption adds an extra constant term to (2.4.2), ` RE M L ( θ ; Y ) = − 1 2 log | Σ ( θ ) | − 1 2 log | B T Σ − 1 ( θ ) B | − 1 2 Y T P ( θ ) Y where the ignored additive constant do es not depend on the parameters ( β , θ ). A REML estimate of θ is an y v alues ˜ θ , that maximize ` RE M L . Once the estimator, ˜ θ , has b een obtained the corresponding estimate of β is obtained through (2.4.3) . Although, REML decreases the bias it may increase the v ariance of the estimation [19]. 2.5 Hiera rchical mo dels The spatial fields are often used as comp onents of hierarc hical models, allo wing for a flexible mo del description and better handling of mo del uncertainties. A hierarc hical mo del is based on the join t distribution of a collection of random v ariables as a series of conditional distributions and a marginal distribution. Assume A, B and C are random v ariables, then one can write the join t distribution as [ A, B , C ] = [ A | B , C ][ B | C ][ C ] where [ C ] is probability distribution of C and [ B | C ] is the conditional distribution of B given C and so on. The simplest hierarc hical model consists of three parts: 1. Data model , [ Data | Pro cess,P arameters ] , 2. Pro cess model , [ Pro cess | P arameters ] , 3. P arameter model , [ Parameters ] . Often one is in terested in the distribution of the pro cess and parameters giv en the data which is called posterior distribution. Using Bay es formula one can write the p osterior as a [ Pro cess, Parameters | Data ] ∝ [ Data | Pro cess , Parameters ][ Process | Parameters ][ P arameters ] In the spatial con text one can specify the hierarchical mo del terms by Data model: Describ es the distribution of the measuremen ts given the laten t pro cess Y | X , θ ∈ N ( B β + AX , I σ 2 ε ) . 12 Spatial Statistics Chapter 2 Laten t process: Describ es ho w the laten t v ariables b ehav e, X | θ ∈ N ( B β , Σ ) . P arameters: Describ es prior knowledge or assumptions regarding the parameters, π ( θ ) . Therefore, the p osterior is π ( X , θ | Y ) ∝ π ( Y | X , θ ) π ( X | θ ) π ( θ ) and the marginal posterior distribution is π ( X | Y ) ∝ Z π ( X | Y , θ ) π ( θ | Y )d θ . A ccounting for parameter uncertain ties the p osterior mean E ( X | Y ) and posterior v ariance V ( X | Y ) pro vide predictions and predictions uncertaint y . Chapter 3 SP A TIO-TEMPORAL ST A TISTICS A spatio-temp oral pro cess is a pro cess which v aries in b oth space and time. Consider a spatial sto chastic pro cesses, { X ( s ) : s ∈ R d } then a spatio-temp oral processes can b e express as { X ( s , t ) : ( s , t ) ∈ R d × R } . Hence, X ( s, t ) is a function of b oth spatial locations, s ∈ R d , and time, t ∈ R . Time can b e considered as an additional co ordinate, and thus the domain of process b ecomes R d +1 = R d × R . Often spatio-temporal dependence can b e modelled b y a spatial process that dynamically c hanges in time [20]. F or example, assume a time discrete, spatially contin uous pro cess X t = { X ( s 1 , t ) , . . . , X ( s n , t ) } , in the simplest case this pro cess can b e modelled as X t = DX t − 1 + ν t ν t ∈ N (0 , Σ ν ) , Y t = C X t + ε t ε t ∈ N (0 , Σ ε ) , where D is the state transition matrix, Σ ν is the co v ariance matrix of the driving spatial process, Y t is the observ ation, C is a sparse observ ation matrix that extracts the appropriate elements from X t and Σ ε is the cov ariance of the observ ations. Note that the cov ariance matrices, Σ ε and Σ ν can dep end on time, pro viding a highly complicated pro cess. T o av oid a to o complex spatio-temp oral process some simplifying assumptions are necessary . F or instance, if a spatio-temp oral co v ariance matrix is separable it can be decomp osed into the pro duct of a purely spatial and a purely temp oral co v ariance function. The assump tion of separabilit y simplifies the construction of the mo del and reduces b oth the n um ber of unknown parameters and the computational time [20]. Spatio-temp oral pro cesses hav e b een applied to different environmen tal problems. F or example, Gelfand et al. [21] used the spatio-temp oral metho ds to mo del rain fall/ precipitation. Cameletti et al. [22] and Sampson et al. [23] applied spatio-temp oral metho ds to mo del the air qualit y and pollution. 13 Chapter 4 NON-ST A TIONARY CO V ARIANCE STRUCTURES So far in this pap er the co v ariance functions hav e b een assumed to b e stationary . F or most en vironmen tal processes the cov ariance structure is non-stationary when considered ov er large enough spatial scales. Ho w ev er, in most cases a non-stationary mo del can b e seen as a combination of small scale stationary comp onents. A main idea when solving non-stationary problems is to consider small/local spatial regions, i.e. to assume that the cov ariance structure is locally stationary . Here, some of these metho ds are reviewed: 1) Pro cess con v olution [24], 2) Deformation approac h [25] and 3) Allo wing for co v ariates in the co v ariance structure [26]. 4.1 Pro cess convolution In the pro cess conv olution metho d, the Gaussian random field, X ( u ) in R d expressed as X ( u ) = Z k ( s, u ) w ( s ) d s (4.1.1) where w is a Gaussian pro cess and k is a con v olution kernel. This can be used to pro duce non-stationary mo dels b y allowing the conv olution kernel to dep end on lo cation. If the pro cess is stationary then k ( s, u ) = k ( s − u ) and the cov ariance function for X ( u ) depends only on h = u − u 0 and is given by r ( h ) = cov ( X ( u ) , X ( u 0 )) = Z k ( s − u ) k ( s − u 0 ) ds = Z k ( s − h ) k ( s ) ds = k ∗ k . Moreo ver, the cov ariance function r and the kernel k are related through F ourier transform suc h that F ( r ) = F ( k ) · F ( k ) (4.1.2) Note that the shape of the k ernel determines the shape of the local spatial co v ariance function. F or example, a Gaussian k ernel corresp ond to a Gaussian co v ariance, and a Matérn k ernel leads to a Matérn cov ariance function. In practice 14 Section 4.2. Defo rmation approach 15 (4.1.1) can b e approximated b y discretizing the area in to interv als, I i cen tred at some fixed lo cations u i ’s and writing (4.1.1) as X ( u ) = X i Z s ∈ I i k ( u − s ) w ( s ) ds and using integral approximation X ( u ) ≈ m X i =1 k ( u − s i ) w i (4.1.3) where m is the num ber of in terv als on R d and w i are independent zero mean Gaussian v ariables with v ariance equal to the area of I i . Simpson et al. [27] sho wed that appro ximation (4.1.3) do es not w ork for general Matérn fields, it only works for fields that hav e smo othness parameter ν c > d/ 2 . It can b e sho wn through the F ourier transform relation (4.1.2) that for any v alue of the smo othness ν k in the Matérn k ernel, the Matérn co v ariance, see T able 2.1, will ha v e a v alue ν c = 2 · ν k + d/ 2 , this implies a lo w er limit on the smoothness of the process, i.e. ν c > d/ 2 in order to satisfy ν k > 0 . If ν k 6 0 the Matérn k ernel is singular. Simpson et al. [27] solv ed the problem by mo difying (4.1.3) to a more general and appropriate discretization X i  1 | I i | Z I i k ( s − u i ) du  w i . Examples of applying conv olution metho ds to environmen tal problems include Higdon et al. [28] and Calder [29], who used the metho d to ground and air p ollution, resp ectiv ely , ov er large areas based on point measurements. 4.2 Defo rmation app roach The spatial deformation approach [30], [25] is a non-parametric metho d to mo del non-stationary and anisotropic co v ariance structures, whic h has b een used , for example, in [31] for air p ollution. The metho d assumes rep eated samples of the sto c hastic field; the rep eated samples are often seen as b eing from different times. The main idea is to relate the spatial co ordinates of the sampling locations to a new set of co ordinates represen ting a stationary spatial cov ariance, i.e. a non-stationary field in the original co ordinates has a stationary representation in the new, or transformed, coordinates. Consider Y it = Y ( u i , t ) observ ations at lo cations u i , i = 1 , . . . , N lo cations and times t = 1 , . . . , T . Assuming a time constan t mean the spatio-temp oral process is written as Y ( u, t ) = µ ( u ) + e ( u, t ) + ε ( u, t ) , where µ ( u ) represen t the mean field, e ( u, t ) is a mean zero spatio-temp oral process and ε ( u, t ) is measuremen t error whic h is independent in space and time, and 16 Non-stationary cova riance structures Chapter 4 indep enden t of e ( u, t ) . Under these assumptions, i.e. e ( u, t ) being non-stationary in u and ha ving indep endent replicates in time, one can expres s the spatial dispersion as D 2 ( u i , u j ) = V  Y ( u i , t ) − Y ( u j , t )  = V  e ( u i , t ) − e ( u j , t )  + V  ε ( u i , t ) − ε ( u j , t )  . where D 2 ( u i , u j ) is a v ariogram, see Definition 2.1.4 and section 2.4. All the common v ariogram models in geostatistical practices can be expressed as a function of Euclidean distances betw een site locations in a bijectiv e transformation of the geographic coordinate system D 2 ( u i , u j ) = g  | f ( u i ) − f ( u j ) |  (4.2.1) where f is a transformation that expresses the spatial Non-stationarity and anisotropy , and g is an appropriate monotone function, i.e. a v ariogram. The space of original geographic co ordinates is called G-space and the space of deformed co ordinates is called D-space with the deformation given b y the mapping f , i.e. u ∈ G f − → f ( u ) ∈ D . T ypically , the measuremen ts are taken in G-space tw o dimensional and f : R 2 |{z} G-space − → R d |{z} D-space . (4.2.2) Deformation approach has been applied on environmen tal data, for instance, Damian et al. [31] used this method to model air pollution. Details on how to choose g and f are given by [30] and [25]. In the next section, one of the metho d of choosing f that allo w s for co v ariates to affect the non-stationarit y will b e explained. 4.3 Allo wing fo r cova riates in the cova riance structure The main idea of this metho d is to consider the cov ariates at each location, u , in the co v ariance structure. This leads to an increase in the dimensionalit y of the latent space and (4.2.2) b ecomes a function from R 2 × R d − 2 to R d . This method arise in order to o v ercome the main problem of the deformation approach, i.e. mapping ma y fold so that t w o differen t p oints in G-space result in one p oint in D-space. Schmidt et al. [26] sho w ed that f in (4.2.1) can be appro ximated b y f ( u ) ∈ N ( µ f ( u ) , Σ f ) where µ f ( u ) is d × n matrix whic h con tains the tw o co ordinates of G-space and d − 2 co v ariates, µ f ( u ) = ( u 1 , u 2 , B 1 ( u ) , . . . , B d − 2 ( u )) T . Clearly , the problem of folding is solv ed by adding d − 2 dimension to G-space. Chapter 5 HANDLING LARGE SP A TIAL D A T A SETS In addition to the issue of non-stationary co v ariance structures so far discussed in this paper, another issue in spatial statistics is the so called "big N problem" , or ho w to handle large spatial data sets. The problem can b e illustrated by studying the log-lik eliho o d, or the reconstruction at unknown sites. Recall that the log-lik elihoo d and the reconstruction are given by ` ( θ | Y ) = constant − 1 2 log | Σ( θ ) | − 1 2 ( Y − µ ( θ )) T Σ( θ ) − 1 ( Y − µ ( θ )) , (5.0.1) and X | Y , θ ∈ N  µ x + Σ xy Σ − 1 y y ( Y − µ x ) , Σ xx − Σ xy Σ − 1 y y Σ T y x  . F or N observ ations, the computational time for ` ( θ | Y ) scales as O ( N 3 ) due to | Σ | and Σ − 1 while the memory requiremen t for Σ scales as O ( N 2 ) . There are several computationally efficien t approac hes for alleviating the big N problem. Hence, a num b er of methods that can be divided in to three main classes will b e discussed in this section: 1) Lo w rank appro ximation : uses exact computations on a reduced rank or simplified v ersion of the field, thus reducing the size of the Σ matrices. These metho ds include: a) Fixed rank kriging [32], b) Predictiv e pro cess [33] and c) Pro cess con v olution [24] whic h was discussed in Section 4.1 page 14. 2) Co v ariance tap ering [34] sets small v alues of the co v ariance, r ( h ) to zero, obtaining a sparse co v ariance matrix, Σ . 3) Fitting Gaussian Mark o v Random Fields ( GMRF ) to Gaussian Random Fields ( GRF ). Note that b oth 2) and 3) modify the cov ariance matrix to obtain a sparse matrix which decreases b oth computational times and storage, see Section 5.2 and 5.3. Large data sets are common in en vironmen tal sciences and application of the metho ds presented here include: Shi and Cressie [9] and Cressie and Johannesson [35] who used fixed rank kriging, respectively on satellite measurements of ozone and aerosols; Latimer et al. [36] used predictive process for modelling of in v asive sp ecies; F urrer et al. [34] used co v ariance tapering for mo delling large climatological precipitation data set; and Cameletti et al. [37] used GMRF for mo delling of particulate matter concentration. 17 18 Handling large spatial data sets Chapter 5 5.1 Lo w rank app ro ximation The main idea in Lo w rank approximations is to express the Gaussian pro cess X ( u ) through a set of, r  n , basis functions { ψ i } r i =1 X ( u ) = r X i =1 ψ i ( u ) w i (5.1.1) where r is fixed and w i ∈ N (0 , Σ w ) . This giv es X = Ψ w ∈ N ( 0 , ΨΣ w Ψ T ) , and (2.3.1) becomes E ( X | Y ) = Σ xy Σ yy − 1 Y = ( Ψ x Σ w Ψ T )( ΨΣ w Ψ T + Σ ε ) − 1 Y (5.1.2) where Σ ε = σ 2 I . The most expensive part of this calculation remain at ( ΨΣ w Ψ T + Σ ε ) − 1 , using the matrix inv ersion lemma this s implifies to ( ΨΣ w Ψ T + Σ ε ) − 1 = Σ − 1 ε − Σ − 1 ε Ψ ( Ψ T Σ − 1 ε Ψ + Σ − 1 w ) − 1 Ψ T Σ − 1 ε where ( Ψ T Σ − 1 ε Ψ + Σ − 1 w ) and Σ w are a r × r matrices. Therefore, the computational time for the in verse matrix is O ( r 3 ) . Since r  n this drastically decreases the computational time. There are many p ossible low rank metho ds and some of the metho ds are stronger in theory than in practice. The Karh unen-Loéve expansion uses eigenfunctions of Σ and is only feasible for the few cases where eigenfunctions can b e found analytically , (see [38]). Here, the fo cus is on methods which are computationally feasible. F or example, the pro cess con v olution, see section 4.1, (4.1.3) can be seen as a lo w rank appro ximation with basis functions ψ i ( u ) = k ( u − s i ) . In the follo wing, tw o other metho ds, fixed rank kriging and predictiv e pro cesses are explained. 5.1.1 Fixed rank kriging Fixed rank kriging [32] uses m ultiresolutional functions, often w a v elets, to construct a low rank, non parametric cov ariance matrix. Allo wing the cov ariance to capture v ariation at several scales. In this case the co v ariance matrices are given b y ΨΣ w Ψ T where Σ w is a p ositiv e definite matrix such that ( ΨΣ w Ψ T + Σ ε ) is a close appro ximation of the empirical co v ariance estimate, ˆ Σ y y . Then ˆ Σ w = R − 1 Q T ( ˆ Σ yy − Σ ε ) Q ( R − 1 ) T where ¯ Ψ = QR is a QR -decomp osition of a matrix containing the wa velets. Strictly , Ψ is a binned version since the ˆ Σ y y estimate is computed using bins similar to those in (2.4.1). F or example, [32] used the local bisquare function at three different resolutions, Ψ i ( l ) ( u ) =     1 −  k u − v i ( l ) k r l  2  2 k u − v i ( l ) k ≤ r l 0 o.w Section 5.2. Cova riance tap ering 19 where v i ( l ) is one of the cen ter p oints of the l th resolution, r l = 1 . 5 · d l and d l is shortest distance b et w een cen ter p oints of the l th resolution. In fixed rank kriging non-parametric co v ariance function is sp ecified and co v ariance is estimated as a non-parametric, lo w -rank approximation to ˆ Σ y y . 5.1.2 Predictive p ro cess The main idea in predictiv e pro cesses [33] is similar to that of fixed rank kriging, ho wev er the difference that predic tiv e pro cess is based on a cov ariance function represen tation of the cov ariance matrix. In the predictive process the cov ariance function of the field is assumed to b e of a parametric form, as described in Section 2.1 and T able 2.1. In this metho d, the Gaussian process in (5.1.1) is replaced with a lo w er rank ed pro cess ˜ w , a predictive pro cess whic h is deriv ed from the paren t pro cess w . Consider a set of knots U ∗ = { u ∗ 1 , · · · , u ∗ r } whic h ma y or ma y not form a subset of the observed locations, then the Gaussian process ev aluated at the knots is w ( u ∗ i ) r i =1 = w ∗ ∈ MV N (0 , Σ ∗ ) , where Σ ∗ = cov ( u ∗ i , u ∗ j ) is a r × r matrix. A ccording to (5.1.2) , the reconstruction at a site u 0 is given by E ( w ( u 0 ) | w ∗ ) = ˜ w ( u 0 ) = Σ T u 0 Σ ∗− 1 w ∗ where Σ u 0 = cov ( u 0 , u ∗ i ) , is the co v ariance betw een each knot and the site u 0 . The reconstruction defines a spatial pro cess ˜ w ( u ) ∈ N (0 , Σ ˜ w ) with co v ariance function Σ ˜ w = Σ T u Σ ∗− 1 Σ u where Σ u = cov ( u, u ∗ i ) . ˜ w is called predictiv e pro cess . It can b e show that ˜ w is the best appro ximation for paren t pro cess w , (see, [33] Section 2.3 ). The result is a pro cess defined on only the r knots that uses kriging to approximate the en tire field. 5.2 Cova riance tap ering T apering is a metho d for appro ximating the co v ariance function of large spatial fields. The basic idea is to introduce zeros in to the co v ariance, r ( h ) , outside of a giv en range, θ , i.e. r ( h ) = 0 if h ≥ θ ; this results in a sparse cov ariance matrix Σ and corresponding speed-up in the ev aluation of (5.0.1) . The tapered co v ariance is defined as r tap ( h ) = r ( h ) T θ ( h ) where T θ ( h ) is a positive definite co v ariance matrix with compact supp ort, | h | < θ . Decreasing θ leads to more zeros in the cov ariance matrix. A basic argument is that T θ ( h ) needs to be of suitable shap e for the estimates to remain consistent. F urrer et al. [34] listed the follo wing possible tapering functions, see T able 5.1, which can be used for Matérn co v ariances. 20 Handling large spatial data sets Chapter 5 T aper T θ ( h ) for h ≥ 0 Spherical max { (1 − h θ ) 2 , 0 } (1 + h 2 θ ) W endland 1 max { (1 − h θ ) 4 , 0 } (1 + 4 h θ ) W endland 2 max { (1 − h θ ) 6 , 0 } (1 + 6 h θ + 35 h 2 3 θ 2 ) T able 5.1: T app erd co v ariance function. 5.3 App ro ximating Gaussian Random Fields with Gaussian Ma rk ov Random Fields First, in this section some useful definitions such as, Mark ov prop erty and neigh- b ourhoo d of an observ ation set are stated and then the section will b e con tin ued with explaining the idea of appro ximating GRF s with GMRF s. F or a time discrete stochastic pro cess x t , the pro cess has a Marko v property if, for an y t , the distribution of x t giv en the entire history is equal to the distribution of x t giv en just x t − 1 , p ( x t | x t − 1 , . . . , x 0 ) = p ( x t | x t − 1 ) . F or the spatial case, the neighbourho o d of a p oin t u i is defined as a set of neighbours N i , { u j , j ∈ N i } whic h are in some senses close to u i . Now a GMRF can be defined. A GRF, z ∈ N ( µ, Σ) is a GMRF if the full conditional distribution for all z i satisfies p ( z i |{ z j ; j 6 = i } ) = p ( z i |{ z j ; j ∈ N i } ) for some neighbourho o d set, {N i } n i =1 , i.e. the distribution of z i giv en the whole field is equal to distribution of z i giv en just the neigh bours in neigh bourho o d set. The Marko v prop ert y with resp ect to a neighbourho o d set leads to a zero-pattern in the precision matrix, Q = Σ − 1 where Q ij = 0 ⇐ ⇒ j / ∈ {N i , i } Using the precision matrix the densit y function (2.1.1) becomes p ( z ) = | Q | 1 / 2 (2 π ) N/ 2 exp  − 1 2 ( z − µ ) T Q ( z − µ )  . and the joint density (2.2.2) b ecomes Z =  X Y  ∈ N  µ x µ y  ,  Q xx Q xy Q yx Q yy  − 1 ! , with conditional distribution X | Y ∈ N  µ x − Q − 1 xx Q xy ( Y − µ y ) , Q − 1 xx  . Section 5.3. App ro ximating Gaussian Random Fields with Gaussian Mark ov Random Fields 21 One can express the p ositiv e definite precision matrix as Q = R T R where R is Cholesky factor. R is a unique upp er triangular matrix with strictly p ositive diagonal elemen ts. If Q is sparse R is often also sparse. Instead of computing | Q | and Q − 1 , one can use inv olving R . The tric k here is that one should nev er calculate R − 1 (the in v erse matrix), but instead solve the triangular equation system Ra = b to obtain b = R − 1 a . Using the sparsity of the precision matrix and its sparse Cholesky factorization leads to efficient computation whic h decreases the computational time. T o use a GMRF one needs to construct a useful and sparse Q matrix. A simple and common method is to use a conditional autoregress iv e mo del ( CAR ). The problem with CAR mo dels is that they are restricted to lattices. Recent work b y Lindgren et al. ( 2011 ) developed an explicit link betw een GMRF s and Matérn co v ariance functions. The method is based on the fact that a GRF s w ith Matérn co v ariance function on R d are solutions to a Sto c hastic Partial Differential Equation ( SPDE ) [39], [40]. ( κ 2 − ∆) α 2 Z ( u ) = τ W ( u ) (5.3.1) where W ( u ) is a Gaussian white noise, ∆ = ∇ T ∇ = ∂ 2 ∂ u 2 x + ∂ 2 ∂ u 2 y is the Laplacian and α = ν + d 2 . Here, a simplified solution sketc h for the case α = 2 is sho wn. The left hand side of (5.3.1) can be written as Kz where K is a finite difference approximation of ( κ 2 − ∆) and z is a discretized v ector of Z ( u ) . Hence, the discretized SPDE b ecomes Kz d = ε where ε is a Gaussian v ector with mean zero and co v ariance matrix τ 2 C . Therefore, z ∈ N (0 , τ 2 K − 1 CK − T ) is a solution to (5.3.1) with Q = 1 τ 2 K T C − 1 K . The matrix K can b e written as K = κ 2 C + G giv en suitable matrices C and G . Therefore Q is sparse if K and C − 1 are sparse. T o obtain a sparse C − 1 one needs to appro ximate C with a diagonal matrix ˜ C . The resulting precision matrix Q = 1 τ 2 K T ˜ C − 1 K is no w sparse sine G is sparse (essentially G only contains finite difference approximation of ∆ ). F or α = 2 ( ν = 1) the lo cal structure of the precision matrix on a regular grid in R 2 is giv en by κ 4 h 2 " 1 # | {z } C +2 κ 2   − 1 − 1 4 − 1 − 1   | {z } ≈− ∆( G ) + 1 h 2       1 2 − 8 2 1 − 8 20 − 8 1 2 − 8 2 1       | {z } ≈ ∆ 2 ( G 2 = G ˜ C − 1 G ) . In conclusion, the approximate, discretized, solution to (5.3.1) is z ∈ N (0 , Q − 1 α,κ ) for differen t in teger v alues of α with 22 Handling large spatial data sets Chapter 5 Q 1 ,κ = 1 τ 2 K = 1 τ 2  κ 2 C + G  α = 1 Q 2 ,κ = 1 τ 2 K C − 1 K = 1 τ 2  κ 4 C + 2 κ 2 G + GC − 1 G | {z } G2  α = 2 Q α,κ = 1 τ 2 K C − 1 Q α − 2 ,κ C − 1 K α = 3 , 4 , . . . No w the mo del defined in (2.2.1) and (2.2.3) can be expressed using a GMRF mo del as Y = AX + ε ε ∈ N ( 0 , I σ 2 ε ) (5.3.2) X = Ψw + B β w ∈ N ( 0 , Q − 1 ) (5.3.3) where w is a GMRF and B β is regression sp ecifying the mean and ε is Gaussian noise. Chapter 6 APPLICA TION TO THE LANDCLIM D A T A 6.1 Intro ducto ry V egetation/Land-co v er is an important part of the climate system with c hanges in vegetation abundance affecting climate and vice v ersa. Man y Global Climate Mo dels (GCMs) or Regional Climate Mo dels (RCMs) consider vegetation as a giv en b oundary condition, making go o d estimates of vegetation is important for accurate climate reconstruction and prediction. Recent studies, [41], [42], [43], [44] sho w the imp ortance of p ollen based estimates of the vegetation abundance in the reconstruction of past v egetation and land-co ver. The main purp ose of the current study is to use the spatial and spatio-temp oral to ols explained in Section 1.1-5 to reconstruct vegetation and land-cov er during differen t time windows. The reconstruction will b e based on estimated v egetation prop ortions pro vided b y pollen data from sp ecific lo cations, see Section 6.2 and Figure 6.1. 6.2 Data The study area cov ers Northw est and W estern Europ e North of the Alps. The region has b een divided in to a spatial grid of 1 ◦ × 1 ◦ ( roughly 111 . 2 × 111 . 2 km 2 ) . The p ollen based reconstructions are a v ailable for three time windows; tw o time windo ws in the past and the modern time; 6000 BP 1 [ BC 4250 − 3750 ≈ 5700 − 6200 BP ] : a perio d with little h uman-induced landscap e op enness, 200 BP [ AD 1600 − 1850 ≈ 100 − 350 BP ] : the classical pre-industrial state widely used as a baseline to compare mo dern h uman-impact in terms of greenhouse gases on climate c hange with past non-h uman impacted climate. 1 BP: Before Present 23 24 Application to the LANDCLIM data Chapter 6 0 BP [ AD 1850 − x 2 ≈ x − 100 BP ] : a modern time window which will be used to v alidate mo dels. 6.2.1 REVEALS, p ollen-based vegetation reconstruction Estimates of vegetation are based on the Reveals model (Regional Estimates of VEgetation Abundance from Large Sites) whic h estimates regional v egetation/land- co ver comp ositions based on p ollen counts in sedimen t cores from large lak es [42]. The Reveals estimates pro vide abundances for 25 taxa which can be grouped in to 10 plant functional t yp es (PFT s) or 3 land-co v er t ypes (LCT s), see T able 6.1. The PFT s are t ypically used in v egetation mo delling while LCT s are applied in climate mo delling. Figure 6.1 shows the av ailable Reveals data (174 lo cations) for the mo dern time windo w co v ering ab out 40% of the study area. Present time window Revels LPJ−guess Figure 6.1: The a v ailable Rev eals data (174 locations) for mo dern time window in the study area, abov e the red line. 2 Date of the most recen t av ailable data Section 6.3. Model 25 PFT Description Plant T axa LCT TBE1 Shade-toleran t-b oreal Pic e a Ev er green TBE2 Shade-toleran t-temp erate Abies IBE Shade-intoleran t-b oreal Pinus TSE T all shrub Junip erus TBS Shade-toleran t-temperate Carpinus, F agus, Tilia , Summer green Ulmus IBS Shade-in tolerant-boreal Alnus, Betula, Corylus , F r axinus, Quer cus TSD T all shrub Salix LSE Lo w evergreen shrub Cal luna vulgaris Op en land GL Grassland - all herbs Artemisia, Rumex ac etosa -t Po ac e ae, Plantago lanc e olata , Cyp eraceae, Plantago montana , Filip endula, Plantago me dia AL Agricultural land- cereals Cerealia-t, Sec ale T able 6.1: shows the classification of 25 taxa in 10 PFT s and 3 LCT s. 6.2.2 Additional data V egetation abundance is strongly related to cov ariates, suc h as elev ation, geographical co ordinates, h uman land-use and natural potential vegetation prop ortions. F or eac h time windows the natural p oten tial vegetation proportions ha v e b een simulated using LPJ-guess (Lund-Potsdam-Jena)-(General Ecosystem Sim ulator). LPJ-guess is a dynamic, pro cess based vegetation mo del that simulates v egetation dynamics base on climate data input [45]. Human land-use is pro vided in form of the KK scenario [46] estimates. In our mo del we use the natural v egetation from LPJ-guess adjusted for the h uman impact estimates from KK scenario. The adjustment is needed since LPJ- guess only mo dels natural openness, without considerations for h uman impact such as farmland and pastures. 6.3 Mo del An important feature of the Reveals data is that it is prop ortions of v egetation. Meaning that observ ations, y i ha ve to sum to one, P i y i = 1 , and b e non-negativ e, y i ≥ 0 , in realit y LCT s are p ositive. Aitc hison [47] mo delled the compositional data b y logistic normal distribution that is a multi-normal distribution via log transformation. The log transformation can b e done in different wa ys dep ends on the data set. In our mo del w e used central log ratio (clr) u i ( s ) = log y i ( s ) p Q i y i ( s ) 26 Application to the LANDCLIM data Chapter 6 where s = 1 , · · · , S are the data lo cations. T o ensure iden tifiability w e ha v e the condition P i u i ( s ) = 0 . Using u i for mo delling is easier than y i due to easier constrain t on u i ; u i can tak e v alues in ( −∞ , ∞ ) with only a sum to zero constrain t. After using u i ( s ) in the mo del, the res ults are transferred bac k to the original space b y y i ( s ) = exp( u i ( s )) P i exp( u i ( s )) where no w P i y i = 1 , y i ≥ 0 . Using the mo del in (5.3.2) w e can reconstruct the latent field using (5.3.3) giv en observ ed data at specific lo cations. The mean v alue of the field can be explain using a regression model. Hence, the first task is to iden tify important explanatory v ariables. Knowing the coefficients of the regression one can reconstruct the mean field. Thereafter, additional spatial structure will b e assessed and a latent field mo del will be used to impro v e the reconstruction. In the Section 6.4, preliminary results from the regression mo del are sho wn. In the regression model we used u i ( s ) as a dep endan t v ariable, as explained in Section 1.1.2 and Definition 1.1.1, u i = X p B p β i , p + e i e i ∈ N ( 0 , σ 2 ) where B con tains our p chosen cov ariates. 6.4 Prelimina ry results The preliminary results from the regression mo dels are a v ailable, and Figure 6.2 sho ws the reconstruction of three different LCT s based on Rev eals data set for the mo dern time. In this model, w e used the adjusted LPJ-guess prop ortions, co ordinates and elev ation as cov ariates. The result sho ws a go o d fit of the mo del to REVEALS data and at the same time keeping the structures of the adjusted LPJ-guess mostly where w e hav e no information from REVEALS, i.e. locations without REVEALS data. Section 6.4. Prelimina ry results 27 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Reveals LPJ.KK+bio ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Evegren.tree ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ.KK10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ−guess ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Reveals LPJ.KK+bio ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Summergreen.tree ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ.KK10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ−guess ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Reveals LPJ.KK+bio ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 Open.land ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ.KK10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −10 0 10 20 30 35 40 45 50 55 60 65 70 LPJ−guess ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 Figure 6.2: Reconstruction of REVEALS data in mo dern time window. First column is the original REVEALS data. Second column is the reconstruction using regression mo del. Third column is the adjusted LPJ-guess with KK scenario and the forth column is the natural p otential vegetation from LPJ-guess. 28 Application to the LANDCLIM data Chapter 6 6.5 F uture w o rk The preliminary results sho w a goo d fit of the mo del to the REVEALS data set. W e plan to con tin ue the study in four main directions: 1) iden tifying and ev aluating co v ariates, 2) assessing spatial structure in the data, 3) attempting to identify deviations from the existing h uman land-use mo dels, and 4) incorp orating the REVEALS reconstructions uncertainties in the mo del. 6.5.1 Identifying and evaluating cova riates As mentioned earlier, the study con tains three con trasting time windo ws in terms of climate and anthropogenic land-cov er c hange, e.g. 6000 BP is considered as a relativ ely w arm climate with low h uman impact. F or eac h different time windo ws w e will assess the main cov ariate and their corresponding coefficients to construct t wo regression models. Comparisons of the (differen t) mo dels obtained for eac h time windo w will allow us to improv e our understanding of land-co v er and vegetation c hanges. In addition the results from mo dern time will b e used to ev aluate models. 6.5.2 Assessing spatial structure Assessing the spatial structures and adding a spatial dependency structure to the mo del to impro v e the reconstruction. The preliminary results from model 2 show the effects of co ordinates on the REVEALS reconstruction. This can b e in terpreted as the existence of spatial dep endencies in the data. T o capture these dependencies w e plan to add a spatial field to our re gression mo del. Expanding the mo del from a linear regression to  u 1 u 2  =  X 1 X 2  + B β , where  X 1 X 2  ∈ N  0 ,  1 ρ ρ 1  ⊗ Q − 1  . (6.5.1) In (6.5.1) , Q is a sparse precision matrix the spatial structure in the field of transformed comp ositional data and ρ is the correlation b et ween the fields. W e hop e that adding spatial dependencies will improv e the reconstruction [48]. 6.5.3 Identifying an indep endent human land-use mo del The LANDCLIM pro ject goals include quan tification of human-induced changes on regional land-cov er. Identifying the h uman impact allows for a separation of the historical pro cess in to: i) climate-driv en changes in v egetation and ii) h uman-induced c hanges in land-cov er. W e will try to identify deviations from existing human land- use mo dels (e.g. KK scenario and HYDE) and use the deviations to construct an impro ved mo del for human impact. Section 6.5. F uture wo rk 29 6.5.4 Inco rp o rating the uncertaint y In addition to the reconstructed vegetation comp osition, the REVEALS data has v arying uncertain t y betw een different lo cations. 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