FOR 496 / 796 Introduction to Dendrochronology. Lab exercise #4: Tree-ring Reconstruction of Precipitation

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1 FOR 496 Iroducio o Dedrochroology Fall 004 FOR 496 / 796 Iroducio o Dedrochroology Lab exercise #4: Tree-rig Recosrucio of Precipiaio Adaped from a exercise developed by M.K. Cleavelad ad David W. Sahle, Tree-Rig Laboraory, Uiversiy of Arkasas. Iroducio This exercise iroduces sudes o he quaiaive recosrucio of climae from reerig daa, usig correlaio ad regressio aalysis. You are supplied wih a ree-rig chroology ad seasoalized climae daa from Califoria (Figure ). Oly 0 years of daa are supplied so you ca acually complee all he compuaios ivolved by had, wih he assisace of a calculaor or, of course, he use of a compuer spreadshee program. Tree-Rig Daa (he predicor): I his exercise, he ree-rig predicor is acually a average of four rig-widh chroologies of Califoria blue oak (Quercus douglasii). Sadard rig-widh chroologies were produced by firs deredig ad sadardizig he idividual rig-widh measureme series. This rasforms each measureme series o a dimesioless idex series wih a mea ad variace ha are relaively sable over ime, ad a ew overall mea of.0, which is he same for all series. These sadardized rig widh idices are he averaged ogeher o a year-by-year basis o complee he sadard chroology. The socalled residual rig-widh chroology is acually used i his exercise, where he low order auocorrelaio (persisece) has bee removed wih auoregressive (AR) modelig procedures [see Fris (976), Cook (985; 987), or Cook ad Kairiuksis (990)]. Climae Daa (he predicad): The climae daa, or predicad, used for his exercise are oal precipiaio (i mm) for he period December of he previous year (-) hrough April of he year cocurre wih growh () averaged for he wo eares climae divisios i Califoria (i.e., divisios 4 ad 5, Karl e al. 983). This December-April seaso was chose because correlaio aalyses wih he mohly daa idicaed ha precipiaio durig his period has he greaes ifluece o blue oak growh. The precipiaio daa were modeled as a AR-0 process, meaig ha successive December- April precipiaio oals are o correlaed (so-called whie oise). For his reaso, we use he residual ree-rig chroology i his exercise, because i is also a AR-0 process ad maches he ime series srucure of he precipiaio daa. Tree-rig recosrucio of precipiaio Page of 4 Lab04_reerig_recosrucio_of_ppe.doc

2 FOR 496 Iroducio o Dedrochroology Fall 004 Correlaio: Correlaio is a basic saisical ool i dedrochroology, widely used o ideify he mohly or seasoal climae variables relaed o rig widh growh. Some of he compuaios ivolved i correlaio aalysis are also used o calculae he slope parameer of regressio models, so we begi wih a correlaio exercise. We firs provide he formula used o compue he Pearso correlaio coefficie (r). We he prese a exercise i Table, where you acually compue he correlaio coefficie for e years of paired ree-rig ad precipiaio daa. We he es he saisical sigificace of he calculaed correlaio coefficie, ad fid he amou of variace shared by climae ad rig-widh. The correlaio coefficie varies bewee +.0 ad -.0. A correlaio of +.0 meas ha he wo series vary perfecly i phase, ha is, whe oe series chages (posiive or egaive aomaly from is mea), he oher chages wih a aomaly of he same sig, ad he chages are always exacly proporioal. A correlaio of -.0 meas ha he wo series are exacly ou of phase ad varyig proporioally. Of course, real world daa rarely achieve perfec correlaio. The Pearso correlaio coefficie (r) is compued as follows: r ( x x)( y ( x x) ( y where x is he ree-rig idex value for year, y is he climae variable (i.e., precipiaio) for year, x ad y are he meas o he ree-rig ad climae values, respecively, ad he subscrip varies from o ( equals he umber of years of observaio, 0 i our simple example). You ca calculae r by fillig i he cells i Table, or by usig ay spreadshee or saisical sofware of your choosig. Table : Sample daa Year Tree-rig idex Precipiaio (mm) The sample daa above produce a r. Sigificace of he Correlaio Coefficie We ca es he sigificace of our derived correlaio coefficie i Table 3. Noe ha we have 0 observaios, bu oly 9 degrees of freedom [degrees of freedom (df) - umber of predicor variables (which always equals i he case of correlaio)]. So i Table 3, read he colum for oe idepede variable ad he row for ie degrees of freedom (uder he colum "Error df"). Noe he magiude of he correlaio coefficie (r) eeded o achieve saisical sigificace a he α 0.05 ad α 0.0 probabiliy levels. These levels are he probabiliy of Tree-rig recosrucio of precipiaio Page of 4 Lab04_reerig_recosrucio_of_ppe.doc

3 FOR 496 Iroducio o Dedrochroology Fall 004 fidig a higher correlaio coefficie bewee wo urelaed series purely by chace. Noe ha if you use he α 0.05, here is a oe i wey chace ha he correlaio bewee wo urelaed variables will be sigifica. If you use he more srige α 0.0, oly oe correlaio i 00 would be expeced o be sigifica jus by radom chace. So, is he correlaio bewee he Califoria rig-widh idex ad precipiaio series sigifica? (Y / N) A wha level? Variace explaied The square of he Pearso correlaio coefficie, kow as he coefficie of deermiaio (R ), is used o esimae he proporio of he variace i oe variable is explaied by he covariace bewee he wo correlaed variables. So, i our 0-year sample, how much of he variace i rig-widh is explaied by he variace i precipiaio. R The perceage variace explaied is R * 00. Perce explaied? Regressio Aalysis We ca use liear regressio aalysis o esimae or predic values of a depede variable (y ) from he correspodig value of a idepede variable (x ), wih he subscrip referrig o observaios of x ad y from he same year. This process is called calibraio. To calibrae a ree-rig chroology wih a climae ime series, we simply make he climae series he depede variable (y), ad use he ree-rig chroology as he idepede variable. Of course, climae is o really "depede" o ree growh, quie he opposie is rue. Bu, we susped logic i his case ad simply use he regressio equaio o obai he quaiaive esimaes of precipiaio from he ree-rig chroology. This regressio-based calibraio model is also referred o as a rasfer fucio. The bes-fiig, sraigh-lie equaio or model for he liear regressio is : ˆ β + β y 0 x where β 0 is he iercep of he bes-fiig sraigh lie, β is he slope of he bes-fiig sraigh lie, ŷ is our esimae or prediced value of he depede variable (i.e., precipiaio) i a give year, based o he correspodig value of he idepede ree-rig variable x, i he same year. I regressio aalysis, he lie of bes fi is he oe which miimizes he sum (i.e., oal) of he squared differeces bewee each observed ad prediced depede variable [i.e.,( y ˆ y ) ]. If you are usig a calculaor, he formula o compue he esimae of he slope of he regressio lie is: ˆ ( xi x)( yi β ( x x) i The formula o compue he iercep (β 0 ) of he regressio lie is simply: Tree-rig recosrucio of precipiaio Page 3 of 4 Lab04_reerig_recosrucio_of_ppe.doc

4 FOR 496 Iroducio o Dedrochroology Fall 004 βˆ 0 y βˆ x You may, if you wish, esimae ˆβ 0 ad ˆβ usig a compuer spreadshee or saisics program. Based o he daa provided i Table, he bes fiig, liear equaio bewee seasoal precipiaio ad ree-rig idices is: ŷ + x Evaluaio of he regressio model There are four ypes of aalysis ha are rouiely performed o evaluae or es he sregh of a regressio model (equaio) likig wo variables, x ad y. These iclude:. Graphical aalysis I: scaerplo. Graphical aalysis II: ime series plo 3. Coefficie of deermiaio (R ) 4. Tesig he saisical sigificace of he slope I his exercise, we will evaluae our 0-year regressio model wih hese echiques. Creae a able similar o Table 4 (eiher o he page provided, or i a spreadshee program) o assis wih he evaluaio.. Graphical aalysis: Scaerplo Usig he graph paper provided (Figure ), or usig a compuer graphics package, prepare a x-y scaerplo by ploig he iersecios of he ree-rig values (x ) as he abscissa (x-axis) ad he climae values ( y ) as he ordiae (y-axis) ake from Table. Nex, add he regressio lie o he scaerplo. The prediced values of precipiaio ( ŷ ) fall exacly o he regressio lie. The regressio lie goes hrough he iersecio of he meas of x ad y, so plo ha poi of Figure. The, if you se x o zero i he regressio model, he prediced value for ŷ equals he iercep (β 0 ). So plo he iercep o Figure. Use a sraigh edge o draw he regressio lie passig hrough hose wo pois (iersecio of he x ad y, ad he iercep). This regressio lie should pass roughly hrough he middle of your scaer of daa pois.. Graphical aalysis: Time series plo Usig he graph paper provided (Figure 3), or usig a compuer graphics package, draw a ime series plo of he observed ( y ) ad recosruced precipiaio daa ( ŷ ). The x-axis will be years used i he aalysis (94-950) ad he y-axis will be precipiaio (i mm). The observed ad recosruced precipiaio daa have he same mea over his period, so he wo ime series should superimpose prey icely o he ime series plo. Use a dashed lie o coec he observed values ad a solid lie o coec he recosruced values (or use differe colours). 3. Coefficie of deermiaio Now we wa o deermie he amou of precipiaio variace is explaied by he rig-widh daa. For his, we deermie he coefficie of deermiaio (R ) from he regressio esimaes. To compue R, we calculae he residuals from regressio ad wo sums of squares i Table 4. The residuals are also referred o as error sice hey represe he differece bewee our prediced ree-rig esimaes of precipiaio ad he acual observed measuremes of precipiaio. These residuals should show up clearly i Figure as he larges deparures from Tree-rig recosrucio of precipiaio Page 4 of 4 Lab04_reerig_recosrucio_of_ppe.doc

5 FOR 496 Iroducio o Dedrochroology Fall 004 he fied regressio lie, ad i Figure 3 as he larges differeces bewee he observed ad recosruced values. The regressio residuals are calculaed as: Residual ( y ˆ y ) Place hese residuals i colum 5 of Table 4, he square each residual (colum 6) ad compue he sum of he squares abou he regressio lie, or Error (SSE) a he boom of colum 6. SSE y yˆ ) ( Nex, compue he sum of he squares due o regressio (SSR) usig colums 7 & 8 of Table 4. SSR ( yˆ Now calculae he oal correced sum of squares (SST) i Table 4 as: SST SSE + SSR This gives us he sums eeded o calculae he coefficie of deermiaio squared (R ), he bes measure of he relaioship bewee he depede ad idepede variables i a regressio model. Usig he values compued i Table 4 ad above, we calculae R as: R SSR / SST R / This R saisic ells us he proporio of variace i he depede variable (y) accoued for, or explaied by, he variace i he idepede variable (x). The perce variace explaied is simply R * Tesig he saisical sigificace of he slope Tesig he ull hypohesis H 0 : β 0 is aoher way o measure he sregh of a regressio model. If he slope is o sigificaly differe ha zero, he he R is also probably low, ad here may be o real relaioship bewee he wo variables. The sig of he slope is also ellig us abou he aure of he physical relaioship bewee he wo variables. Wih ree growh as he depede variable (y), a posiive slope wih precipiaio as he idepede variable (x) simply idicaes ha he greaer he precipiaio, he greaer he ree growh. Bu a regressio bewee ree growh ad emperaure should be egaive o moisure sressed sies, meaig ha he higher he emperaure, he lower he growh. To es he sigificace of he slope (β ), we form he ull ad alerae hypoheses: H 0 : β 0 H a : β > 0 for precipiaio The es saisic is: βˆ 0 se β Tree-rig recosrucio of precipiaio Page 5 of 4 Lab04_reerig_recosrucio_of_ppe.doc

6 FOR 496 Iroducio o Dedrochroology Fall 004 where se β sadard error of β Therefore, SSE /( ) ( x x) i / Lookup he criical values of (-,α) i a -able, where α is he level of sigificace for he aalysis. Deermie if we ca rejec he ull hypohesis ha he slope of he relaioship could be equal o zero a a α Verificaio of he Recosrucio Eve if he regressio model passes all he ess we have discussed above, i is imperaive o check he recosruced values agais idepede climae daa ouside of he calibraio period used o develop he regressio model. The reaso is simple. The regressio procedure fids he bes model possible, ad he mea of he observed ad recosruced series are forced o be he same [i.e., he meas are removed from x ad y i he compuaio of he slope, ad he prediced values are scaled i erms of he depede variable by he iercep]. Bu how well he ree-rig recosrucio acually racks chages i he mea ad variaio of climae ouside he calibraio period is a major quesio, ad ca be evaluaed o some degree wih verificaio aalysis usig idepede climae daa. Oe soluio o he verificaio problem is o use oly par of he climae daa for calibraio ad he res for verificaio (See 977). Because he regressio model eds o improve as he sample size icreases, usig par of he climae daa jus for verificaio may reduce he accuracy of he calibraio. A beer soluio may be o coduc wo calibraio experimes o wo halves of he climae/ree-rig ime series, wih each half used o develop a recosrucio. The each recosrucio ca be verified agais he climae daa o used i is calibraio. If he regressio coefficies i he wo calibraios are o sigificaly differe ad he verificaio saisics are saisfacory, a hird calibraio usig all he climae daa ca be used for he fial recosrucio. The aalyses ad saisics commoly used o es how closely he recosrucio resembles he idepede climae daa iclude:. Graphical aalysis of he observed ad recosruced ime series. Correlaio aalysis I his exercise we will use he addiioal, idepede precipiaio daa from o check he accuracy of our ree-rig recosrucio durig he verificaio. Graphical aalysis of he observed ad recosruced ime series Use Table 3 o firs compue he ree rig recosrucio of precipiaio for , he draw a ime series plo of boh he observed ad recosruced precipiaio daa from i Figure 5. These wo ime series should mach very well if your regressio model or rasfer fucio is valid.. Correlaio Aalysis The correlaio coefficie (r) ca be used o measure how much he observed ad recosruced ime series covary durig he verificaio period (usually a bi lower ha durig he calibraio period). Use Table 5 o compue r bewee he recosruced or prediced ( ŷ ) ad observed (y ) precipiaio durig he verificaio period. Tree-rig recosrucio of precipiaio Page 6 of 4 Lab04_reerig_recosrucio_of_ppe.doc

7 FOR 496 Iroducio o Dedrochroology Fall 004 Criical Thikig Quesios. Wha assumpios are auomaically implied i he climae recosrucio usig ree-rig iformaio (hi: go back ad review some of he priciples of dedrochroology)?. Based o he four aalyical ess used o evaluae he climae recosrucio, comme o he sregh of he observed relaioship bewee seasoal precipiaio ad ree-rig idices. 3. Comme o he sreghs ad weakesses of usig correlaio ad regressio procedures for climae recosrucio usig ree-rig iformaio. Tree-rig recosrucio of precipiaio Page 7 of 4 Lab04_reerig_recosrucio_of_ppe.doc

8 FOR 496 Iroducio o Dedrochroology Fall 004 Table : Calculaio of he Pearso correlaio coefficie (r ) x rig-widh idex y precipiaio (mm) r ( x x)( y ( x x) ( y col(4) col(8) col(7) () () (3) (4)(3)*(3) (5) (6) (7)(6)*(6) (8)(3)*(6) Year x ( x x) ( x x) y ( y Sum (Σ) Mea y ( x x)( y ( y) r ( )( ) r Tree-rig recosrucio of precipiaio Page 8 of 4 Lab04_reerig_recosrucio_of_ppe.doc

9 FOR 496 Iroducio o Dedrochroology Fall 004 Table 3: Sigifica values of r Probabiliy, p df Error Tree-rig recosrucio of precipiaio Page 9 of 4 Lab04_reerig_recosrucio_of_ppe.doc

10 FOR 496 Iroducio o Dedrochroology Fall 004 Table 4: Recosrucio of precipiaio o evaluae he regressio model Regressio model: ˆ β + β x y 0 ŷ + x () () (3) (4) (5) (6) (7) (8) Year x y ŷ Residual y yˆ ) ( ˆ ( y y ) ( y Sum of Squares: SSE SSR ˆ ( yˆ Tree-rig recosrucio of precipiaio Page 0 of 4 Lab04_reerig_recosrucio_of_ppe.doc

11 FOR 496 Iroducio o Dedrochroology Fall 004 Table 5: Verificaio of precipiaio recosrucio usig idepede precipiaio daa from Regressio model: ˆ β + β x y 0 ŷ + x r ( yˆ yˆ)( y ( yˆ yˆ) ( y col(5) col(9) col(8) () () (3) (4) (5) (6) (7) (8) (9) Year x ŷ ( yˆ yˆ ) yˆ ˆ) y ( y ( y Sum (Σ) Mea ( y) y ( yˆ yˆ)( y r ( )( ) r Tree-rig recosrucio of precipiaio Page of 4 Lab04_reerig_recosrucio_of_ppe.doc

12 FOR 496 Iroducio o Dedrochroology Fall 004 Figure : Scaerplo Precipiaio (mm) Tree-Rig Idex Tree-rig recosrucio of precipiaio Page of 4 Lab04_reerig_recosrucio_of_ppe.doc

13 FOR 496 Iroducio o Dedrochroology Fall 004 Figure 3: Time series plo Precipiaio (mm) Year Tree-rig recosrucio of precipiaio Page 3 of 4 Lab04_reerig_recosrucio_of_ppe.doc

14 FOR 496 Iroducio o Dedrochroology Fall 004 Figure 4: Time series plo for verificaio period Precipiaio (mm) Year Tree-rig recosrucio of precipiaio Page 4 of 4 Lab04_reerig_recosrucio_of_ppe.doc