ncl泰勒图（均方根误差、标准差、相关系数）

最近学习了一下泰勒图的做法，对2001年泰勒的文章进行了简单学习，说一点自己的理解：

泰勒图一般是用来评估多个模式对观测数据的模拟能力，包括标准差、中心型均方根误差、相关系数这三部分，这三部分可以构建一个三角关系。相关系数是用来量化模式之间的相似性，但是它有缺点，无法量化变化幅度。比如，当两个空间场变化极其相似，但是变化程度有量级差异，如果不看均方根误差而只看相关系数，那么很容易出现问题。标准差可以看出来模式相对于观测的离散程度。

在ncl的脚本中，官网提供了现成的代码给我们绘制泰勒图，只需要提供模式和观测之间的相关系数，以及模式与观测之间的标准差的比率，就可以画出。因为代码中用到的是标准差比率，根据文章中提到的三角关系，我们图中最终得到的是标准化后的样本标准差和标准化后的中心型均方根误差，这样得到的图更加清晰明了。

如果要对模式进行优选，就可以根据自己的需求，设定标准化后的标准差的范围（标准化后的中心型均方根误差类似），还可以对相关系数进行显著性检验等等。

下面是官网提供的泰勒图代码：


function taylor_diagram (wks:graphic ,RATIO[*][*]:numeric, CC[*][*]:numeric \,rOpts:logical)
;--------------------------------------------------------------------
; This version of taylor_diagram supports "paneling"
; It requires NCL version 4.2.0.a034 because it uses "gsn_create_legend"
;--------------------------------------------------------------------;
; Generate a Taylor Diagram:
; Generate Multiple Aspects of Model Performance in a Single Diagram
; Taylor, K. E., J. Geophys. Res., 106, D7, 7183-7192, 2001
;
; An example:
; http://www.grida.no/climate/ipcc_tar/wg1/fig8-4.htm
;
; This expects one or more datasets. The left dimension
; is the number of datasets. The rightmost is the number of pts.
;
; Markers are at:
; http://www.ncl.ucar.edu/Document/Graphics/Resources/gs.shtml
;
; By default, the function can handle up to 10 variable comparisons..
; To expand ...  modify the 'Colors' and 'Markers' attributes.
; The user can change / add some default settings.
;
; The defaults that the user can modify:
;
; rOpts                 = True
;                                  ; 'made-up' resources
; rOpts@Colors          =  (/ "blue" , "red", "green", "cyan", "black" \
;                           , "turquoise", "brown", "yellow"/)
; rOpts@Markers         =  (/ 2, 3, 6, 14, 9, 12, 7, 4/) ; Marker Indices
; rOpts@markerTxOffset  = 0.0175   ; offset for text above marker
; rOpts@stnRad          = (/ 1. /) ;  (/ 0.50, 0.75, 1.5 /)
; rOpts@centerDiffRMS   = False    ;  True mean draw additional radii from REF
; rOpts@caseLabelsFontHeightF = 0.05
; rOpts@varLabelsFontHeightF  = 0.013
; rOpts@varLabelsYloc         = 0.65
; rOpts@legendWidth           = 0.015
; rOpts@legendHeight          = 0.030*nCase
; rOpts@taylorDraw            = True
; rOpts@taylorFrame           = True
;
;                                  ; standard NCL resources
; rOpts@tiMainString    = "Taylor" ; not using title makes plot bigger
; rOpts@gsMarkerSizeF   = 0.0085   ; marker size
; rOpts@gsMarkerThicknessF = 1.0
; rOpts@txFontHeightF   = 0.0125   ; text size
; rOpts@tiMainFontHeightF = 0.0225 ; tiMainString size
;
; It returns to the user a graphic object containing the
; Taylor background and plotted x/y pts.
; This graphic object contains a simple Taylor background appropriate
; for standardized data and the markers for the datasets.
; ==================================================================
; This version allows paneling:
;      The 'cumbersome' "dum" variables were added by
;      Adam Phillips to allow paneling via "gsn_add_?".
; ==================================================================
begindimR                  = dimsizes(RATIO)nCase                 = dimR(0)    ; # of cases [models] nVar                  = dimR(1)    ; # of variables; x/y coordinates for plottingX    = new ( (/nCase,nVar/) , typeof(RATIO) )Y    = new ( (/nCase,nVar/) , typeof(RATIO) )do nc=0,nCase-1angle      = acos( CC(nc,:) )   ; array operation                                    X(nc,:)    = RATIO(nc,:)*cos( angle )     Y(nc,:)    = RATIO(nc,:)*sin( angle )    end doxyMin                 = 0.  xyOne                 = 1.00xyMax                 = 1.65xyMax_Panel           = xyMax+ 0.10            ; paneling purposesif (rOpts .and. isatt(rOpts,"txFontHeightF"))  then FontHeightF       = rOpts@txFontHeightF    ; user wants to specify sizeelseFontHeightF       = 0.0175end if; ----------------------------------------------------------------
; Part 1:
; base plot: Based upon request of Mark Stevens
; basic x-y and draw the 1.0 observed and the outer curve at 1.65
; ----------------------------------------------------------------rxy                   = True       rxy@gsnDraw           = Falserxy@gsnFrame          = Falserxy@vpHeightF         = 0.65rxy@vpWidthF          = 0.65rxy@tmYLBorderOn      = Falserxy@tmXBBorderOn      = Falserxy@tiYAxisString     = "Standardized Deviations (Normalized)"rxy@tiYAxisFontHeightF= FontHeightF                        ; default=0.025 rxy@tmXBMode          = "Explicit" rxy@tmXBValues        = (/0.0,0.25,0.50,0.75,1.00,1.25,1.5/)    ; major tm; default  "OBS" or "REF";rxy@tmXBLabels        = (/"0.00","0.25","0.50","0.75","REF" ,"1.25","1.50"/)rxy@tmXBLabels        = (/"    ","0.25","0.50","0.75","REF" ,"1.25","1.50"/)if (rOpts .and. isatt(rOpts,"OneX") )  then                     ; eg: rOpts@OneX="1.00" ;rxy@tmXBLabels        = (/"0.00","0.25","0.50","0.75",rOpts@OneX,"1.25","1.50"/)rxy@tmXBLabels        = (/"    ","0.25","0.50","0.75",rOpts@OneX,"1.25","1.50"/)end ifrxy@tmXBMajorLengthF  = 0.015      ; default=0.02 for a vpHeightF=0.6rxy@tmXBLabelFontHeightF = FontHeightFrxy@tmXBMinorOn       = Falserxy@trXMaxF           = xyMax_Panelrxy@tmYLMode          = "Manual"rxy@tmYLMinorOn       = Falserxy@tmYLMajorLengthF  = rxy@tmXBMajorLengthFrxy@tmYLLabelFontHeightF = FontHeightFrxy@tmYLMode          = "Explicit" rxy@tmYLValues        = (/0.0, .25,0.50, 0.75, 1.00, 1.25, 1.5/) ; major tmrxy@tmYLLabels        = (/"0.00","0.25","0.50","0.75","1.00","1.25","1.50"/);rxy@tmYLLabels        = (/"    ","0.25","0.50","0.75","1.00","1.25","1.50"/)rxy@trYMaxF           = xyMax_Panelrxy@tmYRBorderOn      = Falserxy@tmYROn            = False      ; Turn off right tick marks.rxy@tmXTBorderOn      = Falserxy@tmXTOn            = False      ; Turn off right tick marks.rxy@xyDashPatterns    = (/ 0 /)    ; line characteristics (dash,solid)rxy@xyLineThicknesses = (/ 2./)    ; choose line thicknessrxy@gsnFrame          = False      ; Don't advance the frame.; create outer 'correlation axis'npts    = 200                        ; arbitraryxx      = fspan(xyMin,xyMax,npts) yy      = sqrt(xyMax^2 - xx^2    )   ; outer correlation line (xyMax)sLabels = (/"0.0","0.1","0.2","0.3","0.4","0.5","0.6" \ ; correlation labels,"0.7","0.8","0.9","0.95","0.99","1.0"     /); also, major tmcLabels = stringtofloat(sLabels)rad     = 4.*atan(1.0)/180.angC    = acos(cLabels)/rad                     ; angles: correlation labelsif (rOpts .and. isatt(rOpts,"tiMainString")) thenrxy@tiMainString      = rOpts@tiMainString;rxy@tiMainOffsetYF    = 0.015               ; default  0.0if (isatt(rOpts,"tiMainFontHeightF")) thenrxy@tiMainFontHeightF = rOpts@tiMainFontHeightFelserxy@tiMainFontHeightF = 0.0225         ; default  0.025              end ifend if
;;if (rOpts .and. isatt(rOpts,"gsnCenterString")) then
;;    rxy@gsnCenterString  = rOpts@gsnCenterString      ; only gsn_csm_xy
;;end iftaylor  = gsn_xy(wks,xx,yy,rxy)                 ; Create and draw XY plot.rsrRes  = TruersrRes@gsLineThicknessF  = rxy@xyLineThicknesses(0)  ; line thicknessrsrRes@gsLineDashPattern = 0                    ; solid line pattern; draw x and y to xyMaxdum0 = gsn_add_polyline(wks,taylor,(/0.,  0. /),(/0.,xyMax/), rsrRes)dum1 = gsn_add_polyline(wks,taylor,(/0.,xyMax/),(/0.,  0. /), rsrRes)xx   = fspan(xyMin, xyOne ,npts)                ; draw 1.0 standard radiusyy   = sqrt(xyOne - xx^2)   rsrRes@gsLineDashPattern = 1                    ; dashed line patternrsrRes@gsLineThicknessF  = rxy@xyLineThicknesses(0)  ; line thicknessdum2 = gsn_add_polyline(wks,taylor,xx,yy, rsrRes)delete(xx)delete(yy)if (rOpts .and. isatt(rOpts,"stnRad") ) thenrsrRes@gsLineThicknessF  = 1   ; rxy@xyLineThicknesses(0)  nStnRad = dimsizes(rOpts@stnRad)dum3  = new(nStnRad,graphic)do n=0,nStnRad-1rr = rOpts@stnRad(n)xx = fspan(xyMin, rr ,npts) yy = sqrt(rr^2   - xx^2)   dum3(n) = gsn_add_polyline(wks,taylor,xx,yy, rsrRes)end dotaylor@$unique_string("dum")$ = dum3delete(xx)delete(yy)end ifgetvalues taylor                                ; get style info from taylor"tmYLLabelFont"        : tmYLLabelFont        ; use for correlation axis"tmYLLabelFontHeightF" : tmYLLabelFontHeightFend getvalues; ----------------------------------------------------------------
; Part 2:
; Correlation labels
; ----------------------------------------------------------------radC    = xyMax                                  ; for correlation labelsxC      = radC*cos(angC*rad)yC      = radC*sin(angC*rad)
; added to get some separationxC      = xC + 0.020*cos(rad*angC)yC      = yC + 0.060*sin(rad*angC)txRes               = True                      ; text mods desiredtxRes@txFontHeightF = FontHeightF               ; match YL txRes@tmYLLabelFont = tmYLLabelFont             ; match YLtxRes@txAngleF      = -45.if (.not.isatt(rOpts,"drawCorLabel") .or. rOpts@drawCorLabel) then dum4 = gsn_add_text(wks,taylor,"Correlation",1.30,1.30,txRes)taylor@$unique_string("dum")$ = dum4end iftxRes@txAngleF      = 0.0 txRes@txFontHeightF = FontHeightF*0.50          ; bit smaller;;dum0 = gsn_add_text(wks,taylor,"OBSERVED",1.00,0.075,txRes)plRes               = TrueplRes@gsLineThicknessF = 2.txRes@txJust        = "CenterLeft"              ; Default="CenterCenter".txRes@txFontHeightF = FontHeightF               ; match YL ;txRes@txBackgroundFillColor = "white"tmEnd = 0.975radTM = xyMax*tmEnd                             ; radius end: major TM xTM   = new( 2 , "float")yTM   = new( 2 , "float")dum5 = new(dimsizes(sLabels),graphic)dum6 = dum5do i=0,dimsizes(sLabels)-1                      ; Loop to draw stringstxRes@txAngleF = angC(i)dum5(i) = gsn_add_text(wks, taylor, sLabels(i),xC(i),yC(i),txRes) ; cor labelxTM(0)   = xyMax*cos(angC(i)*rad)             ; major tickmarks atyTM(0)   = xyMax*sin(angC(i)*rad)             ; correlation labelsxTM(1)   = radTM*cos(angC(i)*rad)             yTM(1)   = radTM*sin(angC(i)*rad)dum6(i) = gsn_add_polyline(wks,taylor,xTM,yTM,plRes)end do; minor tm locationsmTM     = (/0.05,0.15,0.25,0.35,0.45,0.55,0.65 \ ,0.75,0.85,0.91,0.92,0.93,0.94,0.96,0.97,0.98  /)angmTM  = acos(mTM)/rad                         ; angles: correlation labelsradmTM  = xyMax*(1.-(1.-tmEnd)*0.5)             ; radius end: minor TM dum7 = new(dimsizes(mTM),graphic)do i=0,dimsizes(mTM)-1                          ; manually add tmxTM(0)   = xyMax*cos(angmTM(i)*rad)           ; minor tickmarksyTM(0)   = xyMax*sin(angmTM(i)*rad)xTM(1)   = radmTM*cos(angmTM(i)*rad)          yTM(1)   = radmTM*sin(angmTM(i)*rad)dum7(i)  = gsn_add_polyline(wks,taylor,xTM,yTM,plRes)end do; added for Wanliif (rOpts .and. isatt(rOpts,"ccRays") ) thenangRL = acos(rOpts@ccRays)/rad             ; angles: radial linesrlRes = TruerlRes@gsLineDashPattern= 2  ; line patternrlRes@gsLineThicknessF = 1  ; choose line thicknessif (isatt(rOpts,"ccRays_color")) thenrlRes@gsLineColor    =  "LightGray"end ifdum8 = new(dimsizes(angRL),graphic)do i=0,dimsizes(angRL)-1xRL     = xyMax*cos(angRL(i)*rad)yRL     = xyMax*sin(angRL(i)*rad)dum8(i) = gsn_add_polyline(wks,taylor,(/0, xRL /),(/0,  yRL  /),rlRes)end dotaylor@$unique_string("dum")$ = dum8end if; ----------------------------------------------------------------
; Part 3:
; Concentric about 1.0 on XB axis
; ----------------------------------------------------------------if (rOpts .and. isatt(rOpts,"centerDiffRMS") \.and. rOpts@centerDiffRMS) thenrespl                    = True                ; polyline mods desiredrespl@xyLineThicknessF   = 1.0                 ; line thicknessrespl@xyLineDashPattern  = 2                   ; short dash linesrespl@gsLineColor        = "Black"             ; line color     if (isatt(rOpts,"centerDiffRMS_color")) thenrespl@gsLineColor    =  "LightGray"end ifdx   = 0.25ncon = 4                                       ; 0.75, 0.50, 0.25, 0.0npts = 100                                     ; arbitraryang  = fspan(180,360,npts)*raddum9 = new(ncon,graphic)do n=1,ncon rr  = n*dx            ; radius from 1.0 [OBS] abscissaxx  = 1. + rr*cos(ang)yy  = fabs( rr*sin(ang) )if (n.le.2) thendum9(n-1) = gsn_add_polyline(wks,taylor,xx,yy,respl)end ifif (n.eq.3) thenn3 = floattointeger( 0.77*npts ) dum9(n-1) = gsn_add_polyline(wks,taylor,xx(0:n3),yy(0:n3),respl)end ifif (n.eq.4) thenn4 = floattointeger( 0.61*npts ) dum9(n-1) = gsn_add_polyline(wks,taylor,xx(0:n4),yy(0:n4),respl)end ifend dodelete(ang)delete(xx)delete(yy)taylor@$unique_string("dum")$ = dum9end if
; ---------------------------------------------------------------
; Part 4:
; generic resources that will be applied to all users data points
; of course, these can be changed
; http://www.ncl.ucar.edu/Document/Graphics/Resources/gs.shtml
; ---------------------------------------------------------------if (rOpts .and. isatt(rOpts,"Markers")) thenMarkers = rOpts@MarkerselseMarkers = (/ 4, 6, 8,  0, 9, 12, 7, 2, 11, 16/) ; Marker Indicesend ifif (rOpts .and. isatt(rOpts,"Colors")) thenColors  = rOpts@ColorselseColors  = (/ "red", "blue", "green", "cyan", "orange" \, "turquoise", "brown", "yellow", "purple", "black"/)end ifif (rOpts .and. isatt(rOpts,"gsMarkerThicknessF")) thengsMarkerThicknessF = rOpts@gsMarkerThicknessFelsegsMarkerThicknessF = 1.0end ifif (rOpts .and. isatt(rOpts,"gsMarkerSizeF")) thengsMarkerSizeF      = rOpts@gsMarkerSizeFelsegsMarkerSizeF      = 0.0085                  ; Default: 0.007end ifgsRes = TruegsRes@gsMarkerThicknessF = gsMarkerThicknessF      ; default=1.0gsRes@gsMarkerSizeF      = gsMarkerSizeF           ; Default: 0.007 ptRes = True                        ; text options for pointsptRes@txJust             = "BottomCenter"; Default="CenterCenter".ptRes@txFontThicknessF   = 1.2      ; default=1.00ptRes@txFontHeightF      = 0.0125   ; default=0.05if (rOpts .and. isatt(rOpts,"txFontHeightF")) thenptRes@txFontHeightF  = rOpts@txFontHeightF  end ifmarkerTxYOffset          = 0.0175   ; defaultif (rOpts .and. isatt(rOpts,"markerTxYOffset")) thenmarkerTxYOffset = rOpts@markerTxYOffset             ; user defined offsetend ifdum10 = new((nCase*nVar),graphic)dum11 = dum10do n=0,nCase-1gsRes@gsMarkerIndex   = Markers(n)             ; marker style (+)gsRes@gsMarkerColor   = Colors(n)              ; marker colorptRes@txFontColor     = gsRes@gsMarkerColordo i=0,nVar-1dum10(n*nVar+i) = gsn_add_polymarker(wks,taylor,X(n,i),Y(n,i),gsRes) dum11(n*nVar+i) = gsn_add_text(wks,taylor,(i+1),X(n,i),Y(n,i)+markerTxYOffset,ptRes)end doend do; ---------------------------------------------------------------
; Part 5: ; add case legend and variable labels
; ---------------------------------------------------------------if (rOpts .and. isatt(rOpts,"caseLabels")) then if (isatt(rOpts,"caseLabelsFontHeightF")) thencaseLabelsFontHeightF = rOpts@caseLabelsFontHeightFelsecaseLabelsFontHeightF = 0.05  end iflgres                    = Truelgres@lgMarkerColors     = Colors        ; colors of markerslgres@lgMarkerIndexes    = Markers       ; Markers lgres@lgMarkerSizeF      = gsMarkerSizeF ; Marker sizelgres@lgItemType         = "Markers"     ; draw markers onlylgres@lgLabelFontHeightF = caseLabelsFontHeightF  ; font height of legend case labelsif (isatt(rOpts,"legendWidth")) thenlgres@vpWidthF       = rOpts@legendWidthelselgres@vpWidthF       = 0.15           ; width of legend (NDC)end ifif (isatt(rOpts,"legendHeight")) thenlgres@vpHeightF      = rOpts@legendHeightelse   lgres@vpHeightF      = 0.030*nCase   ; height of legend (NDC)end iflgres@lgPerimOn          = False         ; turn off perimeternModel                   = dimsizes( rOpts@caseLabels )lbid = gsn_create_legend(wks,nModel,rOpts@caseLabels,lgres)amres = Trueamres@amParallelPosF     =  0.35           amres@amOrthogonalPosF   = -0.35             annoid1 = gsn_add_annotation(taylor,lbid,amres)   ; add legend to plotend ifif (rOpts .and. isatt(rOpts,"varLabels")) then nVar    = dimsizes(rOpts@varLabels)if (isatt(rOpts,"varLabelsFontHeightF")) thenvarLabelsFontHeightF = rOpts@varLabelsFontHeightFelsevarLabelsFontHeightF = 0.013end iftxres = Truetxres@txFontHeightF = varLabelsFontHeightFtxres@txJust = "CenterLeft"              ; justify to the center left;delta_y = 0.02       delta_y = 0.06   if (rOpts .and. isatt(rOpts,"varLabelsYloc")) thenys  = rOpts@varLabelsYloc            ; user specifiedelseys  = max( (/nVar*delta_y , 0.30/) )end ifdo i = 1,nVar     if (i.eq.1) thendum12 = new(nVar,graphic)end ifdum12(i-1) = gsn_add_text(wks,taylor,i+" - "+rOpts@varLabels(i-1), .125,ys,txres)ys = ys- delta_yend dotaylor@$unique_string("dum")$ = dum12end iftaylor@$unique_string("dum")$ = dum0   ; x-axistaylor@$unique_string("dum")$ = dum1   ; y-axistaylor@$unique_string("dum")$ = dum2   ; 1.0 std curvetaylor@$unique_string("dum")$ = dum5   ; labels [COR]taylor@$unique_string("dum")$ = dum6   ; major tm [COR]taylor@$unique_string("dum")$ = dum7   ; minor tmtaylor@$unique_string("dum")$ = dum10  ; markerstaylor@$unique_string("dum")$ = dum11  ; textif (.not.isatt(rOpts,"taylorDraw") .or. \(isatt(rOpts,"taylorDraw") .and. rOpts@taylorDraw)) then draw(taylor)end ifif (.not.isatt(rOpts,"taylorFrame") .or. \(isatt(rOpts,"taylorFrame") .and. rOpts@taylorFrame)) then frame(wks)end ifreturn(taylor)
end

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