Spatial Analysis in Landscape Ecology by yew20072

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									                                                    Landscape ecology: concerns
                                                       • pattern and process
                                                       • pattern and scale
            Spatial Analysis in                                              propagation
            Landscape Ecology
                     Dean Urban                               process                      pattern
                                                              process                      pattern
         The Nicholas School, Duke University


                                                                              constraint




Landscapes: scale and pattern                       Scale and pattern
   “ the problem of pattern and scale is               Analytic strategy:
     the central problem in ecology,                   1. Detect and characterize pattern
     unifying population biology and                   2. Pose hypotheses about cause
     ecosystems science, and marrying
     basic and applied ecology”                        3. Test hypotheses
                                     Levin (1992)




Data types in spatial analysis                      Data types in spatial analysis
   • Spatial point patterns                            • Spatial point patterns
   • Geostatistical samples                              • Bivariate K & extensions
   • Lattices                                          • Geostatistical samples
                                                       • Lattices
Data types in spatial analysis                                Data types in spatial analysis
   • Spatial point patterns                                      • Spatial point patterns
   • Geostatistical samples                                      • Geostatistical samples
      • neighbor-based regression (AR, CAR)                      • Lattices
      • distance-based regression (Mantel)                         • vector/raster maps: inferences based
   • Lattices                                                       on landscape metrics (FragStats)
                                                                   • raster lattices: wavelet analysis
                                                                   • graphs: various algorithms




Data type conversions                                         Data type conversions
                                             raster
  vector

                   rasterize                                                     rasterize



                   keep centroids,                                               keep centroids,
                       edges                                                         edges

                                                                                                         regular
                                     graph                                                               geostatistical
                                                                                                         data




Data type conversions                                         Agenda (schedule)
                                                                 1. Scale-specific inference using
                                                                    wavelet analysis
                   rasterize                                     2. Graph-theoretic analyses of
                                                                    landscape pattern (connectivity)
                   keep centroids,
                                                                 Both: exploratory talks about what
                       edges                                        might prove useful for ecologists
neighbors of                                 regular
geostatistical                               geostatistical
samples                                      data
                                                   Data types in landscape ecology
                                                     • Spatial point patterns:
       Scale-specific inference                          • disease, disturbance locations
              Wavelet analysis                             (not a lot, in practice)
                                                     • Geostatistical samples:
                  with Tim Keitt
                                                         • lots of data; the issue is extending it to
                                                           larger spatial extent [“bottom-up”]
                                                     • Lattices:
                                                         • most data for landscapes (maps, DEMs,
                                                           satellite imagery, …) [“top-down”]




Motivation                                         Objectives
  • Landscape ecology is hugely invested             • Provide an introduction to wavelet
    in “scale” and “pattern” but lacks                   analysis, beyond its utility to describe
    methods that address these                           pattern
    explicitly in an inferential framework           •   Illustrate inferential potential with
                                                         applications to forests of the Sierra
                                                         Nevada
                                                     •   Point to issues for further development




Gradient pattern                                   Terrain and sample transect




              Sequoia-Kings Canyon National Park
Data series: NDVI                                                                                                               Data series: environmental variables




                                                                                                                                                 250
                                                                                                                                sun
220
200
180




                                                                                                                                TCI
160
140
120




                                                                                                                                elev
        4.025*10^6                    4.035*10^6                                 4.045*10^6                       4.055*10^6


                               Transect position (N=1024)
                                                                                                                                                                   Transect position




Wavelet analysis                                                                                                                Wavelet transform
• A wavelet is a
                                                                                                                                        1.5
                                                   1.0




                                                                                                                                        1.0




      shape templet
                                                   0.5




                                                                                                                                        0.5




• A wavelet can be                                                                                         Haar                                                                             level 1
                                                   0.0




                                                                                                                                        0.0




      dilated and
                                                   -0.5




      translated
                                                   -1.0




                                                                                                                                        -1.0




                                                          0.0   0.2        0.4          0.6    0.8   1.0
                                                                                                                                                -40    -20   0     20           40   60

• Wavelet analysis is
                                                                                                                                  220




      the convolution of
                                                                                 1.5




                                                                                                                                  200
                                                                                 1.0




      the wavelet with
                                                                                                                                  180
                                                                                 0.5




      the data, at scales,                                                                                               s8
                                                                                                                                  160
                                                                                 0.0




      for lattice data
                                                                                                                                  140
                                                                                 -1.0




                                                                                                                                  120




                                                                                              -2       0     2      4
                                                                                                                                         4.025*10^6              4.0 35* 10^6             4.045*10^6     4 .055 *10^6




Wavelet transform                                                                                                               Wavelet transform
                                                                                                                                         1.5
            1.5




                                                                                                                                         1.0
            1.0




                                                                                                                                         0.5
            0.5




                                                                                          level 2 dilation                                                                                  level 4 dilation
                                                                                                                                        0.0
            0.0




                                                                                                                                         -1.0
            -1.0




                   -40   -20      0      20           40              60                                                                        -40    -20   0     20           40   60
      220




                                                                                                                                  220
      200




                                                                                                                                  200
      180




                                                                                                                                  180
      160




                                                                                                                                  160
      140




                                                                                                                                  140
      120




                                                                                                                                  120




             4.025*10^6                4.0 35* 10^6                              4.045*10^6                      4 .055 *10^6            4.025*10^6              4.0 35* 10^6             4.045*10^6     4 .055 *10^6
                    Wavelet transform                                                                                    Wavelet analysis: interpretation
                              Alternative algorithm:                                                                        • Large wavelet coefficients occur at scales
                                                                                                                                 and locations where the data match the
                              1. Compute transform at first level
                                                                                                                                 shape of the wavelet
                              2. Average adjacent points, discard                                                           •    The coefficients completely capture the
                                 every other point as redundant                                                                  pattern of the data at each scale
                              3. Repeat …                                                                                   •    This provides the means to work with the
                              Result: the decimated transform                                                                    data at specific scales by working with the
                                                                                                                                 wavelet transform of the data




                    Wavelet coefficients                                                                                 Wavelet coefficients: NDVI
                                                                                                                                 idwt
                                                                                                                                  d1
                        Doppler wave:                                   idwt
                                                                                                                                  d2
                                                                         d1
                                                                                                                                  d3
                 0.4




                                                                                                                                  d4
                                                                                                                         scale
                                                                         d2
                 0.2




                                                                                                                                  d5
doppler signal




                                                                         d3
                                                                                                                          (2k)
                 0.0




                                                                                                                                  d6
                                                                         d4
                 -0.2




                                                                                                                                  d7
                                                                         s4
                                                                                                                                  d8
                 -0.4




                        0.0   0.2       0.4
                                               Time
                                                      0.6   0.8   1.0

                                                                                0.0    0.2   0.4    0.6      0.8   1.0
                                                                                                                                  s8

                                                                                                                                        0   200    400   600    800   1000




                    Wavelet utilities                                                                                    Wavelet analysis: terms
                              • Data compression: discard the                                                               • Wavelet variance = Σ(coefficients)2
                                    smallest coefficients (your choice,                                                     • Average of squared coefficients
                                    level of detail)                                                                             within a level: power at that level
                              •     Noise reduction (discard coefficients                                                   • Proportion of wavelet variance across
                                    near 0, or shrink all toward 0)                                                              levels: energy at each level
                                                                                                                                 (power and energy differ because the
                                     -3 -1 1




                                                                                                                                 number of coefficients varies by
                                                                                                                                 level)
                                                0           200           400         600     800         1000
                                     0 1
                                     -2




                                                0           200           400         600     800         1000
Wavelet variance                                                                   Wavelet scaling
      NDVI                          Level     Scale            N        Energy




                                                                                                  0.15
                                      D1            2      512            0.05
                                      D2            4      256            0.10
                                      D3            8      128            0.13
                                                                                    NDVI
                                                                                    scalogram




                                                                                                  0.10
                                      D4            16         64         0.16
                                      D5           32          32         0.18      (energy)
                                      D6           64          16         0.10




                                                                                                  0.05
                                      D7          128          8          0.10
 Σ(energy) =                          D8          256          4          0.17
 simple variance                      S8          >256         4          0.02                             2        4            6         8
                                                                                                                        Level




Wavelet scalograms                                                                 Multiresolution decomposition
                                                                                       • Compute wavelet transform
       0.15




                                                                                       • Use coefficients at a given scale to
                                       0.4




NDVI                                                                       elev
       0.10




                                                                                          reconstruct the data series
                                       0.2
       0.05




                                                                                       • Result: scale-specific signal in the
                                       0.0




              2   4       6    8              2      4     6        8


                                                                                          data
       0.20




                                       0.20




sun                                                                        TCI
       0.15




                                       0.10
       0.10
       0.05




                                       0.0




              2   4       6    8              2      4     6        8




MRD: NDVI                                                                          MRD & MRA: NDVI
          Data                                                                            Data                 Data
           D1                                                                              D1                  S1
           D2                                                                              D2                  S2
           D3                                                                              D3
                                                                                                               S3
                                                                                           D4
 “details” D4                                                                                                  S4
           D5                                                                     “detail” D5                                                  “smooth”
                                                                                                               S5
           D6                                                                              D6
                                                                                           D7                  S6
           D7
           D8                                                                              D8                  S7
           S8                                                                              S8                  S8

                      0       200      400           600        800       1000              0    400     800     0         400       800
Selective reconstruction                                                     Wavelet cospectra
                                                                                               Level Scale     E(ndvi)    E(sun)    Cospectrum
                                                                                                   D1      2      0.05      0.04           0.00
1




                                                                                                   D2      4      0.10      0.05           0.00
                                                                                                   D3      8      0.13      0.08          -0.01
0




                                                                                                   D4     16      0.16      0.20          -0.07
                                                                                                   D5     32      0.18      0.19          -0.02
-1




                                                                                                   D6     64      0.10      0.23          -0.02
-2




                                                                                                   D7    128      0.10      0.09           0.00
                                                                                                   D8    256      0.17      0.06          -0.07
                                                                                                   S8   >256      0.02      0.05           0.00
         0         200            400       600           800       1000

                       NDVI: levels 4, 5, 7 & up                                                   Σ(cospectrum) = simple covariance




Wavelet regression                                                           Wavelet regression
     • Wavelet coefficients are independent                                         idwt                       idwt
         within scales                                                               d1                         d1
     •   Wavelet correlations can be extended to                                     d2                         d2
                                                                                     d3                         d3
         multiple regression framework:
                                                                                     d4                         d4
             • predict wavelets(Y) on wavelets(X)                            NDVI                                                          Sun
                                                                                     d5                         d5
             • reconstruct predicted values from predicted                           d6                         d6
              wavelets                                                               d7                         d7
     → Result: scale-specific explanation                                            d8                         d8
                                                                                     s8                         s8

                                                                                           0   400      800           0   400      800




Wavelet correlations                                                         Wavelet regression
     Level     Scale    E(ndvi)    E(sun)   Cospectrum    R(wave)        P      • H: ndvi =
         D1
         D2
                  2
                  4
                          0.05
                          0.10
                                    0.04
                                    0.05
                                                  0.00
                                                  0.00
                                                             0.03
                                                            -0.04
                                                                      0.46
                                                                      0.51
                                                                                     f(sun[4])+f(tci[6])+f(elev[8])
         D3       8       0.13      0.08          -0.01     -0.06     0.52      • Results:
                                                                                     • ndvi[4] : sun[4] r = -0.38 (P<0.01,
         D4       16      0.16      0.20          -0.07    -0.38     <0.01
         D5      32       0.18       0.19         -0.02     -0.13     0.46
         D6      64       0.10      0.23          -0.02     -0.18     0.50                 N=64)
         D7
         D8
                 128
                256
                          0.10
                          0.17
                                    0.09
                                    0.06
                                                  0.00
                                                  -0.07
                                                             0.03
                                                            -0.38
                                                                      0.94
                                                                      0.62
                                                                                     • ndvi[5] : tci[5] n.s.
         S8     >256      0.02      0.05          0.00      -0.16     0.84           • ndvi[6] : tci[6] n.s. (should be 5,6:4!)
                                                                                     • ndvi[8] : elev[8] r = -0.93 (P~0.05, N=4)
                   scale-specific correlations
… Wavelet regression                     Conclusions
  More possibilities:                      • Wavelets are capable of providing
  • interaction terms (at same scales)         inference on scale-specific
                                               relationships in ecology
  • cross-scale terms?                     •   Lots of potential for further
  • “modern” regressions (LOESS, GAM)?         development (2D data; data modeling
                                               with wavelets)
                                           •   Some statistical issues need to be
                                               resolved along the way

								
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