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    <h1>Spatial cross correlation</h1>
    <small class="dont-index">Source: <a href='https://github.com/jeffreyevans/spatialEco/blob/master/R/crossCorrelation.R'><code>R/crossCorrelation.R</code></a></small>
    <div class="hidden name"><code>crossCorrelation.Rd</code></div>
    </div>

    <div class="ref-description">
    <p>Calculates univariate or bivariate spatial cross-correlation using
local Moran's-I (LISA), following Chen (2015)</p>
    </div>

    <pre class="usage"><span class='fu'>crossCorrelation</span><span class='op'>(</span>
  <span class='va'>x</span>,
  y <span class='op'>=</span> <span class='cn'>NULL</span>,
  coords <span class='op'>=</span> <span class='cn'>NULL</span>,
  w <span class='op'>=</span> <span class='cn'>NULL</span>,
  type <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='st'>"LSCI"</span>, <span class='st'>"GSCI"</span><span class='op'>)</span>,
  k <span class='op'>=</span> <span class='fl'>999</span>,
  dist.function <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='st'>"inv.power"</span>, <span class='st'>"neg.exponent"</span>, <span class='st'>"none"</span><span class='op'>)</span>,
  scale.xy <span class='op'>=</span> <span class='cn'>TRUE</span>,
  scale.partial <span class='op'>=</span> <span class='cn'>FALSE</span>,
  scale.matrix <span class='op'>=</span> <span class='cn'>FALSE</span>,
  alpha <span class='op'>=</span> <span class='fl'>0.05</span>,
  clust <span class='op'>=</span> <span class='cn'>TRUE</span>,
  return.sims <span class='op'>=</span> <span class='cn'>FALSE</span>
<span class='op'>)</span></pre>

    <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
    <table class="ref-arguments">
    <colgroup><col class="name" /><col class="desc" /></colgroup>
    <tr>
      <th>x</th>
      <td><p>Vector of x response variables</p></td>
    </tr>
    <tr>
      <th>y</th>
      <td><p>Vector of y response variables, if not specified the
univariate  statistic is returned</p></td>
    </tr>
    <tr>
      <th>coords</th>
      <td><p>A matrix of coordinates corresponding to (x,y), only
used if w = NULL. Can also be an sp object with relevant
x,y coordinate slot (ie., points or polygons)</p></td>
    </tr>
    <tr>
      <th>w</th>
      <td><p>Spatial neighbors/weights in matrix format. Dimensions
must match (n(x),n(y)) and be symmetrical. If w is not defined
then a default method is used.</p></td>
    </tr>
    <tr>
      <th>type</th>
      <td><p>c("LSCI","GSCI") Return Local Spatial Cross-correlation Index (LSCI)
or Global Spatial cross-correlation Index (GSCI)</p></td>
    </tr>
    <tr>
      <th>k</th>
      <td><p>Number of simulations for calculating permutation distribution
under the null hypothesis of no spatial autocorrelation</p></td>
    </tr>
    <tr>
      <th>dist.function</th>
      <td><p>("inv.power", "neg.exponent", "none") If w = NULL, the default method
for deriving spatial weights matrix, options are: inverse power
or negative exponent, none is for use with a provided matrix</p></td>
    </tr>
    <tr>
      <th>scale.xy</th>
      <td><p>(TRUE/FALSE) scale the x,y vectors, if FALSE it is assumed that
they are already scaled following Chen (2015)</p></td>
    </tr>
    <tr>
      <th>scale.partial</th>
      <td><p>(FALSE/TRUE) rescale partial spatial autocorrelation statistics</p></td>
    </tr>
    <tr>
      <th>scale.matrix</th>
      <td><p>(FALSE/TRUE) If a neighbor/distance matrix is passed, should it
be scaled using (w/sum(w))</p></td>
    </tr>
    <tr>
      <th>alpha</th>
      <td><p>= 0.05     confidence interval (default is 95 pct)</p></td>
    </tr>
    <tr>
      <th>clust</th>
      <td><p>(FALSE/TRUE) Return approximated lisa clusters</p></td>
    </tr>
    <tr>
      <th>return.sims</th>
      <td><p>(FALSE/TRUE) Return randomizations vector n = k</p></td>
    </tr>
    </table>

    <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>

    <p>When not simulated k=0, a list containing:</p><ul>
<li><p>I Global autocorrelation statistic</p></li>
<li><p>SCI A data.frame with two columns representing the xy and yx autocorrelation</p></li>
<li><p>nsim value of NULL to represent p values were derived from observed data (k=0)</p></li>
<li><p>p Probability based observations above/below confidence interval</p></li>
<li><p>t.test Probability based on t-test</p></li>
<li><p>clusters If "clust" argument TRUE, vector representing LISA clusters</p></li>
</ul>

<p>when simulated (k&gt;0), a list containing:</p><ul>
<li><p>I Global autocorrelation statistic</p></li>
<li><p>SCI A data.frame with two columns representing the xy and yx autocorrelation</p></li>
<li><p>nsim value representing number of simulations</p></li>
<li><p>global.p p-value of global autocorrelation statistic</p></li>
<li><p>local.p Probability based simulated data using successful rejection of t-test</p></li>
<li><p>range.p Probability based on range of probabilities resulting from paired t-test</p></li>
<li><p>clusters If "clust" argument TRUE, vector representing lisa clusters</p></li>
</ul>

    <h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>

    <p>In specifying a distance matrix, you can pass a coordinates matrix or spatial
object to coords or alternately, pass a distance or spatial weights matrix to
the w argument. If the w matrix represents spatial weights dist.function="none"
should be specified. Otherwise, w is assumed to represent distance and will be
converted to spatial weights using inv.power or neg.exponent. The w distances
can represent an alternate distance hypothesis (eg., road, stream, network distance)
Here are example argument usages for defining a matrix.</p><ul>
<li><p>IF coords=x, w=NULL, dist.function= c("inv.power", "neg.exponent")
A distance matrix is derived using the data passed to coords then
spatial weights derived using one of the dist.function options</p></li>
<li><p>IF cords=NULL, w=x, dist.function= c("inv.power", "neg.exponent")
It is expected that the distance matrix specified with w represent
some form of distance then the spatial weights are derived using
one of the dist.function options</p></li>
<li><p>IF cords=NULL, w=x, dist.function="none"
It is assumed that the matrix passed to w already represents
the spatial weights</p></li>
</ul>

    <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>

    <p>Chen, Y.G. (2012) On the four types of weight functions for spatial contiguity
matrix. Letters in Spatial and Resource Sciences 5(2):65-72</p>
<p>Chen, Y.G. (2013) New approaches for calculating Moran’s index of spatial
autocorrelation. PLoS ONE 8(7):e68336</p>
<p>Chen, Y.G. (2015) A New Methodology of Spatial Cross-Correlation Analysis.
PLoS One 10(5):e0126158. doi:10.1371/journal.pone.0126158</p>

    <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
    <pre class="examples"><div class='input'><span class='co'># replicate Chen (2015)</span>
 <span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span><span class='op'>(</span><span class='va'>chen</span><span class='op'>)</span>
<span class='op'>(</span> <span class='va'>r</span> <span class='op'>&lt;-</span> <span class='fu'>crossCorrelation</span><span class='op'>(</span>x<span class='op'>=</span><span class='va'>chen</span><span class='op'>[[</span><span class='st'>"X"</span><span class='op'>]</span><span class='op'>]</span>, y<span class='op'>=</span><span class='va'>chen</span><span class='op'>[[</span><span class='st'>"Y"</span><span class='op'>]</span><span class='op'>]</span>, w <span class='op'>=</span> <span class='va'>chen</span><span class='op'>[[</span><span class='st'>"M"</span><span class='op'>]</span><span class='op'>]</span>,  
                        clust<span class='op'>=</span><span class='cn'>TRUE</span>, type <span class='op'>=</span> <span class='st'>"LSCI"</span>, k<span class='op'>=</span><span class='fl'>0</span>, 
                        dist.function <span class='op'>=</span> <span class='st'>"inv.power"</span><span class='op'>)</span> <span class='op'>)</span> 
</div><div class='output co'>#&gt; <span class='message'>Permutation is not being run, estimated p will be based on observed</span></div><div class='output co'>#&gt; <span class='message'>Calculating spatial weights matrix using inverse power function</span></div><div class='output co'>#&gt; Moran's-I... 
#&gt;   First-order Moran's-I:  0.1566446 
#&gt;   First-order p-value:  
#&gt;  
#&gt; Chen's SCI under randomization assumptions... 
#&gt; 
#&gt;  Summary statistics of local partial cross-correlation [xy] 
#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#&gt; -0.0077674 -0.0005055  0.0013531  0.0054015  0.0045101  0.0589298 
#&gt;  
#&gt;     non-simulated second-order p-value based on 2-tailed t-test:  0.037588 
#&gt;     p-value based on 2-tailed t-test observations above/below CI:  0.06896552 
#&gt; 
#&gt;  Counts of cluster types
#&gt; High.High  High.Low  Low.High   Low.Low 
#&gt;         9         1         7        12 </div><div class='input'>
<span class='co'># \donttest{</span>
  <span class='kw'><a href='https://rdrr.io/r/base/library.html'>library</a></span><span class='op'>(</span><span class='va'><a href='https://github.com/edzer/sp/'>sp</a></span><span class='op'>)</span>
  <span class='kw'><a href='https://rdrr.io/r/base/library.html'>library</a></span><span class='op'>(</span><span class='va'><a href='https://github.com/r-spatial/spdep/'>spdep</a></span><span class='op'>)</span>
</div><div class='output co'>#&gt; <span class='message'>Loading required package: spData</span></div><div class='output co'>#&gt; <span class='message'>To access larger datasets in this package, install the spDataLarge</span>
#&gt; <span class='message'>package with: `install.packages('spDataLarge',</span>
#&gt; <span class='message'>repos='https://nowosad.github.io/drat/', type='source')`</span></div><div class='output co'>#&gt; <span class='message'></span>
#&gt; <span class='message'>Attaching package: 'spData'</span></div><div class='output co'>#&gt; <span class='message'>The following object is masked from 'package:spatialEco':</span>
#&gt; <span class='message'></span>
#&gt; <span class='message'>    elev</span></div><div class='input'>   
  <span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>)</span>
    <span class='fu'><a href='https://rdrr.io/pkg/sp/man/coordinates.html'>coordinates</a></span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>)</span> <span class='op'>&lt;-</span> <span class='op'>~</span><span class='va'>x</span><span class='op'>+</span><span class='va'>y</span>  

  <span class='co'>#### Using a default spatial weights matrix method (inverse power function)</span>
  <span class='op'>(</span> <span class='va'>I</span> <span class='op'>&lt;-</span> <span class='fu'>crossCorrelation</span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>$</span><span class='va'>zinc</span>, <span class='va'>meuse</span><span class='op'>$</span><span class='va'>copper</span>, 
               coords <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/pkg/sp/man/coordinates.html'>coordinates</a></span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>)</span>, k<span class='op'>=</span><span class='fl'>99</span><span class='op'>)</span> <span class='op'>)</span>
</div><div class='output co'>#&gt; <span class='message'>Calculating spatial weights matrix using inverse power function</span></div><div class='output co'>#&gt; <span class='message'></span>
#&gt; <span class='message'> Computing Permutation Distribution </span></div><div class='output co'>#&gt; Moran's-I under randomization assumptions... 
#&gt;   First-order Moran's-I:  0.08906818 
#&gt;   First-order p-value:  0 
#&gt; Chen's SCI under randomization assumptions... 
#&gt; 
#&gt;  Summary statistics of local partial cross-correlation [xy] 
#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#&gt; -2.100e-03 -5.538e-06  3.498e-04  5.746e-04  8.350e-04  7.119e-03 
#&gt;  
#&gt;   p-value based on 2-tailed t-test:  0 
#&gt;   p-value based on 2-tailed t-test observations above/below CI:  0 
#&gt; 
#&gt;  Counts of cluster types
#&gt; High.High  High.Low  Low.High   Low.Low 
#&gt;        44        16        25        70 </div><div class='input'>    <span class='va'>meuse</span><span class='op'>$</span><span class='va'>lisa</span> <span class='op'>&lt;-</span> <span class='va'>I</span><span class='op'>$</span><span class='va'>SCI</span><span class='op'>[</span>,<span class='st'>"lsci.xy"</span><span class='op'>]</span>
      <span class='fu'><a href='https://rdrr.io/pkg/sp/man/spplot.html'>spplot</a></span><span class='op'>(</span><span class='va'>meuse</span>, <span class='st'>"lisa"</span><span class='op'>)</span>
</div><div class='img'><img src='crossCorrelation-1.png' alt='' width='700' height='433' /></div><div class='input'> 
  <span class='co'>#### Providing a distance matrix</span>
   <span class='va'>Wij</span> <span class='op'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/pkg/sp/man/spDistsN1.html'>spDists</a></span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>)</span>   
   <span class='op'>(</span> <span class='va'>I</span> <span class='op'>&lt;-</span> <span class='fu'>crossCorrelation</span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>$</span><span class='va'>zinc</span>, <span class='va'>meuse</span><span class='op'>$</span><span class='va'>copper</span>, w <span class='op'>=</span> <span class='va'>Wij</span>, k<span class='op'>=</span><span class='fl'>99</span><span class='op'>)</span> <span class='op'>)</span>
</div><div class='output co'>#&gt; <span class='message'>Calculating spatial weights matrix using inverse power function</span></div><div class='output co'>#&gt; <span class='message'></span>
#&gt; <span class='message'> Computing Permutation Distribution </span></div><div class='output co'>#&gt; Moran's-I under randomization assumptions... 
#&gt;   First-order Moran's-I:  0.08906818 
#&gt;   First-order p-value:  0 
#&gt; Chen's SCI under randomization assumptions... 
#&gt; 
#&gt;  Summary statistics of local partial cross-correlation [xy] 
#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#&gt; -2.100e-03 -5.538e-06  3.498e-04  5.746e-04  8.350e-04  7.119e-03 
#&gt;  
#&gt;   p-value based on 2-tailed t-test:  0 
#&gt;   p-value based on 2-tailed t-test observations above/below CI:  0 
#&gt; 
#&gt;  Counts of cluster types
#&gt; High.High  High.Low  Low.High   Low.Low 
#&gt;        44        16        25        70 </div><div class='input'>
  <span class='co'>#### Providing an inverse power function weights matrix</span>
   <span class='va'>Wij</span> <span class='op'>&lt;-</span> <span class='fu'><a href='https://rdrr.io/pkg/sp/man/spDistsN1.html'>spDists</a></span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>)</span>
     <span class='va'>Wij</span> <span class='op'>&lt;-</span> <span class='fl'>1</span> <span class='op'>/</span> <span class='va'>Wij</span>
           <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='va'>Wij</span><span class='op'>)</span> <span class='op'>&lt;-</span> <span class='fl'>0</span> 
         <span class='va'>Wij</span> <span class='op'>&lt;-</span> <span class='va'>Wij</span> <span class='op'>/</span> <span class='fu'><a href='https://rdrr.io/r/base/sum.html'>sum</a></span><span class='op'>(</span><span class='va'>Wij</span><span class='op'>)</span> 
   <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='va'>Wij</span><span class='op'>)</span> <span class='op'>&lt;-</span> <span class='fl'>0</span>
   <span class='op'>(</span> <span class='va'>I</span> <span class='op'>&lt;-</span> <span class='fu'>crossCorrelation</span><span class='op'>(</span><span class='va'>meuse</span><span class='op'>$</span><span class='va'>zinc</span>, <span class='va'>meuse</span><span class='op'>$</span><span class='va'>copper</span>, w <span class='op'>=</span> <span class='va'>Wij</span>, 
                          dist.function <span class='op'>=</span> <span class='st'>"none"</span>, k<span class='op'>=</span><span class='fl'>99</span><span class='op'>)</span> <span class='op'>)</span>
</div><div class='output co'>#&gt; <span class='message'>Wij matrix is being left raw</span></div><div class='output co'>#&gt; <span class='message'></span>
#&gt; <span class='message'> Computing Permutation Distribution </span></div><div class='output co'>#&gt; Moran's-I under randomization assumptions... 
#&gt;   First-order Moran's-I:  0.08906818 
#&gt;   First-order p-value:  0 
#&gt; Chen's SCI under randomization assumptions... 
#&gt; 
#&gt;  Summary statistics of local partial cross-correlation [xy] 
#&gt;       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
#&gt; -2.100e-03 -5.538e-06  3.498e-04  5.746e-04  8.350e-04  7.119e-03 
#&gt;  
#&gt;   p-value based on 2-tailed t-test:  0 
#&gt;   p-value based on 2-tailed t-test observations above/below CI:  0 
#&gt; 
#&gt;  Counts of cluster types
#&gt; High.High  High.Low  Low.High   Low.Low 
#&gt;        44        16        25        70 </div><div class='input'><span class='co'># } </span>

</div></pre>
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