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KendallTauRankCorrelation.java
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KendallTauRankCorrelation.java
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/*-
* #%L
* Fiji's plugin for colocalization analysis.
* %%
* Copyright (C) 2009 - 2017 Fiji developers.
* %%
* This program is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as
* published by the Free Software Foundation, either version 3 of the
* License, or (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public
* License along with this program. If not, see
* <http://www.gnu.org/licenses/gpl-3.0.html>.
* #L%
*/
package sc.fiji.coloc.algorithms;
import ij.IJ;
import java.util.Arrays;
import net.imglib2.PairIterator;
import net.imglib2.RandomAccessible;
import net.imglib2.RandomAccessibleInterval;
import net.imglib2.TwinCursor;
import net.imglib2.type.logic.BitType;
import net.imglib2.type.numeric.RealType;
import net.imglib2.view.Views;
import sc.fiji.coloc.gadgets.DataContainer;
import sc.fiji.coloc.results.ResultHandler;
/**
* This algorithm calculates Kendall's Tau-b rank correlation coefficient
* <p>
* According to <a
* href="http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient">this
* article</a>, Tau-b (appropriate if multiple pairs share the same first, or
* second, value), the rank correlation of a set of pairs <tt>(x_1, y_1), ...,
* (x_n, y_n)</tt>:
* </p>
*
* <pre>
* Tau_B = (n_c - n_d) / sqrt( (n_0 - n_1) (n_0 - n_2) )
* </pre>
*
* where
*
* <pre>
* n_0 = n (n - 1) / 2
* n_1 = sum_i t_i (t_i - 1) / 2
* n_2 = sum_j u_j (u_j - 1) / 2
* n_c = #{ i, j; i != j &amp;&amp; (x_i - x_j) * (y_i - y_j) &gt; 0 },
* &nbsp; i.e. the number of pairs of pairs agreeing on the order of x and y, respectively
* n_d = #{ i, j: i != j &amp;&amp; (x_i - x_j) * (y_i - y_j) &lt; 0 },
* &nbsp; i.e. the number of pairs of pairs where x and y are ordered opposite of each other
* t_i = number of tied values in the i-th group of ties for the first quantity
* u_j = number of tied values in the j-th group of ties for the second quantity
* </pre>
*
* @author Johannes Schindelin
* @param <T>
*/
public class KendallTauRankCorrelation<T extends RealType< T >> extends Algorithm<T> {
public KendallTauRankCorrelation() {
super("Kendall's Tau-b Rank Correlation");
}
private double tau;
@Override
public void execute(DataContainer<T> container)
throws MissingPreconditionException
{
RandomAccessible<T> img1 = container.getSourceImage1();
RandomAccessible<T> img2 = container.getSourceImage2();
RandomAccessibleInterval<BitType> mask = container.getMask();
TwinCursor<T> cursor = new TwinCursor<T>(img1.randomAccess(),
img2.randomAccess(), Views.iterable(mask).localizingCursor());
tau = calculateMergeSort(cursor);
}
public static<T extends RealType<T>> double calculateNaive(final PairIterator<T> iterator) {
if (!iterator.hasNext()) {
return Double.NaN;
}
// See http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient
int n = 0, max1 = 0, max2 = 0, max = 255;
int[][] histogram = new int[max + 1][max + 1];
while (iterator.hasNext()) {
iterator.fwd();
T type1 = iterator.getFirst();
T type2 = iterator.getSecond();
double ch1 = type1.getRealDouble();
double ch2 = type2.getRealDouble();
if (ch1 < 0 || ch2 < 0 || ch1 > max || ch2 > max) {
IJ.log("Error: The current Kendall Tau implementation is limited to 8-bit data");
return Double.NaN;
}
n++;
int ch1Int = (int)Math.round(ch1);
int ch2Int = (int)Math.round(ch2);
histogram[ch1Int][ch2Int]++;
if (max1 < ch1Int) {
max1 = ch1Int;
}
if (max2 < ch2Int) {
max2 = ch2Int;
}
}
long n0 = n * (long)(n - 1) / 2, n1 = 0, n2 = 0, nc = 0, nd = 0;
for (int i1 = 0; i1 <= max1; i1++) {
IJ.log("" + i1 + "/" + max1);
int ch1 = 0;
for (int i2 = 0; i2 <= max2; i2++) {
ch1 += histogram[i1][i2];
int count = histogram[i1][i2];
for (int j1 = 0; j1 < i1; j1++) {
for (int j2 = 0; j2 < i2; j2++) {
nc += count * histogram[j1][j2];
}
}
for (int j1 = 0; j1 < i1; j1++) {
for (int j2 = i2 + 1; j2 <= max2; j2++) {
nd += count * histogram[j1][j2];
}
}
}
n1 += ch1 * (long)(ch1 - 1) / 2;
}
for (int i2 = 0; i2 <= max2; i2++) {
int ch2 = 0;
for (int i1 = 0; i1 <= max1; i1++) {
ch2 += histogram[i1][i2];
}
n2 += ch2 * (long)(ch2 - 1) / 2;
}
return (nc - nd) / Math.sqrt((n0 - n1) * (double)(n0 - n2));
}
private static<T extends RealType<T>> double[][] getPairs(final PairIterator<T> iterator) {
// TODO: it is ridiculous that this has to be counted all the time (i.e. in most if not all measurements!).
// We only need an upper bound to begin with, so even the number of pixels in the first channel would be enough!
int capacity = 0;
while (iterator.hasNext()) {
iterator.fwd();
capacity++;
}
double[] values1 = new double[capacity];
double[] values2 = new double[capacity];
iterator.reset();
int count = 0;
while (iterator.hasNext()) {
iterator.fwd();
values1[count] = iterator.getFirst().getRealDouble();
values2[count] = iterator.getSecond().getRealDouble();
count++;
}
if (count < capacity) {
values1 = Arrays.copyOf(values1, count);
values2 = Arrays.copyOf(values2, count);
}
return new double[][] { values1, values2 };
}
/**
* Calculate Tau-b efficiently.
* <p>
* This implementation is based on <a
* href="http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient#Algorithms">this
* description of the merge sort based way to calculate Tau-b</a>. This is
* supposed to be the method described in:
* </p>
* <blockquote>Knight, W. (1966). "A Computer Method for Calculating
* Kendall's Tau with Ungrouped Data". Journal of the American Statistical
* Association 61 (314): 436–439. doi:10.2307/2282833.</blockquote>
* <p>
* but since that article is not available as Open Access, it is
* unnecessarily hard to verify.
* </p>
*
* @param iterator the iterator of the pairs
* @return Tau-b
*/
public static<T extends RealType<T>> double calculateMergeSort(final PairIterator<T> iterator) {
final double[][] pairs = getPairs(iterator);
final double[] x = pairs[0];
final double[] y = pairs[1];
final int n = x.length;
int[] index = new int[n];
for (int i = 0; i < n; i++) {
index[i] = i;
}
// First sort by x as primary key, y as secondary one.
// We use IntroSort here because it is fast and in-place.
IntArraySorter.sort(index, new IntComparator() {
@Override
public int compare(int a, int b) {
double xa = x[a], ya = y[a];
double xb = x[b], yb = y[b];
int result = Double.compare(xa, xb);
return result != 0 ? result : Double.compare(ya, yb);
}
});
// The trick is to count the ties of x (n1) and the joint ties of x and y (n3) now, while
// index is sorted with regards to x.
long n0 = n * (long)(n - 1) / 2;
long n1 = 0, n3 = 0;
for (int i = 1; i < n; i++) {
double x0 = x[index[i - 1]];
if (x[index[i]] != x0) {
continue;
}
double y0 = y[index[i - 1]];
int i1 = i;
do {
double y1 = y[index[i1++]];
if (y1 == y0) {
int i2 = i1;
while (i1 < n && x[index[i1]] == x0 && y[index[i1]] == y0) {
i1++;
}
n3 += (i1 - i2 + 2) * (long)(i1 - i2 + 1) / 2;
}
y0 = y1;
} while (i1 < n && x[index[i1]] == x0);
n1 += (i1 - i + 1) * (long)(i1 - i) / 2;
i = i1;
}
// Now, let's perform that merge sort that also counts S, the number of
// swaps a Bubble Sort would require (and which therefore is half the number
// by which we have to adjust n_0 - n_1 - n_2 + n_3 to obtain n_c - n_d)
final MergeSort mergeSort = new MergeSort(index, new IntComparator() {
@Override
public int compare(int a, int b) {
double ya = y[a];
double yb = y[b];
return Double.compare(ya, yb);
}
});
long S = mergeSort.sort();
index = mergeSort.getSorted();
long n2 = 0;
for (int i = 1; i < n; i++) {
double y0 = y[index[i - 1]];
if (y[index[i]] != y0) {
continue;
}
int i1 = i + 1;
while (i1 < n && y[index[i1]] == y0) {
i1++;
}
n2 += (i1 - i + 1) * (long)(i1 - i) / 2;
i = i1;
}
return (n0 - n1 - n2 + n3 - 2 * S) / Math.sqrt((n0 - n1) * (double)(n0 - n2));
}
private final static class MergeSort {
private int[] index;
private final IntComparator comparator;
public MergeSort(int[] index, IntComparator comparator) {
this.index = index;
this.comparator = comparator;
}
public int[] getSorted() {
return index;
}
/**
* Sorts the {@link #index} array.
* <p>
* This implements a non-recursive merge sort.
* </p>
* @param begin
* @param end
* @return the equivalent number of BubbleSort swaps
*/
public long sort() {
long swaps = 0;
int n = index.length;
// There are merge sorts which perform in-place, but their runtime is worse than O(n log n)
int[] index2 = new int[n];
for (int step = 1; step < n; step <<= 1) {
int begin = 0, k = 0;
for (;;) {
int begin2 = begin + step, end = begin2 + step;
if (end >= n) {
if (begin2 >= n) {
break;
}
end = n;
}
// calculate the equivalent number of BubbleSort swaps
// and perform merge, too
int i = begin, j = begin2;
while (i < begin2 && j < end) {
int compare = comparator.compare(index[i], index[j]);
if (compare > 0) {
swaps += (begin2 - i);
index2[k++] = index[j++];
} else {
index2[k++] = index[i++];
}
}
if (i < begin2) {
do {
index2[k++] = index[i++];
} while (i < begin2);
} else {
while (j < end) {
index2[k++] = index[j++];
}
}
begin = end;
}
if (k < n) {
System.arraycopy(index, k, index2, k, n - k);
}
int[] swapIndex = index2;
index2 = index;
index = swapIndex;
}
return swaps;
}
}
@Override
public void processResults(ResultHandler<T> handler) {
super.processResults(handler);
handler.handleValue("Kendall's Tau-b rank correlation value", tau, 4);
}
}

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