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Posted to commits@systemds.apache.org by ja...@apache.org on 2022/02/12 05:51:04 UTC
[systemds] branch main updated: [SYSTEMDS-3184] Builtin for computing information gain using entropy and gini
This is an automated email from the ASF dual-hosted git repository.
janardhan pushed a commit to branch main
in repository https://gitbox.apache.org/repos/asf/systemds.git
The following commit(s) were added to refs/heads/main by this push:
new 7c3cc82 [SYSTEMDS-3184] Builtin for computing information gain using entropy and gini
7c3cc82 is described below
commit 7c3cc82706da3434f6ec4c4bf8e6c719d7b042e0
Author: morf1us <fl...@student.tugraz.at>
AuthorDate: Sat Feb 12 06:50:54 2022 +0100
[SYSTEMDS-3184] Builtin for computing information gain using entropy and gini
Closes #1520
---
docs/site/builtins-reference.md | 45 ++++
scripts/builtin/impurityMeasures.dml | 139 +++++++++++
.../java/org/apache/sysds/common/Builtins.java | 1 +
.../builtin/part1/BuiltinImpurityMeasuresTest.java | 259 +++++++++++++++++++++
.../scripts/functions/builtin/impurityMeasures.dml | 27 +++
5 files changed, 471 insertions(+)
diff --git a/docs/site/builtins-reference.md b/docs/site/builtins-reference.md
index 1073ed7..6977dad 100644
--- a/docs/site/builtins-reference.md
+++ b/docs/site/builtins-reference.md
@@ -50,6 +50,7 @@ limitations under the License.
* [`img_brightness`-Function](#img_brightness-function)
* [`img_crop`-Function](#img_crop-function)
* [`img_mirror`-Function](#img_mirror-function)
+ * [`impurityMeasures`-Function](#impurityMeasures-function)
* [`imputeByFD`-Function](#imputeByFD-function)
* [`intersect`-Function](#intersect-function)
* [`KMeans`-Function](#KMeans-function)
@@ -1020,6 +1021,50 @@ B = img_mirror(img_in = A, horizontal_axis = TRUE)
```
+## `impurityMeasures`-Function
+
+`impurityMeasures()` computes the measure of impurity for each feature of the given dataset based on the passed method (gini or entropy).
+
+### Usage
+
+```r
+IM = impurityMeasures(X = X, Y = Y, R = R, n_bins = 20, method = "gini");
+```
+
+### Arguments
+
+| Name | Type | Default | Description |
+| :--------- | :-------------- | :------ | :---------- |
+| X | Matrix[Double] | --- | Feature matrix X |
+| Y | Matrix[Double] | --- | Target vector Y containing only 0 or 1 values |
+| R | Matrix[Double] | --- | Row vector R indicating whether a feature is categorical or continuous. 1 denotes a continuous feature, 2 denotes a categorical feature. |
+| n_bins | Integer | `20` | Number of equi-width bins for binning in case of scale features. |
+| method | String | --- | String indicating the method to use; either "entropy" or "gini". |
+
+### Returns
+
+| Name | Type | Description |
+| :--- | :------------- | :---------- |
+| IM | Matrix[Double] | (1 x ncol(X)) row vector containing information/gini gain for each feature of the dataset. In case of gini, the values denote the gini gains, i.e. how much impurity was removed with the respective split. The higher the value, the better the split. In case of entropy, the values denote the information gain, i.e. how much entropy was removed. The higher the information gain, the better the split. |
+
+### Example
+
+```r
+X = matrix("4.0 3.0 2.8 3.5
+ 2.4 1.0 3.4 2.9
+ 1.1 1.0 4.9 3.4
+ 5.0 2.0 1.4 1.8
+ 1.1 3.0 1.0 1.9", rows=5, cols=4)
+Y = matrix("1.0
+ 0.0
+ 0.0
+ 1.0
+ 0.0", rows=5, cols=1)
+R = matrix("1.0 2.0 1.0 1.0", rows=1, cols=4)
+IM = impurityMeasures(X = X, Y = Y, R = R, method = "entropy")
+```
+
+
## `imputeByFD`-Function
The `imputeByFD`-function imputes missing values from observed values (if exist)
diff --git a/scripts/builtin/impurityMeasures.dml b/scripts/builtin/impurityMeasures.dml
new file mode 100644
index 0000000..860bc62
--- /dev/null
+++ b/scripts/builtin/impurityMeasures.dml
@@ -0,0 +1,139 @@
+#-------------------------------------------------------------
+#
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements. See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership. The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied. See the License for the
+# specific language governing permissions and limitations
+# under the License.
+#
+#-------------------------------------------------------------
+
+# This function computes the measure of impurity for the given dataset based on the passed method (gini or entropy).
+# The current version expects the target vector to contain only 0 or 1 values.
+#
+# INPUT PARAMETERS:
+# ----------------------------------------------------------------------------------------------------------------------
+# NAME TYPE DEFAULT MEANING
+# ----------------------------------------------------------------------------------------------------------------------
+# X Matrix[Double] --- Feature matrix.
+# Y Matrix[Double] --- Target vector containing 0 and 1 values.
+# R Matrix[Double] --- Vector indicating whether a feature is categorical or continuous.
+# 1 denotes a continuous feature, 2 denotes a categorical feature.
+# n_bins Integer 20 Number of bins for binning in case of scale features.
+# method String --- String indicating the method to use; either "entropy" or "gini".
+# ----------------------------------------------------------------------------------------------------------------------
+
+# Output(s)
+# ----------------------------------------------------------------------------------------------------------------------
+# NAME TYPE DEFAULT MEANING
+# ----------------------------------------------------------------------------------------------------------------------
+# IM Matrix[Double] --- (1 x ncol(X)) row vector containing information/gini gain for
+# each feature of the dataset.
+# In case of gini, the values denote the gini gains, i.e. how much
+# impurity was removed with the respective split. The higher the
+# value, the better the split.
+# In case of entropy, the values denote the information gain, i.e.
+# how much entropy was removed. The higher the information gain,
+# the better the split.
+# ----------------------------------------------------------------------------------------------------------------------
+
+m_impurityMeasures = function(Matrix[Double] X, Matrix[Double] Y, Matrix[Double] R, Integer n_bins = 20, String method)
+ return (Matrix[Double] IM)
+{
+ if (method != "entropy" & method != "gini") {
+ stop("Please specify the correct method - should be either entropy or gini.")
+ }
+
+ IM = matrix(0.0, rows = 1, cols = ncol(X))
+
+ parfor (i in 1:ncol(X)) {
+ if (as.scalar(R[,i]) == 1) {
+ binned_feature = applyBinning(X[,i], n_bins)
+ IM[,i] = getImpurityMeasure(binned_feature, Y, n_bins, method)
+ } else {
+ IM[,i] = getImpurityMeasure(X[,i], Y, max(X[,i]), method)
+ }
+ }
+}
+
+getImpurityMeasure = function(Matrix[Double] feature, Matrix[Double] Y, Double max_cat, String method)
+ return (Double gain)
+{
+ n_true_labels = sum(Y)
+ n_false_labels = length(Y) - n_true_labels
+ parent_impurity = calcImpurity(n_true_labels, n_false_labels, length(feature), method)
+
+ # calculate the impurity after the split
+ children_impurity = 0
+ for (i in 1:max_cat) {
+ count_true = 0
+ count_false = 0
+ for (j in 1:length(feature)) {
+ if (as.scalar(feature[j,]) == i) {
+ if (as.scalar(Y[j,]) == 0) {
+ count_false += 1
+ } else {
+ count_true += 1
+ }
+ }
+ }
+ if (!(count_true == 0 & count_false == 0)) {
+ children_impurity = children_impurity + calcImpurity(count_true, count_false, length(feature), method)
+ }
+ }
+ gain = parent_impurity - children_impurity
+}
+
+calcImpurity = function(Double n_true, Double n_false, Double n_vars, String method)
+ return (Double impurity)
+{
+ impurity = 0
+ prob_true = n_true / (n_true + n_false)
+ prob_false = n_false / (n_true + n_false)
+ weight = (n_true + n_false) / n_vars
+
+ if (prob_true != 1 & prob_false != 1) { # if there is more than one class, calculate new impurity according to method.
+ if (method == "entropy") { # dividing by log(2) to obtain the information gain in bits
+ impurity = (-1) * weight * (prob_true * log(prob_true)/log(2) + prob_false * log(prob_false)/log(2))
+ } else if (method == "gini") {
+ impurity = weight * (1 - (prob_true^2 + prob_false^2))
+ }
+ }
+}
+
+applyBinning = function(Matrix[Double] feature, Double n_bins)
+ return (Matrix[Double] output_f)
+{
+ # equi-width binning.
+
+ if (length(feature) < n_bins) {
+ n_bins = length(feature)
+ }
+ max_v = max(feature)
+ min_v = min(feature)
+ width = (max_v - min_v) / n_bins
+ output_f = matrix(1, rows = nrow(feature), cols = 1)
+
+ parfor (i in 1:length(feature)) {
+ binned = FALSE
+ j = 1
+ while (binned == FALSE) {
+ if (as.scalar(feature[i,]) <= min_v + j * width) {
+ output_f[i,] = j
+ binned = TRUE
+ }
+ j += 1
+ }
+ }
+}
diff --git a/src/main/java/org/apache/sysds/common/Builtins.java b/src/main/java/org/apache/sysds/common/Builtins.java
index f1e99f8..2fab87b 100644
--- a/src/main/java/org/apache/sysds/common/Builtins.java
+++ b/src/main/java/org/apache/sysds/common/Builtins.java
@@ -159,6 +159,7 @@ public enum Builtins {
IMG_SAMPLE_PAIRING("img_sample_pairing", true),
IMG_INVERT("img_invert", true),
IMG_POSTERIZE("img_posterize", true),
+ IMPURITY_MEASURES("impurityMeasures", true),
IMPUTE_BY_MEAN("imputeByMean", true),
IMPUTE_BY_MEAN_APPLY("imputeByMeanApply", true),
IMPUTE_BY_MEDIAN("imputeByMedian", true),
diff --git a/src/test/java/org/apache/sysds/test/functions/builtin/part1/BuiltinImpurityMeasuresTest.java b/src/test/java/org/apache/sysds/test/functions/builtin/part1/BuiltinImpurityMeasuresTest.java
new file mode 100644
index 0000000..d67768b
--- /dev/null
+++ b/src/test/java/org/apache/sysds/test/functions/builtin/part1/BuiltinImpurityMeasuresTest.java
@@ -0,0 +1,259 @@
+/*
+ * Licensed to the Apache Software Foundation (ASF) under one
+ * or more contributor license agreements. See the NOTICE file
+ * distributed with this work for additional information
+ * regarding copyright ownership. The ASF licenses this file
+ * to you under the Apache License, Version 2.0 (the
+ * "License"); you may not use this file except in compliance
+ * with the License. You may obtain a copy of the License at
+ *
+ * http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing,
+ * software distributed under the License is distributed on an
+ * "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+ * KIND, either express or implied. See the License for the
+ * specific language governing permissions and limitations
+ * under the License.
+ */
+
+package org.apache.sysds.test.functions.builtin.part1;
+
+import java.util.HashMap;
+
+import org.apache.sysds.common.Types;
+import org.apache.sysds.common.Types.ExecType;
+import org.apache.sysds.runtime.matrix.data.MatrixValue;
+import org.apache.sysds.test.AutomatedTestBase;
+import org.apache.sysds.test.TestConfiguration;
+import org.apache.sysds.test.TestUtils;
+import org.junit.Test;
+
+public class BuiltinImpurityMeasuresTest extends AutomatedTestBase {
+ private final static String TEST_NAME = "impurityMeasures";
+ private final static String TEST_DIR = "functions/builtin/";
+ private static final String TEST_CLASS_DIR = TEST_DIR + BuiltinImpurityMeasuresTest.class.getSimpleName() + "/";
+
+ private final static double eps = 1e-10;
+
+ @Override
+ public void setUp() {
+ addTestConfiguration(TEST_NAME, new TestConfiguration(TEST_CLASS_DIR, TEST_NAME, new String[] {"C"}));
+ }
+
+ @Test
+ public void GiniTest1() {
+ double[][] X = {{1, 1}, {2, 2}};
+ double[][] Y = {{1}, {0}};
+ double[][] R = {{2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.5);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.5);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void GiniTest2() {
+ double[][] X = {{1},{1},{1},{1},{1},{1},{2},{2},{2},{2}};
+ double[][] Y = {{0}, {0}, {0}, {0}, {0}, {1}, {1}, {1}, {1}, {1}};
+ double[][] R = {{2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.3333333333);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void GiniTest3() {
+ double[][] X = {{1,1,2,1}, {1,3,1,2}, {2,1,1,2}, {3,2,1,1}, {1,3,2,1}};
+ double[][] Y = {{0}, {0}, {1}, {1}, {1}};
+ double[][] R = {{2, 2, 2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.2133333333);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.0799999999);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.0133333333);
+ expected_m.put(new MatrixValue.CellIndex(1, 4), 0.0133333333);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void GiniPlayTennisTest() {
+ double[][] X = {{1,1,1,1},
+ {1,1,1,2},
+ {2,1,1,1},
+ {3,2,1,1},
+ {3,3,2,1},
+ {3,3,2,2},
+ {2,3,2,2},
+ {1,2,1,1},
+ {1,3,2,1},
+ {3,2,2,1},
+ {1,2,2,2},
+ {2,2,1,2},
+ {2,1,2,1},
+ {3,2,1,2}};
+ double[][] Y = {{0}, {0}, {1}, {1}, {1}, {0}, {1}, {0}, {1}, {1}, {1}, {1}, {1}, {0}};
+ double[][] R = {{2, 2, 2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.1163265306);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.0187074829);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.0918367346);
+ expected_m.put(new MatrixValue.CellIndex(1, 4), 0.0306122448);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void GiniWithContinuousValues1() {
+ double[][] X = {{10.3}, {31.2}, {9.5}, {34.3}};
+ double[][] Y = {{0}, {1}, {0}, {1}};
+ double[][] R = {{1}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.5);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void GiniWithContinuousValues2() {
+ double[][] X = {{1.5, 23.7, 2929.6}, {12.6, 80.2, 2823.3}, {3.4, 238.2, 832.2}, {14.2, 282.1, 23.1}};
+ double[][] Y = {{0}, {1}, {0}, {1}};
+ double[][] R = {{1, 1, 1}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.5);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.0);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.25);
+ String method = "gini";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ // comparing with values from https://planetcalc.com/8421/
+ @Test
+ public void EntropyTest1() {
+ double[][] X = {{1, 1}, {2, 2}};
+ double[][] Y = {{1}, {0}};
+ double[][] R = {{2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 1.0);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 1.0);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void EntropyTest2() {
+ double[][] X = {{1},{1},{1},{1},{1},{1},{2},{2},{2},{2}};
+ double[][] Y = {{0},{0},{0},{0},{0},{1},{1},{1},{1},{1}};
+ double[][] R = {{2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.6099865470);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void EntropyTest3() {
+ double[][] X = {{1,1,2,1}, {1,3,1,2}, {2,1,1,2}, {3,2,1,1}, {1,3,2,1}};
+ double[][] Y = {{0}, {0}, {1}, {1}, {1}};
+ double[][] R = {{2, 2, 2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.4199730940);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.1709505945);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.0199730940);
+ expected_m.put(new MatrixValue.CellIndex(1, 4), 0.0199730940);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void EntropyPlayTennisTest() {
+ double[][] X = {{1,1,1,1},
+ {1,1,1,2},
+ {2,1,1,1},
+ {3,2,1,1},
+ {3,3,2,1},
+ {3,3,2,2},
+ {2,3,2,2},
+ {1,2,1,1},
+ {1,3,2,1},
+ {3,2,2,1},
+ {1,2,2,2},
+ {2,2,1,2},
+ {2,1,2,1},
+ {3,2,1,2}};
+ double[][] Y = {{0}, {0}, {1}, {1}, {1}, {0}, {1}, {0}, {1}, {1}, {1}, {1}, {1}, {0}};
+ double[][] R = {{2, 2, 2, 2}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 0.2467498198);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.0292225657);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.1518355014);
+ expected_m.put(new MatrixValue.CellIndex(1, 4), 0.0481270304);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void EntropyWithContinuousValues1() {
+ double[][] X = {{10.3}, {31.2}, {9.5}, {34.3}};
+ double[][] Y = {{0}, {1}, {0}, {1}};
+ double[][] R = {{1}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 1.0);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ @Test
+ public void EntropyWithContinuousValues2() {
+ double[][] X = {{1.5, 23.7, 2929.6}, {12.6, 80.2, 2823.3}, {3.4, 238.2, 832.2}, {14.2, 282.1, 23.1}};
+ double[][] Y = {{0}, {1}, {0}, {1}};
+ double[][] R = {{1, 1, 1}};
+ HashMap<MatrixValue.CellIndex, Double> expected_m = new HashMap<>();
+ expected_m.put(new MatrixValue.CellIndex(1, 1), 1.0);
+ expected_m.put(new MatrixValue.CellIndex(1, 2), 0.0);
+ expected_m.put(new MatrixValue.CellIndex(1, 3), 0.5);
+ String method = "entropy";
+
+ runImpurityMeasuresTest(ExecType.SPARK, X, Y, R, method, expected_m);
+ }
+
+ private void runImpurityMeasuresTest(ExecType exec_type, double[][] X, double[][] Y, double[][] R, String method, HashMap<MatrixValue.CellIndex, Double> expected_m) {
+ Types.ExecMode platform_old = setExecMode(exec_type);
+
+ try {
+ loadTestConfiguration(getTestConfiguration(TEST_NAME));
+
+ String HOME = SCRIPT_DIR + TEST_DIR;
+ fullDMLScriptName = HOME + TEST_NAME + ".dml";
+ programArgs = new String[] {"-args", input("X"), input("Y"), input("R"), method, output("impurity_measures")};
+
+ writeInputMatrixWithMTD("X", X, true);
+ writeInputMatrixWithMTD("Y", Y, true);
+ writeInputMatrixWithMTD("R", R, true);
+
+ runTest(true, false, null, -1);
+
+ HashMap<MatrixValue.CellIndex, Double> actual_measures = readDMLMatrixFromOutputDir("impurity_measures");
+
+ System.out.println(actual_measures);
+ System.out.println(expected_m);
+ TestUtils.compareMatrices(expected_m, actual_measures, eps, "Expected measures", "Actual measures");
+ }
+ finally {
+ rtplatform = platform_old;
+ }
+ }
+}
diff --git a/src/test/scripts/functions/builtin/impurityMeasures.dml b/src/test/scripts/functions/builtin/impurityMeasures.dml
new file mode 100644
index 0000000..01ab4cb
--- /dev/null
+++ b/src/test/scripts/functions/builtin/impurityMeasures.dml
@@ -0,0 +1,27 @@
+#-------------------------------------------------------------
+#
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements. See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership. The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License. You may obtain a copy of the License at
+#
+# http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied. See the License for the
+# specific language governing permissions and limitations
+# under the License.
+#
+#-------------------------------------------------------------
+
+X = read($1)
+Y = read($2)
+R = read($3)
+IM = impurityMeasures(X = X, Y = Y, R = R, method = $4);
+
+write(IM, $5);