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<HTML> <BODY BGCOLOR="white"> <PRE> <FONT color="green">001</FONT> /*<a name="line.1"></a> <FONT color="green">002</FONT> * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a> <FONT color="green">003</FONT> * contributor license agreements. See the NOTICE file distributed with<a name="line.3"></a> <FONT color="green">004</FONT> * this work for additional information regarding copyright ownership.<a name="line.4"></a> <FONT color="green">005</FONT> * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a> <FONT color="green">006</FONT> * (the "License"); you may not use this file except in compliance with<a name="line.6"></a> <FONT color="green">007</FONT> * the License. You may obtain a copy of the License at<a name="line.7"></a> <FONT color="green">008</FONT> *<a name="line.8"></a> <FONT color="green">009</FONT> * http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a> <FONT color="green">010</FONT> *<a name="line.10"></a> <FONT color="green">011</FONT> * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a> <FONT color="green">012</FONT> * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a> <FONT color="green">013</FONT> * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a> <FONT color="green">014</FONT> * See the License for the specific language governing permissions and<a name="line.14"></a> <FONT color="green">015</FONT> * limitations under the License.<a name="line.15"></a> <FONT color="green">016</FONT> */<a name="line.16"></a> <FONT color="green">017</FONT> package org.apache.commons.math.analysis.interpolation;<a name="line.17"></a> <FONT color="green">018</FONT> <a name="line.18"></a> <FONT color="green">019</FONT> import java.io.Serializable;<a name="line.19"></a> <FONT color="green">020</FONT> import java.util.Arrays;<a name="line.20"></a> <FONT color="green">021</FONT> <a name="line.21"></a> <FONT color="green">022</FONT> import org.apache.commons.math.MathException;<a name="line.22"></a> <FONT color="green">023</FONT> import org.apache.commons.math.analysis.polynomials.PolynomialSplineFunction;<a name="line.23"></a> <FONT color="green">024</FONT> <a name="line.24"></a> <FONT color="green">025</FONT> /**<a name="line.25"></a> <FONT color="green">026</FONT> * Implements the <a href="http://en.wikipedia.org/wiki/Local_regression"><a name="line.26"></a> <FONT color="green">027</FONT> * Local Regression Algorithm</a> (also Loess, Lowess) for interpolation of<a name="line.27"></a> <FONT color="green">028</FONT> * real univariate functions.<a name="line.28"></a> <FONT color="green">029</FONT> * <p/><a name="line.29"></a> <FONT color="green">030</FONT> * For reference, see<a name="line.30"></a> <FONT color="green">031</FONT> * <a href="http://www.math.tau.ac.il/~yekutiel/MA seminar/Cleveland 1979.pdf"><a name="line.31"></a> <FONT color="green">032</FONT> * William S. Cleveland - Robust Locally Weighted Regression and Smoothing<a name="line.32"></a> <FONT color="green">033</FONT> * Scatterplots</a><a name="line.33"></a> <FONT color="green">034</FONT> * <p/><a name="line.34"></a> <FONT color="green">035</FONT> * This class implements both the loess method and serves as an interpolation<a name="line.35"></a> <FONT color="green">036</FONT> * adapter to it, allowing to build a spline on the obtained loess fit.<a name="line.36"></a> <FONT color="green">037</FONT> *<a name="line.37"></a> <FONT color="green">038</FONT> * @version $Revision: 925812 $ $Date: 2010-03-21 11:49:31 -0400 (Sun, 21 Mar 2010) $<a name="line.38"></a> <FONT color="green">039</FONT> * @since 2.0<a name="line.39"></a> <FONT color="green">040</FONT> */<a name="line.40"></a> <FONT color="green">041</FONT> public class LoessInterpolator<a name="line.41"></a> <FONT color="green">042</FONT> implements UnivariateRealInterpolator, Serializable {<a name="line.42"></a> <FONT color="green">043</FONT> <a name="line.43"></a> <FONT color="green">044</FONT> /** Default value of the bandwidth parameter. */<a name="line.44"></a> <FONT color="green">045</FONT> public static final double DEFAULT_BANDWIDTH = 0.3;<a name="line.45"></a> <FONT color="green">046</FONT> <a name="line.46"></a> <FONT color="green">047</FONT> /** Default value of the number of robustness iterations. */<a name="line.47"></a> <FONT color="green">048</FONT> public static final int DEFAULT_ROBUSTNESS_ITERS = 2;<a name="line.48"></a> <FONT color="green">049</FONT> <a name="line.49"></a> <FONT color="green">050</FONT> /**<a name="line.50"></a> <FONT color="green">051</FONT> * Default value for accuracy.<a name="line.51"></a> <FONT color="green">052</FONT> * @since 2.1<a name="line.52"></a> <FONT color="green">053</FONT> */<a name="line.53"></a> <FONT color="green">054</FONT> public static final double DEFAULT_ACCURACY = 1e-12;<a name="line.54"></a> <FONT color="green">055</FONT> <a name="line.55"></a> <FONT color="green">056</FONT> /** serializable version identifier. */<a name="line.56"></a> <FONT color="green">057</FONT> private static final long serialVersionUID = 5204927143605193821L;<a name="line.57"></a> <FONT color="green">058</FONT> <a name="line.58"></a> <FONT color="green">059</FONT> /**<a name="line.59"></a> <FONT color="green">060</FONT> * The bandwidth parameter: when computing the loess fit at<a name="line.60"></a> <FONT color="green">061</FONT> * a particular point, this fraction of source points closest<a name="line.61"></a> <FONT color="green">062</FONT> * to the current point is taken into account for computing<a name="line.62"></a> <FONT color="green">063</FONT> * a least-squares regression.<a name="line.63"></a> <FONT color="green">064</FONT> * <p/><a name="line.64"></a> <FONT color="green">065</FONT> * A sensible value is usually 0.25 to 0.5.<a name="line.65"></a> <FONT color="green">066</FONT> */<a name="line.66"></a> <FONT color="green">067</FONT> private final double bandwidth;<a name="line.67"></a> <FONT color="green">068</FONT> <a name="line.68"></a> <FONT color="green">069</FONT> /**<a name="line.69"></a> <FONT color="green">070</FONT> * The number of robustness iterations parameter: this many<a name="line.70"></a> <FONT color="green">071</FONT> * robustness iterations are done.<a name="line.71"></a> <FONT color="green">072</FONT> * <p/><a name="line.72"></a> <FONT color="green">073</FONT> * A sensible value is usually 0 (just the initial fit without any<a name="line.73"></a> <FONT color="green">074</FONT> * robustness iterations) to 4.<a name="line.74"></a> <FONT color="green">075</FONT> */<a name="line.75"></a> <FONT color="green">076</FONT> private final int robustnessIters;<a name="line.76"></a> <FONT color="green">077</FONT> <a name="line.77"></a> <FONT color="green">078</FONT> /**<a name="line.78"></a> <FONT color="green">079</FONT> * If the median residual at a certain robustness iteration<a name="line.79"></a> <FONT color="green">080</FONT> * is less than this amount, no more iterations are done.<a name="line.80"></a> <FONT color="green">081</FONT> */<a name="line.81"></a> <FONT color="green">082</FONT> private final double accuracy;<a name="line.82"></a> <FONT color="green">083</FONT> <a name="line.83"></a> <FONT color="green">084</FONT> /**<a name="line.84"></a> <FONT color="green">085</FONT> * Constructs a new {@link LoessInterpolator}<a name="line.85"></a> <FONT color="green">086</FONT> * with a bandwidth of {@link #DEFAULT_BANDWIDTH},<a name="line.86"></a> <FONT color="green">087</FONT> * {@link #DEFAULT_ROBUSTNESS_ITERS} robustness iterations<a name="line.87"></a> <FONT color="green">088</FONT> * and an accuracy of {#link #DEFAULT_ACCURACY}.<a name="line.88"></a> <FONT color="green">089</FONT> * See {@link #LoessInterpolator(double, int, double)} for an explanation of<a name="line.89"></a> <FONT color="green">090</FONT> * the parameters.<a name="line.90"></a> <FONT color="green">091</FONT> */<a name="line.91"></a> <FONT color="green">092</FONT> public LoessInterpolator() {<a name="line.92"></a> <FONT color="green">093</FONT> this.bandwidth = DEFAULT_BANDWIDTH;<a name="line.93"></a> <FONT color="green">094</FONT> this.robustnessIters = DEFAULT_ROBUSTNESS_ITERS;<a name="line.94"></a> <FONT color="green">095</FONT> this.accuracy = DEFAULT_ACCURACY;<a name="line.95"></a> <FONT color="green">096</FONT> }<a name="line.96"></a> <FONT color="green">097</FONT> <a name="line.97"></a> <FONT color="green">098</FONT> /**<a name="line.98"></a> <FONT color="green">099</FONT> * Constructs a new {@link LoessInterpolator}<a name="line.99"></a> <FONT color="green">100</FONT> * with given bandwidth and number of robustness iterations.<a name="line.100"></a> <FONT color="green">101</FONT> * <p><a name="line.101"></a> <FONT color="green">102</FONT> * Calling this constructor is equivalent to calling {link {@link<a name="line.102"></a> <FONT color="green">103</FONT> * #LoessInterpolator(double, int, double) LoessInterpolator(bandwidth,<a name="line.103"></a> <FONT color="green">104</FONT> * robustnessIters, LoessInterpolator.DEFAULT_ACCURACY)}<a name="line.104"></a> <FONT color="green">105</FONT> * </p><a name="line.105"></a> <FONT color="green">106</FONT> *<a name="line.106"></a> <FONT color="green">107</FONT> * @param bandwidth when computing the loess fit at<a name="line.107"></a> <FONT color="green">108</FONT> * a particular point, this fraction of source points closest<a name="line.108"></a> <FONT color="green">109</FONT> * to the current point is taken into account for computing<a name="line.109"></a> <FONT color="green">110</FONT> * a least-squares regression.</br><a name="line.110"></a> <FONT color="green">111</FONT> * A sensible value is usually 0.25 to 0.5, the default value is<a name="line.111"></a> <FONT color="green">112</FONT> * {@link #DEFAULT_BANDWIDTH}.<a name="line.112"></a> <FONT color="green">113</FONT> * @param robustnessIters This many robustness iterations are done.</br><a name="line.113"></a> <FONT color="green">114</FONT> * A sensible value is usually 0 (just the initial fit without any<a name="line.114"></a> <FONT color="green">115</FONT> * robustness iterations) to 4, the default value is<a name="line.115"></a> <FONT color="green">116</FONT> * {@link #DEFAULT_ROBUSTNESS_ITERS}.<a name="line.116"></a> <FONT color="green">117</FONT> * @throws MathException if bandwidth does not lie in the interval [0,1]<a name="line.117"></a> <FONT color="green">118</FONT> * or if robustnessIters is negative.<a name="line.118"></a> <FONT color="green">119</FONT> * @see #LoessInterpolator(double, int, double)<a name="line.119"></a> <FONT color="green">120</FONT> */<a name="line.120"></a> <FONT color="green">121</FONT> public LoessInterpolator(double bandwidth, int robustnessIters) throws MathException {<a name="line.121"></a> <FONT color="green">122</FONT> this(bandwidth, robustnessIters, DEFAULT_ACCURACY);<a name="line.122"></a> <FONT color="green">123</FONT> }<a name="line.123"></a> <FONT color="green">124</FONT> <a name="line.124"></a> <FONT color="green">125</FONT> /**<a name="line.125"></a> <FONT color="green">126</FONT> * Constructs a new {@link LoessInterpolator}<a name="line.126"></a> <FONT color="green">127</FONT> * with given bandwidth, number of robustness iterations and accuracy.<a name="line.127"></a> <FONT color="green">128</FONT> *<a name="line.128"></a> <FONT color="green">129</FONT> * @param bandwidth when computing the loess fit at<a name="line.129"></a> <FONT color="green">130</FONT> * a particular point, this fraction of source points closest<a name="line.130"></a> <FONT color="green">131</FONT> * to the current point is taken into account for computing<a name="line.131"></a> <FONT color="green">132</FONT> * a least-squares regression.</br><a name="line.132"></a> <FONT color="green">133</FONT> * A sensible value is usually 0.25 to 0.5, the default value is<a name="line.133"></a> <FONT color="green">134</FONT> * {@link #DEFAULT_BANDWIDTH}.<a name="line.134"></a> <FONT color="green">135</FONT> * @param robustnessIters This many robustness iterations are done.</br><a name="line.135"></a> <FONT color="green">136</FONT> * A sensible value is usually 0 (just the initial fit without any<a name="line.136"></a> <FONT color="green">137</FONT> * robustness iterations) to 4, the default value is<a name="line.137"></a> <FONT color="green">138</FONT> * {@link #DEFAULT_ROBUSTNESS_ITERS}.<a name="line.138"></a> <FONT color="green">139</FONT> * @param accuracy If the median residual at a certain robustness iteration<a name="line.139"></a> <FONT color="green">140</FONT> * is less than this amount, no more iterations are done.<a name="line.140"></a> <FONT color="green">141</FONT> * @throws MathException if bandwidth does not lie in the interval [0,1]<a name="line.141"></a> <FONT color="green">142</FONT> * or if robustnessIters is negative.<a name="line.142"></a> <FONT color="green">143</FONT> * @see #LoessInterpolator(double, int)<a name="line.143"></a> <FONT color="green">144</FONT> * @since 2.1<a name="line.144"></a> <FONT color="green">145</FONT> */<a name="line.145"></a> <FONT color="green">146</FONT> public LoessInterpolator(double bandwidth, int robustnessIters, double accuracy) throws MathException {<a name="line.146"></a> <FONT color="green">147</FONT> if (bandwidth < 0 || bandwidth > 1) {<a name="line.147"></a> <FONT color="green">148</FONT> throw new MathException("bandwidth must be in the interval [0,1], but got {0}",<a name="line.148"></a> <FONT color="green">149</FONT> bandwidth);<a name="line.149"></a> <FONT color="green">150</FONT> }<a name="line.150"></a> <FONT color="green">151</FONT> this.bandwidth = bandwidth;<a name="line.151"></a> <FONT color="green">152</FONT> if (robustnessIters < 0) {<a name="line.152"></a> <FONT color="green">153</FONT> throw new MathException("the number of robustness iterations must " +<a name="line.153"></a> <FONT color="green">154</FONT> "be non-negative, but got {0}",<a name="line.154"></a> <FONT color="green">155</FONT> robustnessIters);<a name="line.155"></a> <FONT color="green">156</FONT> }<a name="line.156"></a> <FONT color="green">157</FONT> this.robustnessIters = robustnessIters;<a name="line.157"></a> <FONT color="green">158</FONT> this.accuracy = accuracy;<a name="line.158"></a> <FONT color="green">159</FONT> }<a name="line.159"></a> <FONT color="green">160</FONT> <a name="line.160"></a> <FONT color="green">161</FONT> /**<a name="line.161"></a> <FONT color="green">162</FONT> * Compute an interpolating function by performing a loess fit<a name="line.162"></a> <FONT color="green">163</FONT> * on the data at the original abscissae and then building a cubic spline<a name="line.163"></a> <FONT color="green">164</FONT> * with a<a name="line.164"></a> <FONT color="green">165</FONT> * {@link org.apache.commons.math.analysis.interpolation.SplineInterpolator}<a name="line.165"></a> <FONT color="green">166</FONT> * on the resulting fit.<a name="line.166"></a> <FONT color="green">167</FONT> *<a name="line.167"></a> <FONT color="green">168</FONT> * @param xval the arguments for the interpolation points<a name="line.168"></a> <FONT color="green">169</FONT> * @param yval the values for the interpolation points<a name="line.169"></a> <FONT color="green">170</FONT> * @return A cubic spline built upon a loess fit to the data at the original abscissae<a name="line.170"></a> <FONT color="green">171</FONT> * @throws MathException if some of the following conditions are false:<a name="line.171"></a> <FONT color="green">172</FONT> * <ul><a name="line.172"></a> <FONT color="green">173</FONT> * <li> Arguments and values are of the same size that is greater than zero</li><a name="line.173"></a> <FONT color="green">174</FONT> * <li> The arguments are in a strictly increasing order</li><a name="line.174"></a> <FONT color="green">175</FONT> * <li> All arguments and values are finite real numbers</li><a name="line.175"></a> <FONT color="green">176</FONT> * </ul><a name="line.176"></a> <FONT color="green">177</FONT> */<a name="line.177"></a> <FONT color="green">178</FONT> public final PolynomialSplineFunction interpolate(<a name="line.178"></a> <FONT color="green">179</FONT> final double[] xval, final double[] yval) throws MathException {<a name="line.179"></a> <FONT color="green">180</FONT> return new SplineInterpolator().interpolate(xval, smooth(xval, yval));<a name="line.180"></a> <FONT color="green">181</FONT> }<a name="line.181"></a> <FONT color="green">182</FONT> <a name="line.182"></a> <FONT color="green">183</FONT> /**<a name="line.183"></a> <FONT color="green">184</FONT> * Compute a weighted loess fit on the data at the original abscissae.<a name="line.184"></a> <FONT color="green">185</FONT> *<a name="line.185"></a> <FONT color="green">186</FONT> * @param xval the arguments for the interpolation points<a name="line.186"></a> <FONT color="green">187</FONT> * @param yval the values for the interpolation points<a name="line.187"></a> <FONT color="green">188</FONT> * @param weights point weights: coefficients by which the robustness weight of a point is multiplied<a name="line.188"></a> <FONT color="green">189</FONT> * @return values of the loess fit at corresponding original abscissae<a name="line.189"></a> <FONT color="green">190</FONT> * @throws MathException if some of the following conditions are false:<a name="line.190"></a> <FONT color="green">191</FONT> * <ul><a name="line.191"></a> <FONT color="green">192</FONT> * <li> Arguments and values are of the same size that is greater than zero</li><a name="line.192"></a> <FONT color="green">193</FONT> * <li> The arguments are in a strictly increasing order</li><a name="line.193"></a> <FONT color="green">194</FONT> * <li> All arguments and values are finite real numbers</li><a name="line.194"></a> <FONT color="green">195</FONT> * </ul><a name="line.195"></a> <FONT color="green">196</FONT> * @since 2.1<a name="line.196"></a> <FONT color="green">197</FONT> */<a name="line.197"></a> <FONT color="green">198</FONT> public final double[] smooth(final double[] xval, final double[] yval, final double[] weights)<a name="line.198"></a> <FONT color="green">199</FONT> throws MathException {<a name="line.199"></a> <FONT color="green">200</FONT> if (xval.length != yval.length) {<a name="line.200"></a> <FONT color="green">201</FONT> throw new MathException(<a name="line.201"></a> <FONT color="green">202</FONT> "Loess expects the abscissa and ordinate arrays " +<a name="line.202"></a> <FONT color="green">203</FONT> "to be of the same size, " +<a name="line.203"></a> <FONT color="green">204</FONT> "but got {0} abscissae and {1} ordinatae",<a name="line.204"></a> <FONT color="green">205</FONT> xval.length, yval.length);<a name="line.205"></a> <FONT color="green">206</FONT> }<a name="line.206"></a> <FONT color="green">207</FONT> <a name="line.207"></a> <FONT color="green">208</FONT> final int n = xval.length;<a name="line.208"></a> <FONT color="green">209</FONT> <a name="line.209"></a> <FONT color="green">210</FONT> if (n == 0) {<a name="line.210"></a> <FONT color="green">211</FONT> throw new MathException("Loess expects at least 1 point");<a name="line.211"></a> <FONT color="green">212</FONT> }<a name="line.212"></a> <FONT color="green">213</FONT> <a name="line.213"></a> <FONT color="green">214</FONT> checkAllFiniteReal(xval, "all abscissae must be finite real numbers, but {0}-th is {1}");<a name="line.214"></a> <FONT color="green">215</FONT> checkAllFiniteReal(yval, "all ordinatae must be finite real numbers, but {0}-th is {1}");<a name="line.215"></a> <FONT color="green">216</FONT> checkAllFiniteReal(weights, "all weights must be finite real numbers, but {0}-th is {1}");<a name="line.216"></a> <FONT color="green">217</FONT> <a name="line.217"></a> <FONT color="green">218</FONT> checkStrictlyIncreasing(xval);<a name="line.218"></a> <FONT color="green">219</FONT> <a name="line.219"></a> <FONT color="green">220</FONT> if (n == 1) {<a name="line.220"></a> <FONT color="green">221</FONT> return new double[]{yval[0]};<a name="line.221"></a> <FONT color="green">222</FONT> }<a name="line.222"></a> <FONT color="green">223</FONT> <a name="line.223"></a> <FONT color="green">224</FONT> if (n == 2) {<a name="line.224"></a> <FONT color="green">225</FONT> return new double[]{yval[0], yval[1]};<a name="line.225"></a> <FONT color="green">226</FONT> }<a name="line.226"></a> <FONT color="green">227</FONT> <a name="line.227"></a> <FONT color="green">228</FONT> int bandwidthInPoints = (int) (bandwidth * n);<a name="line.228"></a> <FONT color="green">229</FONT> <a name="line.229"></a> <FONT color="green">230</FONT> if (bandwidthInPoints < 2) {<a name="line.230"></a> <FONT color="green">231</FONT> throw new MathException(<a name="line.231"></a> <FONT color="green">232</FONT> "the bandwidth must be large enough to " +<a name="line.232"></a> <FONT color="green">233</FONT> "accomodate at least 2 points. There are {0} " +<a name="line.233"></a> <FONT color="green">234</FONT> " data points, and bandwidth must be at least {1} " +<a name="line.234"></a> <FONT color="green">235</FONT> " but it is only {2}",<a name="line.235"></a> <FONT color="green">236</FONT> n, 2.0 / n, bandwidth);<a name="line.236"></a> <FONT color="green">237</FONT> }<a name="line.237"></a> <FONT color="green">238</FONT> <a name="line.238"></a> <FONT color="green">239</FONT> final double[] res = new double[n];<a name="line.239"></a> <FONT color="green">240</FONT> <a name="line.240"></a> <FONT color="green">241</FONT> final double[] residuals = new double[n];<a name="line.241"></a> <FONT color="green">242</FONT> final double[] sortedResiduals = new double[n];<a name="line.242"></a> <FONT color="green">243</FONT> <a name="line.243"></a> <FONT color="green">244</FONT> final double[] robustnessWeights = new double[n];<a name="line.244"></a> <FONT color="green">245</FONT> <a name="line.245"></a> <FONT color="green">246</FONT> // Do an initial fit and 'robustnessIters' robustness iterations.<a name="line.246"></a> <FONT color="green">247</FONT> // This is equivalent to doing 'robustnessIters+1' robustness iterations<a name="line.247"></a> <FONT color="green">248</FONT> // starting with all robustness weights set to 1.<a name="line.248"></a> <FONT color="green">249</FONT> Arrays.fill(robustnessWeights, 1);<a name="line.249"></a> <FONT color="green">250</FONT> <a name="line.250"></a> <FONT color="green">251</FONT> for (int iter = 0; iter <= robustnessIters; ++iter) {<a name="line.251"></a> <FONT color="green">252</FONT> final int[] bandwidthInterval = {0, bandwidthInPoints - 1};<a name="line.252"></a> <FONT color="green">253</FONT> // At each x, compute a local weighted linear regression<a name="line.253"></a> <FONT color="green">254</FONT> for (int i = 0; i < n; ++i) {<a name="line.254"></a> <FONT color="green">255</FONT> final double x = xval[i];<a name="line.255"></a> <FONT color="green">256</FONT> <a name="line.256"></a> <FONT color="green">257</FONT> // Find out the interval of source points on which<a name="line.257"></a> <FONT color="green">258</FONT> // a regression is to be made.<a name="line.258"></a> <FONT color="green">259</FONT> if (i > 0) {<a name="line.259"></a> <FONT color="green">260</FONT> updateBandwidthInterval(xval, weights, i, bandwidthInterval);<a name="line.260"></a> <FONT color="green">261</FONT> }<a name="line.261"></a> <FONT color="green">262</FONT> <a name="line.262"></a> <FONT color="green">263</FONT> final int ileft = bandwidthInterval[0];<a name="line.263"></a> <FONT color="green">264</FONT> final int iright = bandwidthInterval[1];<a name="line.264"></a> <FONT color="green">265</FONT> <a name="line.265"></a> <FONT color="green">266</FONT> // Compute the point of the bandwidth interval that is<a name="line.266"></a> <FONT color="green">267</FONT> // farthest from x<a name="line.267"></a> <FONT color="green">268</FONT> final int edge;<a name="line.268"></a> <FONT color="green">269</FONT> if (xval[i] - xval[ileft] > xval[iright] - xval[i]) {<a name="line.269"></a> <FONT color="green">270</FONT> edge = ileft;<a name="line.270"></a> <FONT color="green">271</FONT> } else {<a name="line.271"></a> <FONT color="green">272</FONT> edge = iright;<a name="line.272"></a> <FONT color="green">273</FONT> }<a name="line.273"></a> <FONT color="green">274</FONT> <a name="line.274"></a> <FONT color="green">275</FONT> // Compute a least-squares linear fit weighted by<a name="line.275"></a> <FONT color="green">276</FONT> // the product of robustness weights and the tricube<a name="line.276"></a> <FONT color="green">277</FONT> // weight function.<a name="line.277"></a> <FONT color="green">278</FONT> // See http://en.wikipedia.org/wiki/Linear_regression<a name="line.278"></a> <FONT color="green">279</FONT> // (section "Univariate linear case")<a name="line.279"></a> <FONT color="green">280</FONT> // and http://en.wikipedia.org/wiki/Weighted_least_squares<a name="line.280"></a> <FONT color="green">281</FONT> // (section "Weighted least squares")<a name="line.281"></a> <FONT color="green">282</FONT> double sumWeights = 0;<a name="line.282"></a> <FONT color="green">283</FONT> double sumX = 0;<a name="line.283"></a> <FONT color="green">284</FONT> double sumXSquared = 0;<a name="line.284"></a> <FONT color="green">285</FONT> double sumY = 0;<a name="line.285"></a> <FONT color="green">286</FONT> double sumXY = 0;<a name="line.286"></a> <FONT color="green">287</FONT> double denom = Math.abs(1.0 / (xval[edge] - x));<a name="line.287"></a> <FONT color="green">288</FONT> for (int k = ileft; k <= iright; ++k) {<a name="line.288"></a> <FONT color="green">289</FONT> final double xk = xval[k];<a name="line.289"></a> <FONT color="green">290</FONT> final double yk = yval[k];<a name="line.290"></a> <FONT color="green">291</FONT> final double dist = (k < i) ? x - xk : xk - x;<a name="line.291"></a> <FONT color="green">292</FONT> final double w = tricube(dist * denom) * robustnessWeights[k] * weights[k];<a name="line.292"></a> <FONT color="green">293</FONT> final double xkw = xk * w;<a name="line.293"></a> <FONT color="green">294</FONT> sumWeights += w;<a name="line.294"></a> <FONT color="green">295</FONT> sumX += xkw;<a name="line.295"></a> <FONT color="green">296</FONT> sumXSquared += xk * xkw;<a name="line.296"></a> <FONT color="green">297</FONT> sumY += yk * w;<a name="line.297"></a> <FONT color="green">298</FONT> sumXY += yk * xkw;<a name="line.298"></a> <FONT color="green">299</FONT> }<a name="line.299"></a> <FONT color="green">300</FONT> <a name="line.300"></a> <FONT color="green">301</FONT> final double meanX = sumX / sumWeights;<a name="line.301"></a> <FONT color="green">302</FONT> final double meanY = sumY / sumWeights;<a name="line.302"></a> <FONT color="green">303</FONT> final double meanXY = sumXY / sumWeights;<a name="line.303"></a> <FONT color="green">304</FONT> final double meanXSquared = sumXSquared / sumWeights;<a name="line.304"></a> <FONT color="green">305</FONT> <a name="line.305"></a> <FONT color="green">306</FONT> final double beta;<a name="line.306"></a> <FONT color="green">307</FONT> if (Math.sqrt(Math.abs(meanXSquared - meanX * meanX)) < accuracy) {<a name="line.307"></a> <FONT color="green">308</FONT> beta = 0;<a name="line.308"></a> <FONT color="green">309</FONT> } else {<a name="line.309"></a> <FONT color="green">310</FONT> beta = (meanXY - meanX * meanY) / (meanXSquared - meanX * meanX);<a name="line.310"></a> <FONT color="green">311</FONT> }<a name="line.311"></a> <FONT color="green">312</FONT> <a name="line.312"></a> <FONT color="green">313</FONT> final double alpha = meanY - beta * meanX;<a name="line.313"></a> <FONT color="green">314</FONT> <a name="line.314"></a> <FONT color="green">315</FONT> res[i] = beta * x + alpha;<a name="line.315"></a> <FONT color="green">316</FONT> residuals[i] = Math.abs(yval[i] - res[i]);<a name="line.316"></a> <FONT color="green">317</FONT> }<a name="line.317"></a> <FONT color="green">318</FONT> <a name="line.318"></a> <FONT color="green">319</FONT> // No need to recompute the robustness weights at the last<a name="line.319"></a> <FONT color="green">320</FONT> // iteration, they won't be needed anymore<a name="line.320"></a> <FONT color="green">321</FONT> if (iter == robustnessIters) {<a name="line.321"></a> <FONT color="green">322</FONT> break;<a name="line.322"></a> <FONT color="green">323</FONT> }<a name="line.323"></a> <FONT color="green">324</FONT> <a name="line.324"></a> <FONT color="green">325</FONT> // Recompute the robustness weights.<a name="line.325"></a> <FONT color="green">326</FONT> <a name="line.326"></a> <FONT color="green">327</FONT> // Find the median residual.<a name="line.327"></a> <FONT color="green">328</FONT> // An arraycopy and a sort are completely tractable here,<a name="line.328"></a> <FONT color="green">329</FONT> // because the preceding loop is a lot more expensive<a name="line.329"></a> <FONT color="green">330</FONT> System.arraycopy(residuals, 0, sortedResiduals, 0, n);<a name="line.330"></a> <FONT color="green">331</FONT> Arrays.sort(sortedResiduals);<a name="line.331"></a> <FONT color="green">332</FONT> final double medianResidual = sortedResiduals[n / 2];<a name="line.332"></a> <FONT color="green">333</FONT> <a name="line.333"></a> <FONT color="green">334</FONT> if (Math.abs(medianResidual) < accuracy) {<a name="line.334"></a> <FONT color="green">335</FONT> break;<a name="line.335"></a> <FONT color="green">336</FONT> }<a name="line.336"></a> <FONT color="green">337</FONT> <a name="line.337"></a> <FONT color="green">338</FONT> for (int i = 0; i < n; ++i) {<a name="line.338"></a> <FONT color="green">339</FONT> final double arg = residuals[i] / (6 * medianResidual);<a name="line.339"></a> <FONT color="green">340</FONT> if (arg >= 1) {<a name="line.340"></a> <FONT color="green">341</FONT> robustnessWeights[i] = 0;<a name="line.341"></a> <FONT color="green">342</FONT> } else {<a name="line.342"></a> <FONT color="green">343</FONT> final double w = 1 - arg * arg;<a name="line.343"></a> <FONT color="green">344</FONT> robustnessWeights[i] = w * w;<a name="line.344"></a> <FONT color="green">345</FONT> }<a name="line.345"></a> <FONT color="green">346</FONT> }<a name="line.346"></a> <FONT color="green">347</FONT> }<a name="line.347"></a> <FONT color="green">348</FONT> <a name="line.348"></a> <FONT color="green">349</FONT> return res;<a name="line.349"></a> <FONT color="green">350</FONT> }<a name="line.350"></a> <FONT color="green">351</FONT> <a name="line.351"></a> <FONT color="green">352</FONT> /**<a name="line.352"></a> <FONT color="green">353</FONT> * Compute a loess fit on the data at the original abscissae.<a name="line.353"></a> <FONT color="green">354</FONT> *<a name="line.354"></a> <FONT color="green">355</FONT> * @param xval the arguments for the interpolation points<a name="line.355"></a> <FONT color="green">356</FONT> * @param yval the values for the interpolation points<a name="line.356"></a> <FONT color="green">357</FONT> * @return values of the loess fit at corresponding original abscissae<a name="line.357"></a> <FONT color="green">358</FONT> * @throws MathException if some of the following conditions are false:<a name="line.358"></a> <FONT color="green">359</FONT> * <ul><a name="line.359"></a> <FONT color="green">360</FONT> * <li> Arguments and values are of the same size that is greater than zero</li><a name="line.360"></a> <FONT color="green">361</FONT> * <li> The arguments are in a strictly increasing order</li><a name="line.361"></a> <FONT color="green">362</FONT> * <li> All arguments and values are finite real numbers</li><a name="line.362"></a> <FONT color="green">363</FONT> * </ul><a name="line.363"></a> <FONT color="green">364</FONT> */<a name="line.364"></a> <FONT color="green">365</FONT> public final double[] smooth(final double[] xval, final double[] yval)<a name="line.365"></a> <FONT color="green">366</FONT> throws MathException {<a name="line.366"></a> <FONT color="green">367</FONT> if (xval.length != yval.length) {<a name="line.367"></a> <FONT color="green">368</FONT> throw new MathException(<a name="line.368"></a> <FONT color="green">369</FONT> "Loess expects the abscissa and ordinate arrays " +<a name="line.369"></a> <FONT color="green">370</FONT> "to be of the same size, " +<a name="line.370"></a> <FONT color="green">371</FONT> "but got {0} abscissae and {1} ordinatae",<a name="line.371"></a> <FONT color="green">372</FONT> xval.length, yval.length);<a name="line.372"></a> <FONT color="green">373</FONT> }<a name="line.373"></a> <FONT color="green">374</FONT> <a name="line.374"></a> <FONT color="green">375</FONT> final double[] unitWeights = new double[xval.length];<a name="line.375"></a> <FONT color="green">376</FONT> Arrays.fill(unitWeights, 1.0);<a name="line.376"></a> <FONT color="green">377</FONT> <a name="line.377"></a> <FONT color="green">378</FONT> return smooth(xval, yval, unitWeights);<a name="line.378"></a> <FONT color="green">379</FONT> }<a name="line.379"></a> <FONT color="green">380</FONT> <a name="line.380"></a> <FONT color="green">381</FONT> /**<a name="line.381"></a> <FONT color="green">382</FONT> * Given an index interval into xval that embraces a certain number of<a name="line.382"></a> <FONT color="green">383</FONT> * points closest to xval[i-1], update the interval so that it embraces<a name="line.383"></a> <FONT color="green">384</FONT> * the same number of points closest to xval[i], ignoring zero weights.<a name="line.384"></a> <FONT color="green">385</FONT> *<a name="line.385"></a> <FONT color="green">386</FONT> * @param xval arguments array<a name="line.386"></a> <FONT color="green">387</FONT> * @param weights weights array<a name="line.387"></a> <FONT color="green">388</FONT> * @param i the index around which the new interval should be computed<a name="line.388"></a> <FONT color="green">389</FONT> * @param bandwidthInterval a two-element array {left, right} such that: <p/><a name="line.389"></a> <FONT color="green">390</FONT> * <tt>(left==0 or xval[i] - xval[left-1] > xval[right] - xval[i])</tt><a name="line.390"></a> <FONT color="green">391</FONT> * <p/> and also <p/><a name="line.391"></a> <FONT color="green">392</FONT> * <tt>(right==xval.length-1 or xval[right+1] - xval[i] > xval[i] - xval[left])</tt>.<a name="line.392"></a> <FONT color="green">393</FONT> * The array will be updated.<a name="line.393"></a> <FONT color="green">394</FONT> */<a name="line.394"></a> <FONT color="green">395</FONT> private static void updateBandwidthInterval(final double[] xval, final double[] weights,<a name="line.395"></a> <FONT color="green">396</FONT> final int i,<a name="line.396"></a> <FONT color="green">397</FONT> final int[] bandwidthInterval) {<a name="line.397"></a> <FONT color="green">398</FONT> final int left = bandwidthInterval[0];<a name="line.398"></a> <FONT color="green">399</FONT> final int right = bandwidthInterval[1];<a name="line.399"></a> <FONT color="green">400</FONT> <a name="line.400"></a> <FONT color="green">401</FONT> // The right edge should be adjusted if the next point to the right<a name="line.401"></a> <FONT color="green">402</FONT> // is closer to xval[i] than the leftmost point of the current interval<a name="line.402"></a> <FONT color="green">403</FONT> int nextRight = nextNonzero(weights, right);<a name="line.403"></a> <FONT color="green">404</FONT> if (nextRight < xval.length && xval[nextRight] - xval[i] < xval[i] - xval[left]) {<a name="line.404"></a> <FONT color="green">405</FONT> int nextLeft = nextNonzero(weights, bandwidthInterval[0]);<a name="line.405"></a> <FONT color="green">406</FONT> bandwidthInterval[0] = nextLeft;<a name="line.406"></a> <FONT color="green">407</FONT> bandwidthInterval[1] = nextRight;<a name="line.407"></a> <FONT color="green">408</FONT> }<a name="line.408"></a> <FONT color="green">409</FONT> }<a name="line.409"></a> <FONT color="green">410</FONT> <a name="line.410"></a> <FONT color="green">411</FONT> /**<a name="line.411"></a> <FONT color="green">412</FONT> * Returns the smallest index j such that j > i && (j==weights.length || weights[j] != 0)<a name="line.412"></a> <FONT color="green">413</FONT> * @param weights weights array<a name="line.413"></a> <FONT color="green">414</FONT> * @param i the index from which to start search; must be < weights.length<a name="line.414"></a> <FONT color="green">415</FONT> * @return the smallest index j such that j > i && (j==weights.length || weights[j] != 0)<a name="line.415"></a> <FONT color="green">416</FONT> */<a name="line.416"></a> <FONT color="green">417</FONT> private static int nextNonzero(final double[] weights, final int i) {<a name="line.417"></a> <FONT color="green">418</FONT> int j = i + 1;<a name="line.418"></a> <FONT color="green">419</FONT> while(j < weights.length && weights[j] == 0) {<a name="line.419"></a> <FONT color="green">420</FONT> j++;<a name="line.420"></a> <FONT color="green">421</FONT> }<a name="line.421"></a> <FONT color="green">422</FONT> return j;<a name="line.422"></a> <FONT color="green">423</FONT> }<a name="line.423"></a> <FONT color="green">424</FONT> <a name="line.424"></a> <FONT color="green">425</FONT> /**<a name="line.425"></a> <FONT color="green">426</FONT> * Compute the<a name="line.426"></a> <FONT color="green">427</FONT> * <a href="http://en.wikipedia.org/wiki/Local_regression#Weight_function">tricube</a><a name="line.427"></a> <FONT color="green">428</FONT> * weight function<a name="line.428"></a> <FONT color="green">429</FONT> *<a name="line.429"></a> <FONT color="green">430</FONT> * @param x the argument<a name="line.430"></a> <FONT color="green">431</FONT> * @return (1-|x|^3)^3<a name="line.431"></a> <FONT color="green">432</FONT> */<a name="line.432"></a> <FONT color="green">433</FONT> private static double tricube(final double x) {<a name="line.433"></a> <FONT color="green">434</FONT> final double tmp = 1 - x * x * x;<a name="line.434"></a> <FONT color="green">435</FONT> return tmp * tmp * tmp;<a name="line.435"></a> <FONT color="green">436</FONT> }<a name="line.436"></a> <FONT color="green">437</FONT> <a name="line.437"></a> <FONT color="green">438</FONT> /**<a name="line.438"></a> <FONT color="green">439</FONT> * Check that all elements of an array are finite real numbers.<a name="line.439"></a> <FONT color="green">440</FONT> *<a name="line.440"></a> <FONT color="green">441</FONT> * @param values the values array<a name="line.441"></a> <FONT color="green">442</FONT> * @param pattern pattern of the error message<a name="line.442"></a> <FONT color="green">443</FONT> * @throws MathException if one of the values is not a finite real number<a name="line.443"></a> <FONT color="green">444</FONT> */<a name="line.444"></a> <FONT color="green">445</FONT> private static void checkAllFiniteReal(final double[] values, final String pattern)<a name="line.445"></a> <FONT color="green">446</FONT> throws MathException {<a name="line.446"></a> <FONT color="green">447</FONT> for (int i = 0; i < values.length; i++) {<a name="line.447"></a> <FONT color="green">448</FONT> final double x = values[i];<a name="line.448"></a> <FONT color="green">449</FONT> if (Double.isInfinite(x) || Double.isNaN(x)) {<a name="line.449"></a> <FONT color="green">450</FONT> throw new MathException(pattern, i, x);<a name="line.450"></a> <FONT color="green">451</FONT> }<a name="line.451"></a> <FONT color="green">452</FONT> }<a name="line.452"></a> <FONT color="green">453</FONT> }<a name="line.453"></a> <FONT color="green">454</FONT> <a name="line.454"></a> <FONT color="green">455</FONT> /**<a name="line.455"></a> <FONT color="green">456</FONT> * Check that elements of the abscissae array are in a strictly<a name="line.456"></a> <FONT color="green">457</FONT> * increasing order.<a name="line.457"></a> <FONT color="green">458</FONT> *<a name="line.458"></a> <FONT color="green">459</FONT> * @param xval the abscissae array<a name="line.459"></a> <FONT color="green">460</FONT> * @throws MathException if the abscissae array<a name="line.460"></a> <FONT color="green">461</FONT> * is not in a strictly increasing order<a name="line.461"></a> <FONT color="green">462</FONT> */<a name="line.462"></a> <FONT color="green">463</FONT> private static void checkStrictlyIncreasing(final double[] xval)<a name="line.463"></a> <FONT color="green">464</FONT> throws MathException {<a name="line.464"></a> <FONT color="green">465</FONT> for (int i = 0; i < xval.length; ++i) {<a name="line.465"></a> <FONT color="green">466</FONT> if (i >= 1 && xval[i - 1] >= xval[i]) {<a name="line.466"></a> <FONT color="green">467</FONT> throw new MathException(<a name="line.467"></a> <FONT color="green">468</FONT> "the abscissae array must be sorted in a strictly " +<a name="line.468"></a> <FONT color="green">469</FONT> "increasing order, but the {0}-th element is {1} " +<a name="line.469"></a> <FONT color="green">470</FONT> "whereas {2}-th is {3}",<a name="line.470"></a> <FONT color="green">471</FONT> i - 1, xval[i - 1], i, xval[i]);<a name="line.471"></a> <FONT color="green">472</FONT> }<a name="line.472"></a> <FONT color="green">473</FONT> }<a name="line.473"></a> <FONT color="green">474</FONT> }<a name="line.474"></a> <FONT color="green">475</FONT> }<a name="line.475"></a> </PRE> </BODY> </HTML>