本讲主要说下逻辑回归的相关问题和具体的实现方法

1. 什么是逻辑回归

逻辑回归是线性回归的一种，那么什么是回归，什么是线性回归

回归指的是公式已知，对公式中的未知参数进行估计，注意公式必须是已知的，否则是没有办法进行回归的

线性回归指的是回归中的公式是一次的，例如z=ax+by

逻辑回归其实就是在线性回归的基础上套了一个sigmoid函数，具体的样子如下

2. 正则化项

3. 最小二乘法和最大似然法

4. java实现梯度下降法

实验：

样本：

-0.017612	14.053064	0
-1.395634	4.662541	1
-0.752157	6.538620	0
-1.322371	7.152853	0
0.423363	11.054677	0
0.406704	7.067335	1
0.667394	12.741452	0
-2.460150	6.866805	1
0.569411	9.548755	0
-0.026632	10.427743	0
0.850433	6.920334	1
1.347183	13.175500	0
1.176813	3.167020	1
-1.781871	9.097953	0
-0.566606	5.749003	1
0.931635	1.589505	1
-0.024205	6.151823	1
-0.036453	2.690988	1
-0.196949	0.444165	1
1.014459	5.754399	1
1.985298	3.230619	1
-1.693453	-0.557540	1
-0.576525	11.778922	0
-0.346811	-1.678730	1
-2.124484	2.672471	1
1.217916	9.597015	0
-0.733928	9.098687	0
-3.642001	-1.618087	1
0.315985	3.523953	1
1.416614	9.619232	0
-0.386323	3.989286	1
0.556921	8.294984	1
1.224863	11.587360	0
-1.347803	-2.406051	1
1.196604	4.951851	1
0.275221	9.543647	0
0.470575	9.332488	0
-1.889567	9.542662	0
-1.527893	12.150579	0
-1.185247	11.309318	0
-0.445678	3.297303	1
1.042222	6.105155	1
-0.618787	10.320986	0
1.152083	0.548467	1
0.828534	2.676045	1
-1.237728	10.549033	0
-0.683565	-2.166125	1
0.229456	5.921938	1
-0.959885	11.555336	0
0.492911	10.993324	0
0.184992	8.721488	0
-0.355715	10.325976	0
-0.397822	8.058397	0
0.824839	13.730343	0
1.507278	5.027866	1
0.099671	6.835839	1
-0.344008	10.717485	0
1.785928	7.718645	1
-0.918801	11.560217	0
-0.364009	4.747300	1
-0.841722	4.119083	1
0.490426	1.960539	1
-0.007194	9.075792	0
0.356107	12.447863	0
0.342578	12.281162	0
-0.810823	-1.466018	1
2.530777	6.476801	1
1.296683	11.607559	0
0.475487	12.040035	0
-0.783277	11.009725	0
0.074798	11.023650	0
-1.337472	0.468339	1
-0.102781	13.763651	0
-0.147324	2.874846	1
0.518389	9.887035	0
1.015399	7.571882	0
-1.658086	-0.027255	1
1.319944	2.171228	1
2.056216	5.019981	1
-0.851633	4.375691	1
-1.510047	6.061992	0
-1.076637	-3.181888	1
1.821096	10.283990	0
3.010150	8.401766	1
-1.099458	1.688274	1
-0.834872	-1.733869	1
-0.846637	3.849075	1
1.400102	12.628781	0
1.752842	5.468166	1
0.078557	0.059736	1
0.089392	-0.715300	1
1.825662	12.693808	0
0.197445	9.744638	0
0.126117	0.922311	1
-0.679797	1.220530	1
0.677983	2.556666	1
0.761349	10.693862	0
-2.168791	0.143632	1
1.388610	9.341997	0
0.317029	14.739025	0

public class LogRegression {

	public static void main(String[] args) {
		
		LogRegression lr = new LogRegression();
		Instances instances = new Instances();
		lr.train(instances, 0.01f, 200, (short)1);
	}
	
	public void train(Instances instances, float step, int maxIt, short algorithm) {
		
		float[][] datas = instances.datas;
		float[] labels = instances.labels;
		int size = datas.length;
		int dim = datas[0].length;
		float[] w = new float[dim];
		
		for(int i = 0; i < dim; i++) {
			w[i] = 1;
		}
		
		switch(algorithm){
		//批量梯度下降
		case 1: 
			for(int i = 0; i < maxIt; i++) {
				//求输出
				float[] out = new float[size];
				for(int s = 0; s < size; s++) {
					float lire = innerProduct(w, datas[s]);
					out[s] = sigmoid(lire);
				}
				for(int d = 0; d < dim; d++) {
					float sum = 0;
					for(int s = 0; s < size; s++) {
						sum  += (labels[s] - out[s]) * datas[s][d];
					}
					w[d] = w[d] + step * sum;
				}
				System.out.println(Arrays.toString(w));
			}
			break;
		//随机梯度下降
		case 2: 
			for(int i = 0; i < maxIt; i++) {
				for(int s = 0; s < size; s++) {
					float lire = innerProduct(w, datas[s]);
					float out = sigmoid(lire);
					float error = labels[s] - out;
					for(int d = 0; d < dim; d++) {
						w[d] += step * error * datas[s][d];
					}
				}
				System.out.println(Arrays.toString(w));
			}
			break;
		}
	}
	
	private float innerProduct(float[] w, float[] x) {
		
		float sum = 0;
		for(int i = 0; i < w.length; i++) {
			sum += w[i] * x[i];
		}
		
		return sum;
	}
	
	private float sigmoid(float src) {
		return (float) (1.0 / (1 + Math.exp(-src)));
	}
}