本文实例讲述了Python利用神经网络解决非线性回归问题。分享给大家供大家参考，具体如下：

问题描述

现在我们通常使用神经网络进行分类，但是有时我们也会进行回归分析。
如本文的问题：
我们知道一个生物体内的原始有毒物质的量，然后对这个生物体进行治疗，向其体内注射一个物质，过一段时间后重新测量这个生物体内有毒物质量的多少。
因此，问题中有两个输入，都是标量数据，分别为有毒物质的量和注射物质的量，一个输出，也就是注射治疗物质后一段时间生物体的有毒物质的量。
数据如下图：

其中Dose of Mycotoxins 就是有毒物质，Dose of QCT就是治疗的药物。
其中蓝色底纹的数字就是输出结果。

一些说明

由于本文是进行回归分析，所以最后一层不进行激活，而直接输出。
本文程序使用sigmoid函数进行激活。
本文程序要求程序有一定的可重复性，隐含层可以指定。

另外，注意到
本文将使用数据预处理，也就是将数据减去均值再除以方差，否则使用sigmoid将会导致梯度消失。
因为数据比较大，比如200，这时输入200，当sigmoid函数的梯度就是接近于0了。
与此同时，我们在每一次激活前都进行BN处理，也就是batch normalize，中文可以翻译成规范化。
否则也会导致梯度消失的问题。与预处理情况相同。

程序

程序包括两部分，一部分是模型框架，一个是训练模型

第一部分：

# coding=utf-8import numpy as npdef basic_forard(x, w, b): x = x.reshape(x.shape[0], -1) out = np.dot(x, w) + b cache = (x, w, b) return out, cachedef basic_backward(dout, cache): x, w, b = cache dout = np.array(dout) dx = np.dot(dout, w.T) # dx = np.reshape(dx, x.shape) # x = x.reshape(x.shape[0], -1) dw = np.dot(x.T, dout) db = np.reshape(np.sum(dout, axis=0), b.shape) return dx, dw, dbdef batchnorm_forward(x, gamma, beta, bn_param): mode = bn_param['mode'] eps = bn_param.get('eps', 1e-5) momentum = bn_param.get('momentum', 0.9) N, D = x.shape running_mean = bn_param.get('running_mean', np.zeros(D, dtype=x.dtype)) running_var = bn_param.get('running_var', np.zeros(D, dtype=x.dtype)) out, cache = None, None if mode == 'train': sample_mean = np.mean(x, axis=0) sample_var = np.var(x, axis=0) x_hat = (x - sample_mean) / (np.sqrt(sample_var + eps)) out = gamma * x_hat + beta cache = (gamma, x, sample_mean, sample_var, eps, x_hat) running_mean = momentum * running_mean + (1 - momentum) * sample_mean running_var = momentum * running_var + (1 - momentum) * sample_var elif mode == 'test': scale = gamma / (np.sqrt(running_var + eps)) out = x * scale + (beta - running_mean * scale) else: raise ValueError('Invalid forward batchnorm mode "%s"' % mode) bn_param['running_mean'] = running_mean bn_param['running_var'] = running_var return out, cachedef batchnorm_backward(dout, cache): gamma, x, u_b, sigma_squared_b, eps, x_hat = cache N = x.shape[0] dx_1 = gamma * dout dx_2_b = np.sum((x - u_b) * dx_1, axis=0) dx_2_a = ((sigma_squared_b + eps) ** -0.5) * dx_1 dx_3_b = (-0.5) * ((sigma_squared_b + eps) ** -1.5) * dx_2_b dx_4_b = dx_3_b * 1 dx_5_b = np.ones_like(x) / N * dx_4_b dx_6_b = 2 * (x - u_b) * dx_5_b dx_7_a = dx_6_b * 1 + dx_2_a * 1 dx_7_b = dx_6_b * 1 + dx_2_a * 1 dx_8_b = -1 * np.sum(dx_7_b, axis=0) dx_9_b = np.ones_like(x) / N * dx_8_b dx_10 = dx_9_b + dx_7_a dgamma = np.sum(x_hat * dout, axis=0) dbeta = np.sum(dout, axis=0) dx = dx_10 return dx, dgamma, dbeta# def relu_forward(x):# out = None# out = np.maximum(0,x)# cache = x# return out, cache### def relu_backward(dout, cache):# dx, x = None, cache# dx = (x >= 0) * dout# return dxdef sigmoid_forward(x): x = x.reshape(x.shape[0], -1) out = 1 / (1 + np.exp(-1 * x)) cache = out return out, cachedef sigmoid_backward(dout, cache): out = cache dx = out * (1 - out) dx *= dout return dxdef basic_sigmoid_forward(x, w, b): basic_out, basic_cache = basic_forard(x, w, b) sigmoid_out, sigmoid_cache = sigmoid_forward(basic_out) cache = (basic_cache, sigmoid_cache) return sigmoid_out, cache# def basic_relu_forward(x, w, b):# basic_out, basic_cache = basic_forard(x, w, b)# relu_out, relu_cache = relu_forward(basic_out)# cache = (basic_cache, relu_cache)## return relu_out, cachedef basic_sigmoid_backward(dout, cache): basic_cache, sigmoid_cache = cache dx_sigmoid = sigmoid_backward(dout, sigmoid_cache) dx, dw, db = basic_backward(dx_sigmoid, basic_cache) return dx, dw, db# def basic_relu_backward(dout, cache):# basic_cache, relu_cache = cache# dx_relu = relu_backward(dout, relu_cache)# dx, dw, db = basic_backward(dx_relu, basic_cache)## return dx, dw, dbdef mean_square_error(x, y): x = np.ravel(x) loss = 0.5 * np.sum(np.square(y - x)) / x.shape[0] dx = (x - y).reshape(-1, 1) return loss, dxclass muliti_layer_net(object): def __init__(self, hidden_dim, input_dim=2, num_classes=2, weight_scale=0.01, dtype=np.float32, seed=None, reg=0.0, use_batchnorm=True): self.num_layers = 1 + len(hidden_dim) self.dtype = dtype self.reg = reg self.params = {} self.weight_scale = weight_scale self.use_batchnorm = use_batchnorm # init all parameters layers_dims = [input_dim] + hidden_dim + [num_classes] for i in range(self.num_layers): self.params['W' + str(i + 1)] = np.random.randn(layers_dims[i], layers_dims[i + 1]) * self.weight_scale self.params['b' + str(i + 1)] = np.zeros((1, layers_dims[i + 1])) if self.use_batchnorm and i < (self.num_layers - 1): self.params['gamma' + str(i + 1)] = np.ones((1, layers_dims[i + 1])) self.params['beta' + str(i + 1)] = np.zeros((1, layers_dims[i + 1])) self.bn_params = [] # list if self.use_batchnorm: self.bn_params = [{'mode': 'train'} for i in range(self.num_layers - 1)] def loss(self, X, y=None): X = X.astype(self.dtype) mode = 'test' if y is None else 'train' # compute the forward data and cache basic_sigmoid_cache = {} layer_out = {} layer_out[0] = X out_basic_forward, cache_basic_forward = {}, {} out_bn, cache_bn = {}, {} out_sigmoid_forward, cache_sigmoid_forward = {}, {} for lay in range(self.num_layers - 1): # print('lay: %f' % lay) W = self.params['W' + str(lay + 1)] b = self.params['b' + str(lay + 1)] if self.use_batchnorm: gamma, beta = self.params['gamma' + str(lay + 1)], self.params['beta' + str(lay + 1)] out_basic_forward[lay], cache_basic_forward[lay] = basic_forard(np.array(layer_out[lay]), W, b) out_bn[lay], cache_bn[lay] = batchnorm_forward(np.array(out_basic_forward[lay]), gamma, beta, self.bn_params[lay]) layer_out[lay + 1], cache_sigmoid_forward[lay] = sigmoid_forward(np.array(out_bn[lay])) # = out_sigmoid_forward[lay] else: layer_out[lay+1], basic_sigmoid_cache[lay] = basic_sigmoid_forward(layer_out[lay], W, b) score, basic_cache = basic_forard(layer_out[self.num_layers-1], self.params['W' + str(self.num_layers)], self.params['b' + str(self.num_layers)]) # print('Congratulations: Loss is computed successfully!') if mode == 'test': return score # compute the gradient grads = {} loss, dscore = mean_square_error(score, y) dx, dw, db = basic_backward(dscore, basic_cache) grads['W' + str(self.num_layers)] = dw + self.reg * self.params['W' + str(self.num_layers)] grads['b' + str(self.num_layers)] = db loss += 0.5 * self.reg * np.sum(self.params['W' + str(self.num_layers)] * self.params['b' + str(self.num_layers)]) dbn, dsigmoid = {}, {} for index in range(self.num_layers - 1): lay = self.num_layers - 1 - index - 1 loss += 0.5 * self.reg * np.sum(self.params['W' + str(lay + 1)] * self.params['b' + str(lay + 1)]) if self.use_batchnorm: dsigmoid[lay] = sigmoid_backward(dx, cache_sigmoid_forward[lay]) dbn[lay], grads['gamma' + str(lay + 1)], grads['beta' + str(lay + 1)] = batchnorm_backward(dsigmoid[lay], cache_bn[lay]) dx, grads['W' + str(lay + 1)], grads['b' + str(lay + 1)] = basic_backward(dbn[lay], cache_basic_forward[lay]) else: dx, dw, db = basic_sigmoid_backward(dx, basic_sigmoid_cache[lay]) for lay in range(self.num_layers): grads['W' + str(lay + 1)] += self.reg * self.params['W' + str(lay + 1)] return loss, gradsdef sgd_momentum(w, dw, config=None): if config is None: config = {} config.setdefault('learning_rate', 1e-2) config.setdefault('momentum', 0.9) v = config.get('velocity', np.zeros_like(w)) v = config['momentum'] * v - config['learning_rate'] * dw next_w = w + v config['velocity'] = v return next_w, configclass Solver(object): def __init__(self, model, data, **kwargs): self.model = model self.X_train = data['X_train'] self.y_train = data['y_train'] self.X_val = data['X_val'] self.y_val = data['y_val'] self.update_rule = kwargs.pop('update_rule', 'sgd_momentum') self.optim_config = kwargs.pop('optim_config', {}) self.lr_decay = kwargs.pop('lr_decay', 1.0) self.batch_size = kwargs.pop('batch_size', 100) self.num_epochs = kwargs.pop('num_epochs', 10) self.weight_scale = kwargs.pop('weight_scale', 0.01) self.print_every = kwargs.pop('print_every', 10) self.verbose = kwargs.pop('verbose', True) if len(kwargs) > 0: extra = ', '.join('"%s"' % k for k in kwargs.keys()) raise ValueError('Unrecognized argements %s' % extra) self._reset() def _reset(self): self.epoch = 100 self.best_val_acc = 0 self.best_params = {} self.loss_history = [] self.train_acc_history = [] self.val_acc_history = [] self.optim_configs = {} for p in self.model.params: d = {k: v for k, v in self.optim_config.items()} self.optim_configs[p] = d def _step(self): loss, grads = self.model.loss(self.X_train, self.y_train) self.loss_history.append(loss) for p, w in self.model.params.items(): dw = grads[p] config = self.optim_configs[p] next_w, next_config = sgd_momentum(w, dw, config) self.model.params[p] = next_w self.optim_configs[p] = next_config return loss def train(self): min_loss = 100000000 num_train = self.X_train.shape[0] iterations_per_epoch = max(num_train / self.batch_size, 1) num_iterations = self.num_epochs * iterations_per_epoch for t in range(int(num_iterations)): loss = self._step() if self.verbose:# print(self.loss_history[-1]) pass if loss < min_loss: min_loss = loss for k, v in self.model.params.items(): self.best_params[k] = v.copy() self.model.params = self.best_params

第二部分

import numpy as np# import datadose_QCT = np.array([0, 5, 10, 20])mean_QCT, std_QCT = np.mean(dose_QCT), np.std(dose_QCT)dose_QCT = (dose_QCT - mean_QCT ) / std_QCTdose_toxins = np.array([0, 0.78125, 1.5625, 3.125, 6.25, 12.5, 25, 50, 100, 200])mean_toxins, std_toxins = np.mean(dose_toxins), np.std(dose_toxins)dose_toxins = (dose_toxins - mean_toxins ) / std_toxinsresult = np.array([[0, 4.037, 7.148, 12.442, 18.547, 25.711, 34.773, 62.960, 73.363, 77.878], [0, 2.552, 4.725, 8.745, 14.436, 21.066, 29.509, 55.722, 65.976, 72.426], [0, 1.207, 2.252, 4.037, 7.148, 11.442, 17.136, 34.121, 48.016, 60.865], [0, 0.663, 1.207, 2.157, 3.601, 5.615, 8.251, 19.558, 33.847, 45.154]])mean_result, std_result = np.mean(result), np.std(result)result = (result - mean_result ) / std_result# create the train datatrain_x, train_y = [], []for i,qct in enumerate(dose_QCT): for j,toxin in enumerate(dose_toxins): x = [qct, toxin] y = result[i, j] train_x.append(x) train_y.append(y)train_x = np.array(train_x)train_y = np.array(train_y)print(train_x.shape)print(train_y.shape)import layers_regressionsmall_data = {'X_train': train_x, 'y_train': train_y, 'X_val': train_x, 'y_val': train_y,}batch_size = train_x.shape[0]learning_rate = 0.002reg = 0model = layers_regression.muliti_layer_net(hidden_dim=[5,5], input_dim=2, num_classes=1, reg=reg, dtype=np.float64)solver = layers_regression.Solver(model, small_data, print_every=0, num_epochs=50000, batch_size=batch_size, weight_scale=1, update_rule='sgd_momentum', optim_config={'learning_rate': learning_rate})print('Please wait several minutes!')solver.train()# print(model.params)best_model = modelprint('Train process is finised')import matplotlib.pyplot as plt# %matplotlib inlineplt.plot(solver.loss_history, '.')plt.title('Training loss history')plt.xlabel('Iteration')plt.ylabel('Training loss')plt.show()# predict the training_datapredict = best_model.loss(train_x)predict = np.round(predict * std_result + mean_result, 1)print('Predict is ')print('{}'.format(predict.reshape(4, 10)))# print('{}'.format(predict))# observe the error between the predict after training with ground truthresult = np.array([[0, 4.037, 7.148, 12.442, 18.547, 25.711, 34.773, 62.960, 73.363, 77.878], [0, 2.552, 4.725, 8.745, 14.436, 21.066, 29.509, 55.722, 65.976, 72.426], [0, 1.207, 2.252, 4.037, 7.148, 11.442, 17.136, 34.121, 48.016, 60.865], [0, 0.663, 1.207, 2.157, 3.601, 5.615, 8.251, 19.558, 33.847, 45.154]])result = result.reshape(4, 10)predict = predict.reshape(4, 10)error = np.round(result - predict, 2)print('error between predict and real data')print(error)print('The absulate error in all data is %f' % np.sum(np.abs(error)))print('The mean error in all data is %f' % np.mean(np.abs(error)))# figure the predict map in 3Dx_1 = (np.arange(0, 20, 0.1) - mean_QCT) / std_QCTx_2 = (np.arange(0, 200, 1) - mean_toxins) / std_toxinsx_test = np.zeros((len(x_1)*len(x_2), 2))index = 0for i in range(len(x_1)): for j in range(len(x_2)): x_test[int(index), 0] = x_1[int(i)] x_test[int(index), 1] = x_2[int(j)] index += 1test_pred = best_model.loss(x_test)predict = np.round(test_pred * std_result + mean_result, 3)from mpl_toolkits.mplot3d import Axes3Dx_1, x_2 = np.meshgrid(x_1 * std_QCT + mean_QCT, x_2 * std_toxins + mean_toxins)figure = plt.figure()ax = Axes3D(figure)predict = predict.reshape(len(x_1), len(x_2))ax.plot_surface(x_1, x_2, predict, rstride=1, cstride=1, cmap='rainbow')plt.show()# 最后本文将进行一些预测，但预测效果不是很好# question 2: predict with givendose_QCT_predict = np.ravel(np.array([7.5, 15]))dose_QCT_predict_ = (dose_QCT_predict - mean_QCT)/ std_QCTdose_toxins_predict = np.array([0, 0.78125, 1.5625, 3.125, 6.25, 12.5, 25, 50, 100, 200])dose_toxins_predict_ = (dose_toxins_predict - mean_toxins) / std_toxinstest = []for i,qct in enumerate(dose_QCT_predict): for j,toxin in enumerate(dose_toxins_predict): x = [qct, toxin] test.append(x)test = np.array(test)print('Please look at the test data:')print(test)test = []for i,qct in enumerate(dose_QCT_predict_): for j,toxin in enumerate(dose_toxins_predict_): x = [qct, toxin] test.append(x)test = np.array(test)test_pred = best_model.loss(test)predict = np.round(test_pred * std_result + mean_result, 1)print(predict.reshape(2, 10))

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