[Watermelon_book] Chapter 3 Linear Model
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[Watermelon_book] Chapter 3 Linear Model
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- Linear Model
- 基本定義
- 線性模型簡單形式的實際編碼
- Task
- Generate data
- Cost function
- Gradient descent
- Training
- Model evaluation
- Experiment 1
- Experiment 2
- Task 2
- Generate data
- Normalization
- Prediction format
- Cost function
- Gradient descent
- Training
- Model evaluation
- 由回歸到分類
- Logistic Regression ( Binary Classification）的實際編碼
- Dataset
- Visulization
- Prediction Format
- Cost function
- Gradient descent
- Training
- Model evaluation
- Visulization
- Mapping probabilities to classes
Linear Model
基本定義
線性模型簡單形式的實際編碼
Task
We have derived the formula of Linear Model, here I want to write codes for it by hand. The specific task is to predict sales based on radios shown as the table below.
The original problem is from https://ml-cheatsheet.readthedocs.io/en/latest/linear_regression.html
| Amazon | 37.8 | 22.1 |
| 39.3 | 10.4 |
Generate data
import numpy as np import scipy import seaborn as sns import matplotlib.pyplot as plt sample_size = 200 np.random.seed(5) radio_sample = 60 * np.random.rand(sample_size) weight_truth = 0.4 bias_truth = -3 np.random.seed(10) nosie_sample = 10 * (np.random.normal(0, 0.1, sample_size)) sales_sample = radio_sample * weight_truth + bias_truth + nosie_sample sales_no_noise = radio_sample * weight_truth + bias_truth plt.scatter(radio_sample, sales_sample, alpha=0.5) plt.plot(radio_sample, sales_no_noise, c="r") plt.xlabel("radio") plt.ylabel("sales") Text(0, 0.5, 'sales')Cost function
f(m,b)=1N∑i=1n(yi?(mxi+b))2f(m, b)=\frac{1}{N} \sum_{i=1}^{n}\left(y_{i}-\left(m x_{i}+b\right)\right)^{2}f(m,b)=N1?∑i=1n?(yi??(mxi?+b))2
def cost_function(radio, sales, weight, bias):'''cost_function for linear modelArgs: radio,sales: is numpy arrayweight,bias: is scalar'''sample_size = len(radio)error = 0.0for i in range(sample_size):error += (sales[i] - (radio[i]*weight + bias))**2error_avg = error/sample_sizereturn error_avgGradient descent
\begin{aligned} f^{\prime}(m, b)=\left[ \begin{array}{c}{\frac{d f}{d m}} \ {\frac{d f}{d b}}\end{array}\right] &=\left[ \begin{array}{c}{\frac{1}{N} \sum-2 x_{i}\left(y_{i}-\left(m x_{i}+b\right)\right)} \ {\frac{1}{N} \sum-2\left(y_{i}-\left(m x_{i}+b\right)\right)}\end{array}\right] \end{aligned}
def update_weight(radio, sales, weight, bias, learning_rate):# initial valueweight_deriv = 0bias_deriv = 0sample_size = len(radio)for i in range(sample_size):# calculate partial derivativesweight_deriv += -2*radio[i]*(sales[i] - (weight*radio[i] + bias))bias_deriv += -2*(sales[i] - (weight*radio[i] + bias))#gradient descentweight = weight - (weight_deriv/sample_size)*learning_ratebias = bias - (bias_deriv/sample_size)*learning_ratereturn weight,biasTraining
def train(radio, sales, weight, bias, learning_rate, iters):cost_history = []for i in range(iters):weight, bias = update_weight(radio, sales, weight, bias, learning_rate)cost = cost_function(radio, sales, weight, bias)cost_history.append(cost)if i % 5 == 0:print("Iter: %d \t Weight: %2f \t Bias: %2f \t Cost: %4f" %(i,weight, bias, cost))return weight, bias, cost_historyModel evaluation
Experiment 1
weight_final, bias_final, cost_history = train(radio_sample, sales_sample, 0, 0, 0.0001, 2000) Iter: 0 Weight: 0.080657 Bias: 0.001851 Cost: 77.703045 Iter: 5 Weight: 0.267103 Bias: 0.005752 Cost: 7.320940 Iter: 10 Weight: 0.312288 Bias: 0.006165 Cost: 3.188519 Iter: 15 Weight: 0.323249 Bias: 0.005734 Cost: 2.945016 Iter: 20 Weight: 0.325918 Bias: 0.005099 Cost: 2.929796 Iter: 25 Weight: 0.326578 Bias: 0.004414 Cost: 2.927978 Iter: 30 Weight: 0.326751 Bias: 0.003717 Cost: 2.926946 Iter: 35 Weight: 0.326806 Bias: 0.003017 Cost: 2.925961 Iter: 40 Weight: 0.326832 Bias: 0.002317 Cost: 2.924980 Iter: 45 Weight: 0.326852 Bias: 0.001617 Cost: 2.923999 Iter: 50 Weight: 0.326869 Bias: 0.000917 Cost: 2.923018 Iter: 55 Weight: 0.326887 Bias: 0.000217 Cost: 2.922038 Iter: 60 Weight: 0.326904 Bias: -0.000482 Cost: 2.921058 Iter: 65 Weight: 0.326921 Bias: -0.001182 Cost: 2.920079 Iter: 70 Weight: 0.326939 Bias: -0.001881 Cost: 2.919101 Iter: 75 Weight: 0.326956 Bias: -0.002580 Cost: 2.918123 Iter: 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Iter: 1955 Weight: 0.333151 Bias: -0.253517 Cost: 2.582740 Iter: 1960 Weight: 0.333167 Bias: -0.254154 Cost: 2.581929 Iter: 1965 Weight: 0.333182 Bias: -0.254790 Cost: 2.581119 Iter: 1970 Weight: 0.333198 Bias: -0.255426 Cost: 2.580308 Iter: 1975 Weight: 0.333214 Bias: -0.256063 Cost: 2.579498 Iter: 1980 Weight: 0.333229 Bias: -0.256699 Cost: 2.578689 Iter: 1985 Weight: 0.333245 Bias: -0.257335 Cost: 2.577880 Iter: 1990 Weight: 0.333261 Bias: -0.257970 Cost: 2.577071 Iter: 1995 Weight: 0.333276 Bias: -0.258606 Cost: 2.576263 iters = np.arange(2000) plt.plot(iters, cost_history) [<matplotlib.lines.Line2D at 0x7fb27cf937f0>]We can see, based on the initial weight=0,bias=0, learning_rate=0.0001,the cost value decrease sharply.
sales_predict = weight_final*radio_sample + bias_final plt.scatter(radio_sample, sales_sample, alpha=0.5) plt.plot(radio_sample, sales_predict, c="r") plt.xlabel("radio") plt.ylabel("sales") Text(0, 0.5, 'sales')Experiment 2
In the last experiment, we set four hyperparameters manually which are w, b, l and i. Let’s change them and see what happen.
weight_final, bias_final, cost_history = train(radio_sample, sales_sample, -1000, 0, 0.0001, 2000) Iter: 0 Weight: -753.190221 Bias: 6.091610 Cost: 700169768.718390 Iter: 5 Weight: -182.698756 Bias: 20.168463 Cost: 41100789.097582 Iter: 10 Weight: -44.478396 Bias: 23.573870 Cost: 2412831.778656 Iter: 15 Weight: -10.989863 Bias: 24.393771 Cost: 141812.898689 Iter: 20 Weight: -2.876041 Bias: 24.587251 Cost: 8501.907289 Iter: 25 Weight: -0.910075 Bias: 24.628960 Cost: 676.341469 Iter: 30 Weight: -0.433628 Bias: 24.633899 Cost: 216.885960 Iter: 35 Weight: -0.318065 Bias: 24.629931 Cost: 189.827967 Iter: 40 Weight: -0.289939 Bias: 24.623806 Cost: 188.152151 Iter: 45 Weight: -0.282997 Bias: 24.617160 Cost: 187.966335 Iter: 50 Weight: -0.281188 Bias: 24.610389 Cost: 187.868027 Iter: 55 Weight: -0.280622 Bias: 24.603589 Cost: 187.774899 Iter: 60 Weight: -0.280358 Bias: 24.596783 Cost: 187.682118 Iter: 65 Weight: -0.280166 Bias: 24.589976 Cost: 187.589402 Iter: 70 Weight: -0.279993 Bias: 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-0.220304 Bias: 22.165423 Cost: 156.028539 Iter: 1935 Weight: -0.220151 Bias: 22.159220 Cost: 155.951540 Iter: 1940 Weight: -0.219998 Bias: 22.153019 Cost: 155.874579 Iter: 1945 Weight: -0.219845 Bias: 22.146819 Cost: 155.797656 Iter: 1950 Weight: -0.219692 Bias: 22.140620 Cost: 155.720771 Iter: 1955 Weight: -0.219539 Bias: 22.134424 Cost: 155.643925 Iter: 1960 Weight: -0.219386 Bias: 22.128228 Cost: 155.567117 Iter: 1965 Weight: -0.219233 Bias: 22.122034 Cost: 155.490347 Iter: 1970 Weight: -0.219080 Bias: 22.115842 Cost: 155.413615 Iter: 1975 Weight: -0.218927 Bias: 22.109652 Cost: 155.336921 Iter: 1980 Weight: -0.218774 Bias: 22.103462 Cost: 155.260266 Iter: 1985 Weight: -0.218622 Bias: 22.097275 Cost: 155.183648 Iter: 1990 Weight: -0.218469 Bias: 22.091089 Cost: 155.107068 Iter: 1995 Weight: -0.218316 Bias: 22.084904 Cost: 155.030527 iters = np.arange(2000) # because the derivation of cost is too large # for visulazation, we need to shrink cost cost_history = np.log(np.array(cost_history)) plt.plot(iters, cost_history) [<matplotlib.lines.Line2D at 0x7fb27c9b99e8>] sales_predict = weight_final*radio_sample + bias_final plt.scatter(radio_sample, sales_sample, alpha=0.5) plt.plot(radio_sample, sales_predict, c="r") plt.xlabel("radio") plt.ylabel("sales") Text(0, 0.5, 'sales')The setting of hyperparameters is very import for gradient descent method. There must be lots of tricks which is another topic.
Task 2
Single feature —> multiple features
scalar —> matrix(vector)
Generate data
sample_size = 200 feature_size = 3 np.random.seed(5) feature_sample = 50*np.random.rand(feature_size)*np.random.rand(sample_size, feature_size)np.random.seed(123) weight_sample = np.random.rand(feature_size).reshape(feature_size, 1)np.random.seed(888) noise = np.random.normal(0, 0.01, sample_size).reshape(sample_size ,1)result_sample = np.dot(feature, weight) + noiseNormalization
We want to shrink the data to reduce the time to change weight. There must be lots of tricks which is another topic.
def normalize(feature):#feature: sample size * feature sizefeature = feature.astype("float64")sample_size, feature_size = feature.shapefor i in range(feature_size):fmean = np.mean(feature[:, i])frange = np.amax(feature[:, i]) - np.amin(feature[:, i])feature[:, i] = (feature[:, i] - fmean) / frangereturn featurePrediction format
For simplifing the problem, here we just think bias is zero.
Ysample?1=Fsample?feature?Wfeature?1Y_{sample*1}=F_{sample*feature} \cdot W_{feature*1}Ysample?1?=Fsample?feature??Wfeature?1?
def predict(features, weight):weight = weight.reshape(len(weight), 1)prediction = np.dot(features, weight)return predictionCost function
MSE=12N∥Ytarget?Ypredict∥22MSE=\frac{1}{2 N} \left\|Y_{target}-Y_{predict}\right\|_2^2MSE=2N1?∥Ytarget??Ypredict?∥22?
def cost_function(features, targets, weight):#here weights should be (feature, 1)weight = weight.reshape(len(weight), 1)targets = targets.reshape(len(targets), 1)error = ((targets - predict(features, weight))**2).sum()return error/(2.0*len(targets))Gradient descent
Just the equation derived in the watermelon book(3.11)
def update_weight(features, targets, weight, learning_rate):weight = weight.reshape(len(weight), 1)targets = targets.reshape(len(targets), 1)gradient = np.dot(features.T, (predict(features, weight) - targets))gradient = gradient / len(targets)weight = weight - gradient*learning_ratereturn weightTraining
def train(features, targets, weight, learning_rate, iters):cost_history = []for i in range(iters):weight = update_weight(features, targets, weight, learning_rate)cost = cost_function(features, targets, weight)cost_history.append(cost)if i % 5 == 0:print("Iter: %d \t \t Cost: %4f" %(i, cost))return weight, cost_historyModel evaluation
weight_initial = np.array([0, 0, 0]) weight, cost_history = train(feature_sample, result_sample, weight_initial, 0.0001, 1000) Iter: 0 Cost: 64.953149 Iter: 5 Cost: 33.915984 Iter: 10 Cost: 18.615390 Iter: 15 Cost: 11.053208 Iter: 20 Cost: 7.296763 Iter: 25 Cost: 5.412361 Iter: 30 Cost: 4.449195 Iter: 35 Cost: 3.939745 Iter: 40 Cost: 3.654115 Iter: 45 Cost: 3.479253 Iter: 50 Cost: 3.359556 Iter: 55 Cost: 3.267672 Iter: 60 Cost: 3.190135 Iter: 65 Cost: 3.120312 Iter: 70 Cost: 3.054928 Iter: 75 Cost: 2.992358 Iter: 80 Cost: 2.931792 Iter: 85 Cost: 2.872817 Iter: 90 Cost: 2.815220 Iter: 95 Cost: 2.758881 Iter: 100 Cost: 2.703732 Iter: 105 Cost: 2.649725 Iter: 110 Cost: 2.596827 Iter: 115 Cost: 2.545010 Iter: 120 Cost: 2.494250 Iter: 125 Cost: 2.444522 Iter: 130 Cost: 2.395806 Iter: 135 Cost: 2.348080 Iter: 140 Cost: 2.301324 Iter: 145 Cost: 2.255518 Iter: 150 Cost: 2.210643 Iter: 155 Cost: 2.166678 Iter: 160 Cost: 2.123606 Iter: 165 Cost: 2.081409 Iter: 170 Cost: 2.040067 Iter: 175 Cost: 1.999565 Iter: 180 Cost: 1.959884 Iter: 185 Cost: 1.921007 Iter: 190 Cost: 1.882919 Iter: 195 Cost: 1.845603 Iter: 200 Cost: 1.809044 Iter: 205 Cost: 1.773225 Iter: 210 Cost: 1.738131 Iter: 215 Cost: 1.703749 Iter: 220 Cost: 1.670062 Iter: 225 Cost: 1.637057 Iter: 230 Cost: 1.604721 Iter: 235 Cost: 1.573038 Iter: 240 Cost: 1.541997 Iter: 245 Cost: 1.511583 Iter: 250 Cost: 1.481784 Iter: 255 Cost: 1.452588 Iter: 260 Cost: 1.423981 Iter: 265 Cost: 1.395952 Iter: 270 Cost: 1.368490 Iter: 275 Cost: 1.341582 Iter: 280 Cost: 1.315217 Iter: 285 Cost: 1.289384 Iter: 290 Cost: 1.264073 Iter: 295 Cost: 1.239272 Iter: 300 Cost: 1.214971 Iter: 305 Cost: 1.191160 Iter: 310 Cost: 1.167828 Iter: 315 Cost: 1.144967 Iter: 320 Cost: 1.122566 Iter: 325 Cost: 1.100617 Iter: 330 Cost: 1.079109 Iter: 335 Cost: 1.058034 Iter: 340 Cost: 1.037383 Iter: 345 Cost: 1.017147 Iter: 350 Cost: 0.997318 Iter: 355 Cost: 0.977888 Iter: 360 Cost: 0.958848 Iter: 365 Cost: 0.940190 Iter: 370 Cost: 0.921908 Iter: 375 Cost: 0.903992 Iter: 380 Cost: 0.886435 Iter: 385 Cost: 0.869231 Iter: 390 Cost: 0.852372 Iter: 395 Cost: 0.835851 Iter: 400 Cost: 0.819660 Iter: 405 Cost: 0.803795 Iter: 410 Cost: 0.788246 Iter: 415 Cost: 0.773010 Iter: 420 Cost: 0.758078 Iter: 425 Cost: 0.743444 Iter: 430 Cost: 0.729104 Iter: 435 Cost: 0.715050 Iter: 440 Cost: 0.701277 Iter: 445 Cost: 0.687779 Iter: 450 Cost: 0.674550 Iter: 455 Cost: 0.661586 Iter: 460 Cost: 0.648880 Iter: 465 Cost: 0.636427 Iter: 470 Cost: 0.624223 Iter: 475 Cost: 0.612262 Iter: 480 Cost: 0.600540 Iter: 485 Cost: 0.589050 Iter: 490 Cost: 0.577790 Iter: 495 Cost: 0.566753 Iter: 500 Cost: 0.555936 Iter: 505 Cost: 0.545334 Iter: 510 Cost: 0.534943 Iter: 515 Cost: 0.524758 Iter: 520 Cost: 0.514775 Iter: 525 Cost: 0.504991 Iter: 530 Cost: 0.495400 Iter: 535 Cost: 0.486000 Iter: 540 Cost: 0.476786 Iter: 545 Cost: 0.467754 Iter: 550 Cost: 0.458901 Iter: 555 Cost: 0.450224 Iter: 560 Cost: 0.441718 Iter: 565 Cost: 0.433380 Iter: 570 Cost: 0.425207 Iter: 575 Cost: 0.417196 Iter: 580 Cost: 0.409342 Iter: 585 Cost: 0.401644 Iter: 590 Cost: 0.394097 Iter: 595 Cost: 0.386700 Iter: 600 Cost: 0.379448 Iter: 605 Cost: 0.372339 Iter: 610 Cost: 0.365369 Iter: 615 Cost: 0.358537 Iter: 620 Cost: 0.351839 Iter: 625 Cost: 0.345273 Iter: 630 Cost: 0.338836 Iter: 635 Cost: 0.332525 Iter: 640 Cost: 0.326338 Iter: 645 Cost: 0.320272 Iter: 650 Cost: 0.314326 Iter: 655 Cost: 0.308495 Iter: 660 Cost: 0.302779 Iter: 665 Cost: 0.297175 Iter: 670 Cost: 0.291680 Iter: 675 Cost: 0.286292 Iter: 680 Cost: 0.281010 Iter: 685 Cost: 0.275831 Iter: 690 Cost: 0.270753 Iter: 695 Cost: 0.265774 Iter: 700 Cost: 0.260892 Iter: 705 Cost: 0.256105 Iter: 710 Cost: 0.251411 Iter: 715 Cost: 0.246808 Iter: 720 Cost: 0.242295 Iter: 725 Cost: 0.237870 Iter: 730 Cost: 0.233530 Iter: 735 Cost: 0.229275 Iter: 740 Cost: 0.225102 Iter: 745 Cost: 0.221010 Iter: 750 Cost: 0.216997 Iter: 755 Cost: 0.213062 Iter: 760 Cost: 0.209202 Iter: 765 Cost: 0.205418 Iter: 770 Cost: 0.201706 Iter: 775 Cost: 0.198066 Iter: 780 Cost: 0.194496 Iter: 785 Cost: 0.190995 Iter: 790 Cost: 0.187561 Iter: 795 Cost: 0.184194 Iter: 800 Cost: 0.180891 Iter: 805 Cost: 0.177651 Iter: 810 Cost: 0.174474 Iter: 815 Cost: 0.171358 Iter: 820 Cost: 0.168301 Iter: 825 Cost: 0.165303 Iter: 830 Cost: 0.162362 Iter: 835 Cost: 0.159477 Iter: 840 Cost: 0.156648 Iter: 845 Cost: 0.153873 Iter: 850 Cost: 0.151150 Iter: 855 Cost: 0.148479 Iter: 860 Cost: 0.145860 Iter: 865 Cost: 0.143290 Iter: 870 Cost: 0.140768 Iter: 875 Cost: 0.138295 Iter: 880 Cost: 0.135869 Iter: 885 Cost: 0.133489 Iter: 890 Cost: 0.131153 Iter: 895 Cost: 0.128862 Iter: 900 Cost: 0.126615 Iter: 905 Cost: 0.124410 Iter: 910 Cost: 0.122246 Iter: 915 Cost: 0.120123 Iter: 920 Cost: 0.118040 Iter: 925 Cost: 0.115997 Iter: 930 Cost: 0.113992 Iter: 935 Cost: 0.112024 Iter: 940 Cost: 0.110094 Iter: 945 Cost: 0.108199 Iter: 950 Cost: 0.106340 Iter: 955 Cost: 0.104516 Iter: 960 Cost: 0.102726 Iter: 965 Cost: 0.100970 Iter: 970 Cost: 0.099246 Iter: 975 Cost: 0.097555 Iter: 980 Cost: 0.095895 Iter: 985 Cost: 0.094265 Iter: 990 Cost: 0.092667 Iter: 995 Cost: 0.091097 iters = np.arange(1000) plt.plot(iters, cost_history) [<matplotlib.lines.Line2D at 0x7fb27b4052b0>]由回歸到分類
In this task, I just use the watermelon dataset which is shown below.
Logistic Regression ( Binary Classification）的實際編碼
Dataset
import numpy as np import seaborn as sns import pandas as pd import matplotlib.pyplot as plt import math def createDataSet():"""創建測試的數據集，里面的數值中具有連續值:return:"""dataSet = [# 1['青綠', '蜷縮', '濁響', '清晰', '凹陷', '硬滑', 0.697, 0.460, '好瓜'],# 2['烏黑', '蜷縮', '沉悶', '清晰', '凹陷', '硬滑', 0.774, 0.376, '好瓜'],# 3['烏黑', '蜷縮', '濁響', '清晰', '凹陷', '硬滑', 0.634, 0.264, '好瓜'],# 4['青綠', '蜷縮', '沉悶', '清晰', '凹陷', '硬滑', 0.608, 0.318, '好瓜'],# 5['淺白', '蜷縮', '濁響', '清晰', '凹陷', '硬滑', 0.556, 0.215, '好瓜'],# 6['青綠', '稍蜷', '濁響', '清晰', '稍凹', '軟粘', 0.403, 0.237, '好瓜'],# 7['烏黑', '稍蜷', '濁響', '稍糊', '稍凹', '軟粘', 0.481, 0.149, '好瓜'],# 8['烏黑', '稍蜷', '濁響', '清晰', '稍凹', '硬滑', 0.437, 0.211, '好瓜'],# ----------------------------------------------------# 9['烏黑', '稍蜷', '沉悶', '稍糊', '稍凹', '硬滑', 0.666, 0.091, '壞瓜'],# 10['青綠', '硬挺', '清脆', '清晰', '平坦', '軟粘', 0.243, 0.267, '壞瓜'],# 11['淺白', '硬挺', '清脆', '模糊', '平坦', '硬滑', 0.245, 0.057, '壞瓜'],# 12['淺白', '蜷縮', '濁響', '模糊', '平坦', '軟粘', 0.343, 0.099, '壞瓜'],# 13['青綠', '稍蜷', '濁響', '稍糊', '凹陷', '硬滑', 0.639, 0.161, '壞瓜'],# 14['淺白', '稍蜷', '沉悶', '稍糊', '凹陷', '硬滑', 0.657, 0.198, '壞瓜'],# 15['烏黑', '稍蜷', '濁響', '清晰', '稍凹', '軟粘', 0.360, 0.370, '壞瓜'],# 16['淺白', '蜷縮', '濁響', '模糊', '平坦', '硬滑', 0.593, 0.042, '壞瓜'],# 17['青綠', '蜷縮', '沉悶', '稍糊', '稍凹', '硬滑', 0.719, 0.103, '壞瓜']]return dataSet dataSet = createDataSet() dataSet = np.array(dataSet)[:, 6:] dataSet[dataSet == '好瓜'] = 1 dataSet[dataSet == '壞瓜'] = 0 dataSet = dataSet.astype('float64') dataSet array([[0.697, 0.46 , 1. ],[0.774, 0.376, 1. ],[0.634, 0.264, 1. ],[0.608, 0.318, 1. ],[0.556, 0.215, 1. ],[0.403, 0.237, 1. ],[0.481, 0.149, 1. ],[0.437, 0.211, 1. ],[0.666, 0.091, 0. ],[0.243, 0.267, 0. ],[0.245, 0.057, 0. ],[0.343, 0.099, 0. ],[0.639, 0.161, 0. ],[0.657, 0.198, 0. ],[0.36 , 0.37 , 0. ],[0.593, 0.042, 0. ],[0.719, 0.103, 0. ]])Visulization
data_in_frame = pd.DataFrame(data=dataSet, columns=["density", "sugar_ratio","label"]) data_in_frame| 0.697 | 0.460 | 1.0 |
| 0.774 | 0.376 | 1.0 |
| 0.634 | 0.264 | 1.0 |
| 0.608 | 0.318 | 1.0 |
| 0.556 | 0.215 | 1.0 |
| 0.403 | 0.237 | 1.0 |
| 0.481 | 0.149 | 1.0 |
| 0.437 | 0.211 | 1.0 |
| 0.666 | 0.091 | 0.0 |
| 0.243 | 0.267 | 0.0 |
| 0.245 | 0.057 | 0.0 |
| 0.343 | 0.099 | 0.0 |
| 0.639 | 0.161 | 0.0 |
| 0.657 | 0.198 | 0.0 |
| 0.360 | 0.370 | 0.0 |
| 0.593 | 0.042 | 0.0 |
| 0.719 | 0.103 | 0.0 |
Prediction Format
\begin{split}p \geq 0.5, class=1 \
p < 0.5, class=0\end{split}
\begin{split}P(class=1) = \frac{1} {1 + e^{-z}}\end{split}
def sigmoid(x, derivative=False):sigm = 1. / (1. + np.exp(-x))if derivative:return sigm * (1. - sigm)return sigm def predict(features, weights):"""features: sample size * feature sizeweights: feature size * 1"""weights = weights.reshape(len(weights),1)z = np.dot(features, weights)return sigmoid(z)How could we do here? From Watermelon_book, we know that we can use MLE to estimate the parameters.
Meanwhile we can also repeat what we did in last experiment where we just included a cost function and optimized the patameters by decrease the cost function.
Cost function
We can still use MSE as a cost function. But here what we use is called cross-entroy function actually. The reason why we abandon the previous one is another topic. Basicly, it’s because of non-linear tranformation.
\begin{split}-{(y\log§ + (1 - y)\log(1 - p))}\end{split}
def cost_function(features, weights, labels):weights = weights.reshape(len(weights),1)labels = labels.reshape(len(labels),1)y = predict(features, weights)class1_cost = -labels*np.log(y)class2_cost = -labels*np.log(y)cost = (class1_cost + class2_cost).sum()/len(labels)return costGradient descent
\begin{align}
s’(z) & = s(z)(1 - s(z))
\end{align}
\begin{split}C’ = x(s(z) - y)\end{split}
def update_weights(features, weights, labels, learning_rate):weights = weights.reshape(len(weights),1)labels = labels.reshape(len(labels),1)p = predict(features, weights)gradient = np.dot(features.T, p - labels) / len(labels)weights = weights - gradient*learning_ratereturn weightsTraining
def training(features, weights, labels, learning_rate, iters):cost_history = []for i in range(iters):weights = update_weights(features, weights, labels, learning_rate)cost = cost_function(features, weights, labels)cost_history.append(cost)if i%1000 == 0:print("iters is %d \t \t cost is %f"%(i, cost))return weights, cost_history weights_initial = np.array([0, 0]) weights, cost_history = training(dataSet[:, :2], weights_initial, dataSet[:, -1], 0.1, 20000) iters is 0 cost is 0.651950 iters is 1000 cost is 0.572514 iters is 2000 cost is 0.546478 iters is 3000 cost is 0.528388 iters is 4000 cost is 0.515437 iters is 5000 cost is 0.505936 iters is 6000 cost is 0.498829 iters is 7000 cost is 0.493431 iters is 8000 cost is 0.489282 iters is 9000 cost is 0.486062 iters is 10000 cost is 0.483545 iters is 11000 cost is 0.481564 iters is 12000 cost is 0.479999 iters is 13000 cost is 0.478757 iters is 14000 cost is 0.477769 iters is 15000 cost is 0.476981 iters is 16000 cost is 0.476351 iters is 17000 cost is 0.475847 iters is 18000 cost is 0.475443 iters is 19000 cost is 0.475119Model evaluation
Visulization
iters = np.arange(20000) plt.plot(iters, cost_history) [<matplotlib.lines.Line2D at 0x7fc819b8eda0>]From the fig abobe, we can think the cost function may still decrease when we update the weights furthur.
Mapping probabilities to classes
def classify(predictions):predictions[predictions >= 0.5 ] = 1predictions[predictions < 0.5 ] = 0return predictions def accuracy(predictions, labels):predictions = predictions.astype('int').reshape(len(predictions,))labels = labels.astype('int')diff = np.abs(predictions - labels)same = len(labels) - diff.sum()return same/len(labels) predicted_label = classify(predict(dataSet[:, :2], weights)) accuracy(predicted_label, dataSet[:, -1]) 0.8235294117647058總結
以上是生活随笔為你收集整理的[Watermelon_book] Chapter 3 Linear Model的全部內容，希望文章能夠幫你解決所遇到的問題。
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