Week9_3Program_Anomaly Detection and Recommender Systems编程解析
1. Anomaly detection分析
1.1 Estimate gaussian parameters 0 / 15
计算公式:
μi=1m∑j=1mx(j)iμi=1m∑j=1mxi(j)
σ2i=1m∑j=1m(x(j)i−μi)2σi2=1m∑j=1m(xi(j)−μi)2
X=(300x2) K=3 centroids=(3x2) idx=(300x1)
在 estimateGaussian.m 中添加
mu = mean(X);
sigma2 = var(X,opt=1);
octave中mean函数的作用:
mean (x) = SUM_i x(i) / N
octave中的var函数的作用:
var (x) = 1/(N-1) SUM_i (x(i) - mean(x))^2
1.2 Select threshold 0 / 15
计算公式:
p(x;μ,σ2)=12πσ2√e−(x−μ)22σ2p(x;μ,σ2)=12πσ2e−(x−μ)22σ2
上述求p的过程己经在 multivariateGaussian.m 中完成了, 直接用就行,程序中是pval
下面是 selectThreshold.m中用到的公式:
precison=tp+fptpprecison=tp+fptp
recall=tp+fntprecall=tp+fntp
F1=prec+rec2∗prec∗recF1=prec+rec2∗prec∗rec
上图是lecture 11中的课件截图
* 将距离u1最近的找出来,取个平均值作为新的u1 *
predictions = (pval < epsilon);
truePositives = sum((predictions == 1) & (yval == 1));
falsePositives = sum((predictions == 1) & (yval == 0));
falseNegatives = sum((predictions == 0) & (yval == 1));
precision = truePositives / (truePositives + falsePositives);
recall = truePositives / (truePositives + falseNegatives);
F1 = (2 * precision * recall) / (precision + recall);
2. Recommender Systems
2.1 Collaborative filtering cost 0 / 20
计算公式:
J(x(1),...,x(nm),θ(1),...,θ(nu))=12∑(i,j):r(i,j)=1((θ(j))Tx(i)−y(i,j))2J(x(1),...,x(nm),θ(1),...,θ(nu))=12∑(i,j):r(i,j)=1((θ(j))Tx(i)−y(i,j))2
X(5x3) θθ(4x3) Y(5x4)
在 cofiCostFunc.m 中添加, 实现没有正则化项的costFunction
error = (X*Theta'-Y) .* R;
J = (1/2)*sum(sum(error .^ 2));
2.2 Collaborative filtering gradient 0 / 30
计算公式:
∂J∂x(i)k=∑j:r(i,j)=1((θ(j))Tx(i)−y(i,j))θ(j)k∂J∂xk(i)=∑j:r(i,j)=1((θ(j))Tx(i)−y(i,j))θk(j)
∂J∂θ(j)k=∑i:r(i,j)=1((θ(j))Tx(i)−y(i,j))x(i)k∂J∂θk(j)=∑i:r(i,j)=1((θ(j))Tx(i)−y(i,j))xk(i)
# 下面两行是2.1中添加的,计算costFunction
error = (X*Theta'-Y) .* R;
J = (1/2)*sum(sum(error .^ 2));
# 下面两行是2.2中的,梯度下降计算
X_grad = error * Theta ;
Theta_grad = error' * X ;
2.3 Regularized cost 0 / 10
计算公式:
JnoReg(x(1),...,x(nm),θ(1),...,θ(nu))=12∑(i,j):r(i,j)=1((θ(j))Tx(i)−y(i,j))2JnoReg(x(1),...,x(nm),θ(1),...,θ(nu))=12∑(i,j):r(i,j)=1((θ(j))Tx(i)−y(i,j))2
reg=λ2∑j=1nu∑k=1n(θ(j)k)2+λ2∑j=1nm∑k=1n(x(j)k)2reg=λ2∑j=1nu∑k=1n(θk(j))2+λ2∑j=1nm∑k=1n(xk(j))2
J=JnoReg+regJ=JnoReg+reg
error = (X*Theta'-Y) .* R;
J_noReg = (1/2)*sum(sum(error .^ 2));
X_grad = error * Theta ;
Theta_grad = error' * X ;
# 下面实现正则化的costFunction
costRegLeft = lambda/2 * sum(sum(Theta.^2));
costRegRight = lambda/2 * sum(sum(X.^2));
Reg = costRegLeft + costRegRight;
J = J_noReg + Reg;
2.4 Gradient with regularization 0 / 10
计算公式:
XgradnoReg=∂J∂x(i)k=∑j:r(i,j)=1((θ(j))Tx(i)−y(i,j))θ(j)kXgradnoReg=∂J∂xk(i)=∑j:r(i,j)=1((θ(j))Tx(i)−y(i,j))θk(j)
ThetagradnoReg=∂J∂θ(j)k=∑i:r(i,j)=1((θ(j))Tx(i)−y(i,j))x(i)kThetagradnoReg=∂J∂θk(j)=∑i:r(i,j)=1((θ(j))Tx(i)−y(i,j))xk(i)
XReg=λx(i)kXReg=λxk(i)
ThetaReg=λθ(i)kThetaReg=λθk(i)
ThetaGrad=ThetagradnoReg+XRegThetaGrad=ThetagradnoReg+XReg
ThetaGrad=ThetagradnoReg+ThetaRegThetaGrad=ThetagradnoReg+ThetaReg
# 下面是计算costFuncton,分两步先计算不带cost的J,再计算reg项
error = (X*Theta'-Y) .* R;
J_noReg = (1/2)*sum(sum(error .^ 2));
costRegLeft = lambda/2 * sum(sum(Theta.^2));
costRegRight = lambda/2 * sum(sum(X.^2));
Reg = costRegLeft + costRegRight;
J = J_noReg + Reg;
# 下面是计算grad,分两步先计算不带reg的grad,再计算reg
X_grad_noReg = error * Theta ;
Theta_grad_noReg = error' * X ;
X_grad = X_grad_noReg + lambda * X;
Theta_grad = Theta_grad_noReg + lambda * Theta;
3. 总结
1 Estimate gaussian parameters 0 / 15
2 Select threshold 0 / 15
3 Collaborative filtering cost 0 / 20
4 Collaborative filtering gradient 0 / 30
5 Regularized cost 0 / 10
6 Gradient with regularization 0 / 10