变限积分求导公式
假设有函数定义为:
\[K(x)=\int_{\phi(x)}^{\Psi(x)}f(t)dt \\ \frac{dK(x)}{dx} = f[\Psi(x)]\Psi(x)^{\prime} - f[\phi(x)]\phi(x)^{\prime} \]量化失真与最优标量量化
对于N个量化区间的失真定义为：
\[ D=\sum_{i=1}^{N}(\int_{t_i}^{t_{i+1}}(x-\hat{x_i})^2f(x)dx) \]当D能取得最小值时，即:
\[\frac{\partial(D)}{\partial{t_i}} = 0 \\ 令 M(x)= \int_{t_{i-1}}^{t_{i}}(x-\hat{x_{i-1}})^2f(x)dx + \int_{t_i}^{t_{i+1}}(x-\hat{x_i})^2f(x)dx \\ 则 \frac{\partial(D)}{\partial{t_i}} = \frac{\partial(M(x))}{\partial{t_i}} = 0 \\ (t_{i} - \hat{x_{i-1}})^2f(t_i)(t_i)^\prime - (t_{i-1} - \hat{x_{i-1}})^2f(t_{i-1})(t_{i-1})^{\prime} + (t_{i+1} - \hat{x_{i}})^2f(t_{i+1})(t_{i+1})^\prime - (t_{i} - \hat{x_{i}})^2f(t_{i})(t_{i})^\prime \\ = (t_{i} - \hat{x_{i-1}})^2f(t_i) - (t_{i} - \hat{x_{i}})^2f(t_{i}) = 0 \]\(\because \qquad t_i < \hat{x_i} \\ \therefore\)
\[ t_i = \frac{\hat{x_{i-1}} + \hat{x_i}}{2} \qquad i = {1,2 ... N} \qquad (1) \]令\(\frac{\partial{D}}{\partial{\hat{x_i}}} = 0\), 对该式展开求导，可得
\[令 \qquad M(x) = \int_{t_i}^{t_{i+1}}(x-\hat{x_i})^2f(x)dx \\ 则 \qquad M(X) =\int_{t_i}^{t_{i+1}}(x^2f(x))dx -\int_{t_i}^{t_{i+1}}(2x\hat{x_i})f(x)dx + \int_{t_i}^{t_{i+1}}(\hat{x_i}^2f(x))dx \\ \frac{\partial{M(x)}}{\partial{\hat{x_i}}} = \\ \int_{t_i}^{t_{i+1}}(2\hat{x_i})f(x)dx - \int_{t_i}^{t_{i+1}}(2xf(x))dx = \\ \int_{t_i}^{t_{i+1}}[2(\hat{x_i} - x)]f(x)dx = 0 \]$\therefore $
\[\qquad\qquad \hat{x_i} = \frac{\int_{t_i}^{t_{i+1}}{xf(x)}dx}{\int_{t_i}^{t_{i+1}}{f(x)}dx} \qquad i = 1,2, ... N-1 \qquad （2） \]式(1) 和 式（2）构成最优标量量化器的必要条件，即影响量化是真的所有变量的偏导数都为0，这变量为: 边界 量化估计值
其中式(2)和物理学中的质心公式形式相同，因此也可以理解成区间的质心
一般来说，直接求解这两个公式比较麻烦，现实中的算法步骤是：
代码实现
def interval_MSE(x,t1,t2):
return integrate.quad(lambda t: ((t - x)**2) * f(t), t1, t2)[0]
该函数用于计算积分:
\[\int_{t1}^{t2}[(t-x)^2f(t)]dt \]def MSE(t,x):
s = interval_MSE(x[0], -float('Inf'), t[0]) + interval_MSE(x[-1], t[-1], float('Inf'))
for i in range(1,len(x)-1):
s = s + interval_MSE(x[i], t[i-1], t[i])
return s
该函数用于计算积分:
\[MSE = \int_{-\infty}^{t_0}[(t-x_0)^2f(t)]dt + \int_{t_n}^{\infty}[(t-x_n)^2f(t)]dt \qquad 首尾特殊处理 \\ MSE += \sum_{i=1}^{N-1}[\int_{t_{i-1}^{t_i}}(t-x_i)^2f(t)dt] \]其中n表示序列中的最后一个元素
def centroid(t1,t2):
if integrate.quad(f, t1, t2)[0] == 0 or t1 == t2:
return 0
else:
return integrate.quad(lambda t:t*f(t), t1, t2)[0] / integrate.quad(f, t1, t2)[0]
该函数用于计算区间质心：
\[\frac{\int_{t_1}^{t_2}(t*f(t))dt}{\int_{t_1}^{t_2}(f(t))dt} \]# error_threshold 误差阈值
# 不过问题是，这个东西如何保证算法一定收敛
def maxlloyd(t,x,error_threshold):
# 计算当前的MSE
e = MSE(t,x)
error = [e]
c = 0
# 控制300次以内，以及误差小于阈值
while e > error_threshold and c < 300:
c = c+1
# 奇数
if c%2 == 1:
# adjust thresholds
# 更新边界
for i in range(len(t)):
t[i] = 0.5 * ( x[i] + x[i+1] )
else:
# adjust levels
# 更新量化估计值
# 负∞和t_0 的质心
x[0] = centroid(-float('Inf'), t[0])
# t_n 和∞的质心
x[-1] = centroid(t[-1], float('Inf'))
# 其余每个区间的质心
for i in range(1,len(x)-1):
x[i] = centroid(t[i-1], t[i])
# 计算误差
e = MSE(t,x)
# 增加误差记录
error.append(e)
print(e)
return x,t,error