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‎extend/backup/遗传算法.md

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# 遗传算法
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## 核心步骤
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- 初始化种群：首先，随机生成一个包含多个个体（解）的初始种群。每个个体都代表问题的一个可能解。
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- 适应度评估：对种群中的每个个体进行适应度评估，即计算每个个体的适应度分数，该分数表示个体对于问题的解决方案的优劣程度。
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- 选择：根据适应度分数选择个体，通常是根据适应度分数的高低进行概率性选择，以便更有可能选择到适应度高的个体。常用的选择方法包括轮盘赌选择、锦标赛选择等。
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- 交叉：从被选择的个体中选取一对（或多对）进行交叉操作，生成新的个体。交叉操作模拟了生物界的基因交换过程，可以通过不同的方式进行，如单点交叉、多点交叉、均匀交叉等。
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- 变异：对交叉后的个体进行变异操作，以保持种群的多样性。变异操作是随机改变个体染色体中的一些基因，以产生新的解。常见的变异操作包括基因翻转、基因位移、基因重组等。
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- 替换：用新生成的个体替换原来的个体，形成新一代种群。替换策略可以是全局替换（直接用新生成的个体替换原有种群）或局部替换（保留一部分原有个体，只替换其中一部分）。
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- 重复迭代：重复执行步骤 2 到步骤 6，直到达到停止条件（如达到最大迭代次数、找到满意的解等）为止。
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- 输出结果：在停止迭代后，输出最终的优化结果，通常是种群中适应度最高的个体所对应的解。
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```mermaid
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graph TD
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A[初始化种群] --> B[适应度评估]
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B --> C[选择]
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C --> D[交叉]
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D --> E[变异]
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E --> F[替换]
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F --> G{满足停止条件?}
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G -->|是| H[输出结果]
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G -->|否| B
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```
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## 案例代码
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```python
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import numpy
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import math
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import random
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# 确定一个目标函数， _max=True表示目标最大值，反之最小值
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def objectiveFunc(x : numpy.ndarray, _max : bool = True) -> numpy.ndarray: # _max表示目标最大值，反之最小值
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d = x * numpy.sin(2 * x) - 5 * x * numpy.cos(x * 2)
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return d if _max else -1 * d
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# 基于均匀分布随机生成种群
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def genPop(size : int, start : float, end : float) -> numpy.ndarray:
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return numpy.random.uniform(start, end, size)
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# 选择操作，根据method指定的策略来，默认是轮盘赌方式，可以拓展
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def select(pop : numpy.ndarray, size : int, method : str = '轮盘赌', _max : bool = True) -> numpy.ndarray:
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obj = objectiveFunc(pop, _max)
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match method:
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case '轮盘赌':
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minObj = numpy.abs(numpy.min(obj)) # 用于修正
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obj += minObj
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posible = obj / numpy.sum(obj)
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return numpy.random.choice(pop, size, p=posible, replace=False) # False无放回抽样
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case _:
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...
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# 编码方式
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def encode(x: float, start: float, end: float, step: int = 100) -> str:
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"""
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把数据平分成step份，然后x落在哪部分，获取对应的二进制值
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:param x:
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:param start:
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:param end:
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:param step:
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:return:
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"""
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# 先确定二进制范围同时也规定了统一二进制长度，不够补全0
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range_bin = math.ceil(math.log2(step)) # 2^0 ~ 2^range_bin继续写
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# 计算每个区间的长度
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interval_length = (end - start) / step
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# 确定 x 属于哪个区间
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bin_index = min(int((x - start) / interval_length), step - 1)
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# 将区间索引转换为二进制字符串
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bin_str = bin(bin_index)[2:].zfill(range_bin)
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return bin_str
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# 与编码方式对应的解码方式，有瑕疵，因为精度问题
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def decode(bin_str: str, start: float, end: float, step: int = 100) -> float:
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"""
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将二进制字符串解码为原始数值
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:param bin_str:
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:param start:
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:param end:
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:param step:
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:return:
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"""
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# 先确定二进制范围同时也规定了统一二进制长度
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range_bin = math.ceil(math.log2(step))
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# 将二进制字符串转换为区间索引
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bin_index = int(bin_str, 2)
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# 计算每个区间的长度
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interval_length = (end - start) / step
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# 根据区间索引计算值
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decoded_value = start + bin_index * interval_length + interval_length / 2
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if decoded_value < -4:
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decoded_value = -4
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elif decoded_value > 4:
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decoded_value = 4
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else:
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pass
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return decoded_value
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# 交叉策略
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# 随机从某点开始连续交叉，用random；交叉默认method方式是的第一个跟第二个交叉，第二个跟第三个交叉，以此类推；
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# 交叉的时候采用格外策略：每两个交叉的时候生成1~5个新数据用于填放到新种群，最后在新种群中调用选择选择算法
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# length也采用随机策略，可以不定个数交叉, range(1, 3)表示1、2、3个交叉的任意一种
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# 而且在生成1~5个新数据是有概率的{1 : 0.75, 2 : 0.2, 3 : 0.03, 4 : 0.015, 5 : 0.005}
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# 注意：起点+length一定不会超过编码长度
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# method指定交叉方式，可以另行拓展
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def cross(
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start: float, end: float, step: int, size: int,
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pop: numpy.ndarray, length: range, method: str = '1-2 2-3 ...', _max : bool = True
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) -> numpy.ndarray:
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newGensCodes = []
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# {1 : 0.75, 2 : 0.2, 3 : 0.03, 4 : 0.015, 5 : 0.005}
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lastGenCode = [encode(g, start, end, step) for g in pop]
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mexBinCodeLen: int = math.ceil(math.log2(step))
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for i in range(len(lastGenCode) - 1):
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p1 = list(lastGenCode[i])
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p2 = list(lastGenCode[1 + i])
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haveGenNumbers = numpy.random.choice(numpy.arange(1, 6), size=1, p=[.75, .2, .03, .015, .005])[0]
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# print(haveGenNumbers)
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for j in range(haveGenNumbers):
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random_location = numpy.random.randint(
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0, mexBinCodeLen - random.randint(length.start, length.stop + 1)
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)
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# print(f'random_location = {random_location}')
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# print(f'p1 = {p1}')
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# 交叉得到的两个父类另起变量名字，根据策略当孩子
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for k in range(0, random_location, 1):
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p1[k], p2[k] = p2[k], p1[k]
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# 二者选一入新生代
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p11 = ''.join(p1)
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p21 = ''.join(p2)
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newGensCodes.append(random.choice([p11, p21]))
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returnewGens = [decode(i, start, end, step) for i in newGensCodes]
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return select(numpy.array(returnewGens), size, _max=_max)
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# 变异策略，这里更改随机位置的基因
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def mutations(pop : numpy.ndarray, start: float, end: float, step: int) -> numpy.ndarray:
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mexBinCodeLen : int = math.ceil(math.log2(step))
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lastGenCodeLists = [list(encode(g, start, end, step)) for g in pop]
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k = []
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g : list
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for g in lastGenCodeLists:
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for i in range(math.ceil(step / pop.size)): # 保证原种群大小不变
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_s = g[random.randint(0, mexBinCodeLen - 1)]
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s = g.copy()
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s[random.randint(0, mexBinCodeLen - 1)] = '0' if _s == '1' else '1'
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k.append(s)
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return numpy.array(
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[decode(d, start, end, step) for d in [''.join(g) for g in k]]
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)
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# 迭代100次试试（即模拟一个种群演变了100代）
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_genPop = genPop(1000, -4, 4)
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for i in range(100):
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_select = select(_genPop, 100, _max=True)
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_cross = cross(-4, 4, 1000, 100, _select, range(1, 3))
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_mutations = mutations(_cross, -4, 4, 1000)
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_genPop = _mutations
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print(_genPop)
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"""
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[-3.428 -3.428 -3.3 -3.428 -3.94 -1.38 -3.94 -1.38 -3.428 -2.404
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-3.364 -3.364 -3.428 -2.34 -3.364 0.732 -3.428 -3.364 -3.38 -3.372
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-3.212 -3.18 0.916 -3.436 -3.18 -2.156 -3.18 -3.18 -3.18 -3.18
176+
-3.052 -3.324 -3.196 -3.052 -3.052 -3.06 -3.052 -3.068 -3.068 -3.052
177+
-3.172 -3.172 -1.124 -3.188 0.924 -3.172 -3.172 -3.18 -3.684 -3.684
178+
-3.148 -3.02 -3.084 -2.988 -3.02 -0.972 -0.972 -3.02 -3.148 -3.276
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-3.444 -3.364 -3.38 -3.38 -3.38 0.716 -1.332 -3.38 -3.38 -3.38
180+
-3.284 -3.284 -3.284 -3.284 -3.284 -3.348 0.812 -3.284 -3.796 -3.284
181+
-3.14 -2.372 0.7 -3.396 -3.364 -3.396 -3.908 0.7 -3.396 -3.404
182+
-3.38 -3.26 -3.252 -3.236 0.844 0.844 -2.228 -3.252 -3.252 -3.252
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185+
-2.996 -3.252 -3.252 -3.252 -3.38 -3.764 -3.252 -3.252 -3.764 -2.996
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-3.556 -0.996 -3.044 -3.044 -3.172 -3.044 -3.044 -3.044 -3.044 -3.052
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-3.476 -3.348 -3.86 -3.348 -3.348 -3.348 -3.348 -3.348 0.748 -1.3
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-3.188 -2.164 -3.22 -3.188 -3.188 -3.188 -3.444 -3.188 -3.188 -3.188
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-3.908 -3.396 0.7 -2.372 -3.396 0.7 0.7 -3.14 -2.372 -3.404
191+
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-1.14 0.908 -3.188 -3.188 -3.188 -3.188 -3.124 -3.188 -3.188 -3.188
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-3.052 -3.3 -3.308 -3.308 -3.3 -3.308 -3.82 -3.308 -3.308 -3.308
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-2.028 -3.18 -3.052 -3.052 -3.564 -3.052 -3.052 1.044 -3.052 -3.052
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200+
-2.02 -3.044 -2.02 -3.076 -3.076 -3.044 -0.996 -3.044 -3.076 -3.044
201+
-3.108 -3.364 -2.34 -3.364 -3.38 -3.364 -3.876 -3.428 -3.364 -1.316
202+
-3.428 -3.46 -3.364 -3.428 -3.3 -3.428 -3.444 -3.428 -3.428 -3.3
203+
-3.244 -2.988 -3.244 -3.244 -3.244 -3.244 -3.244 0.852 -3.244 -3.244
204+
-3.364 -3.372 -2.348 -3.372 -3.372 -2.348 -2.348 -3.388 -3.372 -1.324
205+
-3.364 -3.364 -3.876 -3.876 -3.364 -3.364 0.732 -3.364 -3.372 -3.364
206+
-3.444 -1.268 -3.316 -3.316 -3.316 -3.316 -3.316 -3.252 -2.292 -3.316
207+
-3.412 -3.284 -3.284 -3.252 -3.412 0.812 0.812 -3.284 -2.26 -2.26
208+
-3.372 -3.436 -3.372 -3.116 -3.364 -3.404 -3.372 -3.372 -3.244 -3.372
209+
-3.26 0.844 -3.252 -3.252 0.844 -3.26 -3.252 -3.26 -3.252 -3.252
210+
1.036 -3.092 -3.06 -1.012 -3.188 -3.572 -3.06 -3.06 -3.06 -3.06
211+
-2.98 -2.212 -3.236 -3.236 -3.748 -3.268 -3.236 -3.3 -3.236 -3.244
212+
-3.068 -3.068 -3.068 -3.068 -3.068 -3.324 1.028 1.028 -3.052 -3.068
213+
-3.236 -3.236 -3.236 -2.98 -3.748 -3.236 -3.3 -3.236 -3.236 -2.98
214+
-3.244 -3.372 -3.884 -3.372 -3.884 -3.372 0.724 -3.372 -3.244 -3.372
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-3.06 -3.044 -2.02 -3.172 -3.044 -3.3 -3.556 -3.3 -3.044 -3.044
216+
-3.332 -3.268 -3.268 -3.268 -3.268 -3.268 -3.268 -3.276 -3.396 -3.268
217+
-3.316 -3.316 -3.828 -3.316 -3.316 -3.316 -3.316 -3.316 0.78 -3.316
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-3.268 -3.396 -3.396 -3.396 -3.364 -3.396 -3.412 -3.46 -1.348 -3.46
219+
-3.236 -3.364 -3.236 -3.748 -2.98 -3.236 -3.236 -3.748 -3.236 -3.236
220+
-3.284 -3.292 -3.284 -3.292 -1.244 -3.26 -3.276 -3.292 0.804 -3.292
221+
-3.3 -3.812 -3.3 -3.3 -3.3 -3.3 -3.3 0.796 -3.332 -3.428
222+
-3.5 -3.5 -2.988 -2.988 -3.244 -2.988 -2.988 -2.988 -2.988 -3.02
223+
-3.38 -3.252 -3.252 -3.252 -3.252 -1.204 -3.284 -3.252 -2.996 -3.284
224+
-3.348 -3.316 -3.348 -3.348 -3.348 -3.86 -1.3 -1.3 -3.316 -3.348
225+
-3.836 -3.812 -3.316 -1.78 -3.572 -3.828 -3.764 -3.316 -3.316 -2.804
226+
-3.188 -3.188 -3.444 0.652 -3.38 -2.42 -3.188 -3.316 -2.42 -3.316
227+
-3.316 -3.444 -3.316 -3.324 0.78 -3.3 -2.292 -3.3 -1.268 -3.252
228+
-3.292 0.804 -3.292 -1.244 -1.244 -3.804 -3.276 -3.036 -3.292 -3.292
229+
-3.012 -3.076 -3.028 -3.076 -3.14 -3.524 -2.98 -3.012 -3.012 -1.988
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-3.188 -3.124 0.908 0.908 -3.06 -3.188 -3.188 -3.7 -1.14 -3.188
231+
-3.316 -3.324 -3.252 -3.316 -3.316 -3.316 -3.316 -3.316 -3.324 -3.316
232+
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234+
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235+
-3.428 -3.428 -3.428 -3.3 -3.364 0.668 -3.364 -3.428 -1.38 0.668
236+
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237+
-1.22 -1.284 -1.22 -1.188 -1.348 -1.22 -1.188 -1.22 -1.188 -1.22
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-3.076 -3.076 -1.028 -3.084 -3.076 1.02 -2.052 -3.076 -3.076 -3.076
241+
-3.028 -3.028 -2.996 -3.028 -3.028 -3.54 -2.996 -2.004 -3.028 -0.98
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-3.444 -3.316 -3.316 -2.292 -3.316 -3.828 0.78 -2.292 -3.316 -3.348
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-3.188 -3.188 -3.188 -3.124 -3.188 -2.164 -3.196 -3.188 -3.22 -3.188
244+
-3.076 -3.044 -3.044 -3.044 -3.172 -3.06 -3.044 -3.044 -3.3 -3.556
245+
-3.044 -3.3 -3.3 -3.3 -3.3 -3.3 -3.3 -3.428 -3.3 0.796
246+
-3.188 -3.124 -3.7 -3.188 -2.164 -3.188 -3.7 -1.14 -3.188 -3.444
247+
-3.764 -3.38 -3.236 -3.252 -3.252 -3.316 -2.996 -3.252 -3.38 -3.236
248+
-3.308 -3.3 -3.3 -3.3 -3.3 -3.3 -3.308 -3.3 -3.3 -1.252
249+
-3.18 0.924 0.924 -3.428 -3.172 -3.172 -3.108 -2.148 -2.148 -3.172
250+
-3.252 -2.356 -3.38 -3.892 -3.252 0.716 -3.38 -3.38 -3.38 0.716
251+
-3.188 -3.38 -3.444 -3.956 -3.444 -3.452 -3.428 -3.444 -3.316 0.652
252+
-3.068 -3.068 -3.068 -3.068 1.028 -3.052 -3.068 -3.004 -3.068 -3.068
253+
-3.1 -1.02 -3.068 -3.068 -3.068 -2.044 -3.052 -3.068 -3.068 -3.58
254+
-2.98 -2.98 -3.492 -2.996 -1.956 -3.492 -2.98 -3.108 1.116 -3.044
255+
-3.236 -3.236 -2.212 -3.364 -3.364 -3.236 -3.748 -3.236 -3.748 -3.236
256+
-2.18 -2.18 -3.204 -2.18 -3.172 -3.204 -3.204 -3.204 -2.18 -1.156
257+
-3.324 -3.316 -3.316 -1.268 -3.444 -3.324 -3.828 -3.316 -3.316 -3.316
258+
-3.06 -3.06 -2.996 -3.06 -3.06 -3.092 -3.044 -3.092 -2.036 -3.092
259+
-3.044 0.796 -3.236 -3.3 -3.3 -1.252 -3.308 0.796 -3.3 -3.3
260+
-3.3 -3.3 -3.3 -3.044 -3.044 -3.3 -3.3 -3.3 -3.3 -3.308
261+
-3.236 -3.3 -3.244 -3.236 -3.236 -3.236 -3.252 -3.748 -3.236 -3.3
262+
-2.004 -3.028 -2.996 -3.028 -3.028 -3.028 -2.996 -3.028 -3.028 -3.028
263+
-3.028 -3.036 -2.996 1.068 -0.98 -3.028 -3.028 -2.996 -3.012 -3.156
264+
0.844 -2.228 -3.252 -3.316 -3.26 -3.764 -2.228 -3.252 -3.316 -3.252
265+
-3.092 -3.092 -3.092 -3.092 -3.092 -3.092 1.004 -3.348 -3.076 -2.068
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-3.444 -3.444 -3.444 -3.428 -3.444 -3.38 -3.444 -3.444 -3.452 -1.396
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-3.396 -3.46 -2.372 0.7 -3.396 -3.396 -3.412 -3.412 -3.268 -3.14
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-3.428 -3.364 -3.364 -2.34 -3.396 -3.364 -1.316 -3.364 -3.364 -3.364
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-3.044 -3.044 1.052 -3.3 -3.044 -3.044 -3.044 -3.044 -0.996 -3.044
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-3.332 -3.092 -3.348 -3.348 -3.348 -2.324 -3.316 -3.316 -3.348 -1.3
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-3.06 -2.996 -3.06 -3.572 -3.06 -3.06 -3.188 -2.036 -3.06 -3.188
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-2.988 0.852 -1.196 -3.756 -3.244 -3.244 -3.244 -3.244 -2.22 -2.988]
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"""
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```
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## 可视化结果分布
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```python
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import matplotlib.pyplot as plt
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plt.hist(_genPop, bins=30, color='skyblue', edgecolor='black')
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plt.xlabel('Value X')
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plt.ylabel('Frequency')
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plt.title('Population Distribution')
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plt.grid(True)
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plt.show()
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```
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### 分布图
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### 目标函数图像

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