服从正态分布的正弦函数期望
服从正态分布的正弦函数、余弦函数期望。
如果X服从均值为 μ \mu μ，方差为 σ 2 \sigma^2 σ2的正态分布，计算sin(X)与cos(X)的数学期望。
利用特征函数(Characteristic Function)Wiki-Characteristic Function，我们知道 X ∼ N ( μ , σ 2 ) X\sim N(\mu, \sigma^2) X∼N(μ,σ2)的特征函数为：
φ X ( t ) = E ( e i t X ) = e x p ( i μ t − σ 2 t 2 2 ) = e x p ( − σ 2 t 2 / 2 ) e x p ( i μ t ) \varphi_{X(t)}=E(e^{itX})=exp\left({i\mu t - \dfrac{\sigma^2t^2}{2}}\right)=exp(-\sigma^2t^2/2)exp(i\mu t) φX(t)=E(eitX)=exp(iμt−2σ2t2)=exp(−σ2t2/2)exp(iμt)
根据欧拉公式：
e i x = cos ( x ) + i sin ( x ) e^{ix} = \cos(x)+i\sin(x) eix=cos(x)+isin(x)
E ( e i t X ) = e x p ( i μ t − σ 2 t 2 2 ) = e x p ( − σ 2 t 2 / 2 ) e x p ( i μ t ) = e x p ( − σ 2 t 2 / 2 ) cos ( μ t ) + i sin ( μ t ) = e x p ( − σ 2 t 2 / 2 ) cos ( μ t ) + e x p ( − σ 2 t 2 / 2 ) sin ( μ t ) i = R e a l ( E ( e i t X ) ) + i I m ( E ( e i t X ) ) \begin{aligned} E(e^{itX})&=exp\left({i\mu t - \dfrac{\sigma^2t^2}{2}}\right) \\ &=exp(-\sigma^2t^2/2)exp(i\mu t) \\ &=exp(-\sigma^2t^2/2)\left\\cos(\\mu t)+i\\sin(\\mu t)\\right\\ &=exp(-\sigma^2t^2/2)\cos(\mu t) +exp(-\sigma^2t^2/2)\sin(\mu t)i \\ &=Real(E(e^{itX}))+iIm(E(e^{itX})) \end{aligned} E(eitX)=exp(iμt−2σ2t2)=exp(−σ2t2/2)exp(iμt)=exp(−σ2t2/2)cos(μt)+isin(μt)=exp(−σ2t2/2)cos(μt)+exp(−σ2t2/2)sin(μt)i=Real(E(eitX))+iIm(E(eitX))
上述数学期望变为：
E ( e i t X ) = E ( cos ( t X ) + i sin ( t X ) ) = E ( c o s ( t X ) ) + i E ( s i n ( t X ) ) = R e a l ( E ( e i t X ) ) + i I m ( E ( e i t X ) ) \begin{aligned} E(e^{itX})&=E\left(\cos(tX)+i\sin(tX)\right)\\ &= E(cos(tX)) + iE(sin(tX))\\ &=Real(E(e^{itX}))+iIm(E(e^{itX})) \end{aligned} E(eitX)=E(cos(tX)+isin(tX))=E(cos(tX))+iE(sin(tX))=Real(E(eitX))+iIm(E(eitX))
对比上述的实数部分和虚数部分，得到：
E ( cos ( t X ) ) = e x p ( − σ 2 t 2 / 2 ) cos ( μ t ) E ( sin ( t X ) ) = e x p ( − σ 2 t 2 / 2 ) s i n ( μ t ) E(\cos(tX))=exp(-\sigma^2t^2/2)\cos(\mu t)\\ E(\sin(tX))=exp(-\sigma^2t^2/2)sin(\mu t) E(cos(tX))=exp(−σ2t2/2)cos(μt)E(sin(tX))=exp(−σ2t2/2)sin(μt)
最后，当t=1的时候：
E ( cos ( X ) ) = e x p ( − σ 2 / 2 ) cos ( μ ) E ( sin ( X ) ) = e x p ( − σ 2 / 2 ) s i n ( μ ) E(\cos(X))=exp(-\sigma^2/2)\cos(\mu)\\ E(\sin(X))=exp(-\sigma^2/2)sin(\mu) E(cos(X))=exp(−σ2/2)cos(μ)E(sin(X))=exp(−σ2/2)sin(μ)
参考资料：Mean and variance of Y=cos(bX) when X has a Gaussian distribution
应用：IPE位置编码中，对服从高斯分布正弦函数的数学期望计算：
python
# Code Source:
# https://github.com/liuyuan-pal/NeRO/blob/3b4d421a646097e7d59557c5ea24f4281ab38ef1/network/field.py#L369-L378
def expected_sin(mean, var):
"""Compute the mean of sin(x), x ~ N(mean, var)."""
return torch.exp(-0.5 * var) * torch.sin(mean) # large var -> small value.
def IPE(mean,var,min_deg,max_deg):
scales = 2**torch.arange(min_deg, max_deg)
shape = mean.shape[:-1] + (-1,)
scaled_mean = torch.reshape(mean[..., None, :] * scales[:, None], shape)
scaled_var = torch.reshape(var[..., None, :] * scales[:, None]**2, shape)
return expected_sin(torch.concat([scaled_mean, scaled_mean + 0.5 * np.pi], dim=-1), torch.concat([scaled_var] * 2, dim=-1))