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Diffstat (limited to '2-point.tex')
| -rw-r--r-- | 2-point.tex | 87 |
1 files changed, 73 insertions, 14 deletions
diff --git a/2-point.tex b/2-point.tex index b4cb216..bb1b625 100644 --- a/2-point.tex +++ b/2-point.tex @@ -392,6 +392,50 @@ What about the average for the Hessian terms? &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu(\pmb\sigma_a\mid E_0,\mu_0)\right]\,F(\beta\mid E_1,\mu_1,q,\pmb\sigma_1) \end{aligned} \] +\[ + \sum_a^m(i\hat{\pmb\sigma}_{\pmb\sigma_a}\cdot\partial_a-\hat\beta_0)H(\pmb\sigma_a) + + + \sum_b^n(i\hat{\mathbf s}_{\mathbf s_b}\cdot\partial_b-\hat\beta_1)H(\mathbf s_b) + + + \sum_c^\ell(\mathbf x_c\cdot\partial_{\mathbf s_1})^2H(\mathbf s_1) +\] +\begin{align*} + &\sum_{ab}^\ell(\mathbf x_a\cdot\partial_{\mathbf s_1})^2(\mathbf x_b\cdot\partial_{\mathbf s_1'})^2\overline{H(\mathbf s_1)H(\mathbf s_1')}\\ + &=(\mathbf x_a\cdot\mathbf s_1)^2(\mathbf x_b\cdot\mathbf s_1)^2f''''(1) + +2(\mathbf x_a\cdot\mathbf s_1)(\mathbf x_b\cdot\mathbf s_1)(\mathbf x_a\cdot\mathbf x_b)f'''(1) + +(\mathbf x_a\cdot\mathbf x_b)^2f''(1) \\ + &=f''(1)\sum_{ab}^\ell A_{ab} +\end{align*} +\begin{align*} + &\sum_{a}^\ell\sum_b^n(i\hat{\mathbf s}_{\mathbf s_b}\cdot\partial_b-\hat\beta_1)(\mathbf x_a\cdot\partial_{\mathbf s_1})^2\overline{H(\mathbf s_1)H(\mathbf s_b)}\\ + &=-\hat\beta_1(\mathbf x_a\cdot\mathbf s_b)^2f''(C^{11}_{1b}) + +i(\hat{\mathbf s}_b\cdot\mathbf s_1)(\mathbf x_a\cdot\mathbf s_b)^2f'''(C^{11}_{1b}) + +2i(\hat{\mathbf s}_b\cdot\mathbf x_a)(\mathbf x_a\cdot\mathbf s_b)f''(C^{11}_{1b}) \\ + &= + \sum_{a=1}^\ell\sum_{b=2}^n\left[ + -\hat\beta_1(X^1_{ab})^2f''(C^{11}_{1b})-(X^1_{ab})^2R^{11}_{1b}f'''(C^{11}_{1b}) + -2X^1_{ab}\hat X_{ab}f''(C^{11}_{1b}) + \right] \\ + &=\ell\left[-\hat\beta_1x_1^2\sum_{b=2}^nf''(C^{11}_{1b}) + -x_1^2\sum_{b=2}^nR^{11}_{1b}f'''(C^{11}_{1b}) + -x_1\hat x_1\sum_{b=2}^nf''(C^{11}_{1b}) + \right] +\end{align*} +\begin{align*} + &\sum_{a}^\ell\sum_b^m(i\hat{\pmb\sigma}_b\cdot\partial_b-\hat\beta_0)(\mathbf x_a\cdot\partial_{\mathbf s_1})^2\overline{H(\mathbf s_1)H(\pmb\sigma_b)}\\ + &=-\hat\beta_0(\mathbf x_a\cdot\pmb \sigma_b)^2f''(C^{01}_{1b}) + +i(\hat{\pmb \sigma}_b\cdot\pmb \sigma_1)(\mathbf x_a\cdot\pmb \sigma_b)^2f'''(C^{01}_{1b}) + +2i(\hat{\pmb \sigma}_b\cdot\mathbf x_a)(\mathbf x_a\cdot\pmb \sigma_b)f''(C^{01}_{1b}) \\ + &= + \sum_{a=1}^\ell\sum_{b=1}^m\left[ + -\hat\beta_0(X^0_{ab})^2f''(C^{01}_{1b})-(X^0_{ab})^2R^{01}_{1b}f'''(C^{01}_{1b}) + -2X^0_{ab}\hat X^0_{ab}f''(C^{01}_{1b}) + \right] \\ + &=\ell\left[-\hat\beta_0x_0^2f''(q) + -x_0^2r_{10}f'''(q) + -x_0\hat x_0f''(q) + \right] +\end{align*} \begin{align} &\log\det @@ -523,25 +567,36 @@ What about the average for the Hessian terms? X_0 = \begin{subarray}{l} - \hphantom{[}\begin{array}{ccc}\leftarrow&m&\rightarrow\end{array}\hphantom{\Bigg]}\\ + \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\ \left[ \begin{array}{ccc} x_0&\cdots&x_0\\ + 0&\cdots&0\\ \vdots&\ddots&\vdots\\ - x_0&\cdots&x_0 + 0&\cdots&0 \end{array} \right]\begin{array}{c} - \uparrow\\\ell\\\downarrow + \\\uparrow\\m-1\\\downarrow \end{array}\\ \vphantom{\begin{array}{c}n\end{array}} \end{subarray} && \hat X_0 - =\begin{bmatrix} - \hat x_0&\cdots&\hat x_0\\ - \vdots&\ddots&\vdots\\ - \hat x_0&\cdots&\hat x_0 - \end{bmatrix} + = + \begin{subarray}{l} + \hphantom{[}\begin{array}{ccc}\leftarrow&\ell&\rightarrow\end{array}\hphantom{\Bigg]}\\ + \left[ + \begin{array}{ccc} + \hat x_0&\cdots&\hat x_0\\ + 0&\cdots&0\\ + \vdots&\ddots&\vdots\\ + 0&\cdots&0 + \end{array} + \right]\begin{array}{c} + \\\uparrow\\m-1\\\downarrow + \end{array}\\ + \vphantom{\begin{array}{c}n\end{array}} + \end{subarray} && X_1 = @@ -549,10 +604,10 @@ What about the average for the Hessian terms? \hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\ \left[ \begin{array}{ccc} - q&\cdots&q\\ 0&\cdots&0\\ + x_1&\cdots&x_1\\ \vdots&\ddots&\vdots\\ - 0&\cdots&0 + x_1&\cdots&x_1 \end{array} \right]\begin{array}{c} \\\uparrow\\m-1\\\downarrow @@ -560,15 +615,19 @@ What about the average for the Hessian terms? \vphantom{\begin{array}{c}n\end{array}} \end{subarray} && - D^{01} + \hat X_1 =\begin{bmatrix} - d_{01}&\cdots&d_{01}\\ - 0&\cdots&0\\ + \hat x_1^0&\cdots&\hat x_1^0\\ + \hat x_1^1&\cdots&\hat x_1^1\\ \vdots&\ddots&\vdots\\ - 0&\cdots&0 + \hat x_1^1&\cdots&\hat x_1^1 \end{bmatrix} \end{align} +\[ + 2(A-X^TC^{-1}X)^{-1}X^TC^{-1} +\] + \paragraph{Acknowledgements} \paragraph{Funding information} |