diff --git a/doc/pub/Regression/html/._Regression-bs042.html b/doc/pub/Regression/html/._Regression-bs042.html index 5aed6efbe..525941a4c 100644 --- a/doc/pub/Regression/html/._Regression-bs042.html +++ b/doc/pub/Regression/html/._Regression-bs042.html @@ -407,9 +407,14 @@ MathJax.Hub.Config({ numpy and scipy.linalg at the cost of being slower. ''' U, s, VT = np.linalg.svd(A) +# print('test U') +# print( (np.transpose(U) @ U - U @np.transpose(U))) +# print('test VT') +# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) print(U) print(s) print(VT) + D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): D[i,i]=s[i] diff --git a/doc/pub/Regression/html/._Regression-bs044.html b/doc/pub/Regression/html/._Regression-bs044.html index 932067660..3f96d2e2e 100644 --- a/doc/pub/Regression/html/._Regression-bs044.html +++ b/doc/pub/Regression/html/._Regression-bs044.html @@ -409,7 +409,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).

-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \). +The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).

diff --git a/doc/pub/Regression/html/Regression-reveal.html b/doc/pub/Regression/html/Regression-reveal.html index fb956bca5..8d6fb0c6d 100644 --- a/doc/pub/Regression/html/Regression-reveal.html +++ b/doc/pub/Regression/html/Regression-reveal.html @@ -2035,9 +2035,14 @@ Similarly, Mehta et a numpy and scipy.linalg at the cost of being slower. ''' U, s, VT = np.linalg.svd(A) +# print('test U') +# print( (np.transpose(U) @ U - U @np.transpose(U))) +# print('test VT') +# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) print(U) print(s) print(VT) + D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): D[i,i]=s[i] @@ -2104,7 +2109,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).

-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \). +The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \). diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html index 5f4d11cdd..76339995a 100644 --- a/doc/pub/Regression/html/Regression-solarized.html +++ b/doc/pub/Regression/html/Regression-solarized.html @@ -2039,9 +2039,14 @@ Similarly, Mehta et a numpy and scipy.linalg at the cost of being slower. ''' U, s, VT = np.linalg.svd(A) +# print('test U') +# print( (np.transpose(U) @ U - U @np.transpose(U))) +# print('test VT') +# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) print(U) print(s) print(VT) + D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): D[i,i]=s[i] @@ -2106,7 +2111,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).

-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \). +The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).











diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index d34f9242f..636ef6656 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -2044,9 +2044,14 @@ Similarly, Mehta et a numpy and scipy.linalg at the cost of being slower. ''' U, s, VT = np.linalg.svd(A) +# print('test U') +# print( (np.transpose(U) @ U - U @np.transpose(U))) +# print('test VT') +# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) print(U) print(s) print(VT) + D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): D[i,i]=s[i] @@ -2111,7 +2116,7 @@ with \( \boldsymbol{U}\in {\mathbb{R}}^{n\times n} \), \( \boldsymbol{\Sigma}\in and \( \boldsymbol{V}\in {\mathbb{R}}^{p\times p} \).

-We have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \). +The matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are unitary/orthonormal matrices, that is in case the matrices are real we have \( \boldsymbol{U}^T\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^T=\boldsymbol{I} \) and \( \boldsymbol{V}^T\boldsymbol{V}=\boldsymbol{V}\boldsymbol{V}^T=\boldsymbol{I} \).











diff --git a/doc/pub/Regression/ipynb/Regression.ipynb b/doc/pub/Regression/ipynb/Regression.ipynb index 248e57928..ca9ad0e44 100644 --- a/doc/pub/Regression/ipynb/Regression.ipynb +++ b/doc/pub/Regression/ipynb/Regression.ipynb @@ -2676,9 +2676,14 @@ " numpy and scipy.linalg at the cost of being slower.\n", " '''\n", " U, s, VT = np.linalg.svd(A)\n", + "# print('test U')\n", + "# print( (np.transpose(U) @ U - U @np.transpose(U)))\n", + "# print('test VT')\n", + "# print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))\n", " print(U)\n", " print(s)\n", " print(VT)\n", + "\n", " D = np.zeros((len(U),len(VT)))\n", " for i in range(0,len(VT)):\n", " D[i,i]=s[i]\n", @@ -2748,7 +2753,7 @@ "with $\\boldsymbol{U}\\in {\\mathbb{R}}^{n\\times n}$, $\\boldsymbol{\\Sigma}\\in {\\mathbb{R}}^{n\\times p}$\n", "and $\\boldsymbol{V}\\in {\\mathbb{R}}^{p\\times p}$.\n", "\n", - "We have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n", + "The matrices $\\boldsymbol{U}$ and $\\boldsymbol{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\\boldsymbol{U}^T\\boldsymbol{U}=\\boldsymbol{U}\\boldsymbol{U}^T=\\boldsymbol{I}$ and $\\boldsymbol{V}^T\\boldsymbol{V}=\\boldsymbol{V}\\boldsymbol{V}^T=\\boldsymbol{I}$.\n", "\n", "## Spectral Decomposition of the OLS\n", "\n", diff --git a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz index 46c73b726..0f0f4be9e 100644 Binary files a/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz and b/doc/pub/Regression/ipynb/ipynb-Regression-src.tar.gz differ diff --git a/doc/pub/Regression/pdf/Regression-minted.pdf b/doc/pub/Regression/pdf/Regression-minted.pdf index 6921b5b2d..741f6b264 100644 Binary files a/doc/pub/Regression/pdf/Regression-minted.pdf and b/doc/pub/Regression/pdf/Regression-minted.pdf differ diff --git a/doc/src/Regression/Regression.do.txt b/doc/src/Regression/Regression.do.txt index f7dd7f5e5..646c4abf9 100644 --- a/doc/src/Regression/Regression.do.txt +++ b/doc/src/Regression/Regression.do.txt @@ -1590,9 +1590,14 @@ def SVDinv(A): numpy and scipy.linalg at the cost of being slower. ''' U, s, VT = np.linalg.svd(A) +# print('test U') +# print( (np.transpose(U) @ U - U @np.transpose(U))) +# print('test VT') +# print( (np.transpose(VT) @ VT - VT @np.transpose(VT))) print(U) print(s) print(VT) + D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): D[i,i]=s[i] @@ -1652,7 +1657,7 @@ We have our design matrix with $\bm{U}\in {\mathbb{R}}^{n\times n}$, $\bm{\Sigma}\in {\mathbb{R}}^{n\times p}$ and $\bm{V}\in {\mathbb{R}}^{p\times p}$. -We have $\bm{U}^T\bm{U}=\bm{I}$ and $\bm{V}^T\bm{V}=\bm{V}\bm{V}^T=\bm{I}$. +The matrices $\bm{U}$ and $\bm{V}$ are unitary/orthonormal matrices, that is in case the matrices are real we have $\bm{U}^T\bm{U}=\bm{U}\bm{U}^T=\bm{I}$ and $\bm{V}^T\bm{V}=\bm{V}\bm{V}^T=\bm{I}$. !split ===== Spectral Decomposition of the OLS =====