-Small exercise: perform the standars scaling by including the standard deviation. +Small exercise: perform the standard scaling by including the standard deviation. @@ -1394,13 +1394,13 @@ TrainError = np.zeros(maxdegree) polydegree = np.zeros(maxdegree) x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2) scaler = StandardScaler() -scaler.fit(X_train) +scaler.fit(x_train) x_train_scaled = scaler.transform(x_train) x_test_scaled = scaler.transform(x_test) for degree in range(maxdegree): model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False)) - clf = model.fit(x_train_scale,y_train) + clf = model.fit(x_train_scaled,y_train) y_fit = clf.predict(x_train_scaled) y_pred = clf.predict(x_test_scaled) polydegree[degree] = degree @@ -1719,7 +1719,7 @@ As an example, the above defective matrix can be decomposed as
$$
-\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
+\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 4& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
$$
@@ -1788,31 +1788,28 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
SVD is numerically more stable than the inversion algorithms provided by
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)))
+ 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(S)
print(VT)
D = np.zeros((len(U),len(VT)))
for i in range(0,len(VT)):
- D[i,i]=s[i]
- UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
- return np.matmul(V,np.matmul(invD,UT))
+ D[i,i]=S[i]
+ return U @ D @ VT
-X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+X = np.array([ [1.0,-1.0], [1.0,-1.0]])
print(X)
A = np.transpose(X) @ X
print(A)
-# Brute force inversion of super-collinear matrix
-#B = np.linalg.inv(A)
-#print(B)
C = SVDinv(A)
-print(C)
+# Print the difference between the original matrix and the SVD one
+print(C-A)
The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first diff --git a/doc/pub/week35/html/week35-solarized.html b/doc/pub/week35/html/week35-solarized.html index d5b47d26b..7f77448c1 100644 --- a/doc/pub/week35/html/week35-solarized.html +++ b/doc/pub/week35/html/week35-solarized.html @@ -1421,7 +1421,7 @@ Xscaled = scaler.transform(X) display(XPandas-Xscaled)
-Small exercise: perform the standars scaling by including the standard deviation. +Small exercise: perform the standard scaling by including the standard deviation.
@@ -1487,13 +1487,13 @@ TrainError = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
scaler = StandardScaler()
-scaler.fit(X_train)
+scaler.fit(x_train)
x_train_scaled = scaler.transform(x_train)
x_test_scaled = scaler.transform(x_test)
for degree in range(maxdegree):
model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
- clf = model.fit(x_train_scale,y_train)
+ clf = model.fit(x_train_scaled,y_train)
y_fit = clf.predict(x_train_scaled)
y_pred = clf.predict(x_test_scaled)
polydegree[degree] = degree
@@ -1794,7 +1794,7 @@ $$
As an example, the above defective matrix can be decomposed as
$$
-\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
+\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 4& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
$$
@@ -1862,31 +1862,28 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des SVD is numerically more stable than the inversion algorithms provided by 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))) + 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(S) print(VT) D = np.zeros((len(U),len(VT))) for i in range(0,len(VT)): - D[i,i]=s[i] - UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D) - return np.matmul(V,np.matmul(invD,UT)) + D[i,i]=S[i] + return U @ D @ VT -X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ]) +X = np.array([ [1.0,-1.0], [1.0,-1.0]]) print(X) A = np.transpose(X) @ X print(A) -# Brute force inversion of super-collinear matrix -#B = np.linalg.inv(A) -#print(B) C = SVDinv(A) -print(C) +# Print the difference between the original matrix and the SVD one +print(C-A)
The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first diff --git a/doc/pub/week35/html/week35.html b/doc/pub/week35/html/week35.html index 807f09764..5b00624c0 100644 --- a/doc/pub/week35/html/week35.html +++ b/doc/pub/week35/html/week35.html @@ -1426,7 +1426,7 @@ Xscaled = scaler-Xscaled)
-Small exercise: perform the standars scaling by including the standard deviation. +Small exercise: perform the standard scaling by including the standard deviation.
@@ -1867,31 +1867,28 @@ In general the economy-size SVD leads to less FLOPS and still conserving the des
SVD is numerically more stable than the inversion algorithms provided by
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)))
+ 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(S)
print(VT)
D = np.zeros((len(U),len(VT)))
for i in range(0,len(VT)):
- D[i,i]=s[i]
- UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
- return np.matmul(V,np.matmul(invD,UT))
+ D[i,i]=S[i]
+ return U @ D @ VT
-X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+X = np.array([ [1.0,-1.0], [1.0,-1.0]])
print(X)
A = np.transpose(X) @ X
print(A)
-# Brute force inversion of super-collinear matrix
-#B = np.linalg.inv(A)
-#print(B)
C = SVDinv(A)
-print(C)
+# Print the difference between the original matrix and the SVD one
+print(C-A)
The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first
diff --git a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
index ae4e62b79..6fef7343b 100644
Binary files a/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz and b/doc/pub/week35/ipynb/ipynb-week35-src.tar.gz differ
diff --git a/doc/pub/week35/ipynb/week35.ipynb b/doc/pub/week35/ipynb/week35.ipynb
index 605aaf26d..97f67e309 100644
--- a/doc/pub/week35/ipynb/week35.ipynb
+++ b/doc/pub/week35/ipynb/week35.ipynb
@@ -1631,7 +1631,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Small exercise: perform the standars scaling by including the standard deviation.\n",
+ "Small exercise: perform the standard scaling by including the standard deviation.\n",
"\n",
"## Min-Max Scaling\n",
"\n",
@@ -1720,13 +1720,13 @@
"polydegree = np.zeros(maxdegree)\n",
"x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)\n",
"scaler = StandardScaler()\n",
- "scaler.fit(X_train)\n",
+ "scaler.fit(x_train)\n",
"x_train_scaled = scaler.transform(x_train)\n",
"x_test_scaled = scaler.transform(x_test)\n",
"\n",
"for degree in range(maxdegree):\n",
" model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))\n",
- " clf = model.fit(x_train_scale,y_train)\n",
+ " clf = model.fit(x_train_scaled,y_train)\n",
" y_fit = clf.predict(x_train_scaled)\n",
" y_pred = clf.predict(x_test_scaled) \n",
" polydegree[degree] = degree\n",
@@ -2118,7 +2118,7 @@
"metadata": {},
"source": [
"$$\n",
- "\\boldsymbol{X} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& 1 \\\\ 1& -1\\\\ \\end{bmatrix} \\begin{bmatrix} 2& 0 \\\\ 0& 0\\\\ \\end{bmatrix} \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& -1 \\\\ 1& 1\\\\ \\end{bmatrix}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n",
+ "\\boldsymbol{X} = \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& 1 \\\\ 1& -1\\\\ \\end{bmatrix} \\begin{bmatrix} 4& 0 \\\\ 0& 0\\\\ \\end{bmatrix} \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1& -1 \\\\ 1& 1\\\\ \\end{bmatrix}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T,\n",
"$$"
]
},
@@ -2185,31 +2185,28 @@
" SVD is numerically more stable than the inversion algorithms provided by\n",
" 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",
+ " 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(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",
- " UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)\n",
- " return np.matmul(V,np.matmul(invD,UT))\n",
+ " D[i,i]=S[i]\n",
+ " return U @ D @ VT\n",
"\n",
"\n",
- "X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])\n",
+ "X = np.array([ [1.0,-1.0], [1.0,-1.0]])\n",
"print(X)\n",
"A = np.transpose(X) @ X\n",
"print(A)\n",
- "# Brute force inversion of super-collinear matrix\n",
- "#B = np.linalg.inv(A)\n",
- "#print(B)\n",
"C = SVDinv(A)\n",
- "print(C)"
+ "# Print the difference between the original matrix and the SVD one\n",
+ "print(C-A)"
]
},
{
diff --git a/doc/src/week35/week35.do.txt b/doc/src/week35/week35.do.txt
index 92f3f9690..4c3c5815c 100644
--- a/doc/src/week35/week35.do.txt
+++ b/doc/src/week35/week35.do.txt
@@ -973,7 +973,7 @@ Xscaled = scaler.transform(X)
display(XPandas-Xscaled)
!ec
-Small exercise: perform the standars scaling by including the standard deviation.
+Small exercise: perform the standard scaling by including the standard deviation.
!split
===== Min-Max Scaling =====
@@ -1030,13 +1030,13 @@ TrainError = np.zeros(maxdegree)
polydegree = np.zeros(maxdegree)
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
scaler = StandardScaler()
-scaler.fit(X_train)
+scaler.fit(x_train)
x_train_scaled = scaler.transform(x_train)
x_test_scaled = scaler.transform(x_test)
for degree in range(maxdegree):
model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
- clf = model.fit(x_train_scale,y_train)
+ clf = model.fit(x_train_scaled,y_train)
y_fit = clf.predict(x_train_scaled)
y_pred = clf.predict(x_test_scaled)
polydegree[degree] = degree
@@ -1322,7 +1322,7 @@ As an example, the above defective matrix can be decomposed as
!bt
\[
-\bm{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T,
+\bm{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 4& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\bm{U}\bm{\Sigma}\bm{V}^T,
\]
!et
@@ -1378,32 +1378,28 @@ def SVDinv(A):
SVD is numerically more stable than the inversion algorithms provided by
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)))
+ 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(S)
print(VT)
D = np.zeros((len(U),len(VT)))
for i in range(0,len(VT)):
- D[i,i]=s[i]
- UT = np.transpose(U); V = np.transpose(VT); invD = np.linalg.inv(D)
- return np.matmul(V,np.matmul(invD,UT))
+ D[i,i]=S[i]
+ return U @ D @ VT
-X = np.array([ [1.0, -1.0, 2.0], [1.0, 0.0, 1.0], [1.0, 2.0, -1.0], [1.0, 1.0, 0.0] ])
+X = np.array([ [1.0,-1.0], [1.0,-1.0]])
print(X)
A = np.transpose(X) @ X
print(A)
-# Brute force inversion of super-collinear matrix
-#B = np.linalg.inv(A)
-#print(B)
C = SVDinv(A)
-print(C)
-
+# Print the difference between the original matrix and the SVD one
+print(C-A)
!ec
The matrix $\bm{X}$ has columns that are linearly dependent. The first
@@ -1492,13 +1492,13 @@ TrainError = np
polydegree = np.zeros(maxdegree)
x_train, x_test, y_train, y_test = train_test_split(x, y, test_size=0.2)
scaler = StandardScaler()
-scaler.fit(X_train)
+scaler.fit(x_train)
x_train_scaled = scaler.transform(x_train)
x_test_scaled = scaler.transform(x_test)
for degree in range(maxdegree):
model = make_pipeline(PolynomialFeatures(degree=degree), LinearRegression(fit_intercept=False))
- clf = model.fit(x_train_scale,y_train)
+ clf = model.fit(x_train_scaled,y_train)
y_fit = clf.predict(x_train_scaled)
y_pred = clf.predict(x_test_scaled)
polydegree[degree] = degree
@@ -1799,7 +1799,7 @@ $$
As an example, the above defective matrix can be decomposed as
$$
-\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 2& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
+\boldsymbol{X} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1& 1 \\ 1& -1\\ \end{bmatrix} \begin{bmatrix} 4& 0 \\ 0& 0\\ \end{bmatrix} \frac{1}{\sqrt{2}}\begin{bmatrix} 1& -1 \\ 1& 1\\ \end{bmatrix}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T,
$$