diff --git a/doc/BookChapters/chapter3.do.txt b/doc/BookChapters/chapter3.do.txt index 0ebd9707c..47a3b7d53 100644 --- a/doc/BookChapters/chapter3.do.txt +++ b/doc/BookChapters/chapter3.do.txt @@ -1257,9 +1257,9 @@ This equation does not lead to a nice analytical equation as in either Ridge reg ===== Code for SVD and Inversion of Matrices ===== How do we use the SVD to invert a matrix $\bm{X}^\bm{X}$ which is singular or near singular? -The simple answer is to use the linear algebra function for pseudoinvers, that is +The simple answer is to use the linear algebra function for the pseudoinverse, that is !bc pycod -Ainv = np.linlag.pinv(A) +#Ainv = np.linlag.pinv(A) !ec Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD. diff --git a/doc/LectureNotes/_build/.doctrees/chapter2.doctree b/doc/LectureNotes/_build/.doctrees/chapter2.doctree index f7c32732c..55068749b 100644 Binary files a/doc/LectureNotes/_build/.doctrees/chapter2.doctree and b/doc/LectureNotes/_build/.doctrees/chapter2.doctree differ diff --git a/doc/LectureNotes/_build/.doctrees/environment.pickle b/doc/LectureNotes/_build/.doctrees/environment.pickle index 434f41d71..47d7ee608 100644 Binary files a/doc/LectureNotes/_build/.doctrees/environment.pickle and b/doc/LectureNotes/_build/.doctrees/environment.pickle differ diff --git a/doc/LectureNotes/_build/html/_images/chapter2_287_1.png b/doc/LectureNotes/_build/html/_images/chapter2_287_1.png new file mode 100644 index 000000000..dc3a3ce3e Binary files /dev/null and b/doc/LectureNotes/_build/html/_images/chapter2_287_1.png differ diff --git a/doc/LectureNotes/_build/html/_images/chapter2_289_1.png b/doc/LectureNotes/_build/html/_images/chapter2_289_1.png new file mode 100644 index 000000000..ac2df9e69 Binary files /dev/null and b/doc/LectureNotes/_build/html/_images/chapter2_289_1.png differ diff --git a/doc/LectureNotes/_build/html/_images/chapter2_291_1.png b/doc/LectureNotes/_build/html/_images/chapter2_291_1.png new file mode 100644 index 000000000..9666ad291 Binary files /dev/null and b/doc/LectureNotes/_build/html/_images/chapter2_291_1.png differ diff --git a/doc/LectureNotes/_build/html/_images/chapter2_351_1.png b/doc/LectureNotes/_build/html/_images/chapter2_351_1.png new file mode 100644 index 000000000..07584f6bb Binary files /dev/null and b/doc/LectureNotes/_build/html/_images/chapter2_351_1.png differ diff --git a/doc/LectureNotes/_build/html/_sources/chapter2.ipynb b/doc/LectureNotes/_build/html/_sources/chapter2.ipynb index 1c18a18a0..99ead382c 100644 --- a/doc/LectureNotes/_build/html/_sources/chapter2.ipynb +++ b/doc/LectureNotes/_build/html/_sources/chapter2.ipynb @@ -2133,7 +2133,7 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the pseudoinverse, that is" ] }, { @@ -2145,7 +2145,7 @@ }, "outputs": [], "source": [ - "Ainv = np.linlag.pinv(A)" + "#Ainv = np.linlag.pinv(A)" ] }, { diff --git a/doc/LectureNotes/_build/html/chapter2.html b/doc/LectureNotes/_build/html/chapter2.html index 258b482f4..4958e3ed8 100644 --- a/doc/LectureNotes/_build/html/chapter2.html +++ b/doc/LectureNotes/_build/html/chapter2.html @@ -982,10 +982,10 @@ covariance matrix through the np.linalg.eig() function.

-
-0.01210814993019007
-3.8867467323038865
-[[0.91417278 2.88046462]
- [2.88046462 9.9860803 ]]
+
-0.033037772005753835
+3.7371165871823337
+[[ 1.22443803  3.75195757]
+ [ 3.75195757 12.47441766]]
@@ -1022,10 +1022,10 @@ a more brute force way. Here we scale the mean values for each column of the des
-
0.07423848370736122
-1.725394195434945
-[[1.         0.71606852]
- [0.71606852 1.        ]]
+
0.08665060086846632
+1.6796265324852733
+[[1.        0.6183694]
+ [0.6183694 1.       ]]
@@ -1055,33 +1055,30 @@ this matrix we easily see that it is a positive definite matrix.

-
[[ 0.52687171  0.97622676]
- [-2.35038714 -6.09976319]
- [-0.55707065 -1.51576807]
- [-0.98314755 -2.54096582]
- [-0.46556436 -0.43549028]
- [ 3.28310983  9.01503674]
- [ 0.76388013  2.67714918]
- [-0.49722108 -2.64669382]
- [ 1.20184113  3.80026635]
- [-0.92231203 -3.22999784]]
+
[[ 0.68002363  0.95517094]
+ [-0.53545715  0.03652792]
+ [ 1.33886902  4.84251485]
+ [ 0.20375701 -0.16861772]
+ [-2.04455272 -5.89742056]
+ [ 1.17449733  2.3728892 ]
+ [ 0.03019554  0.07581375]
+ [ 0.60949193  1.95628168]
+ [ 0.0530533   1.07839924]
+ [-1.5098779  -5.2515593 ]]
           0         1
-0  0.526872  0.976227
-1 -2.350387 -6.099763
-2 -0.557071 -1.515768
-3 -0.983148 -2.540966
-4 -0.465564 -0.435490
-5  3.283110  9.015037
-6  0.763880  2.677149
-7 -0.497221 -2.646694
-8  1.201841  3.800266
-9 -0.922312 -3.229998
+0  0.680024  0.955171
+1 -0.535457  0.036528
+2  1.338869  4.842515
+3  0.203757 -0.168618
+4 -2.044553 -5.897421
+5  1.174497  2.372889
+6  0.030196  0.075814
+7  0.609492  1.956282
+8  0.053053  1.078399
+9 -1.509878 -5.251559
           0         1
-0  1.000000  0.988663
-1  0.988663  1.000000
-

-

-

+0  1.000000  0.959247
+1  0.959247  1.000000
@@ -1138,37 +1135,37 @@ this matrix we easily see that it is a positive definite matrix.

     0         1         2         3         4         5         6         7   \
 0   0.0  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000   
-1   0.0  0.075212  0.075261  0.077160  0.075772  0.074134  0.069902  0.068276   
-2   0.0  0.075261  0.077533  0.076062  0.075562  0.074848  0.067985  0.066788   
-3   0.0  0.077160  0.076062  0.084092  0.081886  0.079365  0.078927  0.076730   
-4   0.0  0.075772  0.075562  0.081886  0.080115  0.078049  0.076398  0.074453   
-5   0.0  0.074134  0.074848  0.079365  0.078049  0.076469  0.073567  0.071887   
-6   0.0  0.069902  0.067985  0.078927  0.076398  0.073567  0.075821  0.073483   
-7   0.0  0.068276  0.066788  0.076730  0.074453  0.071887  0.073483  0.071313   
-8   0.0  0.066629  0.065591  0.074488  0.072471  0.070182  0.071096  0.069098   
-9   0.0  0.064964  0.064410  0.072198  0.070456  0.068460  0.068654  0.066835   
-10  0.0  0.061842  0.059571  0.071481  0.068920  0.066092  0.069814  0.067531   
-11  0.0  0.060308  0.058282  0.069516  0.067123  0.064472  0.067777  0.065618   
-12  0.0  0.058805  0.057035  0.067577  0.065355  0.062885  0.065762  0.063727   
-13  0.0  0.057332  0.055831  0.065660  0.063615  0.061330  0.063763  0.061855   
-14  0.0  0.055887  0.054674  0.063759  0.061896  0.059806  0.061774  0.059994   
+1   0.0  0.073902  0.077303  0.072279  0.077386  0.082866  0.063526  0.068073   
+2   0.0  0.077303  0.081480  0.074665  0.080336  0.086465  0.064929  0.069872   
+3   0.0  0.072279  0.074665  0.075754  0.080381  0.085289  0.069635  0.074099   
+4   0.0  0.077386  0.080336  0.080381  0.085597  0.091159  0.073333  0.078279   
+5   0.0  0.082866  0.086465  0.085289  0.091159  0.097452  0.077212  0.082687   
+6   0.0  0.063526  0.064929  0.069635  0.073333  0.077212  0.066047  0.069859   
+7   0.0  0.068073  0.069872  0.074099  0.078279  0.082687  0.069859  0.074096   
+8   0.0  0.073035  0.075289  0.078937  0.083657  0.088660  0.073966  0.078674   
+9   0.0  0.078451  0.081227  0.084181  0.089504  0.095173  0.078388  0.083618   
+10  0.0  0.055085  0.055774  0.062327  0.065204  0.068185  0.060520  0.063670   
+11  0.0  0.058892  0.059865  0.066249  0.069510  0.072910  0.064000  0.067504   
+12  0.0  0.063055  0.064354  0.070517  0.074207  0.078076  0.067769  0.071667   
+13  0.0  0.067610  0.069282  0.075163  0.079333  0.083728  0.071855  0.076189   
+14  0.0  0.072598  0.074696  0.080223  0.084931  0.089916  0.076283  0.081103   
 
           8         9         10        11        12        13        14  
 0   0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  0.000000  
-1   0.066629  0.064964  0.061842  0.060308  0.058805  0.057332  0.055887  
-2   0.065591  0.064410  0.059571  0.058282  0.057035  0.055831  0.054674  
-3   0.074488  0.072198  0.071481  0.069516  0.067577  0.065660  0.063759  
-4   0.072471  0.070456  0.068920  0.067123  0.065355  0.063615  0.061896  
-5   0.070182  0.068460  0.066092  0.064472  0.062885  0.061330  0.059806  
-6   0.071096  0.068654  0.069814  0.067777  0.065762  0.063763  0.061774  
-7   0.069098  0.066835  0.067531  0.065618  0.063727  0.061855  0.059994  
-8   0.067061  0.064982  0.065202  0.063415  0.061652  0.059908  0.058178  
-9   0.064982  0.063097  0.062822  0.061164  0.059531  0.057919  0.056326  
-10  0.065202  0.062822  0.065086  0.063122  0.061176  0.059245  0.057320  
-11  0.063415  0.061164  0.063122  0.061255  0.059406  0.057571  0.055744  
-12  0.061652  0.059531  0.061176  0.059406  0.057654  0.055916  0.054187  
-13  0.059908  0.057919  0.059245  0.057571  0.055916  0.054276  0.052645  
-14  0.058178  0.056326  0.057320  0.055744  0.054187  0.052645  0.051115  
+1   0.073035  0.078451  0.055085  0.058892  0.063055  0.067610  0.072598  
+2   0.075289  0.081227  0.055774  0.059865  0.064354  0.069282  0.074696  
+3   0.078937  0.084181  0.062327  0.066249  0.070517  0.075163  0.080223  
+4   0.083657  0.089504  0.065204  0.069510  0.074207  0.079333  0.084931  
+5   0.088660  0.095173  0.068185  0.072910  0.078076  0.083728  0.089916  
+6   0.073966  0.078388  0.060520  0.064000  0.067769  0.071855  0.076283  
+7   0.078674  0.083618  0.063670  0.067504  0.071667  0.076189  0.081103  
+8   0.083775  0.089300  0.067043  0.071268  0.075865  0.080871  0.086325  
+9   0.089300  0.095472  0.070653  0.075309  0.080387  0.085928  0.091979  
+10  0.067043  0.070653  0.056484  0.059455  0.062659  0.066116  0.069848  
+11  0.071268  0.075309  0.059455  0.062731  0.066272  0.070103  0.074248  
+12  0.075865  0.080387  0.062659  0.066272  0.070188  0.074432  0.079037  
+13  0.080871  0.085928  0.066116  0.070103  0.074432  0.079137  0.084251  
+14  0.086325  0.091979  0.069848  0.074248  0.079037  0.084251  0.089931
@@ -1448,27 +1445,10 @@ C(\boldsymbol{X},\boldsymbol{\beta})=\left\{(\boldsymbol{y}-\boldsymbol{X}\bolds

4.10. Code for SVD and Inversion of Matrices

How do we use the SVD to invert a matrix \(\boldsymbol{X}^\boldsymbol{X}\) which is singular or near singular? -The simple answer is to use the linear algebra function for pseudoinvers, that is

+The simple answer is to use the linear algebra function for the pseudoinverse, that is

-
Ainv = np.linlag.pinv(A)
-
-
-
-
-
---------------------------------------------------------------------------
-AttributeError                            Traceback (most recent call last)
-<ipython-input-6-52d2c51caad1> in <module>
-----> 1 Ainv = np.linlag.pinv(A)
-
-~/opt/anaconda3/lib/python3.8/site-packages/numpy/__init__.py in __getattr__(attr)
-    212                 return Tester
-    213             else:
---> 214                 raise AttributeError("module {!r} has no attribute "
-    215                                      "{!r}".format(__name__, attr))
-    216 
-
-AttributeError: module 'numpy' has no attribute 'linlag'
+
#Ainv = np.linlag.pinv(A)
@@ -1508,6 +1488,24 @@ The simple answer is to use the linear algebra function for pseudoinvers, that i
+
+
[[1 2 3]
+ [2 4 5]
+ [3 5 6]]
+test U
+[[ 2.22044605e-16 -7.77156117e-16 -5.55111512e-16]
+ [-7.77156117e-16  0.00000000e+00 -1.11022302e-16]
+ [-5.55111512e-16 -1.11022302e-16  0.00000000e+00]]
+test VT
+[[ 1.11022302e-16 -2.22044605e-16  1.38777878e-16]
+ [-2.22044605e-16 -1.11022302e-16 -1.11022302e-16]
+ [ 1.38777878e-16 -1.11022302e-16  0.00000000e+00]]
+[[2.35367281e-12 1.70885528e-12 3.20632410e-13]
+ [2.17248441e-12 1.46016532e-12 2.00728323e-13]
+ [6.95443703e-13 4.13891144e-13 2.13162821e-14]]
+
+
+

Although our matrix to invert \(\boldsymbol{X}^T\boldsymbol{X}\) is a square matrix, our matrix may be singular.

The pseudoinverse is the generalization of the matrix inverse for square matrices to @@ -1548,6 +1546,18 @@ It is used for the calculation of the inverse for singular or near singular matr

+
+
[[0.3 0.4]
+ [0.5 0.6]
+ [0.7 0.8]
+ [0.9 1. ]]
+[[-13.   -6.    1.    8. ]
+ [ 11.5   5.5  -0.5  -6.5]]
+[[0. 0. 0. 0.]
+ [0. 0. 0. 0.]]
+
+
+

As you can see from this example, our own decomposition based on the SVD agrees the pseudoinverse algorithm provided by Numpy.

@@ -1894,6 +1904,14 @@ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\ve +
+
[2. 2.]
+Training MSE for OLS
+3.0
+
+
+_images/chapter2_287_1.png +

We see here that we reach a plateau. What is actually happening?

@@ -1957,6 +1975,214 @@ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\ve
+
+
[2. 2.]
+Training MSE for OLS
+3.0
+[1.99995    1.99980002]
+[ 0.50001525 -0.99953475]
+[1.99993978 1.99975913]
+[ 0.50001525 -0.99944272]
+[1.99992746 1.99970988]
+[ 0.50001525 -0.99933188]
+[1.99991263 1.99965056]
+[ 0.50001525 -0.99919837]
+[1.99989476 1.99957911]
+[ 0.50001524 -0.99903755]
+[1.99987324 1.99949306]
+[ 0.50001524 -0.99884384]
+[1.99984732 1.99938942]
+[ 0.50001524 -0.99861053]
+[1.9998161  1.99926459]
+[ 0.50001523 -0.9983295 ]
+[1.99977849 1.99911427]
+[ 0.50001523 -0.99799099]
+[1.9997332  1.99893323]
+[ 0.50001522 -0.99758326]
+[1.99967865 1.99871521]
+[ 0.50001521 -0.99709215]
+[1.99961294 1.99845267]
+[ 0.50001521 -0.99650061]
+[1.99953381 1.99813653]
+[ 0.50001519 -0.99578809]
+[1.9994385  1.99775587]
+[ 0.50001518 -0.99492986]
+[1.9993237  1.99729756]
+[ 0.50001517 -0.99389612]
+[1.99918546 1.9967458 ]
+[ 0.50001515 -0.99265097]
+[1.99901896 1.99608161]
+[ 0.50001512 -0.99115119]
+[1.99881845 1.99528218]
+[ 0.5000151 -0.9893447]
+[1.998577  1.9943201]
+[ 0.50001506 -0.98716878]
+[1.99828624 1.99316252]
+[ 0.50001502 -0.98454786]
+[1.99793613 1.99176998]
+[ 0.50001498 -0.98139097]
+[1.99751458 1.99009525]
+[ 0.50001492 -0.97758848]
+[1.99700706 1.98808176]
+[ 0.50001485 -0.97300836]
+[1.9963961  1.98566191]
+[ 0.50001476 -0.9674916 ]
+[1.99566069 1.98275501]
+[ 0.50001466 -0.96084663]
+[1.9947756  1.97926491]
+[ 0.50001454 -0.95284275]
+[1.99371056 1.97507735]
+[ 0.50001439 -0.94320205]
+[1.99242921 1.97005689]
+[ 0.50001422 -0.93158979]
+[1.99088801 1.9640435 ]
+[ 0.500014   -0.91760278]
+[1.9890348  1.95684892]
+[ 0.50001374 -0.90075537]
+[1.98680716 1.9482527 ]
+[ 0.50001343 -0.88046261]
+[1.98413059 1.93799826]
+[ 0.50001306 -0.85601992]
+[1.98091621 1.92578916]
+[ 0.50001261 -0.8265786 ]
+[1.97705827 1.91128596]
+[ 0.50001207 -0.79111643]
+[1.97243128 1.89410423]
+[ 0.50001142 -0.74840212]
+[1.96688672 1.87381451]
+[ 0.50001063 -0.69695259]
+[1.96024953 1.84994524]
+[ 0.50000969 -0.63498144]
+[1.95231424 1.82198978]
+[ 0.50000855 -0.56033697]
+[1.94284104 1.78941903]
+[ 0.50000718 -0.47042744]
+[1.93155188 1.75170092]
+[ 0.50000553 -0.3621311 ]
+[1.91812702 1.70832814]
+[ 0.50001414 -0.23167717]
+[1.90220243 1.65885453]
+[ 0.50000455 -0.07456491]
+[1.88336879 1.60293962]
+[ 0.47132891 -0.        ]
+[1.86117291 1.54039921]
+[ 0.41433969 -0.        ]
+[1.83512277 1.47125748]
+[ 0.34569596 -0.        ]
+[1.80469739 1.39579407]
+[ 0.26301436 -0.        ]
+[1.76936315 1.31457796]
+[ 0.16342407 -0.        ]
+[1.72859758 1.22847924]
+[ 0.04346721 -0.        ]
+[1.68192193 1.13865173]
+[ 0. -0.]
+[1.62894215 1.04648335]
+[ 0. -0.]
+[1.56939714 0.95351665]
+[ 0. -0.]
+[1.50321091 0.86134827]
+[ 0. -0.]
+[1.43054282 0.77152076]
+[ 0. -0.]
+[1.35182854 0.68542204]
+[ 0. -0.]
+[1.26780278 0.60420593]
+[ 0. -0.]
+[1.17949575 0.52874252]
+[ 0. -0.]
+[1.0881981  0.45960079]
+[ 0. -0.]
+[0.99539415 0.39706038]
+[ 0. -0.]
+[0.90266948 0.34114547]
+[ 0. -0.]
+[0.81160425 0.29167186]
+[ 0. -0.]
+[0.7236674  0.24829908]
+[ 0. -0.]
+[0.64012627 0.21058097]
+[ 0. -0.]
+[0.56198284 0.17801022]
+[ 0. -0.]
+[0.48994188 0.15005476]
+[ 0. -0.]
+[0.42441033 0.12618549]
+[ 0. -0.]
+[0.3655222  0.10589577]
+[ 0. -0.]
+[0.31318084 0.08871404]
+[ 0. -0.]
+[0.26710969 0.07421084]
+[ 0. -0.]
+[0.22690428 0.06200174]
+[ 0. -0.]
+[0.19207979 0.0517473 ]
+[ 0. -0.]
+[0.16211139 0.04315108]
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Another Example, now with a polynomial fit.

@@ -2042,6 +2268,16 @@ C(\boldsymbol{\beta})=(4-2\beta_0)^2+(2-\beta_1)^2+\lambda(\vert\beta_0\vert+\ve
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[ 2.03099776 -0.17917768  5.18029127]
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@@ -2420,6 +2656,22 @@ polynomial fit and that for larger and larger polynomial degrees of freedom, the
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+Test MSE OLS
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+0.001 [ 1.034342   -0.18063928 -0.          0.          0.        ]
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How can we understand this?

Using Bayes’ theorem we can gain a better intuition about Ridge and Lasso regression.

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Clustering Analysis","3. Linear Regression","13. Building a Feed Forward Neural Network","4. Ridge and Lasso Regression","5. Resampling Methods","6. Logistic Regression","7. Support Vector Machines, overarching aims","8. Decision trees, overarching aims","9. Ensemble Methods: From a Single Tree to Many Trees and Extreme Boosting, Meet the Jungle of Methods","10. Basic ideas of the Principal Component Analysis (PCA)","12. Neural networks","Content in Jupyter Book","Applied Data Analysis and Machine Learning, FYS-STK3155/4155 at the University of Oslo, Norway","2. Linear Algebra, Handling of Arrays and more Python Features","Teaching schedule with links to material","1. 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\ No newline at end of file diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb b/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb index ec7aa8c86..9e68b7c50 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb +++ b/doc/LectureNotes/_build/jupyter_execute/chapter2.ipynb @@ -1297,10 +1297,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "-0.01210814993019007\n", - "3.8867467323038865\n", - "[[0.91417278 2.88046462]\n", - " [2.88046462 9.9860803 ]]\n" + "-0.033037772005753835\n", + "3.7371165871823337\n", + "[[ 1.22443803 3.75195757]\n", + " [ 3.75195757 12.47441766]]\n" ] } ], @@ -1340,10 +1340,10 @@ "name": "stdout", "output_type": "stream", "text": [ - "0.07423848370736122\n", - "1.725394195434945\n", - "[[1. 0.71606852]\n", - " [0.71606852 1. ]]\n" + "0.08665060086846632\n", + "1.6796265324852733\n", + "[[1. 0.6183694]\n", + " [0.6183694 1. ]]\n" ] } ], @@ -1398,37 +1398,30 @@ "name": "stdout", "output_type": "stream", "text": [ - "[[ 0.52687171 0.97622676]\n", - " [-2.35038714 -6.09976319]\n", - " [-0.55707065 -1.51576807]\n", - " [-0.98314755 -2.54096582]\n", - " [-0.46556436 -0.43549028]\n", - " [ 3.28310983 9.01503674]\n", - " [ 0.76388013 2.67714918]\n", - " [-0.49722108 -2.64669382]\n", - " [ 1.20184113 3.80026635]\n", - " [-0.92231203 -3.22999784]]\n", + "[[ 0.68002363 0.95517094]\n", + " [-0.53545715 0.03652792]\n", + " [ 1.33886902 4.84251485]\n", + " [ 0.20375701 -0.16861772]\n", + " [-2.04455272 -5.89742056]\n", + " [ 1.17449733 2.3728892 ]\n", + " [ 0.03019554 0.07581375]\n", + " [ 0.60949193 1.95628168]\n", + " [ 0.0530533 1.07839924]\n", + " [-1.5098779 -5.2515593 ]]\n", " 0 1\n", - "0 0.526872 0.976227\n", - "1 -2.350387 -6.099763\n", - "2 -0.557071 -1.515768\n", - "3 -0.983148 -2.540966\n", - "4 -0.465564 -0.435490\n", - "5 3.283110 9.015037\n", - "6 0.763880 2.677149\n", - "7 -0.497221 -2.646694\n", - "8 1.201841 3.800266\n", - "9 -0.922312 -3.229998\n", + "0 0.680024 0.955171\n", + "1 -0.535457 0.036528\n", + "2 1.338869 4.842515\n", + "3 0.203757 -0.168618\n", + "4 -2.044553 -5.897421\n", + "5 1.174497 2.372889\n", + "6 0.030196 0.075814\n", + "7 0.609492 1.956282\n", + "8 0.053053 1.078399\n", + "9 -1.509878 -5.251559\n", " 0 1\n", - "0 1.000000 0.988663\n", - "1 0.988663 1.000000" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n" + "0 1.000000 0.959247\n", + "1 0.959247 1.000000\n" ] } ], @@ -1470,37 +1463,37 @@ "text": [ " 0 1 2 3 4 5 6 7 \\\n", "0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.0 0.075212 0.075261 0.077160 0.075772 0.074134 0.069902 0.068276 \n", - "2 0.0 0.075261 0.077533 0.076062 0.075562 0.074848 0.067985 0.066788 \n", - "3 0.0 0.077160 0.076062 0.084092 0.081886 0.079365 0.078927 0.076730 \n", - "4 0.0 0.075772 0.075562 0.081886 0.080115 0.078049 0.076398 0.074453 \n", - "5 0.0 0.074134 0.074848 0.079365 0.078049 0.076469 0.073567 0.071887 \n", - "6 0.0 0.069902 0.067985 0.078927 0.076398 0.073567 0.075821 0.073483 \n", - "7 0.0 0.068276 0.066788 0.076730 0.074453 0.071887 0.073483 0.071313 \n", - "8 0.0 0.066629 0.065591 0.074488 0.072471 0.070182 0.071096 0.069098 \n", - "9 0.0 0.064964 0.064410 0.072198 0.070456 0.068460 0.068654 0.066835 \n", - "10 0.0 0.061842 0.059571 0.071481 0.068920 0.066092 0.069814 0.067531 \n", - "11 0.0 0.060308 0.058282 0.069516 0.067123 0.064472 0.067777 0.065618 \n", - "12 0.0 0.058805 0.057035 0.067577 0.065355 0.062885 0.065762 0.063727 \n", - "13 0.0 0.057332 0.055831 0.065660 0.063615 0.061330 0.063763 0.061855 \n", - "14 0.0 0.055887 0.054674 0.063759 0.061896 0.059806 0.061774 0.059994 \n", + "1 0.0 0.073902 0.077303 0.072279 0.077386 0.082866 0.063526 0.068073 \n", + "2 0.0 0.077303 0.081480 0.074665 0.080336 0.086465 0.064929 0.069872 \n", + "3 0.0 0.072279 0.074665 0.075754 0.080381 0.085289 0.069635 0.074099 \n", + "4 0.0 0.077386 0.080336 0.080381 0.085597 0.091159 0.073333 0.078279 \n", + "5 0.0 0.082866 0.086465 0.085289 0.091159 0.097452 0.077212 0.082687 \n", + "6 0.0 0.063526 0.064929 0.069635 0.073333 0.077212 0.066047 0.069859 \n", + "7 0.0 0.068073 0.069872 0.074099 0.078279 0.082687 0.069859 0.074096 \n", + "8 0.0 0.073035 0.075289 0.078937 0.083657 0.088660 0.073966 0.078674 \n", + "9 0.0 0.078451 0.081227 0.084181 0.089504 0.095173 0.078388 0.083618 \n", + "10 0.0 0.055085 0.055774 0.062327 0.065204 0.068185 0.060520 0.063670 \n", + "11 0.0 0.058892 0.059865 0.066249 0.069510 0.072910 0.064000 0.067504 \n", + "12 0.0 0.063055 0.064354 0.070517 0.074207 0.078076 0.067769 0.071667 \n", + "13 0.0 0.067610 0.069282 0.075163 0.079333 0.083728 0.071855 0.076189 \n", + "14 0.0 0.072598 0.074696 0.080223 0.084931 0.089916 0.076283 0.081103 \n", "\n", " 8 9 10 11 12 13 14 \n", "0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 \n", - "1 0.066629 0.064964 0.061842 0.060308 0.058805 0.057332 0.055887 \n", - "2 0.065591 0.064410 0.059571 0.058282 0.057035 0.055831 0.054674 \n", - "3 0.074488 0.072198 0.071481 0.069516 0.067577 0.065660 0.063759 \n", - "4 0.072471 0.070456 0.068920 0.067123 0.065355 0.063615 0.061896 \n", - "5 0.070182 0.068460 0.066092 0.064472 0.062885 0.061330 0.059806 \n", - "6 0.071096 0.068654 0.069814 0.067777 0.065762 0.063763 0.061774 \n", - "7 0.069098 0.066835 0.067531 0.065618 0.063727 0.061855 0.059994 \n", - "8 0.067061 0.064982 0.065202 0.063415 0.061652 0.059908 0.058178 \n", - "9 0.064982 0.063097 0.062822 0.061164 0.059531 0.057919 0.056326 \n", - "10 0.065202 0.062822 0.065086 0.063122 0.061176 0.059245 0.057320 \n", - "11 0.063415 0.061164 0.063122 0.061255 0.059406 0.057571 0.055744 \n", - "12 0.061652 0.059531 0.061176 0.059406 0.057654 0.055916 0.054187 \n", - "13 0.059908 0.057919 0.059245 0.057571 0.055916 0.054276 0.052645 \n", - "14 0.058178 0.056326 0.057320 0.055744 0.054187 0.052645 0.051115 \n" + "1 0.073035 0.078451 0.055085 0.058892 0.063055 0.067610 0.072598 \n", + "2 0.075289 0.081227 0.055774 0.059865 0.064354 0.069282 0.074696 \n", + "3 0.078937 0.084181 0.062327 0.066249 0.070517 0.075163 0.080223 \n", + "4 0.083657 0.089504 0.065204 0.069510 0.074207 0.079333 0.084931 \n", + "5 0.088660 0.095173 0.068185 0.072910 0.078076 0.083728 0.089916 \n", + "6 0.073966 0.078388 0.060520 0.064000 0.067769 0.071855 0.076283 \n", + "7 0.078674 0.083618 0.063670 0.067504 0.071667 0.076189 0.081103 \n", + "8 0.083775 0.089300 0.067043 0.071268 0.075865 0.080871 0.086325 \n", + "9 0.089300 0.095472 0.070653 0.075309 0.080387 0.085928 0.091979 \n", + "10 0.067043 0.070653 0.056484 0.059455 0.062659 0.066116 0.069848 \n", + "11 0.071268 0.075309 0.059455 0.062731 0.066272 0.070103 0.074248 \n", + "12 0.075865 0.080387 0.062659 0.066272 0.070188 0.074432 0.079037 \n", + "13 0.080871 0.085928 0.066116 0.070103 0.074432 0.079137 0.084251 \n", + "14 0.086325 0.091979 0.069848 0.074248 0.079037 0.084251 0.089931 \n" ] } ], @@ -2255,7 +2248,7 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the pseudoinverse, that is" ] }, { @@ -2265,22 +2258,9 @@ "collapsed": false, "editable": true }, - "outputs": [ - { - "ename": "AttributeError", - "evalue": "module 'numpy' has no attribute 'linlag'", - "output_type": "error", - "traceback": [ - "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", - "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", - "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mAinv\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinlag\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpinv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", - "\u001b[0;32m~/opt/anaconda3/lib/python3.8/site-packages/numpy/__init__.py\u001b[0m in \u001b[0;36m__getattr__\u001b[0;34m(attr)\u001b[0m\n\u001b[1;32m 212\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mTester\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 213\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 214\u001b[0;31m raise AttributeError(\"module {!r} has no attribute \"\n\u001b[0m\u001b[1;32m 215\u001b[0m \"{!r}\".format(__name__, attr))\n\u001b[1;32m 216\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n", - "\u001b[0;31mAttributeError\u001b[0m: module 'numpy' has no attribute 'linlag'" - ] - } - ], + "outputs": [], "source": [ - "Ainv = np.linlag.pinv(A)" + "#Ainv = np.linlag.pinv(A)" ] }, { @@ -2292,12 +2272,33 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 7, "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[1 2 3]\n", + " [2 4 5]\n", + " [3 5 6]]\n", + "test U\n", + "[[ 2.22044605e-16 -7.77156117e-16 -5.55111512e-16]\n", + " [-7.77156117e-16 0.00000000e+00 -1.11022302e-16]\n", + " [-5.55111512e-16 -1.11022302e-16 0.00000000e+00]]\n", + "test VT\n", + "[[ 1.11022302e-16 -2.22044605e-16 1.38777878e-16]\n", + " [-2.22044605e-16 -1.11022302e-16 -1.11022302e-16]\n", + " [ 1.38777878e-16 -1.11022302e-16 0.00000000e+00]]\n", + "[[2.35367281e-12 1.70885528e-12 3.20632410e-13]\n", + " [2.17248441e-12 1.46016532e-12 2.00728323e-13]\n", + " [6.95443703e-13 4.13891144e-13 2.13162821e-14]]\n" + ] + } + ], "source": [ "import numpy as np\n", "# SVD inversion\n", @@ -2363,12 +2364,27 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 8, "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[0.3 0.4]\n", + " [0.5 0.6]\n", + " [0.7 0.8]\n", + " [0.9 1. ]]\n", + "[[-13. -6. 1. 8. ]\n", + " [ 11.5 5.5 -0.5 -6.5]]\n", + "[[0. 0. 0. 0.]\n", + " [0. 0. 0. 0.]]\n" + ] + } + ], "source": [ "import numpy as np\n", "# SVD inversion\n", @@ -3229,12 +3245,37 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 9, "metadata": { "collapsed": false, "editable": true }, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[2. 2.]\n", + "Training MSE for OLS\n", + "3.0\n" + ] + }, + { + "data": { + "image/png": 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\n", 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\n", 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "filenames": { + "image/png": "/Users/mhjensen/Teaching/MachineLearning/doc/LectureNotes/_build/jupyter_execute/chapter2_351_1.png" + }, + "needs_background": "light" + }, + "output_type": "display_data" + } + ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2.py b/doc/LectureNotes/_build/jupyter_execute/chapter2.py index 59a7cc107..9dfb2e212 100644 --- a/doc/LectureNotes/_build/jupyter_execute/chapter2.py +++ b/doc/LectureNotes/_build/jupyter_execute/chapter2.py @@ -1162,9 +1162,9 @@ This equation does not lead to a nice analytical equation as in either Ridge reg ## Code for SVD and Inversion of Matrices How do we use the SVD to invert a matrix $\boldsymbol{X}^\boldsymbol{X}$ which is singular or near singular? -The simple answer is to use the linear algebra function for pseudoinvers, that is +The simple answer is to use the linear algebra function for the pseudoinverse, that is -Ainv = np.linlag.pinv(A) +#Ainv = np.linlag.pinv(A) Let us first look at a matrix which does not causes problems and write our own function where we just use the SVD. diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2_287_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter2_287_1.png new file mode 100644 index 000000000..dc3a3ce3e Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter2_287_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2_289_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter2_289_1.png new file mode 100644 index 000000000..ac2df9e69 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter2_289_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2_291_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter2_291_1.png new file mode 100644 index 000000000..9666ad291 Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter2_291_1.png differ diff --git a/doc/LectureNotes/_build/jupyter_execute/chapter2_351_1.png b/doc/LectureNotes/_build/jupyter_execute/chapter2_351_1.png new file mode 100644 index 000000000..07584f6bb Binary files /dev/null and b/doc/LectureNotes/_build/jupyter_execute/chapter2_351_1.png differ diff --git a/doc/LectureNotes/chapter2.ipynb b/doc/LectureNotes/chapter2.ipynb index 1c18a18a0..99ead382c 100644 --- a/doc/LectureNotes/chapter2.ipynb +++ b/doc/LectureNotes/chapter2.ipynb @@ -2133,7 +2133,7 @@ "## Code for SVD and Inversion of Matrices\n", "\n", "How do we use the SVD to invert a matrix $\\boldsymbol{X}^\\boldsymbol{X}$ which is singular or near singular?\n", - "The simple answer is to use the linear algebra function for pseudoinvers, that is" + "The simple answer is to use the linear algebra function for the pseudoinverse, that is" ] }, { @@ -2145,7 +2145,7 @@ }, "outputs": [], "source": [ - "Ainv = np.linlag.pinv(A)" + "#Ainv = np.linlag.pinv(A)" ] }, {