Lecture Notes For Machine Learning Generalized Linear Models

Leo Migdal
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lecture notes for machine learning generalized linear models

Predictors \(\boldsymbol{x} \in \mathcal{X}\subseteq{\mathbb R}^p\): recall that in general these will be a function of the actual covariates \(\boldsymbol{z}\), for example \(\boldsymbol{x} = \boldsymbol{\varphi}(\boldsymbol{z})\). The predictors are numerical; either naturally, or because they are an encoding of categorical variables. A response \(y \in \mathcal{Y} \subseteq {\mathbb R}\). This may be numerical; continuous or discrete, or it may be binary. For now, we do not consider nominal responses with more than two values, for example \(Y \in \left\{\text{red, green, blue}\right\}\)70. We suppose that there is some functional relationship between the predictors and the response, i.e.

that \(P_{}\left(y |\boldsymbol{x}, f\right) = P_{}\left(y |f(\boldsymbol{x})\right)\), with \({\mathrm E}[Y |\boldsymbol{x}, f] = f(\boldsymbol{x})\), for some \(f\in\mathcal{F}\), a suitable set of possible functions. One class of models of functional relationships, defined via a set of possible functions \(\mathcal{F}\) and a set of possible probability distributions whose means will be controlled by those functions, are the generalised linear... A GLM is specified through the following components: #�&�s����P��9鈴?zc)��� R_DG�񃐎��IG�g�#�."�%�BG<� ��'(��xݓ2)(�BC�!�) �ߕ�  k#�3}C��0�����I��]舾@� �IM-�Nj��2H)��0�A7V��l15�gv��#%��iW^��� �7z��"�]��[J`��K�AJ�´�0������h!�]�c���ֈM�2��]舾��+я����Ć�e�R�;�����K ���4=�����Oߐ��?���vͻ�}�M� .����E��@)�tCa%�I{����%ֆ�{����#��ȋؒw�#�d�\�1Y�D ���)� ��Jaڃn(����\i� ~�S��z�ش+/`Kޕ�h'+��Ԍ���DOR!E��ؔ6�|��[��Nk��!�� ����]�>�߅�� =)�>��I�ّR(;��R �h �oF����Z#6�̋ؒw�#�x�r�#�x�R�@Rl�*� ��GJ!��+iQ: ��� +��*TG�^���$ [�t�˳+�f�A�%��b�(��")���nH��-�H�/��$���y�TǞg�����& lɻ��Br:� �6B��c�C�AH��B�A7��Н55�Ϣ�uj�� ��v�lɻ�� 'ڛ4OP�@ t����@��@)�tCa%]I{����E��!�7b�μ�]�.tD'�nc�#�,~;E ��� Q)E ����ޞtD'�oI��@)L{� ���hM�����2}CB��ve lɻ�}թ:bkX3%���(��")���n(,�]T1iNI$�D�D���ֈM;�"��]��_�h+�Y�}���f�hk� R���tD�A7VҮ'il*3���T显!�7bˎ��]�.tD��#E<57^:J�WU��� R�x������z�\�<#��t��=Qv�lɻVG�埫TG�}�6+쑰��HR�*Ҏ��Jړ�G��xo��%��z�ش3/bK�_�ͯ�BG����t��^鈪�8��=��Jj�ݳ����[f���I:"쓎�ؒw�#����/Ԫ%0*�&e0+�&�v� ���D��`��X�W�{e[v�lɻ��]����)�7ʗ�2x'�K�B�A7$VR�����m$��1�oH�9b�μ�-y:�ߖƶ����/Q��{W u�6��I)L{� K�i&����m�7$�Flڕ�%�Zѫ���:b�~�#6��:"��t�:bcT?lY��J�hMMP/�=QΎ�Ϥ*����͒��ꈻ�ȩ�:jY��s� /Q���!��~��=��Q31 ;��!�#��M;�"��]�^z/tD��'�O枔AJ�´Ǟdb)�(��3~7b�����!�5bӮ��-y:�O'�����Y�p� !EU�|G�3/b׼ �鋜ä�Ġ��Q 'nB�x������N��S��� �ϭ�,�+@B�AG|V��%�/t�S��,D������`$e��k;I)�= �����G����z�'�=}SB�#��� ؒw�#�Y�����cSpR� ��$�2H)�R(;�’|��Z�{ �#�Y�oJ� :��̔�5�JG܆*o�� �(� �2���%�v� ��4TS)5 ��Ţ�|CB�[v�lɻVGl����*��A��CVGL�pVG�=�:Ն|�55�Y1���:�dU�:�iG��5�JG������u�'ѯ�28&e��B�I7$�Ҟ�h� 8��诔��ZGlّ��%�BG��`������y%%�0���AH��B�A7VR�>���v�Fͳ�M)Z#vڙ�%�BG�?G����X�#%й�#+R�x��=��J�a�&�=b�N�!��M����]舾�%��4wн@ �qnQ)E ʂ��� R�x��� VR�FkjΨoKl �|��)6�� ؒw�#�+#����ڞ���ب:"�o�:"���H�j%޺�R�}\�ʊG��R��C�yVf�����]�6I�뺩y��FJ�/#�I��H)��tCb% ����3(K�ڔ�Oߐ�sĖ�y[��} �.���ѳB���~�Y�R�c�C�Q��XI#i�8���$m����#6�� ؒ�:bK��4W���@g������#+R��H)��tCb)�7ZS��1FX쟾!��-;3��]�=(L+�ǀ(J���+QG>��R(;��J�z�/5����vLߐ�sĦ]y[���%U舾��$�g-)��"�#��z��R��h-M��<�#%�Flٙ)�kޅ��Z1 �bH |��E����.� � �8 �V��/�wǛ��z�+�� �� \ɵTGt���}f=�}f/}3�e A�`�ChT�Ȕ��^G�o� {�3?]C��f��[ DK•��ϕ�� )�����֠ R��@)L;��Jʍ��Ď��*����̊e�#Ү��-y:�O��VKj���@���jP)E

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