├── notebooks ├── images │ └── ravnine3.png └── SU-2015-5-Regresija.ipynb ├── .gitignore ├── README.md ├── SU.py ├── LICENSE └── data ├── titanic-test.csv ├── questions101-2014.csv └── titanic-train.csv /notebooks/images/ravnine3.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/jsnajder/StrojnoUcenje/master/notebooks/images/ravnine3.png -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | # Byte-compiled / optimized / DLL files 2 | __pycache__/ 3 | *.py[cod] 4 | 5 | # C extensions 6 | *.so 7 | 8 | # Distribution / packaging 9 | .Python 10 | env/ 11 | build/ 12 | develop-eggs/ 13 | dist/ 14 | downloads/ 15 | eggs/ 16 | .eggs/ 17 | lib/ 18 | lib64/ 19 | parts/ 20 | sdist/ 21 | var/ 22 | *.egg-info/ 23 | .installed.cfg 24 | *.egg 25 | 26 | # PyInstaller 27 | # Usually these files are written by a python script from a template 28 | # before PyInstaller builds the exe, so as to inject date/other infos into it. 29 | *.manifest 30 | *.spec 31 | 32 | # Installer logs 33 | pip-log.txt 34 | pip-delete-this-directory.txt 35 | 36 | # Unit test / coverage reports 37 | htmlcov/ 38 | .tox/ 39 | .coverage 40 | .coverage.* 41 | .cache 42 | nosetests.xml 43 | coverage.xml 44 | *,cover 45 | 46 | # Translations 47 | *.mo 48 | *.pot 49 | 50 | # Django stuff: 51 | *.log 52 | 53 | # Sphinx documentation 54 | docs/_build/ 55 | 56 | # PyBuilder 57 | target/ 58 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Strojno učenje 2 | 3 | (c) 2015 Jan Šnajder, UNIZG FER 4 | 5 | Interaktivne bilježnice za predavanja iz predmeta [Strojno učenje](http://www.fer.hr/predmet/su) na [UNIZG FER](http://www.fer.hr). 6 | 7 | ### [0. Uvod u SciPy](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-0-SciPy.ipynb) 8 | 9 | ### [1. Uvodni primjeri](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-1-UvodniPrimjeri.ipynb) 10 | 11 | ### [2. Osnovni koncepti](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-2-OsnovniKoncepti.ipynb) 12 | 13 | ### [3. Osnove teorije vjerojatnosti i statistike](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-3-VjerojatnostStatistika.ipynb) 14 | 15 | ### [4. Bayesov klasifikator](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-4-BayesovKlasifikator.ipynb) 16 | 17 | ### [5. Regresija](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-5-Regresija.ipynb) 18 | 19 | ### [6. Linearni diskriminativni modeli](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-6-LinearniDiskriminativniModeli.ipynb) 20 | 21 | ### [7. Logistička regresija](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-7-LogistickaRegresija.ipynb) 22 | 23 | ### [8. Stroj potpornih vektora (SVM)](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-8-SVM.ipynb) 24 | 25 | ### [10. Vrednovanje modela](http://nbviewer.ipython.org/github/jsnajder/StrojnoUcenje/blob/master/notebooks/SU-2015-10-VrednovanjeModela.ipynb) 26 | 27 | -------------------------------------------------------------------------------- /SU.py: -------------------------------------------------------------------------------- 1 | # Sveučilište u Zagrebu 2 | # Fakultet elektrotehnike i računarstva 3 | # 4 | # Strojno ucenje 5 | # http://www.fer.hr/predmet/su 6 | # 7 | # (c) 2015 Jan Snajder 8 | 9 | import pandas as pd 10 | from sklearn.preprocessing import LabelEncoder 11 | from sklearn.pipeline import Pipeline 12 | 13 | class MultiColumnLabelEncoder: 14 | def __init__(self,columns = None): 15 | self.columns = columns # array of column names to encode 16 | self.encoders ={} 17 | 18 | def fit(self,X,y=None): 19 | if self.columns is not None: 20 | for colname in self.columns: 21 | self.encoders[colname] = LabelEncoder().fit(X[colname]) 22 | else: 23 | for colname,col in X.iteritems(): 24 | self.encoders[colname] = LabelEncoder().fit(col) 25 | return self 26 | 27 | def transform(self,X): 28 | ''' 29 | Transforms columns of X specified in self.columns using 30 | LabelEncoder(). If no columns specified, transforms all 31 | columns in X. 32 | ''' 33 | output = X.copy() 34 | for colname in self.encoders.keys(): 35 | output[colname] = self.encoders[colname].transform(output[colname]) 36 | return output 37 | 38 | def fit_transform(self,X,y=None): 39 | return self.fit(X,y).transform(X) 40 | 41 | def encoders(self): 42 | return self.encoders 43 | 44 | from sklearn.linear_model import LinearRegression 45 | from sklearn.preprocessing import PolynomialFeatures 46 | 47 | class PolyRegression: 48 | 49 | def __init__(self, order): 50 | self.order = order 51 | self.h = LinearRegression() 52 | 53 | def fit(self, X, y): 54 | Xt = PolynomialFeatures(self.order).fit_transform(X) 55 | self.h.fit(Xt, y) 56 | return self 57 | 58 | def predict(self, X): 59 | Xt = PolynomialFeatures(self.order).fit_transform(X) 60 | return self.h.predict(Xt) 61 | 62 | def __call__(self, x): 63 | return self.predict(x)[0] 64 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | CC0 1.0 Universal 2 | 3 | Statement of Purpose 4 | 5 | The laws of most jurisdictions throughout the world automatically confer 6 | exclusive Copyright and Related Rights (defined below) upon the creator and 7 | subsequent owner(s) (each and all, an "owner") of an original work of 8 | authorship and/or a database (each, a "Work"). 9 | 10 | Certain owners wish to permanently relinquish those rights to a Work for the 11 | purpose of contributing to a commons of creative, cultural and scientific 12 | works ("Commons") that the public can reliably and without fear of later 13 | claims of infringement build upon, modify, incorporate in other works, reuse 14 | and redistribute as freely as possible in any form whatsoever and for any 15 | purposes, including without limitation commercial purposes. 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Affirmer understands and acknowledges that Creative Commons is not a 112 | party to this document and has no duty or obligation with respect to this 113 | CC0 or use of the Work. 114 | 115 | For more information, please see 116 | 117 | 118 | -------------------------------------------------------------------------------- /data/titanic-test.csv: -------------------------------------------------------------------------------- 1 | PassengerId,Pclass,Name,Sex,Age,SibSp,Parch,Ticket,Fare,Cabin,Embarked 2 | 892,3,"Kelly, Mr. James",male,34.5,0,0,330911,7.8292,,Q 3 | 893,3,"Wilkes, Mrs. James (Ellen Needs)",female,47,1,0,363272,7,,S 4 | 894,2,"Myles, Mr. Thomas Francis",male,62,0,0,240276,9.6875,,Q 5 | 895,3,"Wirz, Mr. Albert",male,27,0,0,315154,8.6625,,S 6 | 896,3,"Hirvonen, Mrs. Alexander (Helga E Lindqvist)",female,22,1,1,3101298,12.2875,,S 7 | 897,3,"Svensson, Mr. Johan Cervin",male,14,0,0,7538,9.225,,S 8 | 898,3,"Connolly, Miss. 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William Henry",male,14.5,8,2,CA. 2343,69.55,,S 363 | 1253,2,"Mallet, Mrs. Albert (Antoinette Magnin)",female,24,1,1,S.C./PARIS 2079,37.0042,,C 364 | 1254,2,"Ware, Mrs. John James (Florence Louise Long)",female,31,0,0,CA 31352,21,,S 365 | 1255,3,"Strilic, Mr. Ivan",male,27,0,0,315083,8.6625,,S 366 | 1256,1,"Harder, Mrs. George Achilles (Dorothy Annan)",female,25,1,0,11765,55.4417,E50,C 367 | 1257,3,"Sage, Mrs. John (Annie Bullen)",female,,1,9,CA. 2343,69.55,,S 368 | 1258,3,"Caram, Mr. Joseph",male,,1,0,2689,14.4583,,C 369 | 1259,3,"Riihivouri, Miss. Susanna Juhantytar Sanni""""",female,22,0,0,3101295,39.6875,,S 370 | 1260,1,"Gibson, Mrs. Leonard (Pauline C Boeson)",female,45,0,1,112378,59.4,,C 371 | 1261,2,"Pallas y Castello, Mr. Emilio",male,29,0,0,SC/PARIS 2147,13.8583,,C 372 | 1262,2,"Giles, Mr. Edgar",male,21,1,0,28133,11.5,,S 373 | 1263,1,"Wilson, Miss. 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Carl Edgar",male,5,4,2,347077,31.3875,,S 382 | 1272,3,"O'Connor, Mr. Patrick",male,,0,0,366713,7.75,,Q 383 | 1273,3,"Foley, Mr. Joseph",male,26,0,0,330910,7.8792,,Q 384 | 1274,3,"Risien, Mrs. Samuel (Emma)",female,,0,0,364498,14.5,,S 385 | 1275,3,"McNamee, Mrs. Neal (Eileen O'Leary)",female,19,1,0,376566,16.1,,S 386 | 1276,2,"Wheeler, Mr. Edwin Frederick""""",male,,0,0,SC/PARIS 2159,12.875,,S 387 | 1277,2,"Herman, Miss. Kate",female,24,1,2,220845,65,,S 388 | 1278,3,"Aronsson, Mr. Ernst Axel Algot",male,24,0,0,349911,7.775,,S 389 | 1279,2,"Ashby, Mr. John",male,57,0,0,244346,13,,S 390 | 1280,3,"Canavan, Mr. Patrick",male,21,0,0,364858,7.75,,Q 391 | 1281,3,"Palsson, Master. Paul Folke",male,6,3,1,349909,21.075,,S 392 | 1282,1,"Payne, Mr. Vivian Ponsonby",male,23,0,0,12749,93.5,B24,S 393 | 1283,1,"Lines, Mrs. Ernest H (Elizabeth Lindsey James)",female,51,0,1,PC 17592,39.4,D28,S 394 | 1284,3,"Abbott, Master. Eugene Joseph",male,13,0,2,C.A. 2673,20.25,,S 395 | 1285,2,"Gilbert, Mr. William",male,47,0,0,C.A. 30769,10.5,,S 396 | 1286,3,"Kink-Heilmann, Mr. Anton",male,29,3,1,315153,22.025,,S 397 | 1287,1,"Smith, Mrs. Lucien Philip (Mary Eloise Hughes)",female,18,1,0,13695,60,C31,S 398 | 1288,3,"Colbert, Mr. Patrick",male,24,0,0,371109,7.25,,Q 399 | 1289,1,"Frolicher-Stehli, Mrs. Maxmillian (Margaretha Emerentia Stehli)",female,48,1,1,13567,79.2,B41,C 400 | 1290,3,"Larsson-Rondberg, Mr. Edvard A",male,22,0,0,347065,7.775,,S 401 | 1291,3,"Conlon, Mr. Thomas Henry",male,31,0,0,21332,7.7333,,Q 402 | 1292,1,"Bonnell, Miss. Caroline",female,30,0,0,36928,164.8667,C7,S 403 | 1293,2,"Gale, Mr. Harry",male,38,1,0,28664,21,,S 404 | 1294,1,"Gibson, Miss. Dorothy Winifred",female,22,0,1,112378,59.4,,C 405 | 1295,1,"Carrau, Mr. Jose Pedro",male,17,0,0,113059,47.1,,S 406 | 1296,1,"Frauenthal, Mr. Isaac Gerald",male,43,1,0,17765,27.7208,D40,C 407 | 1297,2,"Nourney, Mr. Alfred (Baron von Drachstedt"")""",male,20,0,0,SC/PARIS 2166,13.8625,D38,C 408 | 1298,2,"Ware, Mr. William Jeffery",male,23,1,0,28666,10.5,,S 409 | 1299,1,"Widener, Mr. George Dunton",male,50,1,1,113503,211.5,C80,C 410 | 1300,3,"Riordan, Miss. Johanna Hannah""""",female,,0,0,334915,7.7208,,Q 411 | 1301,3,"Peacock, Miss. Treasteall",female,3,1,1,SOTON/O.Q. 3101315,13.775,,S 412 | 1302,3,"Naughton, Miss. Hannah",female,,0,0,365237,7.75,,Q 413 | 1303,1,"Minahan, Mrs. William Edward (Lillian E Thorpe)",female,37,1,0,19928,90,C78,Q 414 | 1304,3,"Henriksson, Miss. Jenny Lovisa",female,28,0,0,347086,7.775,,S 415 | 1305,3,"Spector, Mr. Woolf",male,,0,0,A.5. 3236,8.05,,S 416 | 1306,1,"Oliva y Ocana, Dona. Fermina",female,39,0,0,PC 17758,108.9,C105,C 417 | 1307,3,"Saether, Mr. Simon Sivertsen",male,38.5,0,0,SOTON/O.Q. 3101262,7.25,,S 418 | 1308,3,"Ware, Mr. Frederick",male,,0,0,359309,8.05,,S 419 | 1309,3,"Peter, Master. Michael J",male,,1,1,2668,22.3583,,C 420 | -------------------------------------------------------------------------------- /data/questions101-2014.csv: -------------------------------------------------------------------------------- 1 | # Sveučilište u Zagrebu 2 | # Fakultet elektrotehnike i računarstva 3 | # 4 | # Strojno učenje (http://www.fer.hr/predmet/su) 5 | # 6 | # Skup podataka "101 Questions Survey" 7 | # 8 | Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,Q9,Q10,Q11,Q12,Q13,Q14,Q15,Q16,Q17,Q18,Q19,Q20,Q21,Q22,Q23,Q24,Q25,Q26,Q27,Q28,Q29,Q30,Q31,Q32,Q33,Q34,Q35,Q36,Q37,Q38,Q39,Q40,Q41,Q42,Q43,Q44,Q45,Q46,Q47,Q48,Q49,Q50,Q51,Q52,Q53,Q54,Q55,Q56,Q57,Q58,Q59,Q60,Q61,Q62,Q63,Q64,Q65,Q66,Q67,Q68,Q69,Q70,Q71,Q72,Q73,Q74,Q75,Q76,Q77,Q78,Q79,Q80,Q81,Q82,Q83,Q84,Q85,Q86,Q87,Q88,Q89,Q90,Q91,Q92,Q93,Q94,Q95,Q96,Q97,Q98,Q99,Q100,Q101 9 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Palestina,Arial,50 cent,The Beatles,Jodie Foster,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Pepsi,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Motor,Voće,Sloboda,Volim masline,Pizza,Noć,Čaj,Irska,Znanost,Kerum,Grad,Večer,U boji,Kemijska olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Elektronska glazba,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 10 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Karamarko,Sladoled,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Nedjelja,Čokolada,UK,Ronjenje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Mladost,Volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Proza,Kiparstvo,Rock,Virus,USA,Smartphone,Italija,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 11 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Kina,Riža,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Španjolski,Mercedes,Karamarko,Sladoled,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Nedjelja,Čokolada,UK,Ronjenje,Piletina,Nova TV,Teretana,U registraturi,Pepsi,Android,Bach,Dinamo,Josipović,Levi's,Socijalizam,Hvar,Šibice,Dubrovnik,Dolar,Nogomet,Lidl,Tomato,Latinica,Centimentar,Griz,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Povrće,Mladost,Volim masline,Lasagne,Dan,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Italija,Tenisice,Gospodar prstenova,Vanilija,Burek s mesom,Batman 12 | More,Split,Voda,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Friends,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Rano ustajanje,Španjolski,BMW,Karamarko,Sladoled,Iran,Times New Roman,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Bach,Hajduk,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Motor,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Škotska,Znanost,Kerum,Grad,Jutro,U boji,Obična olovka,Rimljani,Bill Gates,Stanford,UT,Miley Cyrus,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Elektronska glazba,Bakterija,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek s mesom,Batman 13 | More,Zagreb,Voda,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Torta,Iran,Arial,50 cent,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Piletina,RTL Televizija,Teretana,U registraturi,Pepsi,Android,Beethoven,Dinamo,Mesić,Replay,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Rugby,Konzum,Tele2,Ćirilica,Centimentar,Čokolino,David Bowie,Putin,Microsoft IE,Petar Preradović,Google,Ne volim grožđice,Ljetovanje,Film,Motor,Voće,Sloboda,Volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Grci,Bill Gates,Stanford,UT,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Poezija,Kiparstvo,Rock,Virus,Kuba,Smartphone,Italija,Cipele,Gospodar prstenova,Čokolada,Burek s mesom,Batman 14 | More,Zagreb,Sok,Psi,USA,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kino,Star Trek,Japan,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Palestina,Times New Roman,50 cent,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Bach,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Motor,Voće,Mladost,Volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Grad,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Tenisice,Harry Potter,Vanilija,Burek sa sirom,Superman 15 | More,Zagreb,Voda,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Francuski,Mercedes,Karamarko,Sladoled,Iran,Times New Roman,TBF,Rolling Stones,Jodie Foster,Avion,Novac,Subota,Čokolada,Njemačka,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,iPhone,Beethoven,Dinamo,Mesić,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Griz,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Grad,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,Macintosh,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Vanilija,Burek s mesom,Superman 16 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Bijelo vino,Kazalište,Star Trek,Japan,Riža,Fizika,Zima,Linux,Autobus,E-knjiga,Java,Rano ustajanje,Francuski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Nedjelja,Jabuka,Njemačka,Plivanje,Orada,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Bach,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tomato,Latinica,Centimentar,Griz,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Volim masline,Lasagne,Dan,Čaj,Irska,Znanost,Bandić,Grad,Jutro,U boji,Obična olovka,Grci,Bill Gates,Stanford,WoW,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Proza,Kiparstvo,Rock,Bakterija,Kuba,Obični mobitel,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek sa sirom,Superman 17 | Planina,Zagreb,Voda,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Japan,Riža,Fizika,Zima,Linux,Autobus,Papirnata knjiga,C++,Rano ustajanje,Španjolski,Mercedes,Milanović,Sladoled,Palestina,Arial,TBF,Rolling Stones,Jodie Foster,Avion,Novac,Subota,Čokolada,UK,Ronjenje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Socijalizam,Brač,Šibice,Pula,Euro,Rugby,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Google,Ne volim grožđice,Skijanje,Film,Motor,Voće,Mladost,Volim masline,Pizza,Dan,Kava,Irska,Znanost,Kerum,Priroda,Jutro,U boji,Obična olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Žuja,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Superman 18 | More,Zagreb,Sok,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kino,Star Trek,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Španjolski,BMW,Milanović,Torta,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,UK,Ronjenje,Piletina,RTL Televizija,Joga,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Pula,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Kava,Irska,Znanost,Bandić,Grad,Jutro,U boji,Obična olovka,Grci,Bill Gates,MIT,WoW,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Elektronska glazba,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 19 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Krumpir,Kemija,Ljeto,Linux,Tramvaj,E-knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Pepsi,Android,Beethoven,Hajduk,Mesić,Levi's,Socijalizam,Hvar,Upaljač,Pula,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Skijanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Noć,Kava,Škotska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Rimljani,Steve Jobs,MIT,UT,Lady Gaga,Žuja,Macintosh,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek s mesom,Superman 20 | More,Zagreb,Voda,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Bijelo vino,Kino,Star Trek,Kina,Riža,Fizika,Zima,Windows,Autobus,E-knjiga,Java,Kasno lijeganje,Španjolski,Mercedes,Milanović,Sladoled,Palestina,Arial,50 cent,Rolling Stones,Leonardo di Caprio,Avion,Sreća,Subota,Jabuka,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Socijalizam,Brač,Upaljač,Pula,Dolar,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Noć,Čaj,Irska,Znanost,Kerum,Priroda,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Obični mobitel,Finska,Tenisice,Harry Potter,Čokolada,Burek s mesom,Batman 21 | More,Zagreb,Voda,Mačke,USA,Televizija,Ronaldo,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Japan,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Francuski,BMW,Milanović,Torta,Iran,Times New Roman,50 cent,The Beatles,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Kapitalizam,Hvar,Upaljač,Pula,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Grad,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Poezija,Kiparstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek sa sirom,Batman 22 | More,Zagreb,Voda,Psi,Europa,Radio,Ronaldo,Oliver,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Kemija,Ljeto,Windows,Autobus,E-knjiga,Java,Kasno lijeganje,Španjolski,Mercedes,Milanović,Torta,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Orada,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Bach,Dinamo,Mesić,Levi's,Kapitalizam,Brač,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Proza,Kiparstvo,Elektronska glazba,Virus,USA,Smartphone,Italija,Tenisice,Gospodar prstenova,Vanilija,Burek sa sirom,Batman 23 | More,Zagreb,Voda,Psi,USA,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Krumpir,Kemija,Ljeto,Linux,Tramvaj,E-knjiga,Java,Kasno lijeganje,Španjolski,Mercedes,Karamarko,Torta,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,Nova TV,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Kapitalizam,Brač,Upaljač,Pula,Dolar,Rugby,Lidl,Tele2,Ćirilica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Motor,Voće,Sloboda,Volim masline,Lasagne,Noć,Kava,Irska,Znanost,Bandić,Grad,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Miley Cyrus,Žuja,PC,Brad Pitt,Facebook,Proza,Kiparstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek s mesom,Batman 24 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Milanović,Torta,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,UK,Plivanje,Orada,Nova TV,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Povrće,Sloboda,Volim masline,Lasagne,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Rimljani,Steve Jobs,MIT,WoW,Lady Gaga,Karlovačko,Macintosh,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Bakterija,USA,Smartphone,Finska,Cipele,Gospodar prstenova,Čokolada,Burek s mesom,Batman 25 | Planina,Zagreb,Voda,Mačke,Europa,Televizija,Messi,Oliver,FER,Friends,Bijelo vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,Mercedes,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Jodie Foster,Avion,Sreća,Subota,Jabuka,UK,Plivanje,Piletina,Nova TV,Joga,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Socijalizam,Brač,Upaljač,Pula,Dolar,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Skijanje,Knjiga,Bicikl,Voće,Mladost,Volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Bakterija,Kuba,Obični mobitel,Finska,Tenisice,Gospodar prstenova,Vanilija,Burek s mesom,Batman 26 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Oliver,FER,Friends,Crno vino,Kino,Star Trek,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Iran,Arial,TBF,The Beatles,Jodie Foster,Avion,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Joga,U registraturi,Pepsi,iPhone,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Brač,Upaljač,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Ćirilica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Motor,Voće,Mladost,Ne volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Grci,Steve Jobs,MIT,UT,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Harry Potter,Vanilija,Burek sa sirom,Batman 27 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Krumpir,Kemija,Ljeto,Linux,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Francuski,Mercedes,Milanović,Sladoled,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Nedjelja,Jabuka,UK,Plivanje,Piletina,RTL Televizija,Joga,U registraturi,Pepsi,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Brač,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tomato,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Skijanje,Knjiga,Bicikl,Voće,Sloboda,Volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Kemijska olovka,Grci,Bill Gates,Stanford,UT,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Batman 28 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Oliver,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,Mercedes,Karamarko,Torta,Iran,Arial,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Novac,Subota,Jabuka,Njemačka,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,Petar Preradović,Google,Ne volim grožđice,Ljetovanje,Film,Motor,Voće,Mladost,Volim masline,Lasagne,Noć,Kava,Irska,Znanost,Bandić,Grad,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Kiparstvo,Elektronska glazba,Bakterija,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 29 | Planina,Zagreb,Voda,Mačke,USA,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Trek,Japan,Krumpir,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Francuski,Mercedes,Milanović,Torta,Palestina,Arial,50 cent,Rolling Stones,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Coca Cola,iPhone,Bach,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Skijanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Rimljani,Steve Jobs,MIT,UT,Lady Gaga,Karlovačko,Macintosh,Brad Pitt,Facebook,Proza,Slikarstvo,Elektronska glazba,Virus,USA,Smartphone,Finska,Cipele,Gospodar prstenova,Vanilija,Burek s mesom,Batman 30 | Planina,Split,Voda,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,BMW,Milanović,Torta,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Mesić,Replay,Kapitalizam,Brač,Šibice,Dubrovnik,Euro,Rugby,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Skijanje,Film,Motor,Voće,Mladost,Volim masline,Lasagne,Dan,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Rimljani,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Poezija,Slikarstvo,Elektronska glazba,Virus,Kuba,Smartphone,Finska,Cipele,Harry Potter,Čokolada,Burek s mesom,Superman 31 | Planina,Zagreb,Voda,Mačke,USA,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Krumpir,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Novac,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Pepsi,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Dolar,Rugby,Lidl,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Skijanje,Film,Bicikl,Voće,Sloboda,Volim masline,Pizza,Noć,Kava,Škotska,Znanost,Kerum,Grad,Večer,U boji,Kemijska olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 32 | More,Zagreb,Voda,Psi,USA,Televizija,Messi,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Windows,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Francuski,BMW,Milanović,Sladoled,Iran,Arial,50 cent,Rolling Stones,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tele2,Ćirilica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Grad,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Miley Cyrus,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Obični mobitel,Italija,Tenisice,Gospodar prstenova,Vanilija,Burek s mesom,Superman 33 | Planina,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,E-knjiga,C++,Kasno lijeganje,Španjolski,Mercedes,Milanović,Sladoled,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,UK,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,iPhone,Beethoven,Dinamo,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Motor,Voće,Sloboda,Ne volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Karlovačko,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 34 | More,Split,Sok,Psi,USA,Televizija,Ronaldo,Oliver,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,Nova TV,Joga,U registraturi,Coca Cola,iPhone,Beethoven,Hajduk,Josipović,Levi's,Kapitalizam,Brač,Upaljač,Pula,Dolar,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Irska,Znanost,Kerum,Priroda,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,Stanford,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Italija,Tenisice,Harry Potter,Vanilija,Burek sa sirom,Batman 35 | More,Split,Voda,Psi,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Riža,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,C++,Rano ustajanje,Španjolski,Mercedes,Milanović,Sladoled,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Piletina,RTL Televizija,Joga,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Brač,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tele2,Ćirilica,Centimentar,Griz,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Škotska,Znanost,Bandić,Priroda,Jutro,Crno-bijelo,Obična olovka,Grci,Bill Gates,Stanford,WoW,Lady Gaga,Karlovačko,PC,Brad Pitt,Twitter,Proza,Kiparstvo,Rock,Bakterija,Kuba,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Batman 36 | More,Split,Sok,Psi,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Francuski,Mercedes,Karamarko,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Pepsi,Android,Beethoven,Hajduk,Mesić,Levi's,Kapitalizam,Brač,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Kerum,Priroda,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Poezija,Kiparstvo,Rock,Virus,Kuba,Obični mobitel,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek sa sirom,Batman 37 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Kina,Riža,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,UK,Ronjenje,Orada,RTL Televizija,Teretana,U registraturi,Pepsi,Android,Beethoven,Dinamo,Josipović,Replay,Kapitalizam,Brač,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Motor,Voće,Mladost,Volim masline,Pizza,Dan,Kava,Irska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Tenisice,Harry Potter,Čokolada,Burek s mesom,Batman 38 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Friends,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Nedjelja,Čokolada,UK,Plivanje,Piletina,Nova TV,Joga,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Kapitalizam,Brač,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tomato,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Skijanje,Film,Motor,Voće,Mladost,Volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Rimljani,Steve Jobs,MIT,UT,Miley Cyrus,Žuja,Macintosh,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Italija,Tenisice,Harry Potter,Čokolada,Burek s mesom,Superman 39 | More,Zagreb,Sok,Mačke,USA,Televizija,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Krumpir,Kemija,Zima,Linux,Tramvaj,E-knjiga,C++,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Iran,Times New Roman,50 cent,Rolling Stones,Leonardo di Caprio,Avion,Sreća,Subota,Jabuka,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Dinamo,Mesić,Levi's,Socijalizam,Hvar,Upaljač,Pula,Dolar,Rugby,Konzum,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Volim grožđice,Skijanje,Film,Motor,Voće,Sloboda,Volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Grci,Steve Jobs,MIT,UT,Miley Cyrus,Karlovačko,PC,Johnny Depp,Twitter,Proza,Slikarstvo,Elektronska glazba,Virus,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek sa sirom,Batman 40 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Kemija,Ljeto,Linux,Autobus,Papirnata knjiga,C++,Kasno lijeganje,Francuski,Mercedes,Milanović,Torta,Iran,Arial,TBF,The Beatles,Jodie Foster,Avion,Sreća,Subota,Jabuka,Njemačka,Plivanje,Piletina,Nova TV,Joga,U registraturi,Pepsi,Android,Beethoven,Hajduk,Mesić,Levi's,Socijalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Yahoo,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Povrće,Sloboda,Volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Steve Jobs,MIT,WoW,Miley Cyrus,Karlovačko,PC,Johnny Depp,Facebook,Proza,Kiparstvo,Rock,Bakterija,Kuba,Obični mobitel,Finska,Tenisice,Gospodar prstenova,Vanilija,Burek sa sirom,Batman 41 | More,Zagreb,Sok,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Big Bang Theory,Bijelo vino,Kino,Star Trek,Japan,Krumpir,Kemija,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Torta,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,UK,Ronjenje,Piletina,RTL Televizija,Joga,Povratak Filipa Latinovića,Pepsi,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Finska,Cipele,Harry Potter,Vanilija,Burek s mesom,Batman 42 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Krumpir,Fizika,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Rano ustajanje,Španjolski,BMW,Karamarko,Sladoled,Iran,Times New Roman,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Pepsi,Android,Beethoven,Dinamo,Mesić,Replay,Kapitalizam,Hvar,Upaljač,Pula,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Griz,David Bowie,Putin,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Rimljani,Steve Jobs,MIT,WoW,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Cipele,Gospodar prstenova,Vanilija,Burek s mesom,Batman 43 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Friends,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Španjolski,BMW,Milanović,Sladoled,Iran,Arial,50 cent,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Jabuka,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Bach,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Irska,Umjetnost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,Kuba,Smartphone,Italija,Tenisice,Harry Potter,Vanilija,Burek s mesom,Superman 44 | More,Split,Voda,Mačke,Europa,Radio,Messi,Oliver,FER,Friends,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Autobus,Papirnata knjiga,Java,Rano ustajanje,Francuski,BMW,Milanović,Torta,Iran,Arial,TBF,Rolling Stones,Jodie Foster,Avion,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Orada,Nova TV,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Pula,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Motor,Povrće,Mladost,Volim masline,Pizza,Dan,Kava,Irska,Znanost,Kerum,Priroda,Jutro,U boji,Kemijska olovka,Grci,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Cipele,Gospodar prstenova,Čokolada,Burek sa sirom,Batman 45 | More,Zagreb,Sok,Mačke,Europa,Televizija,Messi,Oliver,FER,Friends,Crno vino,Kino,Star Trek,Kina,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,Mercedes,Milanović,Torta,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Vlak,Novac,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Bach,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tomato,Latinica,Centimentar,Čokolino,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Žuja,PC,Brad Pitt,Facebook,Proza,Kiparstvo,Rock,Bakterija,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 46 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Autobus,E-knjiga,C++,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Subota,Čokolada,UK,Ronjenje,Piletina,Nova TV,Joga,U registraturi,Coca Cola,Android,Bach,Dinamo,Josipović,Replay,Socijalizam,Brač,Upaljač,Dubrovnik,Euro,Rugby,Konzum,Tele2,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Mladost,Volim masline,Lasagne,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Batman 47 | More,Zagreb,Voda,Psi,USA,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Riža,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Španjolski,Mercedes,Milanović,Sladoled,Palestina,Arial,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Novac,Subota,Jabuka,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Dolar,Rugby,Konzum,Tomato,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Irska,Znanost,Bandić,Grad,Jutro,U boji,Obična olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Karlovačko,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Virus,USA,Smartphone,Italija,Cipele,Gospodar prstenova,Vanilija,Burek s mesom,Batman 48 | More,Zagreb,Sok,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Karamarko,Torta,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Mesić,Levi's,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Putin,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Cipele,Harry Potter,Čokolada,Burek sa sirom,Batman 49 | More,Zagreb,Voda,Psi,Europa,Radio,Ronaldo,Gibonni,FER,Friends,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Francuski,BMW,Milanović,Sladoled,Palestina,Arial,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Orada,Nova TV,Teretana,Povratak Filipa Latinovića,Pepsi,Android,Bach,Dinamo,Josipović,Levi's,Socijalizam,Brač,Upaljač,Pula,Euro,Nogomet,Lidl,Tomato,Ćirilica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,Petar Preradović,Google,Ne volim grožđice,Ljetovanje,Knjiga,Bicikl,Povrće,Sloboda,Volim masline,Lasagne,Dan,Kava,Irska,Umjetnost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Bill Gates,MIT,UT,Miley Cyrus,Karlovačko,PC,Johnny Depp,Twitter,Proza,Slikarstvo,Rock,Bakterija,Kuba,Smartphone,Finska,Cipele,Harry Potter,Čokolada,Burek sa sirom,Batman 50 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Crno vino,Kazalište,Star Trek,Japan,Krumpir,Kemija,Ljeto,Windows,Autobus,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Nedjelja,Čokolada,UK,Ronjenje,Orada,RTL Televizija,Joga,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Kava,Škotska,Umjetnost,Bandić,Grad,Jutro,Crno-bijelo,Obična olovka,Grci,Bill Gates,MIT,UT,Miley Cyrus,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 51 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Kina,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Karamarko,Torta,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,UK,Plivanje,Orada,RTL Televizija,Teretana,U registraturi,Coca Cola,iPhone,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Motor,Voće,Mladost,Ne volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Večer,Crno-bijelo,Kemijska olovka,Grci,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Elektronska glazba,Virus,USA,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 52 | More,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Francuski,BMW,Karamarko,Torta,Palestina,Arial,TBF,Rolling Stones,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Orada,Nova TV,Teretana,U registraturi,Coca Cola,iPhone,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,Petar Preradović,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Povrće,Sloboda,Volim masline,Lasagne,Dan,Kava,Irska,Znanost,Kerum,Priroda,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Žuja,Macintosh,Brad Pitt,Facebook,Proza,Kiparstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 53 | Planina,Zagreb,Voda,Psi,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kino,Star Trek,Japan,Krumpir,Fizika,Zima,Linux,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Francuski,BMW,Milanović,Sladoled,Palestina,Arial,TBF,The Beatles,Leonardo di Caprio,Avion,Novac,Subota,Jabuka,UK,Plivanje,Piletina,Nova TV,Joga,Povratak Filipa Latinovića,Coca Cola,iPhone,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Šibice,Dubrovnik,Dolar,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Skijanje,Film,Bicikl,Povrće,Sloboda,Volim masline,Lasagne,Dan,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,Stanford,UT,Lady Gaga,Žuja,Macintosh,Johnny Depp,Facebook,Proza,Slikarstvo,Elektronska glazba,Virus,USA,Smartphone,Finska,Cipele,Harry Potter,Čokolada,Burek sa sirom,Batman 54 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Karamarko,Sladoled,Iran,Times New Roman,50 cent,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,Povratak Filipa Latinovića,Pepsi,Android,Bach,Dinamo,Mesić,Levi's,Kapitalizam,Hvar,Upaljač,Dubrovnik,Dolar,Nogomet,Lidl,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Rimljani,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Vanilija,Burek sa sirom,Batman 55 | Planina,Zagreb,Voda,Psi,Europa,Radio,Messi,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Linux,Autobus,Papirnata knjiga,C++,Kasno lijeganje,Francuski,BMW,Karamarko,Torta,Iran,Times New Roman,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Plivanje,Orada,RTL Televizija,Joga,Povratak Filipa Latinovića,Coca Cola,iPhone,Beethoven,Hajduk,Josipović,Replay,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Rugby,Lidl,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Ne volim grožđice,Skijanje,Knjiga,Bicikl,Povrće,Sloboda,Volim masline,Pizza,Dan,Kava,Irska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Grci,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,Macintosh,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Bakterija,Kuba,Smartphone,Italija,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Batman 56 | Planina,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kazalište,Star Trek,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,C++,Rano ustajanje,Francuski,BMW,Milanović,Sladoled,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Avion,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Kapitalizam,Hvar,Upaljač,Pula,Dolar,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Ne volim grožđice,Skijanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Lasagne,Noć,Čaj,Irska,Znanost,Bandić,Grad,Večer,U boji,Kemijska olovka,Grci,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Superman 57 | More,Zagreb,Voda,Mačke,Europa,Radio,Ronaldo,Gibonni,FER,Friends,Bijelo vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Autobus,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,BMW,Karamarko,Torta,Iran,Arial,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Orada,Nova TV,Teretana,U registraturi,Pepsi,Android,Beethoven,Dinamo,Mesić,Replay,Kapitalizam,Brač,Šibice,Pula,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Griz,Elton John,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Volim masline,Pizza,Noć,Čaj,Škotska,Znanost,Bandić,Priroda,Večer,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Rock,Virus,USA,Obični mobitel,Italija,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 58 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Windows,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,BMW,Karamarko,Torta,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Ronjenje,Piletina,RTL Televizija,Joga,Povratak Filipa Latinovića,Pepsi,iPhone,Beethoven,Dinamo,Mesić,Replay,Socijalizam,Brač,Upaljač,Dubrovnik,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Griz,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Skijanje,Film,Bicikl,Voće,Sloboda,Volim masline,Pizza,Noć,Čaj,Škotska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Grci,Steve Jobs,MIT,UT,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Batman 59 | More,Zagreb,Sok,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Bijelo vino,Kino,Star Trek,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,E-knjiga,C++,Rano ustajanje,Španjolski,Mercedes,Karamarko,Torta,Iran,Arial,TBF,Rolling Stones,Leonardo di Caprio,Avion,Novac,Subota,Čokolada,Njemačka,Plivanje,Orada,RTL Televizija,Teretana,Povratak Filipa Latinovića,Pepsi,Android,Beethoven,Dinamo,Mesić,Replay,Kapitalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Volim masline,Lasagne,Dan,Čaj,Irska,Znanost,Bandić,Grad,Jutro,U boji,Obična olovka,Grci,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Tenisice,Harry Potter,Čokolada,Burek sa sirom,Superman 60 | More,Zagreb,Voda,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Vlak,Novac,Subota,Čokolada,Njemačka,Plivanje,Orada,Nova TV,Teretana,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Hvar,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Volim masline,Pizza,Noć,Kava,Irska,Znanost,Bandić,Priroda,Večer,Crno-bijelo,Kemijska olovka,Rimljani,Bill Gates,MIT,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 61 | More,Zagreb,Voda,Mačke,Europa,Radio,Messi,Gibonni,FER,Friends,Crno vino,Kazalište,Star Trek,Japan,Riža,Fizika,Ljeto,Windows,Autobus,E-knjiga,Java,Rano ustajanje,Španjolski,BMW,Karamarko,Torta,Palestina,Arial,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Subota,Čokolada,UK,Ronjenje,Orada,RTL Televizija,Joga,U registraturi,Coca Cola,Android,Beethoven,Dinamo,Josipović,Levi's,Socijalizam,Brač,Upaljač,Pula,Euro,Rugby,Lidl,Tomato,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,Petar Preradović,Google,Volim grožđice,Skijanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Rimljani,Bill Gates,MIT,UT,Miley Cyrus,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Elektronska glazba,Bakterija,Kuba,Smartphone,Finska,Tenisice,Harry Potter,Čokolada,Burek s mesom,Superman 62 | More,Zagreb,Voda,Mačke,USA,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Španjolski,BMW,Milanović,Sladoled,Iran,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Nedjelja,Čokolada,Njemačka,Ronjenje,Orada,Nova TV,Joga,U registraturi,Coca Cola,Android,Bach,Dinamo,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Pula,Dolar,Nogomet,Lidl,Tele2,Latinica,Centimentar,Griz,Elton John,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Kava,Irska,Znanost,Kerum,Grad,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Žuja,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Cipele,Harry Potter,Vanilija,Burek sa sirom,Superman 63 | Planina,Zagreb,Sok,Mačke,Europa,Televizija,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Windows,Autobus,E-knjiga,C++,Kasno lijeganje,Španjolski,BMW,Karamarko,Sladoled,Iran,Arial,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Plivanje,Piletina,Nova TV,Teretana,U registraturi,Coca Cola,Android,Bach,Dinamo,Josipović,Levi's,Socijalizam,Hvar,Upaljač,Dubrovnik,Euro,Rugby,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Putin,Microsoft IE,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Motor,Povrće,Sloboda,Volim masline,Pizza,Noć,Čaj,Irska,Znanost,Bandić,Priroda,Večer,U boji,Obična olovka,Rimljani,Bill Gates,MIT,WoW,Miley Cyrus,Žuja,PC,Johnny Depp,Facebook,Proza,Kiparstvo,Rock,Virus,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek sa sirom,Superman 64 | More,Split,Voda,Psi,Europa,Televizija,Ronaldo,Oliver,FER,Big Bang Theory,Bijelo vino,Kino,Star Trek,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,E-knjiga,Java,Kasno lijeganje,Francuski,Mercedes,Karamarko,Sladoled,Iran,Times New Roman,TBF,The Beatles,Jodie Foster,Vlak,Sreća,Nedjelja,Čokolada,Njemačka,Plivanje,Orada,Nova TV,Teretana,U registraturi,Coca Cola,iPhone,Beethoven,Hajduk,Josipović,Levi's,Kapitalizam,Brač,Upaljač,Dubrovnik,Euro,Nogomet,Konzum,Tele2,Latinica,Centimentar,Čokolino,Elton John,Putin,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Mladost,Ne volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Kemijska olovka,Grci,Steve Jobs,MIT,UT,Miley Cyrus,Karlovačko,PC,Brad Pitt,Facebook,Proza,Kiparstvo,Rock,Virus,USA,Smartphone,Italija,Tenisice,Gospodar prstenova,Vanilija,Burek sa sirom,Superman 65 | More,Split,Voda,Psi,Europa,Radio,Messi,Oliver,FER,Big Bang Theory,Crno vino,Kazalište,Star Trek,Kina,Riža,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,Java,Rano ustajanje,Španjolski,BMW,Karamarko,Sladoled,Iran,Times New Roman,TBF,Rolling Stones,Jodie Foster,Vlak,Sreća,Subota,Jabuka,Njemačka,Ronjenje,Orada,RTL Televizija,Joga,U registraturi,Pepsi,Android,Bach,Hajduk,Josipović,Levi's,Socijalizam,Brač,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tomato,Ćirilica,Centimentar,Čokolino,Elton John,Putin,Microsoft IE,Petar Preradović,Google,Volim grožđice,Ljetovanje,Knjiga,Bicikl,Voće,Sloboda,Volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Bill Gates,Stanford,UT,Lady Gaga,Karlovačko,PC,Johnny Depp,Twitter,Poezija,Slikarstvo,Rock,Bakterija,Kuba,Smartphone,Finska,Cipele,Gospodar prstenova,Čokolada,Burek sa sirom,Batman 66 | More,Zagreb,Sok,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Big Bang Theory,Bijelo vino,Kazalište,Star Wars,Japan,Krumpir,Fizika,Ljeto,Linux,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Francuski,BMW,Milanović,Sladoled,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,UK,Ronjenje,Orada,Nova TV,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Bach,Dinamo,Josipović,Replay,Socijalizam,Brač,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Povrće,Mladost,Volim masline,Lasagne,Noć,Kava,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Kemijska olovka,Rimljani,Bill Gates,MIT,WoW,Lady Gaga,Karlovačko,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Cipele,Harry Potter,Čokolada,Burek sa sirom,Batman 67 | More,Zagreb,Sok,Psi,Europa,Televizija,Messi,Gibonni,FER,Friends,Bijelo vino,Kino,Star Wars,Japan,Krumpir,Fizika,Zima,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Francuski,BMW,Milanović,Torta,Palestina,Times New Roman,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,Njemačka,Plivanje,Orada,Nova TV,Joga,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Hajduk,Mesić,Replay,Socijalizam,Hvar,Šibice,Pula,Euro,Rugby,Konzum,Tele2,Latinica,Centimentar,Čokolino,David Bowie,Putin,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Bicikl,Povrće,Sloboda,Volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Priroda,Jutro,U boji,Obična olovka,Grci,Bill Gates,Stanford,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Poezija,Slikarstvo,Elektronska glazba,Bakterija,Kuba,Smartphone,Finska,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 68 | More,Zagreb,Voda,Mačke,Europa,Televizija,Ronaldo,Gibonni,FER,Friends,Crno vino,Kazalište,Star Trek,Japan,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,Java,Kasno lijeganje,Španjolski,Mercedes,Milanović,Torta,Iran,Arial,TBF,The Beatles,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,UK,Ronjenje,Orada,Nova TV,Joga,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Socijalizam,Brač,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tomato,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Knjiga,Motor,Voće,Sloboda,Volim masline,Lasagne,Dan,Kava,Irska,Umjetnost,Bandić,Priroda,Jutro,Crno-bijelo,Kemijska olovka,Grci,Bill Gates,Stanford,WoW,Lady Gaga,Karlovačko,PC,Johnny Depp,Facebook,Proza,Slikarstvo,Rock,Virus,Kuba,Smartphone,Finska,Cipele,Gospodar prstenova,Čokolada,Burek s mesom,Batman 69 | Planina,Split,Sok,Psi,USA,Televizija,Ronaldo,Oliver,FER,Big Bang Theory,Crno vino,Kino,Star Wars,Japan,Riža,Fizika,Ljeto,Windows,Tramvaj,Papirnata knjiga,C++,Kasno lijeganje,Francuski,Mercedes,Milanović,Sladoled,Iran,Arial,TBF,Rolling Stones,Leonardo di Caprio,Vlak,Sreća,Subota,Čokolada,UK,Plivanje,Piletina,RTL Televizija,Teretana,U registraturi,Coca Cola,Android,Beethoven,Hajduk,Josipović,Levi's,Kapitalizam,Brač,Šibice,Dubrovnik,Euro,Nogomet,Lidl,Tele2,Latinica,Centimentar,Čokolino,Elton John,Putin,Google Chrome,August Šenoa,Google,Ne volim grožđice,Ljetovanje,Film,Bicikl,Voće,Sloboda,Ne volim masline,Pizza,Dan,Čaj,Škotska,Znanost,Kerum,Grad,Večer,U boji,Kemijska olovka,Grci,Bill Gates,MIT,WoW,Lady Gaga,Žuja,PC,Brad Pitt,Facebook,Proza,Slikarstvo,Rock,Virus,USA,Smartphone,Finska,Cipele,Gospodar prstenova,Čokolada,Burek s mesom,Batman 70 | More,Zagreb,Voda,Psi,Europa,Televizija,Ronaldo,Gibonni,FER,Big Bang Theory,Bijelo vino,Kino,Star Wars,Kina,Riža,Fizika,Ljeto,Linux,Autobus,E-knjiga,Java,Kasno lijeganje,Španjolski,Mercedes,Milanović,Torta,Iran,Times New Roman,50 cent,Rolling Stones,Leonardo di Caprio,Avion,Sreća,Subota,Jabuka,UK,Plivanje,Piletina,Nova TV,Teretana,Povratak Filipa Latinovića,Coca Cola,Android,Beethoven,Hajduk,Josipović,Replay,Socijalizam,Brač,Upaljač,Dubrovnik,Dolar,Nogomet,Konzum,Tele2,Latinica,Centimentar,Griz,Elton John,Obama,Google Chrome,August Šenoa,Google,Volim grožđice,Ljetovanje,Film,Motor,Povrće,Sloboda,Volim masline,Pizza,Dan,Čaj,Irska,Znanost,Bandić,Grad,Večer,U boji,Obična olovka,Grci,Steve Jobs,Stanford,WoW,Lady Gaga,Karlovačko,Macintosh,Brad Pitt,Facebook,Poezija,Slikarstvo,Rock,Bakterija,USA,Smartphone,Italija,Tenisice,Gospodar prstenova,Čokolada,Burek s mesom,Batman 71 | -------------------------------------------------------------------------------- /notebooks/SU-2015-5-Regresija.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "slideshow": { 7 | "slide_type": "slide" 8 | } 9 | }, 10 | "source": [ 11 | "Sveučilište u Zagrebu
\n", 12 | "Fakultet elektrotehnike i računarstva\n", 13 | "\n", 14 | "# Strojno učenje\n", 15 | "\n", 16 | "http://www.fer.unizg.hr/predmet/su\n", 17 | "\n", 18 | "Ak. god. 2015./2016.\n", 19 | "\n", 20 | "# Bilježnica 5: Regresija\n", 21 | "\n", 22 | "(c) 2015 Jan Šnajder\n", 23 | "\n", 24 | "Verzija: 0.3 (2015-11-09)" 25 | ] 26 | }, 27 | { 28 | "cell_type": "code", 29 | "execution_count": 1, 30 | "metadata": { 31 | "collapsed": false 32 | }, 33 | "outputs": [ 34 | { 35 | "name": "stdout", 36 | "output_type": "stream", 37 | "text": [ 38 | "Populating the interactive namespace from numpy and matplotlib\n" 39 | ] 40 | } 41 | ], 42 | "source": [ 43 | "import scipy as sp\n", 44 | "import scipy.stats as stats\n", 45 | "import matplotlib.pyplot as plt\n", 46 | "import pandas as pd\n", 47 | "%pylab inline" 48 | ] 49 | }, 50 | { 51 | "cell_type": "markdown", 52 | "metadata": {}, 53 | "source": [ 54 | "### Sadržaj:\n", 55 | "\n", 56 | "* Uvod\n", 57 | "\n", 58 | "* Osnovni pojmovi\n", 59 | "\n", 60 | "* Model, funkcija gubitka i optimizacijski postupak\n", 61 | "\n", 62 | "* Postupak najmanjih kvadrata\n", 63 | "\n", 64 | "* Probabilistička interpretacija regresije\n", 65 | "\n", 66 | "* Poopćeni linearan model regresije\n", 67 | "\n", 68 | "* Odabir modela\n", 69 | "\n", 70 | "* Regularizirana regresija\n", 71 | "\n", 72 | "* Sažetak" 73 | ] 74 | }, 75 | { 76 | "cell_type": "markdown", 77 | "metadata": {}, 78 | "source": [ 79 | "# Osnovni pojmovi\n", 80 | "\n", 81 | "* Označen skup podataka: $\\mathcal{D}=\\{(\\mathbf{x}^{(i)},y^{(i)})\\},\\quad \\mathbf{x}\\in\\mathbb{R}^n,\\quad y\\in\\mathbb{R}$\n", 82 | "\n", 83 | "\n", 84 | "* Hipoteza $h$ aproksimira nepoznatu funkciju $f:\\mathbb{R}^n\\to\\mathbb{R}$\n", 85 | "\n", 86 | "\n", 87 | "* Idealno, $y^{(i)}=f(\\mathbf{x}^{(i)})$, ali zbog šuma: $$y^{(i)}=f(\\mathbf{x}^{(i)})+\\varepsilon$$\n", 88 | "\n", 89 | "\n", 90 | "* $\\mathbf{x}$ - **ulazna varijabla** (nezavisna, prediktorska)\n", 91 | "\n", 92 | "\n", 93 | "* $y$ - **izlazna varijabla** (zavisna, kriterijska)\n", 94 | "\n", 95 | "\n", 96 | "### Vrste regresije\n", 97 | "\n", 98 | "* Broj **ulaznih** (nezavisnih) varijabli:\n", 99 | " * Univarijatna (jednostavna, jednostruka) regresija: $n=1$\n", 100 | " * Multivarijatna (višestruka, multipla) regresija: $n>1$\n", 101 | "\n", 102 | "\n", 103 | "* Broj **izlaznih** (zavisnih) varijabli:\n", 104 | " * Jednoizlazna regresija: $f(\\mathbf{x}) = y$\n", 105 | " * Višeizlazna regresija: $f(\\mathbf{x})=\\mathbf{y}$\n", 106 | " " 107 | ] 108 | }, 109 | { 110 | "cell_type": "markdown", 111 | "metadata": {}, 112 | "source": [ 113 | "# Model, funkcija gubitka i optimizacijski postupak\n", 114 | "\n", 115 | "\n", 116 | "### (1) Model\n", 117 | "\n", 118 | "* **Linearan model regresije**: $h$ je linearna funkcija parametara\n", 119 | "$\\mathbf{w} = (w_0,\\dots,w_n)$\n", 120 | "\n", 121 | "\n", 122 | "* Linearna regresija:\n", 123 | " $$h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1 x_1 + w_2 x_2 + \\dots + w_n x_n$$\n", 124 | "\n", 125 | "\n", 126 | "* Polinomijalna regresija:\n", 127 | " * Univarijatna polinomijalna: $$h(x|\\mathbf{w}) = w_0 + w_1 x + w_2 x^2 + \\dots + w_d x^d\\quad (n=1)$$\n", 128 | " * Multivarijatna polinomijalna: $$h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1 x_1 + w_2 x_2 + w_3 x_1 x_2 + w_4 x_1^2 + w_5 x_2^2\\quad (n=2, d=2)$$\n", 129 | " * Modelira međuovisnost značajki (*cross-terms* $x_1 x_2, \\dots$) \n", 130 | "\n", 131 | "\n", 132 | "* Općenite **bazne funkcije**:\n", 133 | " $$h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1\\phi_1(\\mathbf{x}) + \\dots + w_m\\phi_m(\\mathbf{x})$$" 134 | ] 135 | }, 136 | { 137 | "cell_type": "markdown", 138 | "metadata": {}, 139 | "source": [ 140 | "### (2) Funkcija gubitka (funkcija pogreške)\n", 141 | "\n", 142 | "* Kvadratni gubitak (engl. *quadratic loss*)\n", 143 | "\n", 144 | "$$\n", 145 | "L(y^{(i)},h(\\mathbf{x}^{(i)})) = \\big(y^{(i)}-h(\\mathbf{x}^{(i)})\\big)^2\n", 146 | "$$\n", 147 | "\n", 148 | "* Funkcija pogreške (proporcionalna s empirijskim očekivanjem gubitka):\n", 149 | "$$\n", 150 | "E(h|\\mathcal{D})=\\frac{1}{2}\n", 151 | "\\sum_{i=1}^N\\big(y^{(i)}-h(\\mathbf{x}^{(i)})\\big)^2\n", 152 | "$$" 153 | ] 154 | }, 155 | { 156 | "cell_type": "markdown", 157 | "metadata": {}, 158 | "source": [ 159 | "### (3) Optimizacijski postupak\n", 160 | "\n", 161 | "* Postupak **najmanjih kvadrata** (engl. *least squares*)\n", 162 | "\n", 163 | "$$\n", 164 | "\\mathrm{argmin}_{\\mathbf{w}} E(\\mathbf{w}|\\mathcal{D})\n", 165 | "$$\n", 166 | "\n", 167 | "\n", 168 | "* Rješenje ovog optimizacijskog problema postoji u **zatvorenoj formi**\n", 169 | "\n", 170 | "\n", 171 | "# Postupak najmanjih kvadrata\n", 172 | "\n", 173 | "\n", 174 | "* Razmotrimo najprije linearnu regresiju:\n", 175 | "$$h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1 x_1 + w_2 x_2 + \\dots + w_n x_n = \\sum_{i=1}^n w_i x_i + w_0$$\n", 176 | "\n", 177 | "\n", 178 | "* Izračun je jednostavniji ako pređemo u matrični račun\n", 179 | " * Svaki vektor primjera $\\mathbf{x}^{(i)}$ proširujemo *dummy* značajkom $x^{(i)}_0 = 1$, pa je model onda:\n", 180 | "\n", 181 | "$$h(\\mathbf{x}|\\mathbf{w}) = \\mathbf{w}^\\intercal \\mathbf{x}$$\n", 182 | "\n", 183 | "\n", 184 | "* Skup primjera:\n", 185 | "\n", 186 | "$$\n", 187 | "\\mathbf{X} = \n", 188 | "\\begin{pmatrix}\n", 189 | "1 & x^{(1)}_1 & x^{(1)}_2 \\dots & x^{(1)}_n\\\\\n", 190 | "1 & x^{(2)}_1 & x^{(2)}_2 \\dots & x^{(2)}_n\\\\\n", 191 | "\\vdots\\\\\n", 192 | "1 & x^{(N)}_1 & x^{(N)}_2 \\dots & x^{(N)}_n\\\\\n", 193 | "\\end{pmatrix}_{N\\times (n+1)}\n", 194 | "=\n", 195 | "\\begin{pmatrix}\n", 196 | "1 & (\\mathbf{x}^{(1)})^\\intercal \\\\\n", 197 | "1 & (\\mathbf{x}^{(2)})^\\intercal \\\\\n", 198 | "\\vdots\\\\\n", 199 | "1 & (\\mathbf{x}^{(N)})^\\intercal \\\\\n", 200 | "1 & \\end{pmatrix}_{N\\times (n+1)}\n", 201 | "$$\n", 202 | "* Matricu primjera $\\mathbf{X}$ zovemo **dizajn-matrica**\n", 203 | "\n", 204 | "\n", 205 | "* Vektor izlaznih vrijednosti:\n", 206 | "$$\n", 207 | "\\mathbf{y} = \n", 208 | "\\begin{pmatrix}\n", 209 | "y^{(1)}\\\\\n", 210 | "y^{(2)}\\\\\n", 211 | "\\vdots\\\\\n", 212 | "y^{(N)}\\\\\n", 213 | "\\end{pmatrix}_{N\\times 1}\n", 214 | "$$\n", 215 | "\n", 216 | "### Egzaktno rješenje\n", 217 | "\n", 218 | "* Idealno, tražimo egzaktno rješenje, tj. rješenje za koje vrijedi\n", 219 | "$$\n", 220 | "(\\mathbf{x}^{(i)}, y^{(i)})\\in\\mathcal{D}.\\ h(\\mathbf{x}^{(i)}) = y^{(i)}\n", 221 | "$$\n", 222 | "odnosno\n", 223 | "$$\n", 224 | "(\\mathbf{x}^{(i)}, y^{(i)})\\in\\mathcal{D}.\\ \\mathbf{w}^\\intercal \\mathbf{x} = y^{(i)}\n", 225 | "$$\n", 226 | "\n", 227 | "\n", 228 | "* Možemo napisati kao matričnu jednadžbu ($N$ jednadžbi s $(n+1)$ nepoznanica):\n", 229 | "\n", 230 | "$$\n", 231 | "\\mathbf{X}\\mathbf{w} = \\mathbf{y}\n", 232 | "$$\n", 233 | "\n", 234 | "$$\n", 235 | "\\begin{pmatrix}\n", 236 | "1 & x^{(1)}_1 & x^{(1)}_2 \\dots & x^{(1)}_n\\\\\n", 237 | "1 & x^{(2)}_1 & x^{(2)}_2 \\dots & x^{(2)}_n\\\\\n", 238 | "\\vdots\\\\\n", 239 | "1 & x^{(N)}_1 & x^{(N)}_2 \\dots & x^{(N)}_n\\\\\n", 240 | "\\end{pmatrix}\n", 241 | "\\cdot\n", 242 | "\\begin{pmatrix}\n", 243 | "w_0\\\\\n", 244 | "w_1\\\\\n", 245 | "\\vdots\\\\\n", 246 | "w_n\\\\\n", 247 | "\\end{pmatrix}\n", 248 | "=\n", 249 | "\\begin{pmatrix}\n", 250 | "y^{(1)}\\\\\n", 251 | "y^{(2)}\\\\\n", 252 | "\\vdots\\\\\n", 253 | "y^{(N)}\\\\\n", 254 | "\\end{pmatrix}\n", 255 | "$$\n", 256 | "\n", 257 | "* Egzaktno rješenje ovog sustava jednadžbi je\n", 258 | "\n", 259 | "$$\n", 260 | "\\mathbf{w} = \\mathbf{X}^{-1}\\mathbf{y}\n", 261 | "$$\n", 262 | "\n", 263 | "Međutim, rješenja nema ili ono nije jedinstveno ako:\n", 264 | "\n", 265 | "* (1) $\\mathbf{X}$ nije kvadratna, pa nema inverz. U pravilu:\n", 266 | " * $N>(n+1)$
\n", 267 | " $\\Rightarrow$ sustav je **preodređen** (engl. *overdetermined*) i nema rješenja\n", 268 | " * $N<(n+1)$
\n", 269 | " $\\Rightarrow$ sustav je **pododređen** (engl. *underdetermined*) i ima višestruka rješenja\n", 270 | " \n", 271 | "* (2) $\\boldsymbol{X}$ jest kvadratna (tj. $N=(n+1)$), ali ipak nema inverz (ovisno o rangu matrice)
$\\Rightarrow$ sustav je **nekonzistentan**\n", 272 | "\n", 273 | "\n", 274 | "### Rješenje najmanjih kvadrata\n", 275 | "\n", 276 | "\n", 277 | "* Približno rješenje sustava $\\mathbf{X}\\mathbf{w}=\\mathbf{y}$\n", 278 | "\n", 279 | "\n", 280 | "* Funkcija pogreške: \n", 281 | "$$\n", 282 | "E(\\mathbf{w}|\\mathcal{D})=\\frac{1}{2}\n", 283 | "\\sum_{i=1}^N\\big(\\mathbf{w}^\\intercal\\mathbf{x}^{(i)} - y^{(i)}\\big)^2\n", 284 | "$$\n", 285 | "\n", 286 | "\n", 287 | "* Matrični oblik:\n", 288 | "\\begin{align*}\n", 289 | "E(\\mathbf{w}|\\mathcal{D}) \n", 290 | "=& \n", 291 | "\\frac{1}{2} (\\mathbf{X}\\mathbf{w} - \\mathbf{y})^\\intercal (\\mathbf{X}\\mathbf{w} - \\mathbf{y})\\\\\n", 292 | "=&\n", 293 | "\\frac{1}{2}\n", 294 | "(\\mathbf{w}^\\intercal\\mathbf{X}^\\intercal\\mathbf{X}\\mathbf{w} - \\mathbf{w}^\\intercal\\mathbf{X}^\\intercal\\mathbf{y} - \\mathbf{y}^\\intercal\\mathbf{X}\\mathbf{w} + \\mathbf{y}^\\intercal\\mathbf{y})\\\\\n", 295 | "=&\n", 296 | "\\frac{1}{2}\n", 297 | "(\\mathbf{w}^\\intercal\\mathbf{X}^\\intercal\\mathbf{X}\\mathbf{w} - 2\\mathbf{y}^\\intercal\\mathbf{X}\\mathbf{w} + \\mathbf{y}^\\intercal\\mathbf{y})\n", 298 | "\\end{align*}\n", 299 | "\n", 300 | "> Jednakosti linearne algebre:\n", 301 | "> * $(A^\\intercal)^\\intercal = A$\n", 302 | "> * $(AB)^\\intercal = B^\\intercal A^\\intercal$\n", 303 | "\n", 304 | "* Minimizacija pogreške:\n", 305 | "$$\n", 306 | "\\begin{align*}\n", 307 | "\\nabla_{\\mathbf{w}}E &= \n", 308 | "\\frac{1}{2}\\Big(\\mathbf{w}^\\intercal\\big(\\mathbf{X}^\\intercal\\mathbf{X}+(\\mathbf{X}^\\intercal\\mathbf{X})^\\intercal\\big) -\n", 309 | "2\\mathbf{y}^\\intercal\\mathbf{X}\\Big) = \n", 310 | "\\mathbf{X}^\\intercal\\mathbf{X}\\mathbf{w} - \\mathbf{X}^\\intercal\\mathbf{y} = \\mathbf{0}\n", 311 | "\\end{align*}\n", 312 | "$$\n", 313 | "\n", 314 | "\n", 315 | "> Jednakosti linearne algebre:\n", 316 | "> * $\\frac{\\mathrm{d}}{\\mathrm{d}x}x^\\intercal A x=x^\\intercal(A+A^\\intercal)$\n", 317 | "> * $\\frac{\\mathrm{d}}{\\mathrm{d}x}A x=A$\n", 318 | "\n", 319 | "\n", 320 | "* Dobivamo sustav tzv. **normalnih jednadžbi**:\n", 321 | "$$\n", 322 | "\\mathbf{X}^\\intercal\\mathbf{X}\\mathbf{w} = \\mathbf{X}^\\intercal\\mathbf{y}\n", 323 | "$$\n", 324 | "\n", 325 | "\n", 326 | "* Rješenje:\n", 327 | "$$\n", 328 | "\\mathbf{w} = (\\mathbf{X}^\\intercal\\mathbf{X})^{-1}\\mathbf{X}^\\intercal\\mathbf{y} = \\color{red}{\\mathbf{X}^{+}}\\mathbf{y}\n", 329 | "$$\n", 330 | "\n", 331 | "\n", 332 | "* Matrica $\\mathbf{X}^{+}=(\\mathbf{X}^\\intercal\\mathbf{X})^{-1}\\mathbf{X}^\\intercal$ je **pseudoinverz** (Moore-Penroseov inverz) matrice $\\mathbf{X}$\n", 333 | "\n", 334 | "\n", 335 | "* **Q:** Kojih je dimenzija matrica $(\\mathbf{X}^\\intercal\\mathbf{X})^{-1}$?\n", 336 | "* **Q:** Što utječe na složenost izračuna inverza matrice: broj primjera $N$ ili broj dimenzija $n$?" 337 | ] 338 | }, 339 | { 340 | "cell_type": "markdown", 341 | "metadata": {}, 342 | "source": [ 343 | "# Probabilistička interpretacija regresije" 344 | ] 345 | }, 346 | { 347 | "cell_type": "markdown", 348 | "metadata": {}, 349 | "source": [ 350 | "* Ograničimo se BSO na univarijatnu ($n=1$) linearnu regresiju:\n", 351 | "\n", 352 | "$$\n", 353 | "h(x|w_0, w_1) = w_0 + w_1 x\n", 354 | "$$\n", 355 | "\n", 356 | "\n", 357 | "* Zbog šuma u $\\mathcal{D}$:\n", 358 | "$$\n", 359 | " y^{(i)} = f(x^{(i)}) + \\color{red}{\\varepsilon}\n", 360 | "$$\n", 361 | "\n", 362 | "* Prepostavka:\n", 363 | "$$\n", 364 | " \\color{red}{\\varepsilon}\\ \\sim\\ \\mathcal{N}(0, \\sigma^2)\n", 365 | "$$\n", 366 | "\n", 367 | "* Posjedično:\n", 368 | "$$\n", 369 | " \\color{red}{y|x}\\ \\sim\\ \\mathcal{N}\\big(f(x), \\sigma^2\\big)\n", 370 | "$$\n", 371 | "odnosno\n", 372 | "$$\n", 373 | " \\color{red}{p(y|x)} = \\mathcal{N}\\big(f(x), \\sigma^2\\big)\n", 374 | "$$\n", 375 | "\n", 376 | "* Vrijedi \n", 377 | "$$\\mathbb{E}[y|x] = \\mu = f(x)$$\n", 378 | "\n", 379 | "\n", 380 | "* Naš cilj je: $h(x|\\mathbf{w}) = f(x)$\n", 381 | "\n", 382 | "\n", 383 | "* [Skica]\n", 384 | "\n", 385 | "\n", 386 | "* $p(y^{(i)}|x^{(i)})$ je vjerojatnost da je $f(x^{(i)})$ generirala vrijednost $y^{(i)}$\n", 387 | " * (Formulacija nije baš točna, jer je $x$ kontinuirana varijabla a $p$ je gustoća vjerojatnosti.)\n", 388 | " \n", 389 | "### Log-izglednost\n", 390 | "\n", 391 | "$$\n", 392 | "\\begin{align*}\n", 393 | "\\ln\\mathcal{L}(\\mathbf{w}|\\mathcal{D}) \n", 394 | "&= \n", 395 | "\\ln p(\\mathcal{D}|\\mathbf{w}) = \n", 396 | "\\ln\\prod_{i=1}^N p(x^{(i)}, y^{(i)}) =\n", 397 | "\\ln\\prod_{i=1}^N p(y^{(i)}|x^{(i)}) p(x^{(i)}) \\\\ \n", 398 | "&= \n", 399 | "\\ln\\prod_{i=1}^N p(y^{(i)}|x^{(i)}) + \\underbrace{\\color{gray}{\\ln\\prod_{i=1}^N p(x^{(i)})}}_{\\text{Ne ovisi o $\\mathbf{w}$}} \\\\\n", 400 | "& \\Rightarrow \\ln\\prod_{i=1}^N p(y^{(i)}|x^{(i)}) =\n", 401 | "\\ln\\prod_{i=1}^N\\mathcal{N}\\big(h(x^{(i)}|\\mathbf{w}),\\sigma^2\\big)\\\\ &= \n", 402 | "\\ln\\prod_{i=1}^N\\frac{1}{\\sqrt{2\\pi}\\sigma}\\exp\\Big\\{-\\frac{\\big(y^{(i)}-h(x^{(i)}|\\mathbf{w})\\big)^2}{2\\sigma^2}\\Big\\}\\\\ \n", 403 | "&=\n", 404 | "\\underbrace{\\color{gray}{-N\\ln(\\sqrt{2\\pi}\\sigma)}}_{\\text{konst.}} -\n", 405 | "\\frac{1}{2\\color{gray}{\\sigma^2}}\\sum_{i=1}^N\\big(y^{(i)}-h(x^{(i)}|\\mathbf{w})\\big)^2\\\\\n", 406 | "& \\Rightarrow\n", 407 | "-\\frac{1}{2}\\sum_{i=1}^N\\big(y^{(i)}-h(x^{(i)}|\\mathbf{w})\\big)^2\n", 408 | "\\end{align*}\n", 409 | "$$\n", 410 | "\n", 411 | "\n", 412 | "* Uz pretpostavku Gaussovog šuma, **maksimizacija izglednosti** odgovara **minimizaciji funkcije pogreške** definirane kao **zbroj kvadratnih odstupanja**:\n", 413 | "\n", 414 | "$$\n", 415 | "\\begin{align*}\n", 416 | "\\mathrm{argmax}_{\\mathbf{w}} \\ln\\mathcal{L}(\\mathbf{w}|\\mathcal{D}) &= \\mathrm{argmin}_{\\mathbf{w}} E(\\mathbf{w}|\\mathcal{D})\\\\\n", 417 | "E(h|\\mathcal{D}) &=\\frac{1}{2} \\sum_{i=1}^N\\big(y^{(i)}-h(x^{(i)}|\\mathbf{w})\\big)^2\\\\\n", 418 | "L\\big(y,h(x|\\mathbf{w})\\big)\\ &\\propto\\ \\big(y - h(x|\\mathbf{w})\\big)^2\n", 419 | "\\end{align*}\n", 420 | "$$\n", 421 | "\n", 422 | "\n", 423 | "* $\\Rightarrow$ Probabilističko opravdanje za kvadratnu funkciju gubitka\n", 424 | "\n", 425 | "\n", 426 | "* Rješenje MLE jednako je rješenju koje daje postupak najmanjih kvadrata!\n", 427 | "\n" 428 | ] 429 | }, 430 | { 431 | "cell_type": "markdown", 432 | "metadata": {}, 433 | "source": [ 434 | "# Poopćeni linearan model regresije" 435 | ] 436 | }, 437 | { 438 | "cell_type": "markdown", 439 | "metadata": {}, 440 | "source": [ 441 | "* Zanima nas poopćenje na $n>1$ koje obuhvaća sve multivarijatne linearne modele regresije: univarijatna regresija, linearna regresija, polinomijalna regresija, ...\n", 442 | " * $h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1 x_1 + w_2 x_2 + \\dots + w_n x_n$\n", 443 | " * $h(x|\\mathbf{w}) = w_0 + w_1 x + w_2 x^2 + \\dots + w_d x^d$\n", 444 | " * $h(\\mathbf{x}|\\mathbf{w}) = w_0 + w_1 x_1 + w_2 x_2 + w_3 x_1 x_2 + w_4 x_1^2 + w_5 x_2^2$\n", 445 | " * ...\n", 446 | "\n", 447 | "\n", 448 | "* Uvodimo fiksan skup **baznih funkcija** (nelinearne funkcije ulaznih varijabli):\n", 449 | "$$\n", 450 | " \\{\\phi_0, \\phi_1, \\phi_2, \\dots, \\phi_m\\}\n", 451 | "$$ \n", 452 | "gdje $\\phi_j:\\mathbb{R}^n\\to\\mathbb{R}$\n", 453 | "\n", 454 | "\n", 455 | "* Dogovorno: $\\phi_0(\\mathbf{x}) = 1$\n", 456 | "\n", 457 | "\n", 458 | "* Svaki vektor primjera u $n$-dimenzijskom originalnom ulaznom prostoru (engl. *input space*) $\\mathcal{X}$:\n", 459 | "$$\n", 460 | "\\mathbf{x} = (x_1, x_2, \\dots, x_n)\n", 461 | "$$\n", 462 | "preslikavamo u nov, $m$-dimenzijski prostor, tzv. **prostor značajki** (engl. *feature space*):\n", 463 | "$$\n", 464 | "\\boldsymbol{\\phi}(\\mathbf{x}) = \\big(\\phi_0(\\mathbf{x}), \\phi_1(\\mathbf{x}), \\dots, \\phi_m(\\mathbf{x})\\big)\n", 465 | "$$\n", 466 | "\n", 467 | "\n", 468 | "* **Funkija preslikavanja** (vektor baznih funkcija)\n", 469 | "$$\n", 470 | "\\begin{align*}\n", 471 | "\\boldsymbol{\\phi}&:\\mathbb{R}^n\\to\\mathbb{R}^m:\\\\\n", 472 | "\\boldsymbol{\\phi}(\\mathbf{x}) &= \\big(\\phi_0(\\mathbf{x}),\\dots,\\phi_m(\\mathbf{x})\\big)\\\\\n", 473 | "\\end{align*}\n", 474 | "$$\n", 475 | "\n", 476 | "\n", 477 | "* Poopćen linearan model:\n", 478 | "$$\n", 479 | " h(\\mathbf{x}|\\mathbf{w}) = \\sum_{j=0}^m w_j\\phi_j(\\mathbf{x}) = \\mathbf{w}^\\intercal\\boldsymbol{\\phi}(\\mathbf{x})\n", 480 | "$$\n", 481 | "\n", 482 | "\n", 483 | "### Uobičajene funkcije preslikavanja\n", 484 | "\n", 485 | "\n", 486 | "* Linearna regresija:\n", 487 | "$$\n", 488 | "\\boldsymbol{\\phi}(\\mathbf{x}) = (1,x_1,x_2,\\dots,x_n)\n", 489 | "$$\n", 490 | "\n", 491 | "\n", 492 | "* Univarijatna polinomijalna regresija: \n", 493 | "$$\n", 494 | "\\boldsymbol{\\phi}(x) = (1,x,x^2,\\dots,x^m)\n", 495 | "$$\n", 496 | "\n", 497 | "\n", 498 | "* Polinomijalna regresija drugog stupnja: \n", 499 | "$$\n", 500 | "\\boldsymbol{\\phi}(\\mathbf{x}) = (1,x_1,x_2,x_1 x_2, x_1^2, x_2^2)\n", 501 | "$$\n", 502 | "\n", 503 | "\n", 504 | "* Gaussove bazne funkcije (RBF):\n", 505 | "$$\n", 506 | "\\phi_j(x) = \\exp\\Big\\{-\\frac{(x-\\mu_j)^2}{2\\sigma^2}\\Big\\}\n", 507 | "$$\n", 508 | "\n", 509 | "\n", 510 | "* [Skica: RBF] \n", 511 | "\n", 512 | "### Prostor značajki\n", 513 | "\n", 514 | "\n", 515 | "* **Funkcija preslikavanja značajki** $\\mathbf{\\phi} : \\mathbb{R}^n \\to \\mathbb{R}^m $ preslikava primjere iz $n$-dimenzijskog ulaznog prostora u $m$-dimenzijski prostor značajki\n", 516 | "\n", 517 | "\n", 518 | "* Tipično je $m>n$\n", 519 | "\n", 520 | "\n", 521 | "* Tada je funkcija koja je linearna u prostoru značajki **nelinearna u ulaznom prostoru**\n", 522 | "\n", 523 | "\n", 524 | "* Dakle, možemo koristiti linearan model za nelinearne probleme\n", 525 | "\n", 526 | "\n", 527 | "* Imamo unificiran postupak, neovisno koju funkciju $\\boldsymbol{\\phi}$ odaberemo" 528 | ] 529 | }, 530 | { 531 | "cell_type": "markdown", 532 | "metadata": {}, 533 | "source": [ 534 | "### Primjer: Preslikavanje iz ulaznog prostora u prostor značajki\n", 535 | "\n", 536 | "* $\\mathcal{X} = \\mathbb{R}$\n", 537 | "* $n=1$, $m=3$\n", 538 | "* $\\boldsymbol{\\phi} : \\mathbb{R} \\to \\mathbb{R}^3$\n", 539 | "* $\\boldsymbol{\\phi}(x) = (1,x,x^2)$\n", 540 | "* [Skica]" 541 | ] 542 | }, 543 | { 544 | "cell_type": "code", 545 | "execution_count": 261, 546 | "metadata": { 547 | "collapsed": true 548 | }, 549 | "outputs": [], 550 | "source": [ 551 | "def f(x) : return 3*(x - 2)**2 + 1\n", 552 | "\n", 553 | "x1 = 1\n", 554 | "x2 = 2\n", 555 | "x3 = 3" 556 | ] 557 | }, 558 | { 559 | "cell_type": "code", 560 | "execution_count": 262, 561 | "metadata": { 562 | "collapsed": false 563 | }, 564 | "outputs": [ 565 | { 566 | "data": { 567 | "image/png": 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TSGOopy0Nds01sGCB33dZkqW01J8BqoJdeFS0C1jr1jBuHPzyl6GTSENUV8MvfgG33BI6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\n", 568 | "text/plain": [ 569 | "" 570 | ] 571 | }, 572 | "metadata": {}, 573 | "output_type": "display_data" 574 | } 575 | ], 576 | "source": [ 577 | "xs = linspace(0, 4)\n", 578 | "y = f(xs)\n", 579 | "plt.ylim(0,5)\n", 580 | "plt.plot(xs, y)\n", 581 | "plt.plot(x1, f(x1), 'ro')\n", 582 | "plt.plot(x2, f(x2), 'go')\n", 583 | "plt.plot(x3, f(x3), 'bo')\n", 584 | "plt.show()" 585 | ] 586 | }, 587 | { 588 | "cell_type": "code", 589 | "execution_count": 263, 590 | "metadata": { 591 | "collapsed": false 592 | }, 593 | "outputs": [], 594 | "source": [ 595 | "def phi(x): return sp.array([1, x, x**2])" 596 | ] 597 | }, 598 | { 599 | "cell_type": "code", 600 | "execution_count": 264, 601 | "metadata": { 602 | "collapsed": false 603 | }, 604 | "outputs": [ 605 | { 606 | "data": { 607 | "text/plain": [ 608 | "array([1, 1, 1])" 609 | ] 610 | }, 611 | "execution_count": 264, 612 | "metadata": {}, 613 | "output_type": "execute_result" 614 | } 615 | ], 616 | "source": [ 617 | "phi(x1)" 618 | ] 619 | }, 620 | { 621 | "cell_type": "code", 622 | "execution_count": 265, 623 | "metadata": { 624 | "collapsed": false 625 | }, 626 | "outputs": [ 627 | { 628 | "data": { 629 | "text/plain": [ 630 | "array([1, 2, 4])" 631 | ] 632 | }, 633 | "execution_count": 265, 634 | "metadata": {}, 635 | "output_type": "execute_result" 636 | } 637 | ], 638 | "source": [ 639 | "phi(x2)" 640 | ] 641 | }, 642 | { 643 | "cell_type": "code", 644 | "execution_count": 266, 645 | "metadata": { 646 | "collapsed": false 647 | }, 648 | "outputs": [ 649 | { 650 | "data": { 651 | "text/plain": [ 652 | "array([1, 3, 9])" 653 | ] 654 | }, 655 | "execution_count": 266, 656 | "metadata": {}, 657 | "output_type": "execute_result" 658 | } 659 | ], 660 | "source": [ 661 | "phi(x3)" 662 | ] 663 | }, 664 | { 665 | "cell_type": "code", 666 | "execution_count": 267, 667 | "metadata": { 668 | "collapsed": false 669 | }, 670 | "outputs": [], 671 | "source": [ 672 | "xs1 = linspace(0, 5)\n", 673 | "xs2 = linspace(0, 10)\n", 674 | "X1, X2 = np.meshgrid(xs1, xs2)" 675 | ] 676 | }, 677 | { 678 | "cell_type": "code", 679 | "execution_count": 268, 680 | "metadata": { 681 | "collapsed": true 682 | }, 683 | "outputs": [], 684 | "source": [ 685 | "phi_X = 3*X2 - 12*X1 + 13" 686 | ] 687 | }, 688 | { 689 | "cell_type": "code", 690 | "execution_count": 286, 691 | "metadata": { 692 | "collapsed": false 693 | }, 694 | "outputs": [ 695 | { 696 | "data": { 697 | "image/png": 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698 | "text/plain": [ 699 | "" 700 | ] 701 | }, 702 | "metadata": {}, 703 | "output_type": "display_data" 704 | } 705 | ], 706 | "source": [ 707 | "plt.contour(X, Y, phi_X, levels=[1,4])\n", 708 | "plt.scatter(phi(x1)[1], phi(x1)[2], c='r')\n", 709 | "plt.scatter(phi(x2)[1], phi(x2)[2], c='g')\n", 710 | "plt.scatter(phi(x3)[1], phi(x3)[2], c='b')\n", 711 | "plt.legend()\n", 712 | "plt.show()" 713 | ] 714 | }, 715 | { 716 | "cell_type": "markdown", 717 | "metadata": {}, 718 | "source": [ 719 | "### Optimizacijski postupak\n", 720 | "\n", 721 | "* Ništa se ne mijenja u odnosu na ono što smo već izveli, samo umjesto $\\mathbf{X}$ imamo dizajn-matricu $\\boldsymbol{\\Phi}$\n", 722 | "\n", 723 | "\n", 724 | "* Dizajn-matrica:\n", 725 | "$$\n", 726 | "\\boldsymbol{\\Phi} = \n", 727 | "\\begin{pmatrix}\n", 728 | "1 & \\phi_1(\\mathbf{x}^{(1)}) & \\dots & \\phi_m(\\mathbf{x}^{(1)})\\\\\n", 729 | "1 & \\phi_1(\\mathbf{x}^{(2)}) & \\dots & \\phi_m(\\mathbf{x}^{(2)})\\\\\n", 730 | "\\vdots\\\\\n", 731 | "1 & \\phi_1(\\mathbf{x}^{(N)}) & \\dots & \\phi_m(\\mathbf{x}^{(N)})\\\\\n", 732 | "\\end{pmatrix}_{N\\times m}\n", 733 | "=\n", 734 | "\\begin{pmatrix}\n", 735 | "\\mathbf{\\phi}(\\mathbf{x}^{(1)})^\\intercal \\\\\n", 736 | "\\mathbf{\\phi}(\\mathbf{x}^{(2)})^\\intercal \\\\\n", 737 | "\\vdots\\\\\n", 738 | "\\mathbf{\\phi}(\\mathbf{x}^{(N)})^\\intercal \\\\\n", 739 | "\\end{pmatrix}_{N\\times m}\n", 740 | "$$\n", 741 | "\n", 742 | "* Prije smo imali:\n", 743 | "$$\n", 744 | "\\mathbf{w} = (\\mathbf{X}^\\intercal\\mathbf{X})^{-1}\\mathbf{X}^\\intercal\\mathbf{y} = \\color{red}{\\mathbf{X}^{+}}\\mathbf{y}\n", 745 | "$$\n", 746 | "a sada imamo:\n", 747 | "$$\n", 748 | "\\mathbf{w} = (\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi})^{-1}\\boldsymbol{\\Phi}^\\intercal\\mathbf{y} = \\color{red}{\\boldsymbol{\\Phi}^{+}}\\mathbf{y}\n", 749 | "$$\n", 750 | "gdje\n", 751 | "$$\n", 752 | "\\boldsymbol{\\Phi}^{+}=(\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi})^{-1}\\boldsymbol{\\Phi}^\\intercal\n", 753 | "$$" 754 | ] 755 | }, 756 | { 757 | "cell_type": "markdown", 758 | "metadata": {}, 759 | "source": [ 760 | "# Odabir modela" 761 | ] 762 | }, 763 | { 764 | "cell_type": "markdown", 765 | "metadata": {}, 766 | "source": [ 767 | "* Poopćeni linearan model regresije ima jedan **hiperparametar**: funkciju preslikavanje $\\boldsymbol{\\phi}$\n", 768 | "\n", 769 | "\n", 770 | "* Alternativno, možemo reći da se radi o dva hiperparametra:\n", 771 | " * izgled baznih funkcija $\\phi_j$\n", 772 | " * broj baznih funkcija $m$ (dimenzija prostora značajki)\n", 773 | "\n", 774 | "\n", 775 | "* Hiperparametre treba namjestiti tako da odgovaraju podatcima, odnosno treba\n", 776 | "dobro **odabrati model**\n", 777 | "\n", 778 | "\n", 779 | "* U suprotnom model može biti **podnaučen** ili **prenaučen**\n", 780 | "\n", 781 | "\n", 782 | "* Ako model ima mnogo parametra, lako ga je prenaučiti\n", 783 | "\n", 784 | "\n", 785 | "* Sprečavanje prenaučenosti:\n", 786 | " 1. Koristiti više primjera za učenje\n", 787 | " 2. Odabrati model unakrsnom provjerom\n", 788 | " 3. **Regularizacija**\n", 789 | " 4. Bayesovska regresija (bayesovski odabir modela) $\\Rightarrow$ nećemo raditi\n", 790 | "\n" 791 | ] 792 | }, 793 | { 794 | "cell_type": "markdown", 795 | "metadata": {}, 796 | "source": [ 797 | "# Regularizirana regresija\n", 798 | "\n", 799 | "\n", 800 | "### Ideja\n", 801 | "\n", 802 | "* Opažanje: kod linearnih modela, što je model složeniji, to ima veće vrijednosti parametara $\\mathbf{w}$\n", 803 | "\n", 804 | "\n", 805 | "* Prenaučeni linearni modeli imaju:\n", 806 | " * ukupno previše parametara (težina) i/ili\n", 807 | " * prevelike vrijednosti pojedinačnih parametara\n", 808 | "\n", 809 | "\n", 810 | "* Ideja: **ograničiti rast vrijednosti parametara** kažnjavanjem hipoteza s visokim vrijednostima parametara\n", 811 | "\n", 812 | "\n", 813 | "* Time ostvarujemo **kompromis** između točnosti i jednostavnosti modela i to već **pri samom učenju** modela\n", 814 | "\n", 815 | "\n", 816 | "* Efektivno se **graničava složenost** modela i sprečava se prenaučenost\n", 817 | "\n", 818 | "\n", 819 | "* Cilj: što više parametara (težina) pritegnuti na nulu $\\Rightarrow$ **rijetki modeli** (engl. *sparse models*)\n", 820 | "\n", 821 | "\n", 822 | "* Rijetki modeli su:\n", 823 | " * teži za prenaučiti\n", 824 | " * računalno jednostavniji\n", 825 | " * interpretabilniji\n" 826 | ] 827 | }, 828 | { 829 | "cell_type": "markdown", 830 | "metadata": {}, 831 | "source": [ 832 | "### Regularizacija\n", 833 | "\n", 834 | "* U funkciju pogreške (koju minimiziramo) ugrađujemo mjeru složenosti modela:\n", 835 | "\n", 836 | "$$\n", 837 | " E' = \\textrm{empirijska pogreška} + \\color{red}{\\lambda\\times\\textrm{složenost modela}}\n", 838 | "$$\n", 839 | "\n", 840 | "$$\n", 841 | " E'(\\mathbf{w}|\\mathcal{D}) = E(\\mathbf{w}|\\mathcal{D}) + \\underbrace{\\color{red}{\\lambda E_R(\\mathbf{w})}}_{\\text{reg. izraz}}\n", 842 | "$$\n", 843 | "\n", 844 | "* $\\lambda$ je **regularizacijski faktor**\n", 845 | " * $\\lambda=0\\ \\Rightarrow$ neregularizirana funkcija pogreške\n", 846 | " * Veća vrijednost regularizacijskog faktora $\\lambda$ uzrokuje smanjuje efektivne složenost modela\n", 847 | "\n", 848 | "\n", 849 | "* [Skica: Regularizirana regresija]\n", 850 | "\n", 851 | "\n", 852 | "* Općenit regularizacijski izraz: **p-norma vektora težina**\n", 853 | "$$\n", 854 | " E_R(\\mathbf{w}) = \\|\\mathbf{w}\\|_p = \\Big(\\sum_{j=\\color{red}{1}}^m |w_j|^p\\Big)^{\\frac{1}{p}}\n", 855 | "$$\n", 856 | "\n", 857 | "\n", 858 | "* L2-norma ($p=2$):\n", 859 | "$$\\|\\mathbf{w}\\|_2 = \\sqrt{\\sum_{j=\\color{red}{1}}^m w_j^2} = \\sqrt{\\mathbf{w}^\\intercal\\mathbf{w}}$$\n", 860 | "\n", 861 | "\n", 862 | "* L1-norma ($p=1$):\n", 863 | "$$\\|\\mathbf{w}\\|_1 = \\sum_{j=\\color{red}{1}}^m |w_j|$$\n", 864 | "\n", 865 | "\n", 866 | "* L0-norma ($p=0$):\n", 867 | "$$\\|\\mathbf{w}\\|_0 = \\sum_{j=\\color{red}{1}}^m \\mathbf{1}\\{w_j\\neq 0\\}$$\n", 868 | "\n", 869 | "\n", 870 | "* **NB:** Težina $w_0$ se ne regularizira\n", 871 | " * **Q:** Zašto?\n", 872 | " \n", 873 | " \n", 874 | "\n", 875 | "\n", 876 | "\n", 877 | "\n" 878 | ] 879 | }, 880 | { 881 | "cell_type": "markdown", 882 | "metadata": {}, 883 | "source": [ 884 | "### Regularizirani linearni model regresije\n", 885 | " \n", 886 | "* **L2-regularizacija** ili Tikhononova regularizacija $\\Rightarrow$ **Ridge regression**:\n", 887 | "$$\n", 888 | "E(\\mathbf{w}|\\mathcal{D})=\\frac{1}{2}\n", 889 | "\\sum_{i=1}^N\\big(\\mathbf{w}^\\intercal\\boldsymbol{\\phi}(\\mathbf{x}^{(i)}) - y^{(i)}\\big)^2\n", 890 | "+ \\color{red}{\\frac{\\lambda}{2}\\|\\mathbf{w}\\|^2_2}\n", 891 | "$$\n", 892 | " * ima rješenje u zatvorenoj formi\n", 893 | " \n", 894 | "\n", 895 | "* **L1-regularizacija** $\\Rightarrow$ **LASSO regularization** (engl. *least absolute shrinkage and selection operator*)\n", 896 | "$$\n", 897 | "E(\\mathbf{w}|\\mathcal{D})=\\frac{1}{2}\n", 898 | "\\sum_{i=1}^N\\big(\\mathbf{w}^\\intercal\\boldsymbol{\\phi}(\\mathbf{x}^{(i)}) - y^{(i)}\\big)^2\n", 899 | "+ \\color{red}{\\frac{\\lambda}{2}\\|\\mathbf{w}\\|_1}\n", 900 | "$$\n", 901 | " * nema rješenje u zatvorenoj formi!\n", 902 | "\n", 903 | "\n", 904 | "* **L0-regularizacija**\n", 905 | "$$\n", 906 | "E(\\mathbf{w}|\\mathcal{D})=\\frac{1}{2}\n", 907 | "\\sum_{i=1}^N\\big(\\mathbf{w}^\\intercal\\mathbf{\\phi}(\\mathbf{x}^{(i)}) - y^{(i)}\\big)^2\n", 908 | "+ \\color{red}{\\frac{\\lambda}{2}\\sum_{j=1}^m\\mathbf{1}\\{w_j\\neq0\\}}\n", 909 | "$$\n", 910 | " * NP-potpun problem!\n" 911 | ] 912 | }, 913 | { 914 | "cell_type": "markdown", 915 | "metadata": {}, 916 | "source": [ 917 | "### L2-regularizacija\n", 918 | "\n", 919 | "* Linearna regresija sa L2-regularizacijom ima rješenje u zatvorenoj formi:\n", 920 | "\n", 921 | "$$\n", 922 | "\\begin{align*}\n", 923 | "E'(\\mathbf{w}|\\mathcal{D}) &= \\frac{1}{2}\n", 924 | "(\\boldsymbol{\\Phi}\\mathbf{w} - \\mathbf{y})^\\intercal\n", 925 | "(\\boldsymbol{\\Phi}\\mathbf{w} - \\mathbf{y}) + \\color{red}{\\frac{\\lambda}{2}\\mathbf{w}^\\intercal\\mathbf{w}}\\\\\n", 926 | "&=\n", 927 | "\\frac{1}{2}\n", 928 | "(\\mathbf{w}^\\intercal\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi}\\mathbf{w} - 2\\mathbf{y}^\\intercal\\boldsymbol{\\Phi}\\mathbf{w} + \\mathbf{y}^\\intercal\\mathbf{y}\n", 929 | "+ \\color{red}{\\lambda\\mathbf{w}^\\intercal\\mathbf{w}})\\\\\n", 930 | "\\nabla_{\\mathbf{w}}E' &= \n", 931 | "\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi}\\mathbf{w} - \\boldsymbol{\\Phi}^\\intercal\\mathbf{y} + \\color{red}{\\lambda\\mathbf{w}} \\\\\n", 932 | "&=\n", 933 | "(\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi} + \\color{red}{\\lambda\\mathbf{I}})\\mathbf{w} - \\boldsymbol{\\Phi}^\\intercal\\mathbf{y} = 0 \\\\\n", 934 | "\\mathbf{w} &= (\\boldsymbol{\\Phi}^\\intercal\\boldsymbol{\\Phi} + \\color{red}{\\lambda\\mathbf{I}})^{-1}\\boldsymbol{\\Phi}^\\intercal\\mathbf{y}\\\\\n", 935 | "\\end{align*}\n", 936 | "$$" 937 | ] 938 | }, 939 | { 940 | "cell_type": "markdown", 941 | "metadata": {}, 942 | "source": [ 943 | "### Napomene\n", 944 | "\n", 945 | "* Iznos parametra $w_j$ odgovara važnosti značajke, a predznak upućuje na njezin utjecaj (pozitivan ili negativan) na izlaznu vrijednost\n", 946 | "\n", 947 | "\n", 948 | "* Regularizacija smanjuje složenost modela na način da prigušuje vrijednosti pojedinih značajki, odnosno efektivno ih izbacuje (kada $w_j\\to0$)\n", 949 | " * Ako je model nelinearan, to znači smanjivanje nelinearnosti\n", 950 | " \n", 951 | " \n", 952 | "* Težinu $w_0$ treba izuzeti iz regularizacijskog izraza (jer ona definira pomak) ili treba centrirati podatke tako da $\\overline{y}=0$, jer onda $w_0\\to0$\n", 953 | "\n", 954 | "\n", 955 | "* L2-regularizacija kažnjava težine proporcionalno njihovom iznosu (velike težine više, a manje težine manje) Teško će parametri biti pritegnuti baš na nulu. Zato **L2-regularizacija ne rezultira rijetkim modelima**\n", 956 | "\n", 957 | "\n", 958 | "* L1-regularizirana regresija rezultira rijetkim modelima, ali nema rješenja u zatvorenoj formi (međutim mogu se koristiti iterativni optimizacijski postupci\n", 959 | "\n", 960 | "\n", 961 | "* Regularizacija je korisna kod modela s puno parametara, jer je takve modele lako prenaučiti\n", 962 | "\n", 963 | "\n", 964 | "* Regularizacija smanjuje mogućnost prenaučenosti, ali ostaje problem odabira hiperparametra $\\lambda$\n", 965 | " * Taj se odabir najčešće radi **unakrsnom provjerom**\n", 966 | " \n", 967 | " \n", 968 | "* **Q:** Koju optimalnu vrijednost za $\\lambda$ bismo dobili kada bismo optimizaciju radili na skupu za učenje?\n" 969 | ] 970 | }, 971 | { 972 | "cell_type": "markdown", 973 | "metadata": {}, 974 | "source": [ 975 | "# Sažetak\n", 976 | "\n", 977 | "\n", 978 | "* **Linearan model regresije** linearan je u parametrima\n", 979 | "\n", 980 | "\n", 981 | "* Parametri linearnog modela uz kvadratnu funkciju gubitka imaju rješenje u zatvorenoj formi u obliku **pseudoinverza dizajn-matrice**\n", 982 | "\n", 983 | "\n", 984 | "* Nelinearnost regresijske funkcije ostvaruje se uporabom nelinearnih **baznih funkcija** (preslikavanjem ulaznog prostora u prostor značajki\n", 985 | "\n", 986 | "\n", 987 | "* Uz pretpostavku normalno distribuiranog šuma, **MLE je istovjetan postupku najmanjih kvadrata**, što daje probabilističko opravdanje za uporabu kvadratne funkcije gubitka\n", 988 | "\n", 989 | "\n", 990 | "* **Regularizacija smanjuje prenaučenost** ugradnjom dodatnog izraza u funkciju pogreške kojim se kažnjava složenost modela\n" 991 | ] 992 | } 993 | ], 994 | "metadata": { 995 | "kernelspec": { 996 | "display_name": "Python 2", 997 | "language": "python", 998 | "name": "python2" 999 | }, 1000 | "language_info": { 1001 | "codemirror_mode": { 1002 | "name": "ipython", 1003 | "version": 2 1004 | }, 1005 | "file_extension": ".py", 1006 | "mimetype": "text/x-python", 1007 | "name": "python", 1008 | "nbconvert_exporter": "python", 1009 | "pygments_lexer": "ipython2", 1010 | "version": "2.7.6" 1011 | } 1012 | }, 1013 | "nbformat": 4, 1014 | "nbformat_minor": 0 1015 | } 1016 | -------------------------------------------------------------------------------- /data/titanic-train.csv: -------------------------------------------------------------------------------- 1 | PassengerId,Survived,Pclass,Name,Sex,Age,SibSp,Parch,Ticket,Fare,Cabin,Embarked 2 | 1,0,3,"Braund, Mr. Owen Harris",male,22,1,0,A/5 21171,7.25,,S 3 | 2,1,1,"Cumings, Mrs. John Bradley (Florence Briggs Thayer)",female,38,1,0,PC 17599,71.2833,C85,C 4 | 3,1,3,"Heikkinen, Miss. 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