Probabilistic and Bayesian Analytics


 Aubrie Baker
 1 years ago
 Views:
Transcription
1 Probabilistic and Bayesian Analytics Note to other teachers and users of these slides. Andrew would be delighted if you found this source aterial useful in giing your own lectures. Feel free to use these slides erbati, or to odify the to fit your own needs. PowerPoint originals are aailable. If you ake use of a significant portion of these slides in your own lecture, please include this essage, or the following link to the source repository of Andrew s tutorials: Coents and corrections gratefully receied. Andrew W. Moore Professor School of Coputer Science Carnegie Mellon Uniersity Copyright Andrew W. Moore Slide Probability The world is a ery uncertain place 3 years of Artificial Intelligence and Database research danced around this fact And then a few AI researchers decided to use soe ideas fro the eighteenth century Copyright Andrew W. Moore Slide 2
2 What we re going to do We will reiew the fundaentals of probability. It s really going to be worth it In this lecture, you ll see an exaple of probabilistic analytics in action: Bayes Classifiers Copyright Andrew W. Moore Slide 3 Discrete Rando Variables A is a Booleanalued rando ariable if A denotes an eent, and there is soe degree of uncertainty as to whether A occurs. Exaples A The US president in 223 will be ale A ou wake up toorrow with a headache A ou hae Ebola Copyright Andrew W. Moore Slide 4 2
3 Probabilities We write A as the fraction of possible worlds in which A is true We could at this point spend 2 hours on the philosophy of this. But we won t. Copyright Andrew W. Moore Slide 5 Visualizing A Eent space of all possible worlds Worlds in which A is true A Area of reddish oal Its area is Worlds in which A is False Copyright Andrew W. Moore Slide 6 3
4 The Axios Of Probabi lity Copyright Andrew W. Moore Slide 7 The Axios of Probability < A < True False A or B A + B  A and B Where do these axios coe fro? Were they discoered? Answers coing up later. Copyright Andrew W. Moore Slide 8 4
5 Interpreting the axios < A < True False A or B A + B  A and B The area of A can t get any saller than And a zero area would ean no world could eer hae A true Copyright Andrew W. Moore Slide 9 Interpreting the axios < A < True False A or B A + B  A and B The area of A can t get any bigger than And an area of would ean all worlds will hae A true Copyright Andrew W. Moore Slide 5
6 Interpreting the axios < A < True False A or B A + B  A and B A B Copyright Andrew W. Moore Slide Interpreting the axios < A < True False A or B A + B  A and B A B A or B A and B B Siple addition and subtraction Copyright Andrew W. Moore Slide 2 6
7 These Axios are Not to be Trifled With There hae been attepts to do different ethodologies for uncertainty Fuzzy Logic Threealued logic DepsterShafer Nononotonic reasoning But the axios of probability are the only syste with this property: If you gable using the you can t be unfairly exploited by an opponent using soe other syste [di Finetti 93] Copyright Andrew W. Moore Slide 3 Theores fro the Axios < A <, True, False A or B A + B  A and B Fro these we can proe: not A ~A A How? Copyright Andrew W. Moore Slide 4 7
8 Side Note I a inflicting these proofs on you for two reasons:. These kind of anipulations will need to be second nature to you if you use probabilistic analytics in depth 2. Suffering is good for you Copyright Andrew W. Moore Slide 5 Another iportant theore < A <, True, False A or B A + B  A and B Fro these we can proe: A A ^ B + A ^ ~B How? Copyright Andrew W. Moore Slide 6 8
9 Multialued Rando Variables Suppose A can take on ore than 2 alues A is a rando ariable with arity k if it can take on exactly one alue out of {, 2,.. k } Thus A P A A A if i A i j ( 2 k j Copyright Andrew W. Moore Slide 7 An easy fact about Multialued Rando Variables: Using the axios of probability < A <, True, False A or B A + B  A and B And assuing that A obeys A P A It s easy to proe that A A if i A i j ( 2 k j A A 2 A i A i j j Copyright Andrew W. Moore Slide 8 9
10 An easy fact about Multialued Rando Variables: Using the axios of probability < A <, True, False A or B A + B  A and B And assuing that A obeys A P A It s easy to proe that A A if i A i j ( 2 k j A A 2 A And thus we can proe k j A j i A i j j Copyright Andrew W. Moore Slide 9 Another fact about Multialued Rando Variables: Using the axios of probability < A <, True, False A or B A + B  A and B And assuing that A obeys A P A It s easy to proe that A A if i A i j ( 2 k j B [ A A 2 A ] i B A i j j Copyright Andrew W. Moore Slide 2
11 Another fact about Multialued Rando Variables: Using the axios of probability < A <, True, False A or B A + B  A and B And assuing that A obeys A P A It s easy to proe that A A if i A i j ( 2 k j B [ A A 2 A ] And thus we can proe B k j B A i B A i j j j Copyright Andrew W. Moore Slide 2 Eleentary Probability in Pictures ~A + A Copyright Andrew W. Moore Slide 22
12 Eleentary Probability in Pictures B B ^ A + B ^ ~A Copyright Andrew W. Moore Slide 23 Eleentary Probability in Pictures k j A j Copyright Andrew W. Moore Slide 24 2
13 Eleentary Probability in Pictures B k j B A j Copyright Andrew W. Moore Slide 25 Conditional Probability A B Fraction of worlds in which B is true that also hae A true H Hae a headache F Coing down with Flu F H / F /4 H F /2 H Headaches are rare and flu is rarer, but if you re coing down with flu there s a 55 chance you ll hae a headache. Copyright Andrew W. Moore Slide 26 3
14 Conditional Probability F H F Fraction of fluinflicted worlds in which you hae a headache H Hae a headache F Coing down with Flu H / F /4 H F /2 H #worlds with flu and headache #worlds with flu Area of H and F region Area of F region H ^ F F Copyright Andrew W. Moore Slide 27 Definition of Conditional Probability A ^ B A B B Corollary: The Chain Rule A ^ B A B B Copyright Andrew W. Moore Slide 28 4
15 Probabilistic Inference F H Hae a headache F Coing down with Flu H H / F /4 H F /2 One day you wake up with a headache. ou think: Drat! 5% of flus are associated with headaches so I ust hae a 55 chance of coing down with flu Is this reasoning good? Copyright Andrew W. Moore Slide 29 Probabilistic Inference F H Hae a headache F Coing down with Flu H H / F /4 H F /2 F ^ H F H Copyright Andrew W. Moore Slide 3 5
16 Another way to understand the intuition Thanks to Jahanzeb Sherwani for contributing this explanation: Copyright Andrew W. Moore Slide 3 What we just did A ^ B A B B B A A A This is Bayes Rule Bayes, Thoas (763 An essay towards soling a proble in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:3748 Copyright Andrew W. Moore Slide 32 6
17 Using Bayes Rule to Gable $. The Win enelope has a dollar and four beads in it The Lose enelope has three beads and no oney Triial question: soeone draws an enelope at rando and offers to sell it to you. How uch should you pay? Copyright Andrew W. Moore Slide 33 Using Bayes Rule to Gable $. The Win enelope has a dollar and four beads in it The Lose enelope has three beads and no oney Interesting question: before deciding, you are allowed to see one bead drawn fro the enelope. Suppose it s black: How uch should you pay? Suppose it s red: How uch should you pay? Copyright Andrew W. Moore Slide 34 7
18 Calculation $. Copyright Andrew W. Moore Slide 35 More General Fors of Bayes Rule B A A A B B A A + B ~ A ~ A B A X A X A B X B X Copyright Andrew W. Moore Slide 36 8
19 9 Copyright Andrew W. Moore Slide 37 More General Fors of Bayes Rule n A k k k i i i P A A P B P A A P B B P A ( ( ( ( ( Copyright Andrew W. Moore Slide 38 Useful Easytoproe facts ( ( + B A P B A P ( n A k k B A P
20 The Joint Distribution Recipe for aking a joint distribution of M ariables: Exaple: Boolean ariables A, B, C Copyright Andrew W. Moore Slide 39 The Joint Distribution Recipe for aking a joint distribution of M ariables:. Make a truth table listing all cobinations of alues of your ariables (if there are M Boolean ariables then the table will hae 2 M rows. A Exaple: Boolean ariables A, B, C B C Copyright Andrew W. Moore Slide 4 2
21 The Joint Distribution Recipe for aking a joint distribution of M ariables:. Make a truth table listing all cobinations of alues of your ariables (if there are M Boolean ariables then the table will hae 2 M rows. 2. For each cobination of alues, say how probable it is. A Exaple: Boolean ariables A, B, C B C Prob Copyright Andrew W. Moore Slide 4 The Joint Distribution Recipe for aking a joint distribution of M ariables:. Make a truth table listing all cobinations of alues of your ariables (if there are M Boolean ariables then the table will hae 2 M rows. 2. For each cobination of alues, say how probable it is. 3. If you subscribe to the axios of probability, those nubers ust su to. A A Exaple: Boolean ariables A, B, C B.5 C..25 Prob C.3 B. Copyright Andrew W. Moore Slide 42 2
22 Using the Joint One you hae the JD you can ask for the probability of any logical expression inoling your attribute E rows atching E row Copyright Andrew W. Moore Slide 43 Using the Joint Poor Male.4654 E rows atching E row Copyright Andrew W. Moore Slide 44 22
23 Using the Joint Poor.764 E rows atching E row Copyright Andrew W. Moore Slide 45 Inference with the Joint E E 2 E E2 E 2 rows atching E and E rows atching E row 2 2 row Copyright Andrew W. Moore Slide 46 23
24 Inference with the Joint E E 2 E E2 E 2 rows atching E and E rows atching E row 2 2 row Male Poor.4654 / Copyright Andrew W. Moore Slide 47 Inference is a big deal I e got this eidence. What s the chance that this conclusion is true? I e got a sore neck: how likely a I to hae eningitis? I see y lights are out and it s 9p. What s the chance y spouse is already asleep? Copyright Andrew W. Moore Slide 48 24
25 Inference is a big deal I e got this eidence. What s the chance that this conclusion is true? I e got a sore neck: how likely a I to hae eningitis? I see y lights are out and it s 9p. What s the chance y spouse is already asleep? Copyright Andrew W. Moore Slide 49 Inference is a big deal I e got this eidence. What s the chance that this conclusion is true? I e got a sore neck: how likely a I to hae eningitis? I see y lights are out and it s 9p. What s the chance y spouse is already asleep? There s a thriing set of industries growing based around Bayesian Inference. Highlights are: Medicine, Phara, Help Desk Support, Engine Fault Diagnosis Copyright Andrew W. Moore Slide 5 25
26 Where do Joint Distributions coe fro? Idea One: Expert Huans Idea Two: Sipler probabilistic facts and soe algebra Exaple: Suppose you knew A.7 B A.2 B ~A. C A^B. C A^~B.8 C ~A^B.3 C ~A^~B. Then you can autoatically copute the JD using the chain rule Ax ^ By ^ Cz Cz Ax^ By By Ax Ax In another lecture: Bayes Nets, a systeatic way to do this. Copyright Andrew W. Moore Slide 5 Where do Joint Distributions coe fro? Idea Three: Learn the fro data! Prepare to see one of the ost ipressie learning algoriths you ll coe across in the entire course. Copyright Andrew W. Moore Slide 52 26
27 Learning a joint distribution Build a JD table for your attributes in which the probabilities are unspecified A B C Prob???????? Fraction of all records in which A and B are True but C is False The fill in each row with records atching row Pˆ (row total nuber of records A B C Prob Copyright Andrew W. Moore Slide 53 Exaple of Learning a Joint This Joint was obtained by learning fro three attributes in the UCI Adult Census Database [Kohai 995] Copyright Andrew W. Moore Slide 54 27
28 Where are we? We hae recalled the fundaentals of probability We hae becoe content with what JDs are and how to use the And we een know how to learn JDs fro data. Copyright Andrew W. Moore Slide 55 Density Estiation Our Joint Distribution learner is our first exaple of soething called Density Estiation A Density Estiator learns a apping fro a set of attributes to a Probability Input Attributes Density Estiator Probability Copyright Andrew W. Moore Slide 56 28
29 Density Estiation Copare it against the two other ajor kinds of odels: Input Attributes Classifier Prediction of categorical output Input Attributes Density Estiator Probability Input Attributes Regressor Prediction of realalued output Copyright Andrew W. Moore Slide 57 Ealuating Density Estiation Testset criterion for estiating perforance on future data* * See the Decision Tree or Cross Validation lecture for ore detail Input Attributes Classifier Prediction of categorical output Test set Accuracy Input Attributes Density Estiator Probability? Input Attributes Regressor Prediction of realalued output Test set Accuracy Copyright Andrew W. Moore Slide 58 29
30 Ealuating a density estiator Gien a record x, a density estiator M can tell you how likely the record is: Pˆ ( x M Gien a dataset with R records, a density estiator can tell you how likely the dataset is: (Under the assuption that all records were independently generated fro the Density Estiator s JD Pˆ (dataset M Pˆ( x R x2 K xr M Pˆ( xk M k Copyright Andrew W. Moore Slide 59 A sall dataset: Miles Per Gallon pg odelyear aker 92 Training Set Records good 75to78 asia bad 7to74 aerica bad 75to78 europe bad 7to74 aerica bad 7to74 aerica bad 7to74 asia bad 7to74 asia bad 75to78 aerica : : : : : : : : : bad 7to74 aerica good 79to83 aerica bad 75to78 aerica good 79to83 aerica bad 75to78 aerica good 79to83 aerica good 79to83 aerica bad 7to74 aerica good 75to78 europe bad 75to78 europe Fro the UCI repository (thanks to Ross Quinlan Copyright Andrew W. Moore Slide 6 3
31 A sall dataset: Miles Per Gallon pg odelyear aker 92 Training Set Records good 75to78 asia bad 7to74 aerica bad 75to78 europe bad 7to74 aerica bad 7to74 aerica bad 7to74 asia bad 7to74 asia bad 75to78 aerica : : : : : : : : : bad 7to74 aerica good 79to83 aerica bad 75to78 aerica good 79to83 aerica bad 75to78 aerica good 79to83 aerica good 79to83 aerica bad 7to74 aerica good 75to78 europe bad 75to78 europe Copyright Andrew W. Moore Slide 6 A sall dataset: Miles Per Gallon pg odelyear aker 92 Training Set Records Pˆ(dataset M good 75to78 asia bad 7to74 aerica bad 75to78 europe bad 7to74 aerica bad 7to74 aerica bad 7to74 asia bad 7to74 asia bad 75to78 aerica : : : : : : : : : bad 7to74 aerica good 79to83 aerica bad 75to78P ˆ( aerica x good 79to83 aerica bad 75to78 aerica good 79to83 aerica good 79to83 (in this aerica bad 7to74 aerica good 75to78 europe bad 75to78 europe R x2 K x M k 23 case 3.4 R Pˆ( xk M Copyright Andrew W. Moore Slide 62 3
32 Log Probabilities Since probabilities of datasets get so sall we usually use log probabilities log Pˆ(dataset M log R k R Pˆ( xk M log Pˆ( xk M k Copyright Andrew W. Moore Slide 63 A sall dataset: Miles Per Gallon pg odelyear aker 92 Training Set Records log Pˆ(dataset good 75to78 asia bad 7to74 aerica bad 75to78 europe bad 7to74 aerica bad 7to74 aerica bad 7to74 asia bad 7to74 asia bad 75to78 aerica : : : : : : : : : bad 7to74 aerica good 79to83 aerica bad M75to78 aerica log good 79to83 aerica bad 75to78 aerica 79to83 aerica good 79to83 (in this aerica bad 7to74 aerica good 75to78 europe bad 75to78 europe R Pˆ( xk M log Pˆ( xk M k k case R Copyright Andrew W. Moore Slide 64 32
33 Suary: The Good News We hae a way to learn a Density Estiator fro data. Density estiators can do any good things Can sort the records by probability, and thus spot weird records (anoaly detection Can do inference: E E2 Autoatic Doctor / Help Desk etc Ingredient for Bayes Classifiers (see later Copyright Andrew W. Moore Slide 65 Suary: The Bad News Density estiation by directly learning the joint is triial, indless and dangerous Copyright Andrew W. Moore Slide 66 33
34 Using a test set An independent test set with 96 cars has a worse log likelihood (actually it s a billion quintillion quintillion quintillion quintillion ties less likely.density estiators can oerfit. And the full joint density estiator is the oerfittiest of the all! Copyright Andrew W. Moore Slide 67 Oerfitting Density Estiators If this eer happens, it eans there are certain cobinations that we learn are ipossible R log Pˆ(testset M log Pˆ( xk M log Pˆ( xk M k k if for any k Pˆ( x M k R Copyright Andrew W. Moore Slide 68 34
35 Using a test set The only reason that our test set didn t score infinity is that y code is hardwired to always predict a probability of at least one in 2 We need Density Estiators that are less prone to oerfitting Copyright Andrew W. Moore Slide 69 Naïe Density Estiation The proble with the Joint Estiator is that it just irrors the training data. We need soething which generalizes ore usefully. The naïe odel generalizes strongly: Assue that each attribute is distributed independently of any of the other attributes. Copyright Andrew W. Moore Slide 7 35
36 Independently Distributed Data Let x[i] denote the i th field of record x. The independently distributed assuption says that for any i,, u u 2 u i u i+ u M x[ i] x[] u, x[2] u2, Kx[ i ] ui, x[ i + ] ui+, Kx[ M ] u x[ i] Or in other words, x[i] is independent of {x[],x[2],..x[i], x[i+], x[m]} This is often written as x[ i] { x[], x[2], K x[ i ], x[ i + ], Kx[ M ]} M Copyright Andrew W. Moore Slide 7 A note about independence Assue A and B are Boolean Rando Variables. Then A and B are independent if and only if A B A A and B are independent is often notated as A B Copyright Andrew W. Moore Slide 72 36
37 Independence Theores Assue A B A Then A^B Assue A B A Then B A A B B Copyright Andrew W. Moore Slide 73 Independence Theores Assue A B A Then ~A B Assue A B A Then A ~B ~A A Copyright Andrew W. Moore Slide 74 37
38 Multialued Independence For ultialued Rando Variables A and B, A B if and only if u, : A u B A u fro which you can then proe things like u, : A u B A u B u, : B A B Copyright Andrew W. Moore Slide 75 Back to Naïe Density Estiation Let x[i] denote the i th field of record x: Naïe DE assues x[i] is independent of {x[],x[2],..x[i], x[i+], x[m]} Exaple: Suppose that each record is generated by randoly shaking a green dice and a red dice Dataset : A red alue, B green alue Dataset 2: A red alue, B su of alues Dataset 3: A su of alues, B difference of alues Which of these datasets iolates the naïe assuption? Copyright Andrew W. Moore Slide 76 38
39 Using the Naïe Distribution Once you hae a Naïe Distribution you can easily copute any row of the joint distribution. Suppose A, B, C and D are independently distributed. What is A^~B^C^~D? Copyright Andrew W. Moore Slide 77 Using the Naïe Distribution Once you hae a Naïe Distribution you can easily copute any row of the joint distribution. Suppose A, B, C and D are independently distributed. What is A^~B^C^~D? A ~B^C^~D ~B^C^~D A ~B^C^~D A ~B C^~D C^~D A ~B C^~D A ~B C ~D ~D A ~B C ~D Copyright Andrew W. Moore Slide 78 39
40 Naïe Distribution General Case Suppose x[], x[2], x[m] are independently distributed. M, x[2] u2, x[ M ] um x[ k] u k k x[] u K So if we hae a Naïe Distribution we can construct any row of the iplied Joint Distribution on deand. So we can do any inference But how do we learn a Naïe Density Estiator? Copyright Andrew W. Moore Slide 79 Learning a Naïe Density Estiator Pˆ ( x[ i] u # records in which x[ i] u total nuber of records Another triial learning algorith! Copyright Andrew W. Moore Slide 8 4
41 Joint DE Contrast Naïe DE Can odel anything No proble to odel C is a noisy copy of A Gien records and ore than 6 Boolean attributes will screw up badly Can odel only ery boring distributions Outside Naïe s scope Gien records and, ultialued attributes will be fine Copyright Andrew W. Moore Slide 8 Epirical Results: Hopeless The hopeless dataset consists of 4, records and 2 Boolean attributes called a,b,c, u. Each attribute in each record is generated 55 randoly as or. Aerage test set log probability during folds of kfold crossalidation* Described in a future Andrew lecture Despite the ast aount of data, Joint oerfits hopelessly and does uch worse Copyright Andrew W. Moore Slide 82 4
42 Epirical Results: Logical The logical dataset consists of 4, records and 4 Boolean attributes called a,b,c,d where a,b,c are generated 55 randoly as or. D A^~C, except that in % of records it is flipped The DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 83 Epirical Results: Logical The logical dataset consists of 4, records and 4 Boolean attributes called a,b,c,d where a,b,c are generated 55 randoly as or. D A^~C, except that in % of records it is flipped The DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 84 42
43 Epirical Results: MPG The MPG dataset consists of 392 records and 8 attributes A tiny part of the DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 85 Epirical Results: MPG The MPG dataset consists of 392 records and 8 attributes A tiny part of the DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 86 43
44 Epirical Results: Weight s. MPG Suppose we train only fro the Weight and MPG attributes The DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 87 Epirical Results: Weight s. MPG Suppose we train only fro the Weight and MPG attributes The DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 88 44
45 Weight s. MPG : The best that Naïe can do The DE learned by Joint The DE learned by Naie Copyright Andrew W. Moore Slide 89 Reinder: The Good News We hae two ways to learn a Density Estiator fro data. *In other lectures we ll see astly ore ipressie Density Estiators (Mixture Models, Bayesian Networks, Density Trees, Kernel Densities and any ore Density estiators can do any good things Anoaly detection Can do inference: E E2 Autoatic Doctor / Help Desk etc Ingredient for Bayes Classifiers Copyright Andrew W. Moore Slide 9 45
46 Bayes Classifiers A foridable and sworn eney of decision trees Input Attributes Classifier Prediction of categorical output DT BC Copyright Andrew W. Moore Slide 9 How to build a Bayes Classifier Assue you want to predict output which has arity n and alues, 2, ny. Assue there are input attributes called X, X 2, X Break dataset into n saller datasets called DS, DS 2, DS ny. Define DS i Records in which i For each DS i, learn Density Estiator M i to odel the input distribution aong the i records. Copyright Andrew W. Moore Slide 92 46
47 How to build a Bayes Classifier Assue you want to predict output which has arity n and alues, 2, ny. Assue there are input attributes called X, X 2, X Break dataset into n saller datasets called DS, DS 2, DS ny. Define DS i Records in which i For each DS i, learn Density Estiator M i to odel the input distribution aong the i records. M i estiates X, X 2, X i Copyright Andrew W. Moore Slide 93 How to build a Bayes Classifier Assue you want to predict output which has arity n and alues, 2, ny. Assue there are input attributes called X, X 2, X Break dataset into n saller datasets called DS, DS 2, DS ny. Define DS i Records in which i For each DS i, learn Density Estiator M i to odel the input distribution aong the i records. M i estiates X, X 2, X i Idea: When a new set of input alues (X u, X 2 u 2,. X u coe along to be ealuated predict the alue of that akes X, X 2, X i ost likely predict Is this a good idea? argax X u L X u Copyright Andrew W. Moore Slide 94 47
48 How to build a Bayes Classifier Assue you want to predict output which has arity n and alues, 2, ny. Assue there are input attributes This called is a Maxiu X, X 2, X Likelihood classifier. Break dataset into n saller datasets called DS, DS 2, DS ny. Define DS i Records in which It i can get silly if soe s are For each DS i, learn Density Estiator M i to odel the input ery unlikely distribution aong the i records. M i estiates X, X 2, X i Idea: When a new set of input alues (X u, X 2 u 2,. X u coe along to be ealuated predict the alue of that akes X, X 2, X i ost likely predict Is this a good idea? argax X u L X u Copyright Andrew W. Moore Slide 95 How to build a Bayes Classifier Assue you want to predict output which has arity n and alues, 2, ny. Assue there are input attributes called X, X 2, X Break dataset into n saller datasets called DS, DS 2, DS ny. Define DS i Records in which i Much Better Idea For each DS i, learn Density Estiator M i to odel the input distribution aong the i records. M i estiates X, X 2, X i Idea: When a new set of input alues (X u, X 2 u 2,. X u coe along to be ealuated predict the alue of that akes i X, X 2, X ost likely predict argax X u L X u Is this a good idea? Copyright Andrew W. Moore Slide 96 48
49 Terinology MLE (Maxiu Likelihood Estiator: predict argax X u L X u MAP (Maxiu APosteriori Estiator: predict argax X u L X u Copyright Andrew W. Moore Slide 97 Getting what we need predict argax X u L X u Copyright Andrew W. Moore Slide 98 49
50 5 Copyright Andrew W. Moore Slide 99 Getting a posterior probability n j j j P u X u X P P u X u X P u X u X P P u X u X P u X u X P ( ( ( ( ( ( ( ( L L L L L Copyright Andrew W. Moore Slide Bayes Classifiers in a nutshell ( ( argax ( argax predict P u X u X P u X u X P L L. Learn the distribution oer inputs for each alue. 2. This gies X, X 2, X i. 3. Estiate i. as fraction of records with i. 4. For a new prediction:
51 Bayes Classifiers in a nutshell. Learn the distribution oer inputs for each alue. 2. This gies X, X 2, X i. 3. Estiate i. as fraction of records with i. 4. For a new prediction: We can use our faorite Density Estiator here. predict argax argax X u L X X u u L X Right now we hae two options: u Joint Density Estiator Naïe Density Estiator Copyright Andrew W. Moore Slide Joint Density Bayes Classifier predict argax X u L X u In the case of the joint Bayes Classifier this degenerates to a ery siple rule: predict the ost coon alue of aong records in which X u, X 2 u 2,. X u. Note that if no records hae the exact set of inputs X u, X 2 u 2,. X u, then X, X 2, X i for all alues of. In that case we just hae to guess s alue Copyright Andrew W. Moore Slide 2 5
52 Joint BC Results: Logical The logical dataset consists of 4, records and 4 Boolean attributes called a,b,c,d where a,b,c are generated 55 randoly as or. D A^~C, except that in % of records it is flipped The Classifier learned by Joint BC Copyright Andrew W. Moore Slide 3 Joint BC Results: All Irreleant The all irreleant dataset consists of 4, records and 5 Boolean attributes called a,b,c,d..o where a,b,c are generated 55 randoly as or. (output with probability.75, with prob.25 Copyright Andrew W. Moore Slide 4 52
53 53 Copyright Andrew W. Moore Slide 5 Naïe Bayes Classifier ( ( argax predict P u X u X P L In the case of the naie Bayes Classifier this can be siplified: n j j j u X P P predict ( ( argax Copyright Andrew W. Moore Slide 6 Naïe Bayes Classifier ( ( argax predict P u X u X P L In the case of the naie Bayes Classifier this can be siplified: n j j j u X P P predict ( ( argax Technical Hint: If you hae, input attributes that product will underflow in floating point ath. ou should use logs: + n j j j u X P P predict ( log ( log argax
54 BC Results: XOR The XOR dataset consists of 4, records and 2 Boolean inputs called a and b, generated 55 randoly as or. c (output a XOR b The Classifier learned by Joint BC The Classifier learned by Naie BC Copyright Andrew W. Moore Slide 7 Naie BC Results: Logical The logical dataset consists of 4, records and 4 Boolean attributes called a,b,c,d where a,b,c are generated 55 randoly as or. D A^~C, except that in % of records it is flipped The Classifier learned by Naie BC Copyright Andrew W. Moore Slide 8 54
55 Naie BC Results: Logical The logical dataset consists of 4, records and 4 Boolean attributes called a,b,c,d where a,b,c are generated 55 randoly as or. D A^~C, except that in % of records it is flipped This result surprised Andrew until he had thought about it a little The Classifier learned by Joint BC Copyright Andrew W. Moore Slide 9 Naïe BC Results: All Irreleant The all irreleant dataset consists of 4, records and 5 Boolean attributes called a,b,c,d..o where a,b,c are generated 55 randoly as or. (output with probability.75, with prob.25 The Classifier learned by Naie BC Copyright Andrew W. Moore Slide 55
56 BC Results: MPG : 392 records The Classifier learned by Naie BC Copyright Andrew W. Moore Slide BC Results: MPG : 4 records Copyright Andrew W. Moore Slide 2 56
57 More Facts About Bayes Classifiers Many other density estiators can be slotted in*. Density estiation can be perfored with realalued inputs* Bayes Classifiers can be built with realalued inputs* Rather Technical Coplaint: Bayes Classifiers don t try to be axially discriinatiethey erely try to honestly odel what s going on* Zero probabilities are painful for Joint and Naïe. A hack (justifiable with the agic words Dirichlet Prior can help*. Naïe Bayes is wonderfully cheap. And suries, attributes cheerfully! *See future Andrew Lectures Copyright Andrew W. Moore Slide 3 Probability What you should know Fundaentals of Probability and Bayes Rule What s a Joint Distribution How to do inference (i.e. E E2 once you hae a JD Density Estiation What is DE and what is it good for How to learn a Joint DE How to learn a naïe DE Copyright Andrew W. Moore Slide 4 57
58 What you should know Bayes Classifiers How to build one How to predict with a BC Contrast between naïe and joint BCs Copyright Andrew W. Moore Slide 5 Interesting Questions Suppose you were ealuating NaieBC, JointBC, and Decision Trees Inent a proble where only NaieBC would do well Inent a proble where only Dtree would do well Inent a proble where only JointBC would do well Inent a proble where only NaieBC would do poorly Inent a proble where only Dtree would do poorly Inent a proble where only JointBC would do poorly Copyright Andrew W. Moore Slide 6 58
Online Bagging and Boosting
Abstract Bagging and boosting are two of the ost wellknown enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationExample: Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationMachine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationCRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS
641 CRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS Marketa Zajarosova 1* *Ph.D. VSB  Technical University of Ostrava, THE CZECH REPUBLIC arketa.zajarosova@vsb.cz Abstract Custoer relationship
More informationSoftware Quality Characteristics Tested For Mobile Application Development
Thesis no: MGSE201502 Software Quality Characteristics Tested For Mobile Application Developent Literature Review and Epirical Survey WALEED ANWAR Faculty of Coputing Blekinge Institute of Technology
More informationSOME APPLICATIONS OF FORECASTING Prof. Thomas B. Fomby Department of Economics Southern Methodist University May 2008
SOME APPLCATONS OF FORECASTNG Prof. Thoas B. Foby Departent of Econoics Southern Methodist University May 8 To deonstrate the usefulness of forecasting ethods this note discusses four applications of forecasting
More informationarxiv:0805.1434v1 [math.pr] 9 May 2008
Degreedistribution stability of scalefree networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of
More informationHow Secure are Secure Interdomain Routing Protocols?
How Secure are Secure Interdoain Routing Protocols? Sharon Goldberg Microsoft Research Michael Schapira Yale & UC Berkeley Peter Huon AT&T Labs Jennifer Rexford Princeton ABSTRACT In response to highprofile
More informationAnalyzing Methods Study of Outer Loop Current Sharing Control for Paralleled DC/DC Converters
Analyzing Methods Study of Outer Loop Current Sharing Control for Paralleled DC/DC Conerters Yang Qiu, Ming Xu, Jinjun Liu, and Fred C. Lee Center for Power Electroni Systes The Bradley Departent of Electrical
More informationFUTURE LIFETABLES BASED ON THE LEECARTER METHODOLOGY AND THEIR APPLICATION TO CALCULATING THE PENSION ANNUITIES 1
ACA UNIVERSIAIS LODZIENSIS FOLIA OECONOMICA 250, 20 Agnieszka Rossa * FUURE LIFEABLES BASED ON HE LEECARER MEHODOLOGY AND HEIR APPLICAION O CALCULAING HE PENSION ANNUIIES Suary. In the paper a new recursie
More informationExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network
2013 European Control Conference (ECC) July 1719, 2013, Zürich, Switzerland. ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona
More informationSearching strategy for multitarget discovery in wireless networks
Searching strategy for ultitarget discovery in wireless networks Zhao Cheng, Wendi B. Heinzelan Departent of Electrical and Coputer Engineering University of Rochester Rochester, NY 467 (585) 75{878,
More informationManaging Complex Network Operation with Predictive Analytics
Managing Coplex Network Operation with Predictive Analytics Zhenyu Huang, Pak Chung Wong, Patrick Mackey, Yousu Chen, Jian Ma, Kevin Schneider, and Frank L. Greitzer Pacific Northwest National Laboratory
More informationProject Evaluation Roadmap. Capital Budgeting Process. Capital Expenditure. Major Cash Flow Components. Cash Flows... COMM2501 Financial Management
COMM501 Financial Manageent Project Evaluation 1 (Capital Budgeting) Project Evaluation Roadap COMM501 Financial Manageent Week 7 Week 7 Project dependencies Net present value ethod Relevant cash flows
More informationDecision Trees. Andrew W. Moore Professor School of Computer Science Carnegie Mellon University. www.cs.cmu.edu/~awm awm@cs.cmu.
Decision Trees Andrew W. Moore Professor School of Computer Science Carnegie Mellon University www.cs.cmu.edu/~awm awm@cs.cmu.edu 422687599 Copyright Andrew W. Moore Slide Decision Trees Decision trees
More informationKinetic Molecular Theory of Ideal Gases
ecture /. Kinetic olecular Theory of Ideal Gases ast ecture. IG is a purely epirical law  solely the consequence of eperiental obserations Eplains the behaior of gases oer a liited range of conditions.
More informationQuality evaluation of the modelbased forecasts of implied volatility index
Quality evaluation of the odelbased forecasts of iplied volatility index Katarzyna Łęczycka 1 Abstract Influence of volatility on financial arket forecasts is very high. It appears as a specific factor
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis  Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationDoppler effect, moving sources/receivers
Goals: Lecture 29 Chapter 20 Work with a ew iportant characteristics o sound waves. (e.g., Doppler eect) Chapter 21 Recognize standing waves are the superposition o two traveling waves o sae requency Study
More informationRECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure
RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationFactored Models for Probabilistic Modal Logic
Proceedings of the TwentyThird AAAI Conference on Artificial Intelligence (2008 Factored Models for Probabilistic Modal Logic Afsaneh Shirazi and Eyal Air Coputer Science Departent, University of Illinois
More informationThis paper studies a rental firm that offers reusable products to price and qualityofservice sensitive
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523464 eissn 5265498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 453167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More informationInformation Processing Letters
Inforation Processing Letters 111 2011) 178 183 Contents lists available at ScienceDirect Inforation Processing Letters www.elsevier.co/locate/ipl Offline file assignents for online load balancing Paul
More informationSmall Business ebook. 5 Steps to a killer social media strategy
Sall Business ebook 5 Steps to a killer social edia strategy About the authors John Keepax and Frank Irias offer ore than 32 years of cobined experience in the areas of John Keepax Creative Director /
More informationMedia Adaptation Framework in Biofeedback System for Stroke Patient Rehabilitation
Media Adaptation Fraework in Biofeedback Syste for Stroke Patient Rehabilitation Yinpeng Chen, Weiwei Xu, Hari Sundara, Thanassis Rikakis, ShengMin Liu Arts, Media and Engineering Progra Arizona State
More informationPreferencebased Search and Multicriteria Optimization
Fro: AAAI02 Proceedings. Copyright 2002, AAAI (www.aaai.org). All rights reserved. Preferencebased Search and Multicriteria Optiization Ulrich Junker ILOG 1681, route des Dolines F06560 Valbonne ujunker@ilog.fr
More informationCrossvalidation for detecting and preventing overfitting
Crossvalidation for detecting and preventing overfitting Note to other teachers and users of these slides. Andrew would be delighted if ou found this source material useful in giving our own lectures.
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationOpenGamma Documentation Bond Pricing
OpenGaa Docuentation Bond Pricing Marc Henrard arc@opengaa.co OpenGaa Docuentation n. 5 Version 2.0  May 2013 Abstract The details of the ipleentation of pricing for fixed coupon bonds and floating rate
More informationAn Approach to Combating Freeriding in PeertoPeer Networks
An Approach to Cobating Freeriding in PeertoPeer Networks Victor Ponce, Jie Wu, and Xiuqi Li Departent of Coputer Science and Engineering Florida Atlantic University Boca Raton, FL 33431 April 7, 2008
More informationand that of the outgoing water is mv f
Week 6 hoework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign ersions of these probles, arious details hae been changed, so that the answers will coe out differently. The ethod to find the solution is
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More informationFuzzy Sets in HR Management
Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,
More informationAUC Optimization vs. Error Rate Minimization
AUC Optiization vs. Error Rate Miniization Corinna Cortes and Mehryar Mohri AT&T Labs Research 180 Park Avenue, Florha Park, NJ 0793, USA {corinna, ohri}@research.att.co Abstract The area under an ROC
More informationWORK TRAVEL MODE CHOICE MODELING USING DATA MINING: DECISION TREES AND NEURAL NETWORKS
TRB Paper No. 034348 WORK TRAVEL MODE CHOICE MODELING USING DATA MINING: DECISION TREES AND NEURAL NETWORKS Chi Xie Research Assistant Departent of Civil and Environental Engineering University of Massachusetts,
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationDon t Run With Your Retirement Money
Don t Run With Your Retireent Money Understanding Your Resources and How Best to Use The A joint project of The Actuarial Foundation and WISER, the Woen s Institute for a Secure Retireent WISER THE WOMEN
More informationIs PayasYouDrive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes?
Is PayasYouDrive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes? By Ian W.H. Parry Despite concerns about US dependence on a volatile world oil arket, greenhouse gases fro fuel cobustion,
More informationInvention of NFV Technique and Its Relationship with NPV
International Journal of Innovation and Applied Studies ISSN 20289324 Vol. 9 No. 3 Nov. 2014, pp. 11881195 2014 Innovative Space of Scientific Research Journals http://www.ijias.issrjournals.org/ Invention
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationAlgorithmica 2001 SpringerVerlag New York Inc.
Algorithica 2001) 30: 101 139 DOI: 101007/s0045300100030 Algorithica 2001 SpringerVerlag New York Inc Optial Search and OneWay Trading Online Algoriths R ElYaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationINTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS
Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE
More informationCLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY
CLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY Y. T. Chen Departent of Industrial and Systes Engineering Hong Kong Polytechnic University, Hong Kong yongtong.chen@connect.polyu.hk
More informationEnergy Proportionality for Disk Storage Using Replication
Energy Proportionality for Disk Storage Using Replication Jinoh Ki and Doron Rote Lawrence Berkeley National Laboratory University of California, Berkeley, CA 94720 {jinohki,d rote}@lbl.gov Abstract Energy
More informationGenerating Certification Authority Authenticated Public Keys in Ad Hoc Networks
SECURITY AND COMMUNICATION NETWORKS Published online in Wiley InterScience (www.interscience.wiley.co). Generating Certification Authority Authenticated Public Keys in Ad Hoc Networks G. Kounga 1, C. J.
More informationComment on On Discriminative vs. Generative Classifiers: A Comparison of Logistic Regression and Naive Bayes
Coent on On Discriinative vs. Generative Classifiers: A Coparison of Logistic Regression and Naive Bayes JingHao Xue (jinghao@stats.gla.ac.uk) and D. Michael Titterington (ike@stats.gla.ac.uk) Departent
More informationREQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES
REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES Charles Reynolds Christopher Fox reynolds @cs.ju.edu fox@cs.ju.edu Departent of Coputer
More informationADJUSTING FOR QUALITY CHANGE
ADJUSTING FOR QUALITY CHANGE 7 Introduction 7.1 The easureent of changes in the level of consuer prices is coplicated by the appearance and disappearance of new and old goods and services, as well as changes
More informationEndogenous CreditCard Acceptance in a Model of Precautionary Demand for Money
Endogenous CreditCard Acceptance in a Model of Precautionary Deand for Money Adrian Masters University of Essex and SUNY Albany Luis Raúl RodríguezReyes University of Essex March 24 Abstract A creditcard
More informationA Study on the Chain Restaurants Dynamic Negotiation Games of the Optimization of Joint Procurement of Food Materials
International Journal of Coputer Science & Inforation Technology (IJCSIT) Vol 6, No 1, February 2014 A Study on the Chain estaurants Dynaic Negotiation aes of the Optiization of Joint Procureent of Food
More informationModeling operational risk data reported above a timevarying threshold
Modeling operational risk data reported above a tievarying threshold Pavel V. Shevchenko CSIRO Matheatical and Inforation Sciences, Sydney, Locked bag 7, North Ryde, NSW, 670, Australia. eail: Pavel.Shevchenko@csiro.au
More informationEvaluating Software Quality of Vendors using Fuzzy Analytic Hierarchy Process
IMECS 2008 92 March 2008 Hong Kong Evaluating Software Quality of Vendors using Fuzzy Analytic Hierarchy Process Kevin K.F. Yuen* Henry C.W. au Abstract This paper proposes a fuzzy Analytic Hierarchy
More informationPerformance Evaluation of Machine Learning Techniques using Software Cost Drivers
Perforance Evaluation of Machine Learning Techniques using Software Cost Drivers Manas Gaur Departent of Coputer Engineering, Delhi Technological University Delhi, India ABSTRACT There is a treendous rise
More informationInsurance Spirals and the Lloyd s Market
Insurance Spirals and the Lloyd s Market Andrew Bain University of Glasgow Abstract This paper presents a odel of reinsurance arket spirals, and applies it to the situation that existed in the Lloyd s
More informationData Streaming Algorithms for Estimating Entropy of Network Traffic
Data Streaing Algoriths for Estiating Entropy of Network Traffic Ashwin Lall University of Rochester Vyas Sekar Carnegie Mellon University Mitsunori Ogihara University of Rochester Jun (Ji) Xu Georgia
More informationThe Research of Measuring Approach and Energy Efficiency for Hadoop Periodic Jobs
Send Orders for Reprints to reprints@benthascience.ae 206 The Open Fuels & Energy Science Journal, 2015, 8, 206210 Open Access The Research of Measuring Approach and Energy Efficiency for Hadoop Periodic
More informationThe Design and Implementation of an Enculturated WebBased Intelligent Tutoring System
The Design and Ipleentation of an Enculturated WebBased Intelligent Tutoring Syste Phaedra Mohaed Departent of Coputing and Inforation Technology The University of the West Indies phaedra.ohaed@gail.co
More informationON SELFROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 778433128
ON SELFROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 7788 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET
ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de
More informationDynamic Placement for Clustered Web Applications
Dynaic laceent for Clustered Web Applications A. Karve, T. Kibrel, G. acifici, M. Spreitzer, M. Steinder, M. Sviridenko, and A. Tantawi IBM T.J. Watson Research Center {karve,kibrel,giovanni,spreitz,steinder,sviri,tantawi}@us.ib.co
More informationResearch Article Performance Evaluation of Human Resource Outsourcing in Food Processing Enterprises
Advance Journal of Food Science and Technology 9(2): 964969, 205 ISSN: 20424868; eissn: 20424876 205 Maxwell Scientific Publication Corp. Subitted: August 0, 205 Accepted: Septeber 3, 205 Published:
More informationPERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO
Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No.  0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:
More informationYOUNG CHILDREN CONTINUE TO REINVENT ARITHMETIC 3rd Grade IMPLICATIONS OF PIAGET'S THEORY
YOUNG CHILDREN CONTINUE TO REINVENT ARITHMETIC 3rd Grade IMPLICATIONS OF PIAGET'S THEORY Constance Kaii with Sally Jones Livingston Published by Teachers College Press, 1234 Asterda Avenue New York, NY
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 66 2 Nonprobability Sapling and Bias 66 Stratified Rando Sapling 62 6 4 Cluster Sapling 64 6 5 Systeatic Sapling 69 6 6 Nonresponse 62 6 7 Suary and Review of
More informationIntroduction to Learning & Decision Trees
Artificial Intelligence: Representation and Problem Solving 538 April 0, 2007 Introduction to Learning & Decision Trees Learning and Decision Trees to learning What is learning?  more than just memorizing
More informationThe AGA Evaluating Model of Customer Loyalty Based on Ecommerce Environment
6 JOURNAL OF SOFTWARE, VOL. 4, NO. 3, MAY 009 The AGA Evaluating Model of Custoer Loyalty Based on Ecoerce Environent Shaoei Yang Econoics and Manageent Departent, North China Electric Power University,
More informationAutoHelp. An 'Intelligent' CaseBased Help Desk Providing. WebBased Support for EOSDIS Customers. A Concept and ProofofConcept Implementation
//j yd xd/_ ' Year One Report ":,/_i',:?,2... i" _.,.j _,._".;/._. ","/ AutoHelp An 'Intelligent' CaseBased Help Desk Providing WebBased Support for EOSDIS Custoers A Concept and ProofofConcept Ipleentation
More informationPosition Auctions and Nonuniform Conversion Rates
Position Auctions and Nonunifor Conversion Rates Liad Blurosen Microsoft Research Mountain View, CA 944 liadbl@icrosoft.co Jason D. Hartline Shuzhen Nong Electrical Engineering and Microsoft AdCenter
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and BlackMertonScholes
More informationDELTAV AS A MEASURE OF TRAFFIC CONFLICT SEVERITY
DELTAV AS A MEASURE OF TRAFFIC CONFLICT SEVERITY Steen G. Shelby Senior Research Engineer, Econolite Control Products, Inc., Tucson, AZ, USA, eail: sshelby@econolite.co Subitted to the 3 rd International
More informationAn Optimal Task Allocation Model for System Cost Analysis in Heterogeneous Distributed Computing Systems: A Heuristic Approach
An Optial Tas Allocation Model for Syste Cost Analysis in Heterogeneous Distributed Coputing Systes: A Heuristic Approach P. K. Yadav Central Building Research Institute, Rooree 247667, Uttarahand (INDIA)
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationExpected Value. 24 February 2014. Expected Value 24 February 2014 1/19
Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery
More informationAN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES
Int. J. Appl. Math. Coput. Sci., 2014, Vol. 24, No. 1, 133 149 DOI: 10.2478/acs20140011 AN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES PIOTR KULCZYCKI,,
More informationAdaptive Modulation and Coding for Unmanned Aerial Vehicle (UAV) Radio Channel
Recent Advances in Counications Adaptive odulation and Coding for Unanned Aerial Vehicle (UAV) Radio Channel Airhossein Fereidountabar,Gian Carlo Cardarilli, Rocco Fazzolari,Luca Di Nunzio Abstract In
More informationReconnect 04 Solving Integer Programs with Branch and Bound (and Branch and Cut)
Sandia is a ultiprogra laboratory operated by Sandia Corporation, a Lockheed Martin Copany, Reconnect 04 Solving Integer Progras with Branch and Bound (and Branch and Cut) Cynthia Phillips (Sandia National
More informationAn Integrated Approach for Monitoring Service Level Parameters of SoftwareDefined Networking
International Journal of Future Generation Counication and Networking Vol. 8, No. 6 (15), pp. 1974 http://d.doi.org/1.1457/ijfgcn.15.8.6.19 An Integrated Approach for Monitoring Service Level Paraeters
More informationOn the Mutual Coefficient of Restitution in Two Car Collinear Collisions
//006 On the Mutual Coefficient of Restitution in Two Car Collinear Collisions Milan Batista Uniersity of Ljubljana, Faculty of Maritie Studies and Transportation Pot poorscako 4, Sloenia, EU ilan.batista@fpp.edu
More informationMathematical Model for GlucoseInsulin Regulatory System of Diabetes Mellitus
Advances in Applied Matheatical Biosciences. ISSN 8998 Volue, Nuber (0), pp. 9 International Research Publication House http://www.irphouse.co Matheatical Model for GlucoseInsulin Regulatory Syste of
More informationCalculusBased Physics I by Jeffrey W. Schnick
Chapter Matheatical Prelude Calculusased Physics I by Jeffrey W. Schnick cbphysicsia8.doc Copyright 005008, Jeffrey W. Schnick, Creatie Coons Attribution ShareAlike License 3.0. You can copy, odify,
More informationA Scalable Application Placement Controller for Enterprise Data Centers
W WWW 7 / Track: Perforance and Scalability A Scalable Application Placeent Controller for Enterprise Data Centers Chunqiang Tang, Malgorzata Steinder, Michael Spreitzer, and Giovanni Pacifici IBM T.J.
More informationCOMBINING CRASH RECORDER AND PAIRED COMPARISON TECHNIQUE: INJURY RISK FUNCTIONS IN FRONTAL AND REAR IMPACTS WITH SPECIAL REFERENCE TO NECK INJURIES
COMBINING CRASH RECORDER AND AIRED COMARISON TECHNIQUE: INJURY RISK FUNCTIONS IN FRONTAL AND REAR IMACTS WITH SECIAL REFERENCE TO NECK INJURIES Anders Kullgren, Maria Krafft Folksa Research, 66 Stockhol,
More informationStandards and Protocols for the Collection and Dissemination of Graduating Student Initial Career Outcomes Information For Undergraduates
National Association of Colleges and Eployers Standards and Protocols for the Collection and Disseination of Graduating Student Initial Career Outcoes Inforation For Undergraduates Developed by the NACE
More informationImplementation of Active Queue Management in a Combined Input and Output Queued Switch
pleentation of Active Queue Manageent in a obined nput and Output Queued Switch Bartek Wydrowski and Moshe Zukeran AR Special Research entre for UltraBroadband nforation Networks, EEE Departent, The University
More information 265  Part C. Property and Casualty Insurance Companies
Part C. Property and Casualty Insurance Copanies This Part discusses proposals to curtail favorable tax rules for property and casualty ("P&C") insurance copanies. The syste of reserves for unpaid losses
More informationA WISER Guide. Financial Steps for Caregivers: What You Need to Know About Money and Retirement
WISER WOMEN S INSTITUTE FOR A SECURE RETIREMENT A WISER Guide Financial Steps for Caregivers: What You Need to Know About Money and Retireent This booklet was prepared under a grant fro the Adinistration
More informationModeling Cooperative Gene Regulation Using Fast Orthogonal Search
8 The Open Bioinforatics Journal, 28, 2, 889 Open Access odeling Cooperative Gene Regulation Using Fast Orthogonal Search Ian inz* and ichael J. Korenberg* Departent of Electrical and Coputer Engineering,
More informationAnalyzing Spatiotemporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy
Vol. 9, No. 5 (2016), pp.303312 http://dx.doi.org/10.14257/ijgdc.2016.9.5.26 Analyzing Spatioteporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy Chen Yang, Renjie Zhou
More informationResource Allocation in Wireless Networks with Multiple Relays
Resource Allocation in Wireless Networks with Multiple Relays Kağan Bakanoğlu, Stefano Toasin, Elza Erkip Departent of Electrical and Coputer Engineering, Polytechnic Institute of NYU, Brooklyn, NY, 0
More informationAn Improved Decisionmaking Model of Human Resource Outsourcing Based on Internet Collaboration
International Journal of Hybrid Inforation Technology, pp. 339350 http://dx.doi.org/10.14257/hit.2016.9.4.28 An Iproved Decisionaking Model of Huan Resource Outsourcing Based on Internet Collaboration
More informationAirline Yield Management with Overbooking, Cancellations, and NoShows JANAKIRAM SUBRAMANIAN
Airline Yield Manageent with Overbooking, Cancellations, and NoShows JANAKIRAM SUBRAMANIAN Integral Developent Corporation, 301 University Avenue, Suite 200, Palo Alto, California 94301 SHALER STIDHAM
More informationTowards Change Management Capability Assessment Model for Contractors in Building Project
MiddleEast Journal of Scientific Research 23 (7): 13271333, 2015 ISSN 19909233 IDOSI Publications, 2015 DOI: 10.5829/idosi.ejsr.2015.23.07.120 Towards Change Manageent Capability Assessent Model for
More informationAn improved TFIDF approach for text classification *
Zhang et al. / J Zheiang Univ SCI 2005 6A(1:4955 49 Journal of Zheiang University SCIECE ISS 10093095 http://www.zu.edu.cn/zus Eail: zus@zu.edu.cn An iproved TFIDF approach for text classification
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationLeak detection in open water channels
Proceedings of the 17th World Congress The International Federation of Autoatic Control Seoul, Korea, July 611, 28 Leak detection in open water channels Erik Weyer Georges Bastin Departent of Electrical
More information