cs345a data mining jure leskovec and anand rajaraman j
play

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford - PowerPoint PPT Presentation

Jan 05, 2024 •135 likes •700 views

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Homework 2 is out: Homework 2 is out: Due Monday 15 th at midnight! Submit PDFs Submit PDFs Talk: http://rain.stanford.edu Wed at 12:30 in Terman 453

  1. CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University

  2. Homework 2 is out: Homework 2 is out:  Due Monday 15 th at midnight!  Submit PDFs  Submit PDFs Talk:  http://rain.stanford.edu  Wed at 12:30 in Terman 453  Yehuda Koren – Winner of the Netflix challenge! g 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 2

  3.  Text ‐ LSI: find ‘concepts’ T t LSI fi d ‘ t ’ 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 3

  4.  Compress / reduce dimensionality  10 6 rows; 10 3 columns; no updates  random access to any cell(s); small error: OK 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 4

  5. 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 5

  6. 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 6

  7. A [n x m] = U [n x r]   r x r] ( V [m x r] ) T  ) T (  A : n x m matrix  A : n x m matrix (eg., n documents, m terms)  U : n x r matrix (n documents, r concepts)   : r x r diagonal matrix (strength of each ‘concept’) (strength of each concept ) (r : rank of the matrix)  V : m x r matrix (m terms, r concepts) 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 7

  8. n n    V T m m m A A U U 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 8

  9. n  1 u 1  v 1  2 u 2  v 2 A   + m A 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 9

  10. THEOREM [P THEOREM [Press+92]: always possible to 92] l ibl decompose matrix A into A = U  V T , where  U  V : unique  U,  V : unique  U , V : column orthonormal:  U T U = I ; V T V = I (I: identity matrix ) ; ( y )  (Cols. are orthogonal unit vectors)   : diagonal  Entries (singular values) are positive, and sorted in decreasing order 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 10

  11.  A = U  V T ‐ example: U  V T A l retrieval inf . brainlung g brain d data 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0.71 0.71 0 71 0 71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 11

  12.  A = U  V T ‐ example: U  V T A l retrieval CS ‐ concept inf . brainlung g MD ‐ concept p brain d data 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0.71 0.71 0 71 0 71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 12

  13.  A = U  V T ‐ example: U  V T A l d doc ‐ to ‐ concept t t similarity matrix retrieval CS ‐ concept inf . brainlung g MD ‐ concept p brain data d 0.18 0 1 1 1 0 0 0 36 0 0.36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 13

  14.  A = U  V T ‐ example: U  V T A l retrieval inf . brainlung g brain data d ‘strength’ of CS ‐ concept 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0.71 0.71 0 71 0 71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 14

  15.  A = U  V T ‐ example: U  V T A l term ‐ to ‐ concept retrieval similarity matrix inf . brainlung g brain d data 0.18 0 CS ‐ concept 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0.71 0.71 0 71 0 71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 15

  16.  A = U  V T ‐ example: U  V T A l term ‐ to ‐ concept retrieval similarity matrix inf . brainlung g brain d data 0.18 0 CS ‐ concept 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 CS 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 MD 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0.71 0.71 0 71 0 71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 16

  17. ‘d ‘documents’, ‘terms’ and ‘concepts’: t ’ ‘t ’ d ‘ t ’  U : document ‐ to ‐ concept similarity matrix  V : term ‐ to ‐ concept sim. matrix   : its diagonal elements:  it di l l t ‘strength’ of each concept 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 17

  18. SVD: gives best axis to project first singular  best axis to vector project on: p j (‘best’ = min sum of squares v1 of projection of projection errors)  minimum RMS error 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 18

  19.  A = U  V T ‐ example: U  V T A l 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 v 1 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 19

  20.  A = U  V T ‐ example: U  V T A l variance (‘spread’) p on the v 1 axis 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 20

  21.  A = U  V T ‐ example: U  V T A l  U  gives the coordinates of the points in the projection axis points in the projection axis 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0 27 0.27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 21

  22. M More details d t il  Q: how exactly is dim. reduction done? 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0.27 0 27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 22

  23. M More details d t il  Q: how exactly is dim. reduction done?  A: set the smallest singular values to zero:  A: set the smallest singular values to zero: 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x = 0 5.29 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0.27 0 27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 23

  24. 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x ~ 0 0 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0.27 0 27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 24

  25. 0.18 0 1 1 1 0 0 0.36 0 0 36 0 2 2 2 2 2 2 0 0 0 0 9.64 0 0.18 0 1 1 1 0 0 x x ~ 0 0 5 5 5 0 0 0.90 0 0 0 0 2 2 0 0.53 0 0 0 3 3 0 0.80 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 0.27 0 27 0 0 0 0 0 0 0 71 0 71 0.71 0.71 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 25

  26. 0.18 1 1 1 0 0 0 36 0.36 2 2 2 2 2 2 0 0 0 0 9.64 0.18 1 1 1 0 0 x x ~ 5 5 5 0 0 0.90 0 0 0 2 2 0 0 0 0 3 3 0 0.58 0.58 0.58 0 0 0 0 0 1 1 0 0 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 26

  27. 1 1 1 0 0 1 1 1 0 0 2 2 2 0 0 2 2 2 2 2 2 0 0 0 0 1 1 1 0 0 1 1 1 0 0 5 5 5 0 0 ~ 5 5 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 3 3 0 0 0 0 0 0 0 0 1 1 2/9/2010 Jure Leskovec & Anand Rajaraman, Stanford CS345a: Data Mining 27

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Mining Data Streams (Part 1) CS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford

Mining Data Streams (Part 1) CS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford

Mining Data Streams (Part 1) CS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford University In many data mining situations, we know the entire data set in advance Sometimes the input rate is controlled externally Google

502 views • 34 slides
CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Would like to do

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Would like to do

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Would like to do prediction: Would like to do prediction: learn a function: y = f(x) Where y can be: h b Real: Regression Categorical: Classification

586 views • 39 slides
CS345a: Data Mining Jure Leskovec Stanford University Instructors: Instructors: Jure

CS345a: Data Mining Jure Leskovec Stanford University Instructors: Instructors: Jure

CS345a: Data Mining Jure Leskovec Stanford University Instructors: Instructors: Jure Leskovec A Anand Rajaraman d R j TAs: Abhishek Gupta Abhi h k G t Roshan Sumbaly Reach us at cs345a win0910 staff@ R

1.04k views • 51 slides
CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University HW3 is out HW3

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University HW3 is out HW3

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University HW3 is out HW3 is out Poster session is on last day of classes: Thu March 11 at 4:15 Thu March 11 at 4:15 Reports are due March 14 Final is

461 views • 43 slides
Clustering Algorithms CS345a: Data Mining Jure Leskovec and Anand

Clustering Algorithms CS345a: Data Mining Jure Leskovec and Anand

Clustering Algorithms CS345a: Data Mining Jure Leskovec and Anand Rajaraman Stanford University Given a set of data points, group them into a clusters so

670 views • 46 slides
CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Friday 5:30 at

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Friday 5:30 at

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Friday 5:30 at Gates B12 5:30 7:30pm Friday 5:30 at Gates B12 5:30 7:30pm You will learn and get hands on experience on: Login to Amazon EC2 and

312 views • 29 slides
CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Instead of generic

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Instead of generic

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Instead of generic popularity can we measure Instead of generic popularity, can we measure popularity within a topic? E.g., computer science, health Bias the

624 views • 49 slides
CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Feature selection:

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Feature selection:

CS345a: Data Mining Jure Leskovec and Anand Rajaraman j Stanford University Feature selection: Feature selection: Given a set of features X 1 , X n Want to predict Y from a subset A = (X Want to predict Y from a subset A =

571 views • 42 slides
CS345a: Data Mining Jure Leskovec Stanford University CPU Machine Learning, Statistics Memory

CS345a: Data Mining Jure Leskovec Stanford University CPU Machine Learning, Statistics Memory

CS345a: Data Mining Jure Leskovec Stanford University CPU Machine Learning, Statistics Memory Memory Classical Data Mining Disk 1/7/2010 Jure Leskovec, Stanford CS345a: Data Mining 2 20+ billion web pages x 20KB = 400+ TB 20+

784 views • 54 slides
DATA MINING LECTURE 15 The Map-Reduce Computational Paradigm Most of the slides are taken from:

DATA MINING LECTURE 15 The Map-Reduce Computational Paradigm Most of the slides are taken from:

DATA MINING LECTURE 15 The Map-Reduce Computational Paradigm Most of the slides are taken from: Mining of Massive Datasets Jure Leskovec, Anand Rajaraman, Jeff Ullman Stanford University http://www.mmds.org J. Leskovec, A. Rajaraman, J.

990 views • 51 slides
Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2019) Part 6: Data Mining (3/4)

Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2019) Part 6: Data Mining (3/4)

Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2019) Part 6: Data Mining (3/4) November 5, 2019 Ali Abedi Thanks to Jure Leskovec, Anand Rajaraman, Jeff Ullman (Stanford University) These slides are available at

757 views • 52 slides
http://cs246.stanford.edu Instructor: Jure Leskovec TAs: Aditya Parameswaran

http://cs246.stanford.edu Instructor: Jure Leskovec TAs: Aditya Parameswaran

CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu Instructor: Jure Leskovec TAs: Aditya Parameswaran Bahman Bahmani Peyman Kazemian 1/3/2011 Jure Leskovec, Stanford C246: Mining

522 views • 35 slides
Jeffrey D. Ullman You can download a free copy of Mining of Massive Datasets , by Jure

Jeffrey D. Ullman You can download a free copy of Mining of Massive Datasets , by Jure

Finding Similar Sets Application to Document Similarity Shingling Minhashing Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman You can download a free copy of Mining of Massive Datasets , by Jure Leskovec, Anand

856 views • 43 slides
http://www.mmds.org #1: C4.5 Decision Tree - Classification (61 votes) #2: K-Means -

http://www.mmds.org #1: C4.5 Decision Tree - Classification (61 votes) #2: K-Means -

Slide source: Mining of Massive Datasets Jure Leskovec, Anand Rajaraman, Jeff Ullman Stanford University http://www.mmds.org #1: C4.5 Decision Tree - Classification (61 votes) #2: K-Means - Clustering (60 votes) #3: SVM

445 views • 33 slides
End-toEnd In-memory Graph Analytics Jure Leskovec (@jure) Including joint work with Rok Sosic,

End-toEnd In-memory Graph Analytics Jure Leskovec (@jure) Including joint work with Rok Sosic,

End-toEnd In-memory Graph Analytics Jure Leskovec (@jure) Including joint work with Rok Sosic, Deepak Narayanan, Yonathan Perez, et al. Jure Leskovec, Stanford 1 Background & Motivation My research at Stanford: Mining large social and

890 views • 70 slides
Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2020) Part 4: Analyzing Graphs

Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2020) Part 4: Analyzing Graphs

Data-Intensive Distributed Computing CS 431/631 451/651 (Fall 2020) Part 4: Analyzing Graphs (2/2) Ali Abedi Thanks to Jure Leskovec, Anand Rajaraman, Jeff Ullman (Stanford University) These slides are available at

869 views • 49 slides
Spanning Trees Recall the definitions of: - graphs, the vertex set V={0,1,2,,n-1}, the edge

Spanning Trees Recall the definitions of: - graphs, the vertex set V={0,1,2,,n-1}, the edge

[Section 1.1] Spanning Trees Recall the definitions of: - graphs, the vertex set V={0,1,2,,n-1}, the edge set E - n = |V|, m = |E| - adjacency matrix - spanning tree [Section 1.1] Spanning Trees Thm 1.1 (Kirchhoff): Let A be the

502 views • 10 slides
The Revised Simplex Method Combinatorial Problem Solving (CPS) Javier Larrosa Albert Oliveras

The Revised Simplex Method Combinatorial Problem Solving (CPS) Javier Larrosa Albert Oliveras

The Revised Simplex Method Combinatorial Problem Solving (CPS) Javier Larrosa Albert Oliveras Enric Rodr guez-Carbonell April 17, 2020 Tableau Simplex Method The simplex method we have seen so far is called tableau simplex method

1.17k views • 41 slides
Clustering and Dimensionality Reduction Stony Brook University CSE545, Fall 2017 Goal:

Clustering and Dimensionality Reduction Stony Brook University CSE545, Fall 2017 Goal:

Clustering and Dimensionality Reduction Stony Brook University CSE545, Fall 2017 Goal: Generalize to new data Model New Data? Original Data Does the model accurately reflect new data? Supervised vs. Unsupervised Supervised

882 views • 49 slides
E-PUBS @ OECD Organisation for Economic Cooperation and Development Toby Green Public Affairs

E-PUBS @ OECD Organisation for Economic Cooperation and Development Toby Green Public Affairs

E-PUBS @ OECD Organisation for Economic Cooperation and Development Toby Green Public Affairs and Communications Directorate E-Pub Summit Brussels, March 2017 Better policies for better lives: the evidence Our digital stuff . . . . . . for:

298 views • 10 slides
http://cs246.stanford.edu High-dimension == many features Find concepts/topics/genres:

http://cs246.stanford.edu High-dimension == many features Find concepts/topics/genres:

CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu High-dimension == many features Find concepts/topics/genres: Documents: Features: thousands of words, millions of word pairs Surveys

928 views • 54 slides
Implementation of Xen PVHVM drivers in OpenBSD Mike Belopuhov Esdenera Networks GmbH

Implementation of Xen PVHVM drivers in OpenBSD Mike Belopuhov Esdenera Networks GmbH

Implementation of Xen PVHVM drivers in OpenBSD Mike Belopuhov Esdenera Networks GmbH mike@esdenera.com Tokyo, March 12 2016 The goal Produce a minimal well-written and well-understood code base to be able to run in Amazon EC2 and fix

1.32k views • 100 slides
11/14/2016 Disclosures Research grants to my institution from Merck and GSK Tuberculosis:

11/14/2016 Disclosures Research grants to my institution from Merck and GSK Tuberculosis:

11/14/2016 Disclosures Research grants to my institution from Merck and GSK Tuberculosis: Whats New for the HIV Provider Susan Swindells MBBS University of Nebraska Medical Center Omaha, NE December 2016 Case #1 ARS Question #1

308 views • 10 slides
MATH 12002 - CALCULUS I 5.2: Laws of Logarithms Professor Donald L. White Department of

MATH 12002 - CALCULUS I 5.2: Laws of Logarithms Professor Donald L. White Department of

MATH 12002 - CALCULUS I 5.2: Laws of Logarithms Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 4 Laws of Logarithms The familiar Laws of Logarithms all follow

223 views • 4 slides

More recommend