Phd. studies at Mälardalen University

Transcription

1 Phd. studies at Mälardalen University Christopher Engström Mälardalen University, School of Education, Culture and Communication (UKK), Mathematics/Applied mathematics

2 Contents What do you do as a Phd. student?

3 Contents What do you do as a Phd. student? My education before Phd. studies.

4 Contents What do you do as a Phd. student? My education before Phd. studies. Examples of some mathematical problems.

5 What do you do as a Phd. Student? 80/20 Phd studies/faculty work

6 What do you do as a Phd. Student? 80/20 Phd studies/faculty work Research

7 What do you do as a Phd. Student? 80/20 Phd studies/faculty work Research Studies

8 What do you do as a Phd. Student? 80/20 Phd studies/faculty work Research Studies Teaching

9 Courses Mathematical Modelling Mathematical Modelling, (A) Biological Systems Optimization Applied Mathematics Engineering Mathematics Num. Methods for Differential Eq. Discrete Mathematics Simulation Tools Time Series Analysis Finite Element Method Monte Carlo and Empirical Methods for Statistical Inference Calculus in One Variable Calculus in Several Variables Linear Algebra Matrix theory Mathematical Statistics Markov Processes Mathematical Communication Mathematical Structures Algebra Stationary Stochastic Processes Systems and Transforms Analytic Functions

10 Courses Signals and Communications Biology, Introductory Course Image Analysis Multi-spectral Imaging Microeconomics for Technicians Automatic Control, Basic Course Physics Electromagnetic Field Theory Numerical Methods in CAGD Programming Algorithms and Data structures Algorithm Implementation C programming Algorithm Theory Database Technology Functional programming Computer Graphics

11 Natural language processing Problem formulation: Given a collection of abstracts of scientific articles in some field of research: Find words or short phrases which convey knowledge of the domain. Annotation done manually by experts: time consuming and costly.

12 Natural language processing Problem formulation: Given a collection of abstracts of scientific articles in some field of research: Find words or short phrases which convey knowledge of the domain. Annotation done manually by experts: time consuming and costly. Annotation done automatically: not good enough.

13 Natural language processing Problem formulation: Given a collection of abstracts of scientific articles in some field of research: Find words or short phrases which convey knowledge of the domain. Annotation done manually by experts: time consuming and costly. Annotation done automatically: not good enough. Applications in for example medicine.

14 Natural language processing Problem formulation: Given a collection of abstracts of scientific articles in some field of research: Find words or short phrases which convey knowledge of the domain. Annotation done manually by experts: time consuming and costly. Annotation done automatically: not good enough. Applications in for example medicine. Work together with Thierry Hamon, Paris-Nord University, Computer Science Laboratory for Mechanics and Engineering Sciences.

15 Example of annotated abstract To address potential mechanisms by which estrogens suppress erythropoiesis, we have examined their effects on GATA 1, an erythroid transcription factor that participates in the regulation of the majority of erythroid cell specific genes and is necessary for full maturation of erythrocytes.

16 Natural language processing Texts Term extraction rule-based approaches Candidate terms My work in the Term ranking step: Term ranking frequency term length C-Value frequency: How often does a term appear? Ranked candidate terms Validation by terminologists

17 Natural language processing Texts Term extraction rule-based approaches Candidate terms My work in the Term ranking step: Term ranking frequency term length C-Value frequency: How often does a term appear? term length: Longer terms less important? Ranked candidate terms Validation by terminologists

18 Natural language processing Texts Term extraction rule-based approaches Candidate terms My work in the Term ranking step: Term ranking frequency term length C-Value frequency: How often does a term appear? term length: Longer terms less important? Ranked candidate terms Validation by terminologists C-value: Long terms not part of other terms are preferred.

19 Parsing tree of a term modifier component head component modifier component full (of) head component maturation erythrocytes Figure: Parsing tree of the term full maturation of erythrocytes

20 C-Value Original definition [Frantzi et al. 1997]: log 2 ( t + 1) f (t) if t is not included in a term C Value(t) = log 2 ( t + 1) (f (t) 1 f (t )) otherwise P(T t ) t T t

21 C-Value Original definition [Frantzi et al. 1997]: log 2 ( t + 1) f (t) if t is not included in a term C Value(t) = log 2 ( t + 1) (f (t) 1 f (t )) otherwise P(T t ) t T t Parametrised C-Value ( ) t + 1 log 2 t α f (t), if t is not included in a term (Roo R ( ) t + 1 log 2 t α f (t) c H H 1/β H f (t ) βh, C Value t T t = if t is a Head term ( ) t + 1 log 2 t α f (t) c M M 1/β M f (t ) βm, t T t if t is a Modifier term

22 F-measure and average precision for different set of parameters Frequency and the C-Value: better average precision for the very first terms Then, all the C-Value* models outperform the frequency and the C-Value ranking Similar ranking for all the C-Value* models

23 Network analysis in cancer research, background Out of inferred gene expression networks, can we predict which genes are important for the development of a cancer? Work together with (mainly) Holger Weishaupt, Uppsala University, Rudbecklaboratoriet.

24 Network analysis in cancer research, background Out of inferred gene expression networks, can we predict which genes are important for the development of a cancer? Work together with (mainly) Holger Weishaupt, Uppsala University, Rudbecklaboratoriet. Finding relevant genes makes it probable that you could develop medication for that cancer type rather than using other more intrusive methods.

25 Network analysis in cancer research Omics data Samples Gene correlations Genes Adjacency matrix Genes Genes Genes Genes Centralities Inferred network

26 Network analysis in cancer research Problems: Low quality and bad conservation of centralities for generated networks compared to reference networks.

27 Network analysis in cancer research Problems: Low quality and bad conservation of centralities for generated networks compared to reference networks. What centrality mesures can be used to find driver genes for the cancer?

28 Network analysis in cancer research Problems: Low quality and bad conservation of centralities for generated networks compared to reference networks. What centrality mesures can be used to find driver genes for the cancer? Low amount of good reference networks for training and verification.

29 PageRank Used to rank homepages based on the webs linkstructure.

30 PageRank Used to rank homepages based on the webs linkstructure. Needs to be both very fast but also of good quality.

31 PageRank Used to rank homepages based on the webs linkstructure. Needs to be both very fast but also of good quality.

32 PageRank Used to rank homepages based on the webs linkstructure. Needs to be both very fast but also of good quality. Our main considerations: Can the rank of different types of graphs and/or subgraphs be calculated more efficiently?

33 PageRank Used to rank homepages based on the webs linkstructure. Needs to be both very fast but also of good quality. Our main considerations: Can the rank of different types of graphs and/or subgraphs be calculated more efficiently? Generalizations and variations of the definition of PageRank.

34 PageRank Used to rank homepages based on the webs linkstructure. Needs to be both very fast but also of good quality. Our main considerations: Can the rank of different types of graphs and/or subgraphs be calculated more efficiently? Generalizations and variations of the definition of PageRank. How does the rank change if you introduce some changes to the graph?

35 PageRank Defined as the (right) eigenvector of matrix M: M = c(a + gw ) + (1 c)we (1)

36 PageRank Defined as the (right) eigenvector of matrix M: M = c(a + gw ) + (1 c)we (1) Defines as a sum of random walks on a graph: R j = W j + ( ) W i P(v i v j ) (P(v j v j )) k v i V,v i v j k=0 (2)

37 Calculating PageRank Power iteration: R n+1 = MR n

38 Calculating PageRank Power iteration: Better: R n+1 = MR n R n+1 = ca R n + dw d = 1 ca R n 1

39 Calculating PageRank Power iteration: Better: R n+1 = MR n R n+1 = ca R n + dw d = 1 ca R n 1 Power series: { Pn+1 = cap n = W P 0 R (3) n = n k=0 P k

40 Different types of vertices Definition For the vertices of a simple directed graph with no loops we define 5 distinct groups G 1, G 2,..., G 5 1. G 1 : Vertices with no outgoing or incoming edges.

41 Different types of vertices Definition For the vertices of a simple directed graph with no loops we define 5 distinct groups G 1, G 2,..., G 5 1. G 1 : Vertices with no outgoing or incoming edges. 2. G 2 : Vertices with no outgoing edges and at least one incoming edge (also called dangling nodes).

42 Different types of vertices Definition For the vertices of a simple directed graph with no loops we define 5 distinct groups G 1, G 2,..., G 5 1. G 1 : Vertices with no outgoing or incoming edges. 2. G 2 : Vertices with no outgoing edges and at least one incoming edge (also called dangling nodes). 3. G 3 : Vertices with at least one outgoing edge, but no incoming edges (also called root nodes).

43 Different types of vertices Definition For the vertices of a simple directed graph with no loops we define 5 distinct groups G 1, G 2,..., G 5 1. G 1 : Vertices with no outgoing or incoming edges. 2. G 2 : Vertices with no outgoing edges and at least one incoming edge (also called dangling nodes). 3. G 3 : Vertices with at least one outgoing edge, but no incoming edges (also called root nodes). 4. G 4 : Vertices with at least one outgoing and incoming edge, but which is not part of any directed cycle (no path from the vertex back to itself).

44 Different types of vertices Definition For the vertices of a simple directed graph with no loops we define 5 distinct groups G 1, G 2,..., G 5 1. G 1 : Vertices with no outgoing or incoming edges. 2. G 2 : Vertices with no outgoing edges and at least one incoming edge (also called dangling nodes). 3. G 3 : Vertices with at least one outgoing edge, but no incoming edges (also called root nodes). 4. G 4 : Vertices with at least one outgoing and incoming edge, but which is not part of any directed cycle (no path from the vertex back to itself). 5. G 5 : Vertices that is part of at least one cycle.

45 PageRank for different types of vertices Vertices with no incomming or outgoing edges: Theorem We let e g G 1 and e i be any other vertex, then R g = W g, and P(v g v i ) = 0. (3)

46 PageRank for different types of vertices Vertices with no incomming or outgoing edges: Theorem We let e g G 1 and e i be any other vertex, then R g = W g, and P(v g v i ) = 0. (3) Vertices with no outgoing edges: Theorem We let e g G 2 and e i be any other vertex, then R g = v i G, v i v g W i P(v i v g ) + v g, and P(v g v i ) = 0. (4)

47 PageRank for different types of vertices Vertices with no incomming edges: Theorem We let e g G 3 and e i be any other vertex, then R g = W g, and v i G, v i v g P(v g v i ) > 0. (5)

48 PageRank for different types of vertices Useful when calculating the rank of other vertices: Theorem Given R g = W g, v g G 3. We can write the PageRank of another general vertex e i as R i = W i + W g ca gi + v j G v j v i,v g ( ) (P(v i v i )) k k=0 (W j + W g ca gj )P(v j v i ) where ca gi is the one-step probability to go from v g to v i.

49 PageRank, graph reordering Figure: Non-zero values of adjacency matrix for the Web graph before and after sorting vertices according to level and component.

50 PageRank iterations/component Figure: Number of iterations needed per SCC. The horizontal line denotes the result where the whole graph is considered a single component.

51 Thank you all for listening! Questions?