Using a signature-based machine learning model to analyse a psychiatric stream of data

Transcription

1 Using a signature-based machine learning model to analyse a psychiatric stream of data Imanol Perez (Joint work with T. Lyons, K. Saunders and G. Goodwin) Mathematical Institute University of Oxford Rough Paths in Toulouse

2 Signature of a path Continuous paths with finite p-variation Given p 1 and X C([s, t], R d ) with s < t we define X p,[s,t] := sup {t i } i [s,t] ( ) 1/p X ti X ti 1 p. i Rough Paths in Toulouse Signatures and psychiatric streams of data 2

3 Signature of a path Continuous paths with finite p-variation Given p 1 and X C([s, t], R d ) with s < t we define X p,[s,t] := sup {t i } i [s,t] ( ) 1/p X ti X ti 1 p. V p ([s, t], R d ) := {X C([s, t], R d ) : X p,[s,t] < }. i Rough Paths in Toulouse Signatures and psychiatric streams of data 3

4 Signature of a path Definition (Signature of a continuous path) Let X V 1 ([0, T ], R d ). The signature of X is defined as where X n = S(X ) = (1, X 1, X 2,...)... 0<u 1 <u 2 <...<u n<t (R d ) n n=0 dx u1... dx un n 1. Rough Paths in Toulouse Signatures and psychiatric streams of data 4

5 Signature of a path Definition (Truncated signature of a continuous path) Similarly, we define, for n 0, S n (X ) := (1, X 1, X 2,..., X n ). Rough Paths in Toulouse Signatures and psychiatric streams of data 5

6 Signature of a stream of data Definition (Time-joined transformation) Let {(t i, X ti )} N i=0 R+ R be a stream of data. Its time-joined transformation is defined as the path Y : [0, 2N + 1] R + R that is given by (t 0, X t0 t) for t [0, 1) Y t := (t i + (t i+1 t i )(t 2i 1), X ti ) for t [2i + 1, 2i + 2), (t i+1, X ti + (X i+1 X ti )(t 2i 2)) for t [2i + 2, 2i + 3) for 0 i N 1. Rough Paths in Toulouse Signatures and psychiatric streams of data 6

7 Signature of a stream of data Figure: IBM stock price from October to November Figure: Time-joined transformation of the path. Rough Paths in Toulouse Signatures and psychiatric streams of data 7

8 Signature of a stream of data Definition (Signature of a stream of data) The signature of a stream of data {(t i, X ti )} N i=0, which with some abuse of notation will be denoted by S({(t i, X ti )} N i=0 ), is defined as the signature of its time-joined transformation. Rough Paths in Toulouse Signatures and psychiatric streams of data 8

9 Signatures and machine learning Supervised learning We have two data sets: a known set of known input-output pairs (the training set), {(X i, Y i )} i, which is used to train the model, and a set of inputs that is used for testing (the out-of-sample set). Rough Paths in Toulouse Signatures and psychiatric streams of data 9

10 Signatures and machine learning Supervised learning We have two data sets: a known set of known input-output pairs (the training set), {(X i, Y i )} i, which is used to train the model, and a set of inputs that is used for testing (the out-of-sample set). Features play an important role in machine learning. Rough Paths in Toulouse Signatures and psychiatric streams of data 10

11 Signatures and machine learning Signatures as features: uniqueness Theorem (B. Hambly, T. Lyons) The signature of a path with bounded variation is unique up to tree-like equivalence. Rough Paths in Toulouse Signatures and psychiatric streams of data 11

12 Signatures and machine learning Signatures as features: estimate Rough Paths in Toulouse Signatures and psychiatric streams of data 12

13 Signatures and machine learning Signatures as features: estimate Theorem Let X V 1 ([0, T ], R d ) be a path with bounded variation. Then, given 1 i 1, i 2,..., i n d we have... 0<u 1 <u 2 <...<u n<t dx i X n 1 u 1... dxu in n n! 1,[0,T ]. Rough Paths in Toulouse Signatures and psychiatric streams of data 13

14 Signatures and machine learning The model Given a training set {(R i, Y i )} N i=0, of input-output pairs, where R i = {(t ij, r ij )} j is a stream of data, construct a new set {(X i, Y i )} N i=0 with X i V 1. Rough Paths in Toulouse Signatures and psychiatric streams of data 14

15 Signatures and machine learning The model Given a training set {(R i, Y i )} N i=0, of input-output pairs, where R i = {(t ij, r ij )} j is a stream of data, construct a new set {(X i, Y i )} N i=0 with X i V 1. Compute {(S n (X i ), Y i )} N i=0 for some n N. Rough Paths in Toulouse Signatures and psychiatric streams of data 15

16 Signatures and machine learning The model Given a training set {(R i, Y i )} N i=0, of input-output pairs, where R i = {(t ij, r ij )} j is a stream of data, construct a new set {(X i, Y i )} N i=0 with X i V 1. Compute {(S n (X i ), Y i )} N i=0 for some n N. Apply regression against the truncated signature. Rough Paths in Toulouse Signatures and psychiatric streams of data 16

17 Application to psychiatric data The problem Rough Paths in Toulouse Signatures and psychiatric streams of data 17

18 Application to psychiatric data The problem Rough Paths in Toulouse Signatures and psychiatric streams of data 18

19 Application to psychiatric data The problem Given some information about a participant, can we tell if he or she was diagnosed to have bipolar disorder, borderline personality disorder or to be healthy? Rough Paths in Toulouse Signatures and psychiatric streams of data 19

20 Application to psychiatric data The problem Given some information about a participant, can we tell if he or she was diagnosed to have bipolar disorder, borderline personality disorder or to be healthy? Given a participant and information about the last few days, can we predict the mood the following day? Rough Paths in Toulouse Signatures and psychiatric streams of data 20

21 Application to psychiatric data Methodology Rough Paths in Toulouse Signatures and psychiatric streams of data 21

22 Application to psychiatric data Methodology Rough Paths in Toulouse Signatures and psychiatric streams of data 22

23 Application to psychiatric data Methodology Figure: Normalised path for anxiety scores. Rough Paths in Toulouse Signatures and psychiatric streams of data 23

24 Application to psychiatric data Methodology Rough Paths in Toulouse Signatures and psychiatric streams of data 24

25 Application to psychiatric data Predicting if a person is healthy, has bipolar disorder or has borderline disorder {(t i, S ti )} 19 i=0 ( 1, 1), if the partcipant is healthy ( 1, 1), if the participant is bipolar. (1, 0), if the participant is borderline. Rough Paths in Toulouse Signatures and psychiatric streams of data 25

26 Application to psychiatric data Predicting if a person is healthy, has bipolar disorder or has borderline disorder Order Correct guesses 2nd 75% 3rd 70% 4th 69% Table: Percentage of people correctly classified in the three clinical groups. Rough Paths in Toulouse Signatures and psychiatric streams of data 26

27 Application to psychiatric data Predicting if a person is healthy, has bipolar disorder or has borderline disorder Rough Paths in Toulouse Signatures and psychiatric streams of data 27

28 Application to psychiatric data Predicting if a person is healthy, has bipolar disorder or has borderline disorder Rough Paths in Toulouse Signatures and psychiatric streams of data 28

29 Application to psychiatric data Predicting if a person is healthy, has bipolar disorder or has borderline disorder Rough Paths in Toulouse Signatures and psychiatric streams of data 29

30 Application to psychiatric data Predicting the future mood {(t i, S ti )} 19 i=0 S {1,..., 7} 6 where S is the scores of the participant the following observation. Rough Paths in Toulouse Signatures and psychiatric streams of data 30

31 Application to psychiatric data Predicting the future mood Category Healthy Bipolar Borderline Anxious 98% 82% 73% Elated 89% 86% 78% Sad 93% 84% 70% Angry 98% 90% 70% Irritable 97% 84% 70% Energetic 89% 82% 75% Table: Percentage of correct guesses for mood predictions Rough Paths in Toulouse Signatures and psychiatric streams of data 31

32 Thank you! Thank you! Rough Paths in Toulouse Signatures and psychiatric streams of data 32