Using kernels to detect abrupt changes in time series
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Title 
Using kernels to detect abrupt changes in time series

Title of Series  
Part Number 
2

Number of Parts 
10

Author 

License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

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Subject Area  
Abstract 
In this talk we discuss the changepoint detection problem when dealing with complex data. Our goal is to present a new procedure involving positive semidefinite kernels and allowing to detect abrupt changes arising in the full distribution of the observations along the time (and not only in their means). This twostage procedure is based first on dynamic programming, and second on a new l 0type penalty derived from a nonasympsotic model selection result applying to vectors in a reproducing kernel Hilbert space. Since our procedure relies on the dynamic programming algorithm, which induces a high computational complexity at the first step, we will also discuss an improved version of this first step allowing to achieve a complexity of O(n^2) in time and O(n) in space. Finally, we will illustrate the behavior of our kernel changepoint procedure on a wide range of simulated data. In particular we empirically validate our penalty since the resulting penalized criterion recovers the true (number of) changepoints with high probability. We also infer the influence of the kernel on the final results in practice.

00:00
Regression analysis
Musical ensemble
Statistics
Multiplication sign
Point (geometry)
Definite quadratic form
Curve
Sampling (statistics)
Time series
Mereology
Functional (mathematics)
Equivalence relation
Number
Mathematics
Position operator
Mathematics
Sample (statistics)
Phase transition
Module (mathematics)
Position operator
Local ring
02:40
Point (geometry)
Group action
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Moment (mathematics)
Distribution (mathematics)
Range (statistics)
Algebraic structure
Insertion loss
Grand Unified Theory
Sequence
Mathematics
Arithmetic mean
Object (grammar)
Distribution (mathematics)
Subtraction
04:17
Mathematics
Computer animation
Divisor
State of matter
Set (mathematics)
Multiplication sign
Time series
Object (grammar)
Distribution (mathematics)
05:17
Point (geometry)
Existence
Curvature
Multiplication sign
Characteristic polynomial
Distribution (mathematics)
Mathematical model
Infinity
Distance
Expected value
Frequency
Mathematics
Term (mathematics)
Subtraction
Associative property
Logical constant
Product (category theory)
Spacetime
Element (mathematics)
Algebraic structure
Set (mathematics)
Sequence
Time domain
Maxima and minima
Mathematics
Arithmetic mean
Sample (statistics)
Computer animation
Noise
Element (mathematics)
Stochastic kernel estimation
Object (grammar)
Distribution (mathematics)
10:07
Goodness of fit
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Square number
Stochastic kernel estimation
Measurement
Square number
Linear map
11:06
Point (geometry)
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Complex (psychology)
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Letterpress printing
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Dressing (medical)
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Natural number
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Selectivity (electronic)
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Limit (category theory)
Open set
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Metric tensor
Maxima and minima
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Causality
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Computer animation
Function (mathematics)
Hausdorff dimension
Stochastic kernel estimation
Ranking
Mathematical optimization
Matrix (mathematics)
17:38
Point (geometry)
Computer programming
Frobenius method
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Latin square
Distribution (mathematics)
1 (number)
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Distance
Discrete element method
Event horizon
Runge's theorem
Number
Frequency
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Matrix (mathematics)
Thermodynamisches System
Term (mathematics)
Ideal (ethics)
Subtraction
Wiener filter
Position operator
Distribution (mathematics)
Linear map
Chisquared distribution
Newton's law of universal gravitation
Area
Point (geometry)
Line (geometry)
Metric tensor
Maxima and minima
Partition (number theory)
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Arithmetic mean
Computer animation
Function (mathematics)
Hausdorff dimension
Sheaf (mathematics)
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Resultant
22:48
Histogram
Frobenius method
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Euler angles
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Binary file
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Mathematics
Partition (number theory)
Frequency
Category of being
Arithmetic mean
Mathematics
Computer animation
Graph coloring
Term (mathematics)
Subtraction
Position operator
Distribution (mathematics)
Linear map
Chisquared distribution
25:10
Point (geometry)
Histogram
Mathematics
Frobenius method
Computer animation
Point (geometry)
Algebraic structure
Haar measure
Vector potential
Chisquared distribution
Newton's law of universal gravitation
26:21
Axiom of choice
Standard deviation
Multiplication sign
Equaliser (mathematics)
Mathematical model
Mereology
Discrete element method
Event horizon
Order of magnitude
Number
Expected value
Group representation
Maxima and minima
Strategy game
Term (mathematics)
Forest
Square number
Selectivity (electronic)
Associative property
Subtraction
Data transmission
Linear map
Condition number
Logical constant
Spacetime
Coalition
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Projective plane
Moment (mathematics)
Parameter (computer programming)
Term (mathematics)
Sequence
Functional (mathematics)
Estimator
Maxima and minima
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Function (mathematics)
Theorem
Stochastic kernel estimation
Ideal (ethics)
Orthogonality
31:09
Logical constant
Point (geometry)
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Free group
1 (number)
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Icosahedron
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Number
Frequency
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Hausdorff dimension
Series (mathematics)
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Addition
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3 (number)
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Surface of revolution
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Right angle
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Family
Resultant
35:29
Point (geometry)
Computer programming
Complex (psychology)
Pulse (signal processing)
Musical ensemble
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Momentum
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Sigmaalgebra
Characteristic polynomial
Distribution (mathematics)
Graph (mathematics)
Discrete element method
Mathematics
Hausdorff dimension
Distribution (mathematics)
Position operator
Addition
Spacetime
Moment (mathematics)
Approximation
Category of being
Computer animation
Distribution (mathematics)
Spacetime
39:30
Area
Direction (geometry)
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Characteristic polynomial
Mathematical analysis
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Discrete element method
Event horizon
Food energy
Number
Power (physics)
Maxima and minima
Band matrix
Mathematics
Summation
Computer animation
Meeting/Interview
Term (mathematics)
Element (mathematics)
Normal (geometry)
Stochastic kernel estimation
Orthogonality
Resultant
Position operator
43:01
Basis (linear algebra)
Icosahedron
Mereology
Open set
Number
Frequency
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Hausdorff dimension
Units of measurement
Physical system
Addition
Link (knot theory)
Theory of relativity
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Term (mathematics)
Open set
Mathematics
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Theorem
Stochastic kernel estimation
Distribution (mathematics)
Mathematical optimization
Matrix (mathematics)
Directed graph
45:36
Logical constant
Point (geometry)
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2 (number)
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Length
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Process (computing)
Consistency
Moment (mathematics)
Curve
Algebraic structure
Independence (probability theory)
Estimator
Hausdorff dimension
Right angle
Resultant
00:02
so the time and you have to and to so my morning the light of his more focused by band 1 and it is lawyers indicated to a parade of detecting metabolic change .period arising in time series and remain on the aspect of this work is to introduce the positive semidefinite Keller's in his missionary and to be able to detect popular changes and we talk about Martin interest against so it is a joint crew equivalency follows a luxury we Dimling and the so it's not looking too the I'm trying talk so of that 1st time introduce overnight modulating samples and also a framework we use in Sequeira and particularly the candles we use then I give details about the agreement foreign detecting change bonds which is called case simply and the sewer discussed some of the competition aspects about this algorithm and then will discuss in Vermont and Maine New statistical Baltic stock which is divided into 2 parts the 1st 1 is what locals with the problem of defending the assuming that the the number of segments is known aware of the change .period ended in a 2nd time and how many such change .period 2 1 true detect OK so let us stop with the simplest problem which is that detecting changes in demeanor of a signal so you observe the curve and the air in this example of is has been generated phase 2 of the regression function which is black here and that it is a piecewise consent and then that the problem is that the use of other changes in the bluecollar and the main question is would come no waiver of his changes or related to a number of changes in the Russian function or not and so here it's the case some of them are power plants difficult to recognize and the other was all awful clearing forced positives so I had to do this In dramatic so that what all the
02:43
purpose of this work the 1st one is to be able to detect changes in the distribution the injured or distribution of the observations so in Budokan opponent demeanor as I would just represented and for instance you have menacing losses here it is a signal coming from Barigye women due to major about that but anyway the fight the fact you can observer invited issue a computer look a range of this signal you will always be everywhere staying something which is equal to onehalf of Amin and so over classical techniques which are able to detect changes in the meaning of this overseeing all completely useless with such strictures of structural signal so the 2nd point is that it would like to be able to detect it to deal with the complex data and so I spilled them into 2 2 different kinds of data of highdimensional once that would up highdimensional majors or killers and most ritual objects such as the video sequences blasts audience sequences for instance we'd like
03:57
to be able to work to do to detect the sequences in which given action is happening for instance someone is playing music or people rush perusing or someone is no there there there are discussing here for instance and so the main
04:18
factor here is that you have such an object for instance usually usually usually people summarized observations or the majors by use of is to run so he ditched instead cubes on which to run yesterday to detect changes from this it is to run the state's reasons to run a cake and
04:38
another of the structural object we would like to be able to do with it is for instance a time series of grass so obviously from time to time the rough changes but the question is to know whether this change rises by 1 from them or if it's a related to structural change in this mission that has generated a raft so I would like to be able to detect abrupt changes in such stances of Ross for instance looking and of course I would like to be able to even the ideally to retreat to deal with some suggested did with such different types of common over the top and the provide efficient algorithms as usual
05:18
so how to do that I thought into
05:20
Reuss the framework on camera so I assume you know or what follows that I independent and identically observations that run by worsening Aswan examined in a given sector computer access and that this sudden in particular has no plot structure the only requirement I make it on it is that I can define a positive solution can indicate which is replacing income in the meaning of introduce over can introduced by or on our around in 1915 and so as long as you have an idea such again or you can have the The Associated Press :colon and also because the chemical future fight their defined this way and the idea simply 2 users can encourage you not to be able to to match the obsession initial space and to very oncologists wherever is Victor structure and you can explain the structure so it's a visit to tour and the commitment to to deal with very different kinds of data and our goal is to provide a unified and resist the urge to deal with the very different types of data on the use of cameras OK so existence is which are a classic or a friend but you have national in 4 of them there and with the first one is encounter which is where defined this way and the 2nd 1 which is the fastest is called the case "quotation mark can and it is defined as expectations of minus the square distance between these 2 runs with IBM's we will use it to his admits a torrent of the talk so what is the reason was for introducing can but it is summarized the value of this fighting so the yield the obvious observation XI and thanks to the work of the curvature of 5 you can write it as a white so now we will prove why I made sure I need and that you can ride back to white light in in that space is simply "quotation mark to its expectation in the arcade chest presses mineral terms which is defined as simply as a difference by construction so but why doing that because I knew I start left to Koshice about that because it's a defined as the expectation of an objective which is that which belongs to an Indian demanding mentioned little space and so we have to take care of what how to define this expectation and so the meaning and knew I'd start is that of X of big side is defined as a unique element in the arcade chess said tract for every effort in the arcade knew the adult collected enough product or cages between realize foreign F is equal to the expectations of the Khandaker Fitch not at .period Big XII and OK so destinations for having but the age has to be tested by war and or so but for the expectation of cable execs as defined in the domain .period which is of interest for us is that the fall collected characteristic cameras for instance well that's a different any difference between the solution they excite and P X J implies a difference between the mean immense so now since Roy interested by detecting changes in the distribution well I've read introduces to detecting changes in the man in the Internet in meant that all the time so that's a reason why we really tried to estimated amends the sequence of beginning of monuments which is assumed to be piecewise constant and that the main point is that we have to to thank you you To keep in mind that in regions where the signaltonoise ratio is too low it will be impossible to recover all that would change .period because of because of the noise of course and so that's the reason why we only try to estimate the meaning the sequence of immense and of course in settings where vicinity promoters ratio is large enough Venice where it would provide us with to change OK so now that gives some
10:11
details about we use so
10:15
far a given segmentation tho in decent uh uh we use it as a great measure of 2 25 quality of a given segmentation at the Sanger of quality as that but when used in ushering in campaign 2007 which is not by or in a pot of tea felt sorry which has referring expressions and that the thing you have to keep in mind about this quantity is that when we consider relying you cannot which is simply the police between the victims in on the vendor this quantity would use this to the classical square empirical I a bit of the American United OK so that we can assure quality of the good instrumentation thanks to his quantity and then
11:07
we use the custody agreement which is defined this way so as an input you have the observations excite extended and the camera to compare them and your seniority asked at 1 for each diner for each number of Sigmund between 1 and given the Max you have to compute the minimum over the missing person or segmentation as we do segments of the on project and I just introduced before also are and how to so we have to note is that this quantity this optimization program is a very hobbled program and it's performed by use of temporary so that's each did we are given with the best segmentation with duties segments so discuss consists of just after and the 2nd step of the grievance is given such collection of the best segmentation for each dimension between 1 and the next we have to design penny rise but the criterion by which is defined as the sum of Orrin Hatch tho the president to turn at us to be presented at us 3 made more precise and that is precisely what we do by use of selection so finery by optimizing this denies attention and will be given with the best of the number of new segments and then to we did will be given the best intentions with the at Siemens 28 so authority to says focuses on the 1st step of the grievance so it is based on dynamic running the update full of their dating programming agreement is a offering 1 so it says that for each number of citizens between 2 Indian lacks the cost of the best implementation in indeed segments but from 1 to 10 is a quartermillion none of the costs of the best indentation induce minus1 segments and 2 did not to tainted dress the cost of only 1 segment between T and N so well From that's the usual strategy so look at the cost of a segment that between tea and then as the following formula and the usual strategy is the 1st to compute this the cosmetics which is of size and things and and ended in storage and so it in use it didn't it induces a space complexity of an square because you have to storms and square matrix and so once this strategy which is quite naive is embedded in the control framework it's the costs in time and into the fall because you have to computer and square tourism C S and each of them realize or involves a party member of of terms which provisions of the Romantics so it's by far too costly to be able to deal with the signals so we've and you met whole raft proposes a small change in this agreement which is summarized by at least 2 crude which is only based on 2 ideas the first one is that we never asked to store because cosmetics because we want to have a space complexity as of the end square and having to look to 2 of them algorithm we see that there remained in the village of must and ensure ensure step is to be able to update to compute over to plus 1 kind of cost matrix from their natural vocal and city so it wasn't a tool to Contra disk with you this :colon from this 1 from the previous 1 and later it arises to avoid storing of whole cosmetics but and so to reduce the space complexity to line into the line during and and the time complexity to be in square at most so it is illustrated in this picture when you see the particulate complexity and complexity of the native implementation and there reduced and complexity of our improvements in the as a communities we are able to deal with summer size of hand 100 thousand observations in about 3 minutes so it's already quite interesting to do things but anyway it can be seen as a limitation because it tho it prevents us from dealing with a very very sits emphasizes and certain of playing question which would be interested to tool 6 2 2 address would be to be able to reduce its competition for instance by use of ferromagnetic approximation we could use a Roman to support nation of 2 of the run metrics and use it with you use this with some the cleaning strategies which are used to actually and to reduce the Commission died of dynamic programming agreement and certainly another point is to be able to quantify what has been lost by this approximation look at the agency out of business said
17:03
was the restoration of the IRA whom I just presented is in big prizes you the exact solutions to problems but had to wait to be able to reduce its commission plan would be to to use an approximation of the origin or metrics I and bear from the print version of this disagrees and the fact is that you have to be able to quantify what has been lost from a statistical point of view by using this approximation but it's not
17:40
have this as so it took about this statistical performance of the state 1 of
17:47
the areas I mean each number of Sigmund we are we we I say that we are able or using their programming to computer best implementation with dissonance by minimizing this fighting OK so what is statistical performance of this quantity of this procedure and to compare the quality of segmentation we introduced to the distances between segmentation the first one is quite drastic or it is the home the distance and the 2nd for the 2nd 1 is the former Miss Distance it is defined as a known between mattresses and the mattress the metrics and is simply a defined this way so and adjacent confusion by a jury of his matrix is simply as "quotation mark 201 if I j belong to the same segment of segmentation divided by the College of this England this so that we visited distances between segmentation we consider Saracen I scenarios to assess from an intricate point of view the quality of this 1st step but when we very very the victories can we use so the 1st scenario and it is so here in this picture you you see an instance of the kind of signal we have to with so fast we choose we find proves inundation of India's star which is equal to difference in so so here you have invited that dashed flank of a position which will be right .period so the event such a troupe of politicians in each segment we randomly chosen solution is among the poorest of of them and that will resume segmentation of these distributions share the same views differ on a different meaning Borens so as long and as soon as you changes from this 2nd to the other was fined 1 but you have to change a and Ivorians so it is quite an ordeal a simple program and here are the results so far on the left and the see performance in terms of the the ideals of distance and therefore we need not for over the again so here kills or plotted versus the number of segments and that is what we what we see is that solution can performs quite well because communication so best performance of the segmentation is obtained with the dimension which is equal to rich ones so destroyers equipped to intervene and if we compare this performance to what is obtained by use of the line you can learn them you see vats for instance if you look at the cost distance is that it is the only way is decreasing and what it means and in particular voted for large dimensions and so what it means it means that even when the dimensions of the numerous England's is by far larger about which 1 you are used to to I'm saying that checkpoints which improving so which are close to true ones and better from the previous ones so Jerry speaking openly about the relying upon petition the changes in the and it is confirmed by the his rocks so here it is frequencies of exact recovery of its position as the change .period much knows Virginia little change or in Latin and you see the results over ocean camel and fullline you want and you serve at over 500 petitions we detect exactly but to change .period at the least 60 per cent of the Wiener repetitions White away whereas with line Canada it is vote the exact recovery frequency is between 10 and 20 per cent if I turn to the 2nd scenario the main difference between the 2nd and the previous 1 is but here when you change from pursuant to the use of firing on the next 1 actually you change the year is emerging in environs so the distribution here and here Schirmer singing and Lawrence some of the Friends of Europe had to higher order of the universe emissions of solutions but OK so here you have an instance of a became seniority opted to be wanted to to deal with and the results are
22:50
quite something not so you have to the same unification which is at for true number of segments in the the conclusion foreign to the line you can work at all similar benefit but that the performance is even worse but which is just to when but and here we see that the exact Pecora frequency forward line you can is on most so you almost never on the board to account could change .period it only puts change us in the region OK well Corsican another category which is the Aamodt 1 because it is related to the the Government Printing Office and the work of the main idea of circular is simply that it is sensitive to revert to changes arising in the dissolution of the 2 of us in the 1st 5 minutes of distributions so the performance in terms of electrical frequency or better abounds with Romania can of course but you survivors to difference between what is provided by the Gaussian coming at the end of it is muster rated 205 and the Gaussian color is a characteristic Keller with religion the case for the OK and the last nite and it says that of the attitude attitude and position we observe and printed means is to run so work in each segment we have randomly chosen by the practical efficiency of this during his usually process and the sole association in in each segment we are able to to generates a list of items we've printed OK so you have a new sense of what you can observed In the 1st 3 coordinates of listing USA and the important
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question is that in this scenario that we or dealing with the structural object because of the some of the kind of used to run also the sun is accord 1
25:10
and so the question is Is it necessary to tool to take into account this structure of the data we are dealing with it is to answer restrictions we of considered good compared to a different can also square 1 I've already introduced earlier and the Goshen can which is here and the the main point is that devotion ignores the structure of the data so he doesn't know much interest to like the conceals what should solution is simply a Victorian or printed and what you see is not about the performance Of the gross income is less accurate that's what you were obtained with a cash grant which explains precisely the structure and it is confirmed by the exacted prolific answers because use saw about 2 performances at the World with dilution Caroline Weaver case go on so there is a potential gain in exploiting the switchover data president Maine contrition for his 1st nite but they case now that
26:22
we're talk about the last part of his talk and so it is mainly concerned with the problem of designing in naughty so that the immunizing the specialized frightened that provides you with an estimated the number of Simmons and this is the life of the selection sir at the moment was the first 1 so each association very awkward chess is equal to its expectation plus another storm and we assume that the media sequence of minimums is piecewise constant so forgive segmentation turn and we can consider very associated with little space which is made up of peace Wisconsin functions of and bit format this segmentation time so it is a debate after all and estimator consider simply from under coalition which is noted by new had style and that it is simply forests deal from a projection on 2 of his little space so what is the best possible choice of penalty to to to find the best intentions and actually it is what we call a the idea penalty and how it is always defined in this way so the idea penalty is what we have to have to bid on the core West to recover but the best possible representation which is called the or representation so fast how would you we do will be a real recovery the best conditions where if we minimize this quantity some which is the very a true of the associated with the same indigent out the issue minimizes quantity over time where you you get the best possible Simanjuntak and so it's simply a matter of writing this In this way but you can define the penalty as the difference between this quantity and this 1 in writing this in another way it is a quarter to apply this summer the project term ended a nineyear terms so I was strategy is offering 1 it's simply to use competition equality is to be able to provide a number of armed with high probability of this but I paid the idea of and so we have provided in here transmission equity for the nineyear term the 1st which is simply based on than shiny quite classical and that never existed and greedy for those who project and so
29:14
it is this 1 so it is stated that under a summer sessions the first one says simply that we assume that the date or bounded invoking chest so it's true that as long as friends and consider abundant cannot or if Lawrence sitting where excites or abounded themselves and we also assume that the notice of indoor cages is bounded by you the next so inarticulate we cannot assume that the date organization in the days until I come from various assumptions so well this research labs you as to a true to deal with it but Bavaria victims and that only picked normally as easy as it used to with ongoing reserves of that kind and so our results so that the project term is a is close to exceed expectations with a derogation term which is related to to the estimation Iraq and that another edition which depends on the next and an important fact is that this dependency on the X here which is related to the probability of a large events on which of his iniquity always full and it has to be simply X and take square in budget writers who the use them some strategies based on that I warmly greedy for instance you would get the next square and the invading our sitting where we have knowledge collections of the segmentation and 6 square prevents you from from getting enough electorate so it said it was important to to get to this order of magnitude for a 2nd edition OK and so that
31:12
thanks to this position equities were able to live in Rocklin greedy and which simply says that under a assumptions but if we defined the best service implementation by minimizing the use of it in their eyes for Italian where nobody is defined this way so you of constancy 1 Institute as did the Dow is simply the number of segments of the segmentation town and then there exists an an event of high probability on which of the performance of the final you get is a remains could almost the same as the best performance you can get with an estimated you consider I have to constants which is not even 1 and the remainder and so being very interesting thing is that in an hour more generous sitting because we consider it a very good victors we were able to recover a penalty which is quite similar to the 1 obtained by the right by by additional sites in France and 1 and and the revolution ascension for instance and we'll victims but so over the years the
32:24
horrible events simply can't be speech into 2 steps of the first one is computed by preventing providing the opportunity the optimal this column this criterion Over over segmentation with dissonance In the 2nd step and find the best number of segments by optimizing this benighted denies president and another in Baltimore and to the point is that that's the 1 answer to our concerns and there have been no and chosen from the top by use of what we call the stroke ballistics they feel that you get with the program greater functional surveillance at the underdogs and I know it's as far as I know it's a really difficult thing it's still progression the affiliation with the ones will that do the work of the Legislative Yuan ultimately and actually from when we knew oxygen as a result we get C1 and C2 we get Bobby pessimistic views of of C 1 and C 2 from Europe because the only life from conversion increases and the constant or not it buys from the head of the various locations in the city as was hot with the real thing I would not turn up because this structure is 6 is based on a protocol arguments which says that in some regions it's is it works quite well to be able to to twist Maine is a console so is not really a difficult problem but in practice so if we compare now that the performance of the open criterion we optimize which is back to the tune of the true risk rent you see that innocent either 1 which is the simplest missing just 1 both for Lucian counterpart and the family can I work in the captures that it is not that bad to be true to to capture of did you know the true risk and communication is all it is the same and you absorb out the exact career frequencies for all the changes that will change .period when we choose to do hot is quite good and that it is even better if we allow a small mistakes an issue however ,comma change .period and you consider that it is true ,comma reissue we now a small airport for incidents of Free position and then a 3rd of these frequencies that blow up to the 70 % so
35:30
it's visited Singleton also true for the 2nd scenario went over the solutions of the same Singaporeans give you see that the
35:41
dissolution of the heart is 32 bit spread more spread by the press 1 that there is that the program actually and I was still dealing
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with to data you just got quite well but they said that summarized left providing and then described him and an agreement to detect changes arising in the dissolution of observations we it is relatively efficient and it it is fair to carry grounded thanks to position great general or so or Akron equity which provides no innocently granted on pursuant to this agreement and arrives to deal with the victims and so a search of data such as graphs and so long as long as you can define camera on and there are also many open programs some of them for instance the 1 I've already mentioned by reducing the inscribed time of course you can use approximations of instead of ordinary programming agreement which warned that the made the key thing is that you that's the most important thing is to to for ways to quantify what has been lost by using this approximation and it is really helpful from a statistical point of view and we have always said to investigate France and between Pietersen Kummer and the type of changes were sensitive so if it did it depends on the kind of change arising in the 1st or 2nd moments aura it in higher moments of the distribution and perhaps of a given country and that's the the best for all of that simultaneously even Jesus which discourages because of injury characteristic of our sensitive to any change in your chair to a change arising in any momentum distributions that's not in the same way if it arises in the 1st went on sale in the higherorder moments OK and the last appointees to work that there is a very slow so you mentioned the Strawberry sick and I had to say that I have to say that using that induces an additional competition and cost because you have to extraordinary sentations of 2 D and as you see you in the Commission complicity or complexities died in time and space or during the Mac's so it's whom portent of future program so it is it would be a good thing to be able to revisit this will proceed to sight competition resources available for investment space and the wife a missing the entire countries thank you if images of women in London the I have a problem pulses of Holyoke Hospital this music at home you could use the taxi or the chairman of the good whereas if you have kids you have the rumors that the of me but different overruns that included estimates that of get the thing of as this and new start to use the new position and thanks to his property the addicted to kick just before before that yes when you
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have a camera which is characteristic than any changes between between solutions induces a changing the monuments and so issue it as years a characteristic well issue of the change between these 2 institutions you can you can bring in pink it or you should be able to take this change in terms of the monuments some amendments should not be the same otherwise the company's defined under Canada's characteristics the new head of the different too forces not were that depends on what it so that the new Toro II is different From the new J. stock because of furious says about but actually when tonight the things so this 1 and this 1 obviously depends on the quantity of U.S. military reviews and so on ensuring that the region's issue of various has a very small number of solutions and new hearts this administrator this quantity and this 1 should be very similar and Sosanna event Yukon detect any change but it's only a matter of personal administration and would say this this an easy getting here also have usually been uneven with you guessed it :colon all the chooses to do so but of course it depends on the bandwidth but in the area not in the book is considered and the dependence is not so strong because except issue choose very very very bad bandwidth because it's for instance I would say you that tend to have a voluminous for and an aura and the other extreme tend to power 10 for instance would accept such extreme administrations it would work that's bad and of course there is enough to more than but or I've version you know are not provided with an optimized by and we of only continent abundance which provides us with another bad results so far is that some money room for improvement but if we were able to precisely which was banned bundling of Internet analysis will eventually have to sign directions and I think you know we don't have any elements to answer little so if I show you recommended toward the we we say that the norm which yes misnomer here it depends on the counter the better because it is the sum of 2
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defined this way sorry for this this way it is this norm which is the sum of Over the positions of the norm normal of the fight in the architecture Spanish age into this over on the calorie of choosing so I I I say that
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because the looking to invest solely
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this is 1 and so that the
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benefits "quotation mark appears here and there here so it's difficult to say that there is a good for instance you knew you cannot hope open optimize are in saying that I would truce H survived to this Applebaum is the smallest as possible because dependence on this quantity and sent to His father was 1 of the restaurant all the units using GTE at least the continent's With the system of the of the head of the Palestinian of here this was the the world so we have to understand that the a recent history that information was given to the all the best of what was going on the useful changes which is 1 of the things that I did there no we will you the it this major foreign source as of course some of the analysis is that would be really more intricate so parts it's possible rebuttal Weaver many technical additional technical adjustments the light thing world consider changes to his office and going home .period announced last month that I was going to be who is still in use relations following you choose the units this approach for example almost In the end he each should be most of them less than the full number you will see have the support end of 1 of the most reliable informed that him that parliament all along the burning of the agree that
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actually I would say except in very extreme situations because the terms of Denmark's is really influential in that state I mean when when I would choose from a date from the time when I would estimate from the top of the consul C 1 and C 2 by use of shoppers the strawberry 6 it's difficult to explain that in a few seconds but yes it says roughly speaking that you can't predict that this type of independence respect to dimension of the curve Of the sanctions which is the chorus and so you can you can we've had probably taken say what should be the duty of the empirical risk as a technician will and all artist is large enough and Is is with this to stop structure stick your exploited the the knowledge of this shape of the Virgin political risk should have the 2 of them and you will learn from this shape you compare this shape to observed the shape of the empirical risk and the comparison provides you with 2 constants and the fact is that this comparisons depending on the length of the you do you will use to make this comparison and if it's too small you really really bad estimates of his 2 quantities and between degrades your results so in practice in our simulations I've chosen D Max according to 100 and assuring really remember just always according to even so that it already provides a really reliable results with a lower over use of the max but it was only my way to work 2 To be the new but it is clear we wanted to thank you mission .period and at the beginning of the world with water and energy here so there is no way I would say it means nothing because there is a again which is in this penalties and so you can decrease the 1 but between critical the other consumed here so it's in light of tradeoffs and and even if you don't want is equal to 1 indictment saying we leave about to the overview of the procedures of the more because there is Logan here and it's not very clear than have a yet results in terms of year however house said this this is not getting the from from a practical point of view of the Association of results from the fury that I would say there is a pledged to the needle of a which is a good vise by a job you ensued on music is exactly that and so he has the consistency results to show that this procedure of the same agreement or almost same that provides consistency for to heart when snitching to stop the truce inclusion in the office this is something you already mentioned this sense of beating the agency said guarantees says this I would say that in pretty but there is no right to Clinton's form bit of Dorfman the probability but this union you this is the home of the only recruiting young soldiers of the the number of for instance because of the enormity of the this moment the question of the moment it's like so there is not 1 of the men and women questions we said that the that fumed as