so today I'm going to be talking about asterisk individual and so this talk is really broken into two parts one about the Milky Way and then what about you and this might afternoon I know everyone here is an expert astronomer so you already know about dark matter and everything but you know it is Friday afternoon so let's go through the basics so dark matter we know it's one of the biggest mysteries in astronomy we know that of all of the matter in the universe it should make up more than links every person no matter so there's still e being dry representing all of the matter in the universe all of the dark matter and one reason we're interested in dark matters because we believe that every galaxy in the universe is sitting with an informed are matter halo and we also see evidence for dark matter halos and clusters of galaxies right and when we look at galaxies in a which when we when we look out into the universe and map all the galaxies there we see this hierarchical structure right and we cannot recreate this structure in the universe unless we include dark matter interesting and so there's all these lines of evidence like dark matter this but we actually don't really understand exactly what kind of subatomic particle is or exactly what its distribution is its spatial distribution is in individual galaxies and so this is why I've been interested in estimating the mass of the Milky Way because the mass of the Milky Way is dominated by its dark matter halo and if we can better understand how much dark matter is there how its spatially distributed and that could tell us something about the nature of dark matter and how it affects galaxy evolution so I'm definitely not the first person to do this people have been estimating a mass but milky way for years and years and this figure I guess is about five years old now but it's a collection of a bunch of different mass estimates of the Milky Way are all normalized to M 200 and you can see that they really don't agree with each other right there's and not only that but their error bars don't always overlap so and that's what the Milky Way is incredibly uncertain and there's a listen Friday reasons for this one is that people use different methods and Ava and they make different assumptions and they put so just a really really difficult problem to tackle we just can't even see this they approach that a lot of people take and I take as well I'm to use what's called kinematic tracers so use their objects orbiting around the galaxies that we can see they might be halo stars it may be a stellar stream in this case I've highlighted globular clusters States if we look at their positions and velocities then there are these maths estimators which can take some sort of weighted average of the positions and velocities might be sort of assumption about the velocity dispersion of these systems and then you can estimate okay but there's lots of challenges with using these kinematic tracers and that's something that I focused on during my PhD and after is sharing so one of the challenges is related to the fact that we do not sit at the center of the galaxy right we're sitting in the dis somewhere maybe in between spiral arms or in one of these little Spurs that's still kind of uncertain we're in the disparate so we have extinction and all sorts of things and not only that but because we're at this position when we look out at say a globular cluster and we measure its position and velocity we're measuring it from our own reference frame our heliocentric and our mathematical models are a much simpler in in the galacto centric reference frame so we have to transform these measurements from heliocentric to Galactus and we can do that if we have the full velocity components measured and we can assume things about the solar motion and the rotational disk and so on you can do that transformation but this is difficult when we have incomplete data so for a long time pre April 2018 we didn't have proper motions for about half of a lot of the clusters and so that makes things more in certain people with either use mass estimators that don't use any of the proper motions that we know and use only line-of-sight velocities or they would throw away any data we have that's incomplete and only focused on the data that was so in either case you're throwing away some of the information that we have another challenges associated with this is that there are measurement uncertainty right and so these measurement uncertainties they can be a function of distance and propagating these measurement uncertainties from heliocentric to galacto Center can be a very nonlinear problem you might not understand another assumption is that we have to assume something about the population so we might assume that most of the tracers are following radial type orbits or most cases are following tangential type orbits or maybe there's some isotropic mixture of the two and the parameter that we usually use to modify this with the velocity and I saw to be parameter rate what looks the ratio between the tangential velocity variance in certain versions so what people have done in the past is they'll assume a value for beta and make better masses but it's been shown that this can lead to Isis okay and the another challenge that I always like to highlight is interpretation because there's obviously different methods that people use not just the kinematic tracers but people always they also report the masks at different distances from the center of the galaxy so on this plot I have mass the mass within some radius in the center and here's the radius horizontal and you can see there's a whole bunch of different mass estimates here and even at 50 kiloparsecs there's a lot of different measurements with different air bars and some of them overlap and some of them don't and as you get further out in the galaxy it's more and more advanced and so how do you interpret these kind of results for these different studies right however what are we supposed to make sense of the total mass of the galaxy we were getting such a variety so what I've been interested in doing is taking a Bayesian approach so I always like to include a theorem here before I'd Rachel this is the posterior distribution of the model parameters given your data and I could go through the regular introduction about Bayes theorem a lot of people here so I'm going to take this opportunity to tell you about a teaching method that I've developed to teach easing analysis to students and this uses M&Ms so the basic idea is that you've introduced Bayes theorem to your students you say here's the posterior distribution is the probability of your model parameters given the data data is your vector model parameters Y is your data and this is proportional to the likelihood times of prior distribution and so the likelihood is where you can put in your model about how you think the data is distributed and the prior distribution is reading from a prior information you might have about the model parameters in that distribution and so with the eminence activity what you can do is ask the students to find the posterior distribution of the percentage of blue M&Ms that are made at the factory using Bayes theorem and this is fun because usually students have eaten M&Ms so they have prior information okay and it's also analytic and you can set this up analytically so you can help the students long as he theta is the percentage of blue M&Ms produced in the factory that's the parameter we're interested in estimating and then imagine that gave you some data that our M&Ms from one from only one package you have a small and then you ask them what's the likelihood what's the card information so you can talk a little bit with them we know okay well when you draw an M&M from the bag it's either blue or it's not blue right so what kind of a likelihood might you want to set up for this binomial exactly so it's a binomial distribution okay and so you can talk about the parameter of the binomial distribution here is data and then you can ask them about their intuition about the percentage of clues and you can try to draw out well how many colors do you think are innovative how often have you seen a blue come up in the past or so you know you talked with them about this and then get them to sketch out what they think the percentage of blue M&Ms is it's kind of a probability distribution and you say okay but what you drew it doesn't have a mathematical form how can we make it Mathi right and so you can say well what's a conjugate distribution to the binomial under your car and that's the beta distribution so after they've done their sketch then you say this is the conjugate prior it has hybri parameters alpha and beta right and so I want you to find alpha beta that match on your sketch that you so this is this is the prior that my classes chosen oh is that when I was at UWA develop this exercise and the class agreed that alpha equals two and beta equal nice represented what they thought were the percentages so you do want this and here I like to hide the fact that I'm actually going to get Momentum's and at this point you get some so here's a posterior distribution you can also ask them once you simplify the likelihood times the prior what what's the kernel of this distribution now right and it takes them a minute but they usually see this is also a beta distribution because you can see there's still the chunk minus one and then this chunk limit so it's still beta distribution is analytic they already have the code to the beta distribution maybe just need to read analyze it so they open their data and they get their results and then they can plot the posterior distribution and so I get this to my class and I check out the first ten M&Ms in my bag to get a really small sample size and four of them were blue so this was the prior distribution and then the dashed line is the posterior distribution did actually change but one thing then then what you can do is collect all the dams in the class and show them how the costume changes when you have more data and so this is the cross posterior distribution and you can see it's much narrower and it's centered around 23 and what's what's interesting here is if you were to naively just look at this and say well it's 25% if I just did but it's in what we actually estimated was more like 22 or 23 percent and that's because the prior is having some influence it's eating into a counterpart so there's a surprise twist and this is my favorite part of the exercise there's actually two factories of mmm that make M&M and they make different distributions of colors so I don't know why New Jersey they make about 25 percent pseudonym and Tennessee makes about 20.7 so you can then do is ask them predict which factory or eminence and so some of them will look on the back of the package New Jersey but not unless you not where the code is the code is actually written near the barcode you'll see either H K P or Co V and so if you look at the package you can see there's the HPP and so our M&M that she did come from in New Jersey so it's a fun little exercise and I've done it a few times with many different groups and they don't think it's funny so I like to share it and also after doing this I was talking to some physicians so if you should write this up as a paper okay and there's my github there's some example code and everything feel free to use this stuff and the emails are given question um watch out for the Canadian on this okay so back to the science what I'm interested in for the Milky Way was the posterior distribution of the mass right and so to do this we can't write at all analytically like we can with this another example so we sample the posterior distribution using some Markov chain this is just a 1d example but what an MCMC algorithm does is it walks through parameter space gather samples until it converges to a distribution of that parameter that is stationary and so when it's you believe it's converged and that stationary distribution should be proportional to the true posterior and then you can get estimates for your parameter and answer though with the Milky Way I use what's called a hierarchical method and this is to overcome these challenges that I mentioned before so one was including measurement uncertainties in a model so instead of treating the positions and velocities is known quantities and not treating the best parameters in the model we can also incorporate incomplete data by marginalizing over the components of the velocity that we don't know so you can use both of these and the data simultaneously you can also estimate that velocity and watch would be parameter it'd be enough to make it take some assumptions about velocity and I started people completely and then in the end we got a posterior distribution for those model parameters which we can use to estimate not just a single wave mass estimate but a cumulative mass profile for them okay which I hope can improve the interpretability [Music] so I'm going to this again in pictures so you think of this hierarchical models having a population level and then having an individual level and then a population level so at the individual level every globular cluster has individual parameters one for the true position and one for the true velocity and then at the population level the prior on that likelihood is that all of these globular clusters are following the gravitational potential and so as a population they share the spatial density of the globular cluster population they share the parameters for the gravitational potential and the velocity and I soccer and then of course because we have parameters for our gravitational potential means that hyper parameter hyper priors on those parameters as well and so this is where we can put in bounds for the model parameters maybe some of them have unphysical meanings that we don't what that curve so this is really how how big hierarchical models that just to go through like the math a little bit we're going to have a likelihood times the prior times by Hitler prior and we are assuming that at the individual level but a lot of the clusters are independent so we treat the we treat the true positions with velocities as unknowns so in this case this is just one example the proper motion measurement is drawn from some normal distribution centered around the true proper motion and constrained by the measurement uncertainty and then the prior on this we use a distribution function of energies and angular momenta that's in the galacto century reference frame and so then we can pass this function H that transforms our parameters in the heliocentric reference frame to the galactose enteric and then use this inner distribution function together probability and then we need suburb site so the parameters for the physical model there's four of them we're using actually really really simple well it's a power law for the gravitational potential so there's a scale factor by not and there's a power law scope gamma and then for the satellite population or these globular clusters we can also find a power law but with a different slope alpha and then there's the velocity and I sought your feet rabitor or the globular cluster competition those are our four parameters remodeled and we need to set prior distributions were those model parameters and so what we have chosen is the uniform prior curve why not with found a normal distribution for gamma gamma gamma distribution for alpha so in the end we're going to get this posterior distribution I'm doing Markov chain Monte Carlo algorithm which uses some techniques that I've brought in from this disease literature so we're using an adaptive tuning algorithm and so what this does is it runs the chain for for a short amount of time and then looks at the covariance matrix of the chain and then updates the proposal distribution East and then you run the chain somewhere and you iteratively do that it will actually help to tune the chain a lot faster now we also use what's called the our hat statistic to moderate monitor convergence so we run multiple chains in parallel in another and then you basically what our hat does that looks at the different looks at the compares with variants within each chain to the variance between the chains and finally we use a hybrid gift sampler and this is because we have a bunch of globular clusters every global cluster has a bunch of parameters itself so we're dealing with over 500 parameters and so using a hybrid gibbs sampler is a more efficient way of sampling I can go okay so on the development and implementation of this page a method is a series of papers and in the end the you know you see what's the mass of the Milky Way so this is our our math rm200 measurement it was 0.87 times 10 to the 12 solar masses and this year is the 95 percent of credible region and then we also can amass an estimate for our 200 days albinism because we have this posterior distribution we can also get a cumulative mass profile so using that posterior distribution we can calculate you can compute the posterior every-every radius every distance from the center and then we can look at this and say okay what is the mad what's our mass estimate within 125 kiloparsecs you can get a median or a mean and get credible retailer plan right away and it also enables us to compare how our model is doing compared to others and other studies and hoping that this provides kind of a more coherent picture of where we aren't understand of course this is this is assuming the model that we have assumed where where is exactly the data that you're using like in radius yeah no that's a great question so this was the globular clusters and so a lot of them are within 50 full of parsecs or so yeah so that's why it's more constrained there it is there's a couple of those clusters out here but so with that when you have things like an esophagus I imagine that might be to families of course I didn't he gave me CD that are theorized and then a few globular clusters are being full again with some of the Moores and Mike represent a very different set of orbitals and statistically might be quite correlated independent yeah that's an excellent point and so we did we did exclude those but we didn't consider you know if some might have a different velocity um so this was actually a good segue into the next expedition it because I thought that's answer and everything and I was kind of like okay but I don't know the true mass of the Milky Way so I would like to do some flying tests on simulated galaxies and my collaborators Ben Keller he does these hedger dynamical simulations of Milky Way galaxies and he had 18 galaxies they were all very different they're complicated definitely more complicated than the model that I'm assuming and so we thought it would be fine if he just gave me the kinematic information of globular cluster type objects in these and didn't tell me what the masculine logicals like are they cause multi-position if it's a much - so um they can't create globular clusters in these kind of simulations yet so what what then did was he went through in fact these globular cluster analogs so each particle has a mass of about 10 to the 5 solar masses which incidentally was actually pretty close to a lot of the clusters they reach ones that were low metallicity and they were really old so our hope was that they were kind of representative and then heated the simulated data which were the galacto centric positions and velocities and he also removed ones that were in the disk and then I took these and I transform them into heliocentric values and also created mock observations so I removed 50% of the proper motions also introduced some observational error in both distance and velocity and then we subsampled them so that we have a number of clusters that were too similar to are actually and so I did this in the the very first galaxies I looked at I I got this prediction and I sent it to Ben and said here's here's our very own max estimate and then this is verbatim when he wrote back he's like look like the method works well shopping well actually hear the numbers direct from these simulations and they matched really well and I was super surprised because the model we're using it so simple and these galaxies are actually really complicated so I was surprised by this but if we look a little bit closer and look at the cumulative mass profile then we can see that it actually doesn't fit the data that well did you test how important those assumptions you made earlier were like you just a certain subset of the dogmatic particles I stood a contestant on your testers if you just pick something else you have done BOTS verse and this question yeah we didn't look at other particles and yeah that's something we do [Music] yeah so so here we can see um even though the barrel mass was estimated quite well right you can see that in the inner regions are the model is having trouble explaining the data so the red line is the true cumulative mass no public right or a busy and so what I think is happening here is that maybe the gravitational potential doesn't actually follow a single power law and so it's making some compromises between the information has in the outer regions at the inner as we repeated this for all 18 galaxies simulated galaxies and the true estimates are the blue diamonds here and our sorry the true values of the blue diamonds and our estimates are the purple dots with 95% and you can see for the most part the true truth values do whywould in the night regions although they seem to be somewhat underestimated there's a couple exceptions this one and the one over here this is the this is the worst one but this one it turns out had a companion next to it that was very very close and so we were sampling tracer particles from that companion that was like in falling so so we did this one test and kind of give me a little bit more confidence in what we're doing but also um gave me maybe a hint that there there could be some bias in our nots estimates that it could be a blower who'd be giving a lower process but it's not conditional on the data that we chose particles but we did this because we knew Gaia was going and I wanted to know how guy would do once we did have proper motion measurements for a bunch of globular clusters so April 25th 2018 and I of dare to a release happened and so we reran our model with the complete velocity information for the globular clusters and got a new assessment and actually it's lower than our previous one and I think part of the reason might be because but what is interesting actually is that when those guys even came out there were a bunch of papers about the master just a bit more now as well but a lot of people reported the mass within about 30 or 40 kiloparsecs and they're all around here and we all agree with one another even though we're using different methods on it different approaches so I think that's really encouraging that having complete data in my best constraint and it kind of makes the case to get more data but still so I talked to leave up globular clusters so that would be the blue blue orbits drawn on this diagram but there's also of course dwarf galaxies and so this is what I've been doing currently eww I have some undergraduates and working it so well I was at UW moneth has an undergraduate so she's looking at the dwarf galaxies proper motions from the Gaia data and using my method to estimated mass we're also collaborating with Morrie Erik Robinson that's all they were working with them because we're using the fire simulations to help set our prior distributions on the spatial distribution of galaxies okay so this is super super preliminary so but it does look like when we look at the dwarf galaxies we're getting a much higher so that is using only the roughly 12 dwarf galaxies not measured by the guy collaboration but there's a bunch more so here's an example from Fritz 2018 if velocity measurements from 39 more galaxies so now we're looking at these and these ones have a bigger range of bigger spatial range so these are this is just to show kind of their spatial distribution we have the proper motions on the vertical axis distance from the center of the galaxy rasaan told the blue dots are all globular clusters and the mid diamonds of the galaxies so they are really probing much further into the Dark Matter halo and one question might be will where you know which work galaxies are going to equilibrium with the Milky Way which are not and is the velocity and a such a fancy constant for these galaxies so what Riley had all have done is they've they've already started this research and made a cut of 100 kiloparsecs and looked at the velocity anisotropy for Murph galaxies beyond 100 and then within oh yeah [Music] and so this is what rally it all have found and that's so the green line on the right here that's the posterior distribution for beta if you only look at the birth counties to be honest it's quite different than a people and so what I think is interesting here is how would this affect our assessment so some of those four galaxies were quite far away right so at some point I guess that's to jointly model it together with the Andromeda galaxy's parameters for in analyzing most common things look at the Milky Way yeah yes it's it's still much closer to the Milky Way but I was thinking that eventually you'll have to think of it as the sum of the two potential problem just yappity the LMC will matter first yeah yeah so and yeah so so one thing that we're doing is asking how might these different velocity and I studies actually affect our math assessments so I have couple undergraduates at the University of Toronto they're doing sensitivity analysis so one is is using this cut that Riley at all excused and it's getting a max estimate for those women outside of a hundred Zander and then Kaplan is going to be doing like a jackknife analysis to see which two workouts [Music] and there's lots of other areas to pursue with this kind of gives me model in terms of the Milky Way I think ultimately we want to be able to combine all different types of data at the same time so that kind of long term who I have so they within some distance from the center galaxy maybe you can assume that everything has fallen with no central Milky Way but they have different spatial distribution separately also I'm interested in doing Bayesian model comparison tests and averaging and because in the literature people are using different models and I don't think there's been a really thorough investigation and then there's also transferrable applications so using this method to estimate mass distributions of globular clusters or an elliptical galaxies or and I just started working on some simulated globular cluster data wait okay so now we're going to change gears and talk about teller stellar time to me and what I hope is an improved method for tender man astronomy and this is going to be an in context of Astro seismology um and you know that time domain astronomy deals with time series analysis in statistics sometimes they call this spectral analysis which could be you think but that's because they're talking about the Fourier domain so the spectral analysis part of it so if I say that sorry I don't mean like spectra okay so just to start from basics again we know that our Sun is a medium to low mass star and we know it has these different regions right and it's a we assume it's in hydrostatic equilibrium so there's radiative and thermal pressure outwards and then you also have a gravitation and then there's all these different zones right you have the convection zone the radiative zone at the core and these different zones create boundary conditions and so through the connection that's going on in the convection zone and these these different boundary layers you can actually set up standing waves at the start and so this is how you can get so I'm really just gonna talk about G modes in this chakra pressure mode oscillations and what's great about these these special waves is that they can actually measure them then they can tell us about the interior from the start so people have done this with a sudden they look at the Sun over time collect some time series data and then look for signals in the frequency domain right that has some periodic signals by looking at the power spectrum and the power spectrum or power spectral density is really just tiny with how is power distributed across the different and so if you have some strong periodic signal and you should see the power spectrum for the Sun has been well studied Aries there it is and what's what's interesting is the spacing between the modes and the fate of the people because the spacing between adjacent modes can actually tell you about the mean density of the star and the peak mode is related to the mass radius and effective temperature this car and so if you have a separate measurement for the effective temperature of the star then you have two equations with two unknowns because the radius and so you can solve for the mass and radius of the star if you measure new maximum and so after seismology is doing the same thing but for other stars and this is really handy because we want to measure these stellar properties that are interior to the star and test our theories about stellar evolution and stars that exercise model just commonly look at our red giant stars and that's because they have this very large convective zone that are that are thought to excite P modes or solar light ops oscillations and they stood help us answer questions like how common are these civilizations of red giants can we identify different evolutionary stages of red giants what are the range of masses radii age you know is there internal rotation what's the depth of the connections oh and all of these things could be answering to Masterson so we can look at stars over time collections time series data look at the subtle changes in the light and get an estimate of our power spectrum and measure these Astro psycho and it's a lot more difficult with stars that are really really far away there's also some measurement errors and we can't always resolve the star and so on so here's an example of a power spectrum of a red giant star and you can see there is a hint of these solar light oscillations there's also a lot of other noise going on and people try to model things like the background model any noise on this in your time series there's granulation on the surface of the store all these things so these challenges investors face terminologies that we can't always resolve the Stars surface another challenge is that the Classical period economy is a poor estimator of the power spectrum and then of course our data in astronomy is often I mean they sampled at times we can't grab all of those time series analysis methods from status leaders a lot of them work for you so why is the classical pair under before estimator of the power spectra and I'm going to show this through an example that I didn't come up with it's in a book by person but what it does it uses an autoregressive process so imagine we we have some some data XT ok and every point XT depends on the points before at XT minus I and this autoregressive process via order one or two or so on you would say an autoregressive process of order three then every xt would depend on the three TS so if i generate some just a toy signal from an autoregressive process of order for so here's my time series data there's a thousand points and then and this is evenly sampled data the true power spectrum of an error per process is known analytically so you can let you solve for that and it looks like this Batman so you're like okay that's the that's the true one and then if we take the power spectrum of our actual data and look at the estimate of the power spectrum that's good but we actually compare it to the truth you can see that there's some bias and so this is one problem with the periodogram this is a biased estimator of the true spectrum when n is small um and the size of n that would make it unbiased is unclear and depend on the process that you it is asymptotically unbiased so as you collect more data so here's one rate generated time series of 10,000 points you can see the by starts go away so that's it is asymptotically unbiased but it's also inconsistent and so what that means is that the variance of this estimator does not go to 0 as n goes to infinity so here's to show that whose love carry out a gram estimate our spectrum estimate for fifty thousand points and you can see that there's a lot of noise and hybrid so these are the two main statistical issues and these come from something called spectral leakage these figures are found on tape and/or classes paper about cargo pair under so in this example we have some time series that is occurring in nature okay and it could go infinitely off in both directions okay if we take the Fourier transform then this is what that signal would look like in these perfect sites but we don't observe objects in space infinite or an infinite amount of time right our telescope is awesome and then it's on again and then we turn it off again so you have some window function that were actually looking at the data so in the time domain we're getting a point wise product of this time series right but the thing is is in the frequency domain this sort of box or hat function in the frequency domain is a sync function so you can see it has all these Wiggles as you go out of the center and so a for us product in the time domain is a convolution in the frequency domain and so you're going to get this spectral leakage that leaks power into adjacent sees and makes your power spectrum estimate Morrissey so what people have done to try to overcome it is they'll taper the data and tapering the data means um you're gonna involve your multiply your data by some function that's going to down wait the discontinuities at the end of your window function and so there's many different tapers and there's a good paper from back in 1978 by Mike Harris and they look at all the different tapers point and compare their performance so I'm gonna show you just as serious as examples but you'll see the window function on the left and then it's Fourier transform so there's a triangular window the Hamming window there's also one called a hand window and they're different this is a two key window you can see days all might have advantages and certain applications over others um but there was a big breakthrough in 1982 by David Thompson he introduced the multi taper estimator he did this while he was at Bell Labs and the story goes was that his supervisor came to him and said find me the best keeper and it turned out that the best taper was multiple papers so this is the multiple papers that you use come from the special set of functions of this discrete prolate spheroidal sequences or DPSS they were discovered by slappy and so they're also called the slippy ins and what's special about these is they come in a bunch of different orders so the lines the first order second order and so on and each order is orthogonal with all of the other orders so what you can do is you can take your time series data and you taper it with the first order and then you get an estimate of your spectrum and then you take your time series again and you taper it with the second order you can get a second estimate your spectrum and so on and because all of those tapers orthogonal with one another that means that all your estimates of the spectrum are independent and then you can multiply them together and get a better estimate of your department and so by doing that this reduces the specter leakage between the GSM frequency and it also reduces the highest area and so this became a super popular approach in signal processing areas but mostly in engineering where they you know have controlled experiments and then have even I think this is why it hasn't really shown up in astronomy and so if we go back to our air for example right this was the estimate of the power spectrum from the regular standard periodogram and then the black line here is the multi taper estimate you can do it you can see it does a lot better at estimating like it said this is for regularly sample choice and we have unevenly sampled data what do we okay so here's an example I need data I start one there's gaps sampled so it'd be nice if we were able to use the multi-tier with this type of data exit look for Astro size but what I struck what we usually do in astronomy is we'll use the long Scargill spectrum estimator because this doesn't you deal with unevenly example of x the lungs cargo is really is the periodic gram for unevenly several times and it has some good properties so it's invariant under time translation and it is also equivalent to just doing a least square least squares fit got a sinusoid to the data so if you're looking for just one signal it's great but I have some not-so-good properties too and when is that it's retro leakage is actually worse than the periodogram and identifying anything other than one dominant frequency can be really difficult because of this so I'm going to go back to Jake bender classes class students test again imagine we have this is a it isn't even a periodic signal but just like a maybe a transient object or some technique we were tilikum the frequency domain would look like this but if we observe this object it's very unevenly unevenly sample times then when we do the point-wise product this would be essentially what we have our window function I'm sorry idea would look like this and so when we go to the frequency domain and we're now convolving rate the Fourier transform of this window function with that signal all of a sudden we get a really nasty looking around and you can see there's all sorts of heat that heats that you might mistake for being intrusive and I believe that Scargill actually warrants about this this reason this is really useful if you have a single frequency but if you're looking for might not be the best so is there a multi chamber for unevenly sampled times and because if there was then maybe it would be useful for looking at things like the pasture assessments there is a generalized solution place from romance 1988 but it's really difficult and expensive to compute and that depends on the length of your data and all sorts of things so it really never got any traction I mean maybe now nowadays but one of Dave Thompson students here at spring furred for his PhD thesis he he came up with another approach to this so when he did was instead of calculating these sleds on their own because this really wasn't expensive on the approximated the sloughing weights with splines and then combine that with a lung Scargill to create the multi taper mom start global approach and so the result isn't optimal for some arbitrary unevenly sample time so you always want to look and sort of the window function of how you sample your x um but it's identical to the multi cheaper estimator if you have a regularly regulars that's a nice feature and it's really fast to commute so if I have this train series data um this was given to us white bedding and it's been reprocessed and everything because I'm not an expert in Kepler data so I didn't even kind of gave us give us this and um and so I went here and I said I think that your month it could be really useful for this so let's try it with the Argo so here I'm comparing this is the lungs garbled periodic gram estimate ash my transparent you can see there's the hint of these extra seismic mode and if we apply them on GT performance car go like this and you can see the the extra static moves a lot here when I saw this shows very exciting it seems like this could be a release and we've we've submitted a paper to AJ got the right food print is positive so we're just you know filling into the things that are everyone is Pattinson really and then [Music] yeah my hope is that this is gonna obviously help us has to make these primaries that we want to know right the spacing between and um in astronomy it's not just a star seismology right there's all sorts of time series coming and there's lots of periodic phenomena in astronomy like stone rotation by naming systems variable stars black hole collisions tillage bonds all sorts of things where maybe this would be useful I don't know I'm not experts in these areas but you are please there's also lots of archival and new time series data to explore with this method and of course we have busy we key transit facility and your ribbon I'm so very excited information astronomy into the twenties and having methods that art has to compute are going to be important in this era of large surveys there's also the comedian hydrogen and sexy not being experiment so this and they look at things that can reverse and stuff so they also have okay so I'm hoping that that take away my talk is that model uncertainty and the math allotted methodological approaches are becoming the limiting factor in astronomy and I believe all of you know that okay and I think that just introduces venerian collaboration really has a lot of potential because these things wouldn't have been possible without collaborating with statisticians and so it has a potential to improve statistically economy and make new discoveries but um it also can inspire new statistical methods I think on both sides [Music] of course translating between disciplines can be really difficult but I am building a narcissistic research team so if you if you know any students they're interested in graduate school or postdocs please send them my way and you know they'll have an opportunity collaborate with statisticians your scientists data scientists and astronomers at U of T we have some great Institute like them up there's the field Institute yes how did you do that like you do olicity orbit so we use ones that have been identified in the literature by other studies [Music] but again I can actually look you deeply I just this is up with these slepping wavefunctions there are solutions to where well Angela they represent the different energy levels original papers and you can use that to generalize and sub new functions right so that's the until right they have swear but it will get wolves right but if you have a different shape potential to different spacings actually those functions [Music] and I'm wondering yeah you essentially yeah so that this is used by the geoscientists Humane Society yeah right you've got your samples distributed around South Africa and you've got the condition and the spatial case they do that I wondered if this [Music] it could be the papers by Frederick Simons localize this unevenly sample papers in geosciences and dust this multi cape or anything exactly that's the same spirit yeah and I wonder if some of that way yeah [Music] but when also changing to play [Music] um I just was gonna ask ya just that including the LMC in masses if you don't include the elements either you're biased hi and I was especially interested in your comments on that given the preliminary work yes yes so that but that's one reason why we how did you ever graduate because I think it's been suggested that there's kind of a radius or the penalty made at which the LMC hasn't really had an effect yet but beyond that might affect on the other cork Nazis and so our model is essentially [Music] yeah and I have the same question but also just adding to that that you could probably do all different scenarios right I see how and I think it sounds like a very exactly like Matt [Music] actually [Music] yeah yeah that's definitely something so I think nothing that you could do is you get different posterior distributions evaluate a little bit of those models lead them accordingly and then I think that might be nice because then you could get a cumulative mass profile the Milky Way that kind of encompasses what all of the literature is saying and I'm guessing the error bars would be very very good but at least then you would [Music] [Applause]