| 00:01 | mm hmm. Mm hmm, mm , mm hmm. Mhm. |
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| 00:13 | Distance now. So so first I black enough since it was coming back |
|
| 00:21 | the end of last lecture. Any complaining about the rest so expensive. |
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| 00:32 | . For me it helps to have picture in my head of what goes |
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| 00:37 | there. Not just a bunch of . So this for me helps a |
|
| 00:41 | bit in terms of understanding for 8000 it means to them depending on what |
|
| 00:48 | choose restoration to get linear combination of that needs to stay in some kind |
|
| 00:54 | a vector space. Yeah. Ah the dimension of it is in this |
|
| 01:00 | a number of roles. Normally, I said, you have when you're |
|
| 01:06 | to do these squares. Yeah. conditions. Make a lot of observations |
|
| 01:14 | something. And we tried to use redundancy basically in all the observations, |
|
| 01:21 | out the super friends. Well, sort of the scenario early on in |
|
| 01:28 | examples of sitting right to measure the tensions and temperature or something. In |
|
| 01:35 | case the boy, this relationship between two variables and 15 Observations. And |
|
| 01:41 | you try to make this fit since , so the rest of linear combinations |
|
| 01:49 | and this space that is span by columns of faith and normally the things |
|
| 01:56 | are financial independence negatives have votes. it stays some space. And then |
|
| 02:04 | minimize this expression minimized at least squares names what's left between a C. |
|
| 02:12 | the thing they wanted. Why this is um basically orthogonal. You cannot |
|
| 02:22 | the terror by A. N. . I want to be the best |
|
| 02:26 | finance. See so that this thing minimized and that means it's kind of |
|
| 02:30 | talking about in space in victory and we went through um this equations and |
|
| 02:36 | got this normally questions start getting out . Are there just from manipulating |
|
| 02:44 | That was that. And then so thing I consult this directly, instances |
|
| 02:51 | will just give to this in the . But the advantage of Sharansky's that's |
|
| 02:59 | double symmetry. So it's kind of there are only represented the triangle of |
|
| 03:06 | because the other triangle. Yes. . And mirror image symmetry. And |
|
| 03:14 | you also have to work so the self factor too in computing space. |
|
| 03:20 | you sent to Alaska supported just go the corruption of the mission that was |
|
| 03:26 | But also the point is don't do . Um because of the condition. |
|
| 03:31 | then said one thing that's working okay do and reduced it a bunch of |
|
| 03:37 | . And that's the use also the that's this way of living solving linear |
|
| 03:42 | . Now it's pretty much in the . So that's why I've included inside |
|
| 03:47 | to remember. And then Alright gentlemen stick it into the eight comes in |
|
| 03:56 | AC on this thing here. What this guy here this because the |
|
| 04:02 | Is not a factor those two things . And then you can play around |
|
| 04:06 | the equations that I've got to be little bit in directed. But in |
|
| 04:11 | end So they just naively put eight x equals y. Um and |
|
| 04:18 | S. You are then you get our concepts equals Y. And then |
|
| 04:24 | fuel is this um for the normal that you get the householder you can |
|
| 04:31 | from the left. But if you and then students to other symptoms of |
|
| 04:36 | becomes identity and what's left is part sex. So I don't have the |
|
| 04:42 | to the right side. But it's trying to resolve best. No one |
|
| 04:47 | on dope, wow, Convert to on this one and the next. |
|
| 04:53 | the next thing was to do using right in their composition instead of |
|
| 04:58 | R decomposition of A. And that what is called immigrations. That's taking |
|
| 05:06 | of the fact that both you and are for a normal selections. |
|
| 05:11 | The beauty thank you is the identity the three T. is identity. |
|
| 05:19 | you're kind of unravel. What's Uh huh. This is where the |
|
| 05:24 | is basically this is the vector A X. Is a vector vector. |
|
| 05:30 | these are normally column vectors. When do this thing best you transform it |
|
| 05:36 | a row vector times common vector. then you kind of get you in |
|
| 05:41 | middle if you do that or take thing and we talked about, I |
|
| 05:46 | you get duty, You or your in the middle that is that that |
|
| 05:52 | haven't changed. And I think that's the beauty and the big question and |
|
| 05:56 | you can just start to buy it and get this thing. And so |
|
| 06:01 | was the identity matrix. You can it in. Yes, doesn't change |
|
| 06:06 | . And then they change their association now the investment have sigma best because |
|
| 06:12 | they're multiplied from the left by T. You get U. |
|
| 06:15 | U. At this identity one from right to be then you can be |
|
| 06:21 | . And besides that. So best this. Yeah. So then you |
|
| 06:29 | it's the work of change notation a bit called Z. Yeah. This |
|
| 06:34 | a matrix and vector vector Z. this is another director call. See |
|
| 06:42 | investor be right because this is the matrix and sigma is you should not |
|
| 06:47 | grants because it comes out of a is typically rectangular matrix. So it's |
|
| 06:54 | best is that the rank is equal the number of columns assuming the number |
|
| 06:59 | roles instead of larger. So that's there's some set here. And then |
|
| 07:04 | signals from zero. So it was first part from there Singular right has |
|
| 07:11 | non zero. And it becomes from trivial because there is a diagonal |
|
| 07:15 | Csk leads from the director of components and and sigma zero. The only |
|
| 07:24 | that's left is to seize. And not now the third day to work |
|
| 07:31 | our transformation instead you kind of do same first step here, but now |
|
| 07:36 | of doing to have cube and then you follow that thinking this is |
|
| 07:42 | this from this, it doesn't make . But to basically end up with |
|
| 07:47 | solution that I had a few So and then the purpose then was |
|
| 07:55 | minimize this thing for the the squares . So basically they try them too |
|
| 08:03 | . Um now see why is the thing you came from A So this |
|
| 08:12 | an city and what's not known is . That comes out of this expression |
|
| 08:18 | it defines the first but trying to this expression then we can recover forget |
|
| 08:24 | accent corresponds to it. That that what was on his next time. |
|
| 08:33 | yeah, Back in the derivative and it to zero standard versions than the |
|
| 08:42 | impression. Fc is simple dispersion between see a single around. So then |
|
| 08:51 | have to see and then you can yes, once you have to |
|
| 08:56 | So this came on what was given matrix A and they So then one |
|
| 09:04 | gets awesome expression for well, I the least worst error is in this |
|
| 09:11 | we've got some of the square and of these things are true an |
|
| 09:17 | Oh, so that is kind of SVB for the sport and stopping their |
|
| 09:29 | , what was how much someone's gonna born. So again, it's yes |
|
| 09:41 | this kind of expensive to compute it also gives you sort of nice |
|
| 09:46 | symptoms of what is remaining. And the problem is small enough than become |
|
| 10:00 | complexity or the work and STD may acceptable. And that's what he used |
|
| 10:05 | come to the graphic stand on the video shows. Yes. Okay. |
|
| 10:15 | I guess we suggest summarizing equations of next step and then there was this |
|
| 10:24 | . It is, um, I it's him do this by hand. |
|
| 10:31 | believe me and verify that yes, , the SPD. And as I |
|
| 10:36 | out last time this is the full , there's only two columns so you |
|
| 10:44 | use and reduce and skip this part skip this column, uh, affect |
|
| 10:50 | . So this is kind of doing work than you actually need to |
|
| 10:54 | Um, but that's what, I know if public again, these are |
|
| 11:03 | things and then the new was invasive the organization that we have here is |
|
| 11:09 | views. Oh, sorry. To east of there. But so in |
|
| 11:19 | case and find out what the seas and then we can find out what |
|
| 11:23 | hell is the texas are in terms the lease scaling of the lead actors |
|
| 11:29 | them. Saint jove under their So we're just plugging in the |
|
| 11:37 | So that's what I talked about last I tried to. Yeah. So |
|
| 11:46 | more things and then I'll talk about weeks. So, um, so |
|
| 11:57 | and talk about that. It was in terms of discrete number is going |
|
| 12:04 | have but prison majesties. So now can also do it with continuous |
|
| 12:14 | And then either way you measure the then is essentially for the integral. |
|
| 12:22 | the sum of errors, transitions kind an infinite song in the sense that |
|
| 12:27 | kind of integration, discrete points and adopted. So, so that's kind |
|
| 12:36 | simple difference in some ways. But think of this is no, they're |
|
| 12:43 | their somebody taking there's influential folks and of them. So now He is |
|
| 12:54 | . The approximation we want to find function and you have some kind of |
|
| 13:02 | functions the Gps can do within their to find something that and the dog |
|
| 13:10 | under the squares is too ah The coefficients for mm hmm. That's |
|
| 13:21 | my two, I still have the year. So. Okay, |
|
| 13:30 | So the other thing also, one also if my wants and sometimes that's |
|
| 13:36 | that the terrorists in the interval between and B are not necessarily of equal |
|
| 13:46 | in the application you have. so in that case function also include |
|
| 13:51 | function. That sort of weight how there are different points in the |
|
| 13:58 | wow matters to respect for others in same influence and we'll wait functional. |
|
| 14:05 | example, the one that was used your vote so otherwise the machinery is |
|
| 14:10 | same. Ah now that you plug and they want to have the |
|
| 14:21 | That I was for this expression You plug this thing in into this |
|
| 14:27 | . Um and then what you get then put the some out there. |
|
| 14:35 | deep functions on the outside and then have the city and then on the |
|
| 14:42 | . Thank you. Take this Take them for everything as usual. |
|
| 14:47 | the minimum and you got on the and yeah. Um these are nice |
|
| 14:58 | of diagonal and this is kind of away but before we get that. |
|
| 15:03 | this is just for every mm Okay function when you do unravel this |
|
| 15:14 | On the side so you get three the integral times the derivative respectively. |
|
| 15:20 | . And so what you have them this equation and it's very similar. |
|
| 15:25 | got the matrix you want for every that do seek to find the minimum |
|
| 15:32 | the rest of to minimize this So it's again, vineyard because of |
|
| 15:40 | way the G overexpose defined is linear . That's why you get this when |
|
| 15:46 | get an equation for everything right? each one of these places function |
|
| 15:53 | Each one of the coefficients. Here go basically. And system of equations |
|
| 15:59 | . Um and then if they were the diagonal pcs that matters, that |
|
| 16:10 | it simple. And then there's the that the jews in the book that |
|
| 16:16 | is the championship polynomial. That's the dysfunction. And for whatever reason in |
|
| 16:23 | book, they choose them to use as your waiting correction some of this |
|
| 16:28 | of their weight. The errors basically the um 10. was 1 -1 |
|
| 16:36 | one. So this is, it's of ended up not being very nicely |
|
| 16:41 | integral and he appointed us to respond zero but it's values um at the |
|
| 16:47 | points are much more heavily weighted than that's great function fundamentally they can figure |
|
| 16:58 | coefficients. Now they call them down instead of No problem. So we're |
|
| 17:06 | color just shows that if you have continuous functions, the only kind of |
|
| 17:12 | that happens compared to the dispute places some sort of place pretty much families |
|
| 17:25 | on this, this first continuous functions non linear and nonlinear is as |
|
| 17:35 | Not so same. So, some years have a good book so |
|
| 17:47 | something that is roughly as E. the X. And you have to |
|
| 17:53 | out what the buyer for accents welcome the time constant. The many examples |
|
| 18:00 | overtime or it's they came right through financial nation. Great. So, |
|
| 18:07 | clinton's also decreases in terms of making field. So they have a bunch |
|
| 18:15 | observations again and you think their behaviors the X. And some flavor. |
|
| 18:21 | then they want to minimize this Now, unlike anything we've talked about |
|
| 18:27 | far the relationship here between Y and . Is not linear because C. |
|
| 18:33 | in the experiment. So now we the derivative respect to see you've got |
|
| 18:40 | other question that is not linear. so that means it's kind of tricky |
|
| 18:51 | do. But one solution that we do and I think it's on the |
|
| 18:56 | side to cheat a little bit. But the achievement works on that for |
|
| 19:02 | problem. That's the news. It be signed that. Sorry. But |
|
| 19:09 | one looks at this, the if you take the log of |
|
| 19:13 | then all of a sudden the relationship the logger Y and C. Is |
|
| 19:21 | . So that's what happens. Science Stone said, oh, I |
|
| 19:25 | the law, natural law, the . And best now you have a |
|
| 19:31 | ship relationship between the longer wise and economic kind of use the machinery used |
|
| 19:41 | and we can figure out a good proceed. But it doesn't solve the |
|
| 19:46 | problem and solved a slightly different But that's sort of one trick that |
|
| 19:53 | non linear. Okay, And uh I think I'm in your case |
|
| 20:03 | that's right. Kind of use different to try to figure out what the |
|
| 20:09 | . Um So I'm just guessing the that this relationship between the observations |
|
| 20:18 | And and X makes sense. So it's kind of linear and a but |
|
| 20:23 | not linear. And so it's a of fall straight forward. This expression |
|
| 20:36 | the approximation we have for a So in the various independent variable that's |
|
| 20:43 | here and look at from the observation of the squares of errors and go |
|
| 20:50 | and take the derivative. And then get this set of heritage. So |
|
| 20:58 | particularly that illustrated? And the work ? Um It's fun. Looks at |
|
| 21:08 | first equation this time on the street working kind of solvent for A. |
|
| 21:17 | it's just all this for A. you got to get on the right |
|
| 21:21 | side, you have the sum of and on the left hand side is |
|
| 21:29 | sign sign, sign, sign, , square. So this is kind |
|
| 21:34 | A. From this equation. And you can do the same with the |
|
| 21:40 | equation and solve for A. And let's get this ratio between the So |
|
| 21:49 | means the areas. So the confines B. For which this equality is |
|
| 21:56 | true. Then one doesn't distinguish. then you can play it on whatever |
|
| 22:02 | what you want to uh choose these and error or something better. Uh |
|
| 22:09 | what the left hand side is and the left hand side is. And |
|
| 22:13 | figured out A B. For which are equal. And once you have |
|
| 22:17 | deed, then you can stick it . So that's what I'm saying. |
|
| 22:24 | like. Um And we talked about England also of significance that have to |
|
| 22:31 | created to find ways in that case reduce the impact of loss of signal |
|
| 22:37 | all linear problems and other things. at the particular problem at hand and |
|
| 22:42 | to figure maybe um, in this it was the same formulation but was |
|
| 22:48 | to the fact Linda and one variable then the other or in this case |
|
| 22:56 | thing saw this like a different problem hope that that's cool. So I |
|
| 23:03 | that's part of what have we got reports, fixed questions on these Squires |
|
| 23:13 | move on. Not. Mm So next thing was four years |
|
| 23:30 | So we'll see how much of this today. And then But then there |
|
| 23:35 | four years years next time. Oh, after the victorious areas that |
|
| 23:42 | continues functions. Next time I will about it's practically most useful. Just |
|
| 23:52 | fourier transforms and fast forward. so here is a little bit of |
|
| 24:01 | of words that people use. Will transform some twinkies? Not much of |
|
| 24:06 | idea. So there is something examples people over and that said years |
|
| 24:14 | The um and what we get in case crystal, Yes, Grayscale image |
|
| 24:26 | looks like the one period but no rings in it. Um kind of |
|
| 24:33 | . See a little bit on the the other rings. It pretty much |
|
| 24:41 | like a total knows. Ah but using fourier transform and a lot of |
|
| 24:51 | actually generates models for this image And in this case, in the |
|
| 25:00 | of the virus. So but mm . So, FFT plays a big |
|
| 25:07 | and converting some something that right, something. Think of work. |
|
| 25:21 | And there's the comments that it's kind useful. They like computing. This |
|
| 25:26 | a very nice application because the normal , if there's something like this is |
|
| 25:31 | increase the signal to noise structure but don't get such amazing image. But |
|
| 25:38 | you had to do it in this , just frying the specimens and you |
|
| 25:41 | absolutely not. Morning, the game is being played in applications of this |
|
| 25:49 | to figure out ways of working with noisy images as you can. You |
|
| 25:54 | , they want to ah walter or of what you're trying to communications and |
|
| 26:04 | . This is nice because the better number computer and the rest of the |
|
| 26:11 | . Oh, interesting for this Vargas was another one. So |
|
| 26:20 | today says here there are solutions. , this picture was about eight own |
|
| 26:30 | among those instructions. 10 to the m. So, on the spot |
|
| 26:43 | from, from a nanometer. So they do a solution for about |
|
| 26:53 | That's what they can look for 50 . Otherwise they are calling TF |
|
| 27:01 | and the department professor settles they This is and it turns out and |
|
| 27:18 | that case um what you're measuring. not or has ever been. They |
|
| 27:27 | get an image out of it Yeah, in verse dependence on this |
|
| 27:36 | observations are made in the frequency So we're trying to get something out |
|
| 27:41 | that inverse fourier transform. Absolutely. then the structure and then they will |
|
| 27:49 | other ones here an example from They also um, the property is |
|
| 27:57 | find the spectrum of frequencies that they do here. The resolution or the |
|
| 28:12 | of the transforms can be very So I have a The side of |
|
| 28:20 | things trying to eat 200 pixels for kinds. They work with signals that |
|
| 28:27 | billions of 4th. So we talked always also try to emphasize scale is |
|
| 28:37 | to many context and in this case definitely important for these fellows the harder |
|
| 28:45 | and the And these are the work other things to do with weather and |
|
| 28:50 | situations and also partial differential equations. turns out sometimes it's easier to find |
|
| 28:58 | solution for your space and the geometric china's space stuff. Okay, |
|
| 29:07 | So this is the opposite to make observations and frequencies places that that's you |
|
| 29:12 | like to see a picture and try understand. So this is just sample |
|
| 29:19 | lot. Oh, you selected So it's important together show the example |
|
| 29:29 | fulfill the strongest the position of my . That tends to be the problem |
|
| 29:38 | . one algorithm that is most likely it's used in similar processing all the |
|
| 29:46 | sciences, everyone. So it's But a fast one. And not before |
|
| 29:55 | transforming to talk about today. What want to talk about today is a |
|
| 30:01 | for this kind. So and then general questions transform. So first thing |
|
| 30:14 | that we have to develop sort of for using four years serious and there's |
|
| 30:19 | bunch of equations on sites but essentially is using um big functions and some |
|
| 30:31 | the times. They are very powerful the sense that it was use this |
|
| 30:38 | mentioned in mpeg standards and the apex early on because turns out from any |
|
| 30:47 | type someone to do compression on. didn't take all that many of these |
|
| 30:53 | . And the cases that we talked . Yes. three functions of the |
|
| 30:59 | function then you didn't need many of that needs to represent the signal of |
|
| 31:05 | image. Just for the few provision the different basis functions so that you |
|
| 31:13 | AIDS and the bees and that was of enough to the present. And |
|
| 31:19 | as strong as ever standard and receiver that this is a proficient for their |
|
| 31:25 | For what three functions then you can that was and we'll talk about the |
|
| 31:36 | problems. Mhm. So this is I like most of the examples of |
|
| 31:48 | talk about. So This is basically for your series approximations to look |
|
| 31:57 | That's the so f previously the function tried approximate. Then we hunted up |
|
| 32:10 | and G. Is the one racist in this example or there's a certain |
|
| 32:20 | and now it's a continuous problem as in one of these squares. So |
|
| 32:27 | we work with an integral instead of . And then given that there will |
|
| 32:34 | strict functions when he and in development in center represents one period minus 55 |
|
| 32:46 | . But This is the chosen a interval around 0.55. It's one period |
|
| 32:55 | the bass frequency. Go on. thing that that's the sign of extra |
|
| 33:03 | . Sign of extra supposed to sign course. Okay, cancer. Ah |
|
| 33:15 | And so the two norm since it's to be in the product defined in |
|
| 33:21 | particular way. So in this case The two norm was the sum of |
|
| 33:31 | ? Okay, thanks. All Square of the mr someone square I |
|
| 33:38 | in this place. And the song a young and replaced by the |
|
| 33:43 | So this case uh The function happens be one. uh the basis function |
|
| 33:50 | one. So university integrate One over center malls and I'll give you |
|
| 33:55 | Oh, tell me why. But important. Now, so again, |
|
| 34:04 | basis functions are here. So we with one. So this is what |
|
| 34:10 | do. Uh huh. Being the or the norm in this space points |
|
| 34:19 | , it tells you that it's not because the body was too far. |
|
| 34:28 | Now then there's our functionality. So have the fund is one of the |
|
| 34:32 | functions and then there's a whole school or trick functions. So look at |
|
| 34:40 | integral Between one of these places functions one and another one. Goldstein protect |
|
| 34:47 | all make it simple. So this kind of integration of one respectively. |
|
| 34:53 | one of the coastline functions and if integrate anyone of the trip functions call |
|
| 35:00 | or sign Over the whole interval from it's there when it goes up and |
|
| 35:07 | . So that's equivalent much about the . Axis as it's been all the |
|
| 35:13 | . Axis regardless of and the multiplier front of their dollars. So many |
|
| 35:20 | johann between minus island high. It's there. It's about things. One |
|
| 35:26 | a stroganoff for every call sign version every sign version. And that's kind |
|
| 35:32 | the first step. Now the other what about sign versus co sign? |
|
| 35:42 | let's see if it was so I can't remember. 560. |
|
| 35:50 | I think that happens on the next . And it turns out that so |
|
| 35:55 | are orthogonal. So one was a with respect to both post and scientists |
|
| 36:01 | of the multiplier. So sine and are also octagonal respect regardless of the |
|
| 36:09 | . And M. Then there's cold of one Flavor vs coastline of another |
|
| 36:15 | And that's also a diagonal uh in sense of them and and it's different |
|
| 36:22 | in zero and otherwise it ends up upon yes, that's the coastline |
|
| 36:33 | So if they're equal, then question negative, yes. Um, private |
|
| 36:40 | positives. That's all positive elements. why it's no Longer zero. But |
|
| 36:47 | turns out to the party and similar the suns that they were talking about |
|
| 36:53 | respect to each other than they are And then they have, they don't |
|
| 36:58 | about it. But it's also far the same sign what? So banish |
|
| 37:08 | front and center here. So the man shows up these beliefs and this |
|
| 37:15 | I think from the exercise of proving true, but it means that |
|
| 37:21 | regardless of energy, you can continue . And these basis functions for 4000 |
|
| 37:28 | checks But they are not normalized because guy was two points. It's not |
|
| 37:34 | and this place is fine enough to care of that. So let's see |
|
| 37:42 | for this. And, and to the work for me is that this |
|
| 37:49 | are zero or five. These are uses strict formulas by breaking it down |
|
| 37:57 | some of Carlson functions instead of a . Then it's easier to integrate each |
|
| 38:03 | of them separately. Uh, and guess it was down here in the |
|
| 38:09 | of the science of this is this that is in the product and do |
|
| 38:14 | integration of this product sign and This expression target in under the |
|
| 38:22 | not this the best the instagram approach sign. So that sign and |
|
| 38:29 | And if again these guys are It's An integral over one or multiple |
|
| 38:38 | . So that means each one of are definitely 20 unless them is important |
|
| 38:44 | them and then it depends. so then, so she said that |
|
| 39:01 | there's someone who's died. Sorry. basically what was said that this was |
|
| 39:08 | high. This part is for a . These guys were a cardinal except |
|
| 39:18 | Eminem is equal centering this. So have this for the whole set and |
|
| 39:26 | were not normalized so one can normalize then bye scaling that each one. |
|
| 39:34 | instead of just having one we have over square root of two parts. |
|
| 39:38 | when you multiply it by itself to them to fire that comes out of |
|
| 39:43 | guy. So we'll take yes. are so that's the way that you |
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| 39:53 | need length. And now all of functions are normalized. Ortho normalized, |
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| 40:00 | and normalized if they go through the exercise and come into this pain |
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| 40:07 | Yeah. Ah mm hmm. The expressions. So this is thank |
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| 40:18 | And when I come to show you terms of confusing for your transform then |
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| 40:23 | will see the scale factor shows up one way or another and it's not |
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| 40:29 | consistent to use someone you use from T routine. If you don't want |
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| 40:36 | yourself, you have to pay attention escape doctors sometimes because it's not it's |
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| 40:52 | exact. Mhm wow. There is inverse fourier transform. So I want |
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| 41:10 | apply the forward and then the inverse transform you should that star functions that |
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| 41:18 | normal um defying and performance and an whatever it is should return whatever it |
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| 41:26 | . But if people don't use the basis functions, that's not true. |
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| 41:33 | if you just use a normalized these after you apply for your transform using |
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| 41:41 | space destruction and we would apply the from your transform also used the same |
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| 41:49 | functions. Then the result is the thing scaled by please street by |
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| 41:58 | So that is the policy that can . That's right. Let's go any |
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| 42:10 | so far. So this is the functions and you have a complete set |
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| 42:16 | the constant Normal is kind of the of the two and then the various |
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| 42:23 | functions of all different witnesses. So that is the four year serious |
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| 42:31 | . I can use that in the . Mhm. Okay then. And |
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| 42:44 | four year series representations of functions there simplifications but in terms of reducing computational |
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| 42:58 | or memory of apartments in terms of many proficient do you need to represent |
|
| 43:06 | signal? So for that one of franchise between what they haven't even function |
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| 43:14 | not but this is an even function it's kind of symmetric with respect to |
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| 43:22 | Y axis. So the first one X deployment intercoastal function for instance to |
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| 43:29 | the same value. Um Cynics is for all that. That's the odd |
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| 43:39 | . That's what it to your The next value for the negative |
|
| 43:44 | Files. We've got the same in but the sign is different. So |
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| 43:48 | that case the differences talking the negative the arguments instead of its positive |
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| 43:55 | The outcome is the same magnitude but office. So what's his name is |
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| 44:04 | one No 1 does content in fact Yes serious expansion of the single. |
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| 44:18 | if I have an even function very one of this kind of expect from |
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| 44:25 | that is symmetric with respect to And take the assignment is what then |
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| 44:34 | whole thing is. Uh huh. yeah and this has sense whether you |
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| 44:42 | X and minus X. When you sign with that it doesn't affect |
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| 44:49 | So if you are on the negative it's the same time it's invested here |
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| 44:53 | the positive side. So that But to sign his opposites in that |
|
| 44:59 | this instagram for any even function will zero. Yeah the values for plus |
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| 45:09 | minus X. On the opposite. but the integral um is zero. |
|
| 45:17 | that means the only thing that demands the glorious series are the coastline terms |
|
| 45:26 | the expansion on the side in terms . So that's what so that means |
|
| 45:34 | only need to complete half as many in this case. The provisions for |
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| 45:40 | all the time basis functions. It one else if they want the same |
|
| 45:47 | half the work. This one that sort of function etc. Similar. |
|
| 45:54 | the function is on that they can an online even they still are. |
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| 46:02 | then in this case it means that that the coastline proficiency No. |
|
| 46:10 | representation then of odd function only includes , Face it Fox. And then |
|
| 46:20 | need to confuse these things. So, this is a sad |
|
| 46:32 | The principle of four years here is of a function looks like this or |
|
| 46:41 | . This is not. So I'm on the next an approximation but it's |
|
| 46:46 | terminated. That's for less than infinity you have an approximation. But it's |
|
| 46:54 | opinion that have an infinite series. when certain conditions, it's an executive |
|
| 47:02 | . And the moment it is a function. This holds true. Then |
|
| 47:07 | can exactly represent X over in for . This expression And here are mr |
|
| 47:16 | base of the beast. They People just talked about Houston being the |
|
| 47:26 | . Mm hmm. And then there the simplification. Still exact representation of |
|
| 47:32 | even and exact representation. It's up one of Our Own. Probably a |
|
| 47:40 | type of basis functions and spaces. what? Um All right. And |
|
| 47:52 | from that. So external recording And that's being used in Dean's loose |
|
| 48:07 | that time for some other packages stay . Yes, known as all that |
|
| 48:13 | we don't use the property. We some other motion capital efficiency to watch |
|
| 48:22 | movie. And then there are the that are going to talk to you |
|
| 48:28 | time. Additional symmetries or just to ourselves reel um uh from so there's |
|
| 48:44 | versions of bullion transforms that produces them to work. But let's talk about |
|
| 48:54 | coarse approximation before I get ahead of too much. Um so now the |
|
| 49:01 | that nose to find that serious in case ah basis functions or at least |
|
| 49:10 | its approximation to go back to that of continuous functions. So chapter minibusses |
|
| 49:17 | on the square Aaron um Overactive against wall. Again, the same thing |
|
| 49:23 | do. So now produce enormous version , then um can look translating structure |
|
| 49:38 | the context of all deciding how many dysfunction. What makes sense in order |
|
| 49:51 | minimize the squared error. Ah and that showed ah the proficiency and this |
|
| 50:00 | of expansion we respected. The basis is basically the senior product on this |
|
| 50:07 | of the integral of if you respect the corresponding basis function divided by the |
|
| 50:18 | hmm value. I think it's not . The scale of the invested function |
|
| 50:28 | . So now if you use these , they were Ortho normalized. So |
|
| 50:33 | This one is the one. So . And so that this reduces to |
|
| 50:41 | have to evaluate this perfect expression because number later is one of the cost |
|
| 50:47 | their own. What the normal principle this was the outcome of the |
|
| 50:54 | I mean for a problem. So the list was approximation over function using |
|
| 51:06 | trig functions as basis that looks something like this. There is kind of |
|
| 51:15 | the first basis function that in normal well as external and then the next |
|
| 51:23 | function which this one normalized and these not proficient just coming out of this |
|
| 51:30 | . The business then why don't even us now? Mm hmm. For |
|
| 51:36 | serious approximation one function. Yes. what your computer is best for the |
|
| 51:44 | grounds function f itself with first of the next basis functions. So these |
|
| 51:53 | the things we need to confuse, get three squares approximation using trig functions |
|
| 52:00 | the function. It's not okay for for Yeah, the more people seem |
|
| 52:15 | find annotation, I think this is just the result of the integral but |
|
| 52:22 | the state factors. And from the that the only thing one simple um |
|
| 52:29 | the A's and the B's difficult one see. And that was kind of |
|
| 52:35 | emotional power. Um just when I the series about Yes. So look |
|
| 52:51 | this expression, Yes, best of A's. Um our coefficient will spend |
|
| 52:58 | respect to even functions if you Even the constant is also very |
|
| 53:05 | And these are the proficiency respect on basis function. So this is kind |
|
| 53:13 | what nothing at the bottom of this . Just making correspondent. So this |
|
| 53:19 | basically now the first set of base in the approximation. Okay, so |
|
| 53:26 | can think of its correspondent for the series, right? It's a bunch |
|
| 53:31 | then there's the rest of it, the character. And in this case |
|
| 53:35 | rest of it. Is that the order advanced dysfunctions from our function? |
|
| 53:47 | , yes. So this is that's first part of the for the expansion |
|
| 53:54 | the basis functions are orthogonal. If want to hire and it doesn't change |
|
| 54:00 | about the producer. You don't need go back and re mark things because |
|
| 54:04 | basis from control of our lives. it's just extending the space and then |
|
| 54:09 | the projections into it additional space that added when nothing you want to paint |
|
| 54:14 | porch. So um but the response , this is them. Okay, |
|
| 54:26 | this I think is now the bottom . So you just kind of the |
|
| 54:31 | of the four year serious stuff. f is continuous or thank you. |
|
| 54:38 | know, piecewise continuous. Including the then before this year is emerges to |
|
| 54:46 | function. If it has jobs, it converges to the midpoint between |
|
| 54:54 | Yeah. So and also accept my says the approach, the jump from |
|
| 55:00 | left and the approach from the I guess approach. Right. Then |
|
| 55:05 | midpoint is that's part of the four series approximation. And the aside from |
|
| 55:13 | new part, it's kind of approximately the limit. So we're just coming |
|
| 55:19 | terms. It will exactly represent the . Yeah. Any questions on |
|
| 55:31 | Oh, what do you want? . Simple person just pointing at our |
|
| 55:45 | . Like when we talk about polynomial the number of integration formulas, I |
|
| 55:51 | using a standard interval and for trig despite the fire, this is convenient |
|
| 55:58 | use but that may not be what have. So that means if you |
|
| 56:03 | another symmetric interval around zero but the in this case the Children of |
|
| 56:14 | but then you can do the transformation variables. So to do this, |
|
| 56:19 | have a new variable. So start this thing that comes between minus pi |
|
| 56:24 | phi that means now the new value electrons between minus 5/2 and Pi over |
|
| 56:36 | . The bounds, you know, got the bucking five, then it |
|
| 56:44 | 5, 2. and similarly on other side we're gonna need to take |
|
| 56:50 | replace this is the exit polls. out what the next derivative of |
|
| 56:59 | All this and some of the new . So it is best to affect |
|
| 57:03 | relatives of the relatives of the new this have to do everything that they |
|
| 57:08 | them that their answer going into this of equations to work with different |
|
| 57:16 | The new one, they need to this VX this trunk, I do |
|
| 57:24 | know quantum and then again here's a equation and we're gonna interval and then |
|
| 57:32 | the variables. Then we have extolled to find on the X. New |
|
| 57:40 | of here instead of just at the . And in terms of interval for |
|
| 57:49 | spice here um The scaling factor that from the scaling from one is quite |
|
| 57:58 | party they sent them off. Come in the show working bombs are you |
|
| 58:06 | have to plug in the consequences um This with a new one. So |
|
| 58:16 | fire you cancel out this price. you got two of the three year |
|
| 58:20 | factor instead of one of the market . So it's just variable substitution from |
|
| 58:29 | straightforward and then they've got something for symmetric interval and I think this one |
|
| 58:37 | been in Over to L five months something but you can also do the |
|
| 58:46 | on the interval. So it's not around zero. You can happen and |
|
| 58:52 | did all that from zero safety too with the previous one. Do the |
|
| 58:57 | sort of variable substitution, contracted the case and forget him. Forget is |
|
| 59:04 | it's something that does not so much think of all from and scale the |
|
| 59:14 | . So we're shifting and scaling or . So we're getting our examples underscore |
|
| 59:29 | questions. So so here's the softwood that so this is honest. And |
|
| 59:39 | continuous function has jobs ah, but it's pretty simple. Um, so |
|
| 59:48 | it's an X. And an interval then have a periodic. So it's |
|
| 59:52 | of jealous In this case the symmetric zero. Don't worry. We're all |
|
| 59:59 | on the table. So now I'm to tell you for your serious approximation |
|
| 60:07 | this sorting function. Want to the and flex and extend. That's an |
|
| 60:14 | function. Um, yes, this a zero right integration of aftereffects And |
|
| 60:25 | basis functions zero is the constant. the question is zero. So this |
|
| 60:32 | all versus even function expensive thought. yeah. Mm hmm. All this |
|
| 60:44 | , ah Am coefficients are zero because is co sign that is even So |
|
| 60:52 | the times. Even if it's on integration of any odd function on the |
|
| 60:59 | the point by its. Oh, . So we only have to evaluate |
|
| 61:07 | B coefficient. So here's the function . And then there is the various |
|
| 61:14 | basis functions. And then we can this special immigration for parks. This |
|
| 61:22 | nothing united. We respect the And the way information support works. |
|
| 61:28 | you remember that presumably is ah the or the anti derivative that fits you |
|
| 61:40 | internet and there is a problem formula how this integration works. I don't |
|
| 61:47 | for me, simple to install is try to verify that. Okay, |
|
| 61:52 | think that everything about this. So actually get this thing here and then |
|
| 61:55 | believe it's true. So the derivative this first and when you have a |
|
| 62:04 | here and it's kind of a general we take the derivative of the first |
|
| 62:12 | France's second and then the first one celebrated on the 2nd 1. The |
|
| 62:18 | with respect to experts. It remains . This derivative. Of course |
|
| 62:25 | And when we take the derivative addition and comes out and becomes one |
|
| 62:29 | And so yes. So that thing's . And what you get this this |
|
| 62:34 | the political benefits. Sorry. I that's the verification if you don't remember |
|
| 62:42 | for the integration and functional. Mm . So then if you believe that |
|
| 62:49 | is indeed the derivative of this Now we need to plug in the |
|
| 62:55 | . Um, so look uh the , this stern 1st looking pie. |
|
| 63:08 | why is here? That's fine. co sign for some multiple of five |
|
| 63:16 | Either plus or -1. Right. huh. So, so that's what |
|
| 63:23 | says here. What uh, has power of like this one? |
|
| 63:30 | the other thing is simple. Anything that is in the zero or |
|
| 63:40 | multiple of spy. It's Sign is zero for 0 for 5 - 5 |
|
| 63:50 | and five. So this one is . So the only thing that |
|
| 63:55 | that's the name of this type. we plug in on the upper |
|
| 64:01 | Quiet. And this part here, of this part of the point goes |
|
| 64:06 | . So we only get the number depending on and was there and the |
|
| 64:13 | of minus pond depending on Yeah, this is basically indiana yeah something next |
|
| 64:26 | . So this is the best stuff the coefficients are all the AIDS of |
|
| 64:32 | and these four we some functions And so this is the approximation of |
|
| 64:45 | function depending upon how far you go just what you get and it's unconscious |
|
| 64:52 | makes sense again because this low sorted far so quite a bit and I |
|
| 65:04 | I have a picture of yes, . Mhm. Just the national number |
|
| 65:09 | the picture about this is let's talk the difference from credible I have said |
|
| 65:14 | this is now instead of having it , I respect that it's equal |
|
| 65:19 | This is shifted some this representation but can see that depending upon how many |
|
| 65:26 | of the check functions or racist functions against both and both to straight found |
|
| 65:34 | between the and the next step is so this is the shifted interval comes |
|
| 65:43 | as well. It's symmetric or on respectable midpoint so it's still not going |
|
| 65:52 | change its proficient. No need to about. So what we're going through |
|
| 65:58 | same machinery again. zero now it and we're trying to uh so |
|
| 66:09 | this is finally in the function was well integrated equation systems turned out from |
|
| 66:19 | previous Exercise and shifting the interval from then do the integration and now it's |
|
| 66:30 | the constant. And that comes from fact, yes, I've been locked |
|
| 66:36 | ah the previous example I have, was But with respect to the midpoint |
|
| 66:43 | was zero. But it was all on, wow, I was |
|
| 66:52 | So we basically had that it was first and then until it's equal zero |
|
| 66:59 | then it was positive. So now difference is not only shifted in the |
|
| 67:05 | axis but it's also shifted in the axis. So as you can see |
|
| 67:12 | , no part the so in this the average value is action 45 for |
|
| 67:23 | . So this The fact that a now is Not zero that it wasn't |
|
| 67:30 | famous one comes from the heart that's shifted sometimes we talked about it as |
|
| 67:40 | ah the statistic carrier or current, know. So this is the same |
|
| 67:53 | average now. Yeah, we have so now it's kind of a little |
|
| 68:04 | more interesting so that any coefficients with to the callsign basis irrespective and even |
|
| 68:15 | around this place. So we know defined even respect yeah, Y axis |
|
| 68:25 | the fact that we work the center tear ah then in fact it becomes |
|
| 68:36 | with respect to the point taxes. in the end it doesn't shift change |
|
| 68:42 | from now and still the case that worked through all the algebra. |
|
| 68:46 | all the coefficients except from the first that parents every single word But the |
|
| 68:54 | of it is zero. So it's only going to be that's fine terms |
|
| 69:02 | fundamentally it's kind of the same things this sort of service just shifted. |
|
| 69:07 | the four years there is still, ends up being just a set of |
|
| 69:13 | functions doing the same integration, respect sign and algebra and then you're finding |
|
| 69:21 | efficient. So Oh no, it's still only signed functions and there is |
|
| 69:33 | . The first one is in different here and what terms to get the |
|
| 69:39 | that they may be discreet but there's except our trick functions. So they're |
|
| 69:46 | a straight line and one of them the more of them you include the |
|
| 69:50 | to the straight line you get and can also see that it's beginning to |
|
| 69:56 | so with the highest electron here and all of them died. So I |
|
| 70:00 | got to the point between for the period on the center mhm time is |
|
| 70:12 | but you can then do a square in this case. The states now |
|
| 70:17 | is um also and I order not again Sign functions because cosigning respected the |
|
| 70:34 | . So the Coast guard and stop and talked about okay, thank |
|
| 70:56 | |
|
