WEBVTT 00:00:00.000 --> 00:00:02.735 align:middle line:90% 00:00:02.735 --> 00:00:04.860 align:middle line:84% ALVIN THONG-JUAK KHO: Transforming data, as well as 00:00:04.860 --> 00:00:06.030 align:middle line:90% modeling and genomics. 00:00:06.030 --> 00:00:08.189 align:middle line:90% It's a very broad category. 00:00:08.189 --> 00:00:12.600 align:middle line:84% And I'm sure that in the other, lectures the people who 00:00:12.600 --> 00:00:16.440 align:middle line:84% are giving them have touched on several aspects 00:00:16.440 --> 00:00:19.840 align:middle line:90% of these same topics, as well. 00:00:19.840 --> 00:00:23.050 align:middle line:84% The lecture outline is as follows. 00:00:23.050 --> 00:00:27.900 align:middle line:84% The first thing we'll do is to go through two 00:00:27.900 --> 00:00:30.780 align:middle line:84% very prototypical study designs, one of which 00:00:30.780 --> 00:00:35.710 align:middle line:84% is a two-way comparison which occurs all the time in studies. 00:00:35.710 --> 00:00:39.390 align:middle line:84% The second one is the time series, or a dosage study. 00:00:39.390 --> 00:00:46.110 align:middle line:84% The idea being that there is a one parameter in the study that 00:00:46.110 --> 00:00:50.010 align:middle line:84% moves on, progresses, and you are making measurements 00:00:50.010 --> 00:00:52.350 align:middle line:90% alongside that one parameter. 00:00:52.350 --> 00:00:57.330 align:middle line:84% And finally, we get to the topic of data representation itself, 00:00:57.330 --> 00:00:58.650 align:middle line:90% properly. 00:00:58.650 --> 00:01:01.260 align:middle line:84% We'll talking about what it is, what 00:01:01.260 --> 00:01:04.620 align:middle line:84% it means to go from a measurement device 00:01:04.620 --> 00:01:07.570 align:middle line:84% to the spreadsheet, because in order to do analysis, 00:01:07.570 --> 00:01:11.070 align:middle line:84% we must somehow end up in the world of numbers. 00:01:11.070 --> 00:01:13.470 align:middle line:84% So how does that mapping actually happen? 00:01:13.470 --> 00:01:18.330 align:middle line:90% We'll touch on it very little. 00:01:18.330 --> 00:01:20.970 align:middle line:84% Then we talk about the numbers themselves. 00:01:20.970 --> 00:01:25.050 align:middle line:84% Do they have a dimensionality attributed to them? 00:01:25.050 --> 00:01:27.400 align:middle line:84% Are there scales attributed to them? 00:01:27.400 --> 00:01:28.890 align:middle line:90% These things actually matter. 00:01:28.890 --> 00:01:33.360 align:middle line:84% Because depending on whether or not it's dimensioned, 00:01:33.360 --> 00:01:36.150 align:middle line:84% certain, fundamental theorems of mathematics 00:01:36.150 --> 00:01:40.890 align:middle line:84% actually do apply in guiding you on how 00:01:40.890 --> 00:01:44.280 align:middle line:84% to formulate, say, a null hypothesis 00:01:44.280 --> 00:01:45.570 align:middle line:90% for your experiments. 00:01:45.570 --> 00:01:49.620 align:middle line:84% What would the outcome be had it been random? 00:01:49.620 --> 00:01:50.710 align:middle line:90% Things like that. 00:01:50.710 --> 00:01:54.090 align:middle line:84% And then we talk about how, once you have the data down, 00:01:54.090 --> 00:01:58.020 align:middle line:84% you transform it so as to uncover 00:01:58.020 --> 00:02:02.560 align:middle line:84% internal or relational structures within the data. 00:02:02.560 --> 00:02:04.980 align:middle line:84% And if we get through to this and we have time left, 00:02:04.980 --> 00:02:09.479 align:middle line:84% we will go through background concepts which crops up 00:02:09.479 --> 00:02:13.080 align:middle line:84% very often when we talk about genomics or microarray 00:02:13.080 --> 00:02:17.640 align:middle line:84% analysis, namely, things or words like noise, 00:02:17.640 --> 00:02:19.230 align:middle line:90% what is a replicate? 00:02:19.230 --> 00:02:21.630 align:middle line:84% What does it mean to be reproducible? 00:02:21.630 --> 00:02:24.270 align:middle line:84% Do you have to normalize your data, and what 00:02:24.270 --> 00:02:28.230 align:middle line:84% is the notion of a fold in the context of, say, 00:02:28.230 --> 00:02:29.550 align:middle line:90% microarray data? 00:02:29.550 --> 00:02:33.240 align:middle line:84% I mean, a fold is a very natural measure of change 00:02:33.240 --> 00:02:34.980 align:middle line:90% when it comes to PCR. 00:02:34.980 --> 00:02:37.230 align:middle line:84% But is it that natural, after all, 00:02:37.230 --> 00:02:40.450 align:middle line:84% when you are dealing with microarray data? 00:02:40.450 --> 00:02:42.810 align:middle line:84% And finally, because by going through 00:02:42.810 --> 00:02:47.400 align:middle line:84% some miscellaneous rules of thumb. 00:02:47.400 --> 00:02:50.530 align:middle line:84% So two prototypical study designs-- 00:02:50.530 --> 00:02:52.200 align:middle line:84% the first is a two-way comparison. 00:02:52.200 --> 00:02:56.820 align:middle line:84% You see this all the time, in fact. 00:02:56.820 --> 00:03:04.410 align:middle line:84% For instance, you want to find molecular gene protein 00:03:04.410 --> 00:03:06.630 align:middle line:84% differences in the white blood cells 00:03:06.630 --> 00:03:09.420 align:middle line:84% of type 2 diabetes human patients 00:03:09.420 --> 00:03:12.480 align:middle line:90% versus normal human patients. 00:03:12.480 --> 00:03:15.510 align:middle line:84% For instance, you have 27 diabetes patients 00:03:15.510 --> 00:03:18.448 align:middle line:84% and 11 normal ones, and you microarray them. 00:03:18.448 --> 00:03:19.990 align:middle line:84% Actually, there's a little typo here. 00:03:19.990 --> 00:03:21.240 align:middle line:90% It should be the word "micro." 00:03:21.240 --> 00:03:24.010 align:middle line:90% 00:03:24.010 --> 00:03:26.730 align:middle line:84% So these studies are often carried out. 00:03:26.730 --> 00:03:31.260 align:middle line:84% And there are very obvious things to keep in mind when 00:03:31.260 --> 00:03:32.430 align:middle line:90% you do these studies. 00:03:32.430 --> 00:03:35.760 align:middle line:84% Namely, before you do them, you should 00:03:35.760 --> 00:03:41.130 align:middle line:84% check for stratification, say, gender, age, 00:03:41.130 --> 00:03:43.620 align:middle line:84% things like that-- other clinical parameters 00:03:43.620 --> 00:03:46.680 align:middle line:84% that could enter later on and bedevil 00:03:46.680 --> 00:03:50.940 align:middle line:90% your subsequence analysis. 00:03:50.940 --> 00:03:55.530 align:middle line:84% Now, a partial mathematical formulation of this problem 00:03:55.530 --> 00:04:00.060 align:middle line:84% would be, say, if you were going to microarray every person, 00:04:00.060 --> 00:04:03.600 align:middle line:84% every patient goes onto a chip, then 00:04:03.600 --> 00:04:07.230 align:middle line:84% the j-th patients, j-th diabetic patient, 00:04:07.230 --> 00:04:10.710 align:middle line:84% you could represent by this symbol, d sub j. 00:04:10.710 --> 00:04:14.970 align:middle line:84% And j normal patient, you could reproduce by this symbol. 00:04:14.970 --> 00:04:21.120 align:middle line:84% And this quantity here is actually a vector or matrix. 00:04:21.120 --> 00:04:23.040 align:middle line:90% It's multi-factorial. 00:04:23.040 --> 00:04:27.480 align:middle line:84% We are measuring, say, 10,000 genes, or proteins, 00:04:27.480 --> 00:04:31.240 align:middle line:90% or RNA of one person. 00:04:31.240 --> 00:04:34.440 align:middle line:84% So it's a multi-dimensional quantity. 00:04:34.440 --> 00:04:37.260 align:middle line:90% 00:04:37.260 --> 00:04:40.240 align:middle line:84% And you go on to do the subsequence analysis. 00:04:40.240 --> 00:04:43.420 align:middle line:84% So this is the first prototypical study design. 00:04:43.420 --> 00:04:45.960 align:middle line:84% The second one is equally common. 00:04:45.960 --> 00:04:49.230 align:middle line:84% It has to do with a time series or dosage study. 00:04:49.230 --> 00:04:52.680 align:middle line:84% For instance, you are observing a developing organ, 00:04:52.680 --> 00:04:56.790 align:middle line:84% and you are assaying that organ for RNA 00:04:56.790 --> 00:05:00.900 align:middle line:84% or protein at different stages of its development 00:05:00.900 --> 00:05:02.640 align:middle line:90% as it unfolds. 00:05:02.640 --> 00:05:05.040 align:middle line:84% Again, there are reality checks that you 00:05:05.040 --> 00:05:09.120 align:middle line:84% should do before you begin, such as that system which 00:05:09.120 --> 00:05:14.100 align:middle line:84% you are assaying, how heterogeneous is it? 00:05:14.100 --> 00:05:16.650 align:middle line:84% Is it one cell type, several cell type? 00:05:16.650 --> 00:05:20.820 align:middle line:84% How is that going to affect your interpretation of the signals 00:05:20.820 --> 00:05:22.950 align:middle line:90% later on? 00:05:22.950 --> 00:05:27.270 align:middle line:84% And again, a partial math formulation of this design 00:05:27.270 --> 00:05:30.660 align:middle line:84% would be let T sub j represent the chip 00:05:30.660 --> 00:05:33.150 align:middle line:84% data of the j-th developmental stage. 00:05:33.150 --> 00:05:35.130 align:middle line:90% Again, it is a vector. 00:05:35.130 --> 00:05:38.280 align:middle line:84% And you can get very fancy pictures 00:05:38.280 --> 00:05:43.530 align:middle line:84% like this of the expression profile as a function of time-- 00:05:43.530 --> 00:05:49.590 align:middle line:84% of one gene and another gene or RNA or protein. 00:05:49.590 --> 00:05:52.770 align:middle line:84% Now, what actually is data representation. 00:05:52.770 --> 00:05:55.800 align:middle line:90% 00:05:55.800 --> 00:05:59.040 align:middle line:84% It's a very commonly-used term, and I 00:05:59.040 --> 00:06:00.510 align:middle line:84% don't think anyone can really agree 00:06:00.510 --> 00:06:01.800 align:middle line:90% on what it means, actually. 00:06:01.800 --> 00:06:06.450 align:middle line:84% But it could go anywhere from a mathematical formulation 00:06:06.450 --> 00:06:09.630 align:middle line:90% of a scientific problem. 00:06:09.630 --> 00:06:12.930 align:middle line:84% Or it could be mapping observations and measurements 00:06:12.930 --> 00:06:14.340 align:middle line:90% into the set of symbols. 00:06:14.340 --> 00:06:16.625 align:middle line:84% Well, it's not just any old set of symbols. 00:06:16.625 --> 00:06:18.000 align:middle line:84% Typically, they are symbols which 00:06:18.000 --> 00:06:22.200 align:middle line:84% you can do an algebra on, say, numbers, integers, 00:06:22.200 --> 00:06:23.520 align:middle line:90% things like that. 00:06:23.520 --> 00:06:25.920 align:middle line:84% It's pointless to map it into a set of symbols 00:06:25.920 --> 00:06:30.130 align:middle line:84% where you can't actually do higher-order operations on, 00:06:30.130 --> 00:06:32.040 align:middle line:90% that being the point. 00:06:32.040 --> 00:06:35.850 align:middle line:84% If you go on a website and type in data representation, 00:06:35.850 --> 00:06:42.480 align:middle line:84% the governmental agency for atmospheric science 00:06:42.480 --> 00:06:44.160 align:middle line:84% gives you this definition here, which 00:06:44.160 --> 00:06:49.530 align:middle line:84% you can read for yourself in this slide print out later. 00:06:49.530 --> 00:06:51.900 align:middle line:84% Another notion of data representation 00:06:51.900 --> 00:06:55.650 align:middle line:84% is something to do with database annotation and standards. 00:06:55.650 --> 00:06:57.420 align:middle line:84% You've heard of Miami, presumably, 00:06:57.420 --> 00:07:00.420 align:middle line:84% in one of the lectures before or to come. 00:07:00.420 --> 00:07:03.390 align:middle line:84% I'm not going to be talking about that at all. 00:07:03.390 --> 00:07:06.120 align:middle line:84% And thirdly, it could be multimedia. 00:07:06.120 --> 00:07:08.130 align:middle line:84% How you present your data-- graphically, 00:07:08.130 --> 00:07:11.190 align:middle line:84% and charts, plots, et cetera, things like that. 00:07:11.190 --> 00:07:14.070 align:middle line:84% What I shall be actually concentrating on 00:07:14.070 --> 00:07:17.500 align:middle line:84% is actually the first two points. 00:07:17.500 --> 00:07:20.670 align:middle line:84% It is arguable that these two points actually have 00:07:20.670 --> 00:07:22.380 align:middle line:90% to do with data representation. 00:07:22.380 --> 00:07:24.660 align:middle line:84% If anything, the second point certainly 00:07:24.660 --> 00:07:28.530 align:middle line:84% does-- mapping observations into a set of symbols. 00:07:28.530 --> 00:07:31.910 align:middle line:90% 00:07:31.910 --> 00:07:36.760 align:middle line:84% Now, so you make an observation, and a device 00:07:36.760 --> 00:07:39.460 align:middle line:90% takes a picture of it. 00:07:39.460 --> 00:07:41.950 align:middle line:84% But eventually, it has to get translated 00:07:41.950 --> 00:07:43.270 align:middle line:90% into the set of symbols. 00:07:43.270 --> 00:07:49.120 align:middle line:84% So it's a very obvious thing that one has to actually 00:07:49.120 --> 00:07:52.960 align:middle line:84% understand at least the basic principles of how 00:07:52.960 --> 00:07:55.930 align:middle line:84% the measurement device works if you are going 00:07:55.930 --> 00:07:58.330 align:middle line:84% to be using it in your undertaking 00:07:58.330 --> 00:08:01.210 align:middle line:90% of any biological endeavor. 00:08:01.210 --> 00:08:03.800 align:middle line:84% You should know the relevance of the scanner setting, things 00:08:03.800 --> 00:08:04.300 align:middle line:90% like that. 00:08:04.300 --> 00:08:06.250 align:middle line:84% I'm not going to go through all of this. 00:08:06.250 --> 00:08:10.070 align:middle line:84% It's quite obvious, a lot of these things. 00:08:10.070 --> 00:08:17.860 align:middle line:84% And you also have to be sure that the journey from an image 00:08:17.860 --> 00:08:19.630 align:middle line:90% to the actual number itself-- 00:08:19.630 --> 00:08:23.680 align:middle line:84% does it go through analytical or statistical processing 00:08:23.680 --> 00:08:24.790 align:middle line:90% software? 00:08:24.790 --> 00:08:29.500 align:middle line:84% How far removed are you from the reality of the observation? 00:08:29.500 --> 00:08:33.520 align:middle line:84% One should check for these things, in fact. 00:08:33.520 --> 00:08:35.409 align:middle line:84% Because subsequent stuff that you're 00:08:35.409 --> 00:08:37.330 align:middle line:84% going to be doing on this numbers 00:08:37.330 --> 00:08:43.850 align:middle line:84% will matter depending on the condition upon these factors. 00:08:43.850 --> 00:08:46.870 align:middle line:84% So you get, in the case of microarrays, basically 00:08:46.870 --> 00:08:49.180 align:middle line:90% a bright spot. 00:08:49.180 --> 00:08:53.140 align:middle line:84% And somehow, you have to translate this brightness. 00:08:53.140 --> 00:08:56.140 align:middle line:90% And the grid here is imaginary. 00:08:56.140 --> 00:09:00.130 align:middle line:84% It doesn't just-- you won't see just very discrete grids 00:09:00.130 --> 00:09:00.700 align:middle line:90% lighting up. 00:09:00.700 --> 00:09:04.210 align:middle line:84% Typically, there is a diffusion of light 00:09:04.210 --> 00:09:06.320 align:middle line:90% from one feature to another. 00:09:06.320 --> 00:09:09.340 align:middle line:84% So you have to basically translate this whole mess here 00:09:09.340 --> 00:09:14.440 align:middle line:84% down to a simplistic-looking table of numbers. 00:09:14.440 --> 00:09:18.970 align:middle line:84% So you have genes measured across different samples 00:09:18.970 --> 00:09:21.340 align:middle line:84% and their, quote, unquote, "intensity." 00:09:21.340 --> 00:09:23.950 align:middle line:84% And next thing we'll do is to understand 00:09:23.950 --> 00:09:27.490 align:middle line:84% if these intensities themselves have a dimension associated 00:09:27.490 --> 00:09:32.710 align:middle line:84% with them, say, centimeters, Fahrenheit, things like that, 00:09:32.710 --> 00:09:38.100 align:middle line:84% or they have no dimensions at all. 00:09:38.100 --> 00:09:41.160 align:middle line:90% So dimensionality and scales. 00:09:41.160 --> 00:09:49.020 align:middle line:84% And I guess we can concentrate for a while on microarray data. 00:09:49.020 --> 00:09:53.040 align:middle line:84% In the case of two-channel or competitive hybridization 00:09:53.040 --> 00:09:58.260 align:middle line:84% microarrays where you have two dyes, a size 3 and size 5, 00:09:58.260 --> 00:10:00.960 align:middle line:90% competing-- 00:10:00.960 --> 00:10:04.290 align:middle line:84% being hybridized onto different tissue types 00:10:04.290 --> 00:10:08.760 align:middle line:84% and put onto one array, the readout 00:10:08.760 --> 00:10:10.830 align:middle line:84% is arguably dimensionless because it 00:10:10.830 --> 00:10:12.030 align:middle line:90% has to do with a ratio. 00:10:12.030 --> 00:10:15.300 align:middle line:84% And arithmetic tells you that the ratio operation 00:10:15.300 --> 00:10:19.530 align:middle line:84% nullifies or cancels out any dimension there is in the data. 00:10:19.530 --> 00:10:22.680 align:middle line:90% 00:10:22.680 --> 00:10:25.020 align:middle line:84% Second type species of microarrays 00:10:25.020 --> 00:10:27.160 align:middle line:84% are, say, oligonucleotide microarrays 00:10:27.160 --> 00:10:28.410 align:middle line:90% where there is no competition. 00:10:28.410 --> 00:10:31.920 align:middle line:84% You have, essentially, one tissue type. 00:10:31.920 --> 00:10:37.650 align:middle line:84% You tag it with some fluorescence, 00:10:37.650 --> 00:10:38.790 align:middle line:90% and then you hybridize it. 00:10:38.790 --> 00:10:40.590 align:middle line:84% There is no competition whatsoever. 00:10:40.590 --> 00:10:45.780 align:middle line:84% And the registered intensities have some kind 00:10:45.780 --> 00:10:47.820 align:middle line:90% of units associated with them. 00:10:47.820 --> 00:10:51.780 align:middle line:84% If anything, you don't have to know the actual units. 00:10:51.780 --> 00:10:55.350 align:middle line:84% It may be sufficient just to if units exist 00:10:55.350 --> 00:10:58.140 align:middle line:84% or not-- dimensions or dimensionless. 00:10:58.140 --> 00:10:59.610 align:middle line:90% Why? 00:10:59.610 --> 00:11:01.830 align:middle line:84% Because different math techniques actually 00:11:01.830 --> 00:11:05.670 align:middle line:84% will apply depending on whether the quantity is a dimension 00:11:05.670 --> 00:11:07.110 align:middle line:90% quantity or dimensionless one. 00:11:07.110 --> 00:11:11.790 align:middle line:84% For example, if you are having intense-- 00:11:11.790 --> 00:11:15.600 align:middle line:84% you are dealing with a set of numbers which have to do with 00:11:15.600 --> 00:11:19.890 align:middle line:84% radiation measurements-- clearly phosphor images or radiation-- 00:11:19.890 --> 00:11:23.610 align:middle line:84% then physics tells you that the underlying, 00:11:23.610 --> 00:11:26.530 align:middle line:84% or the most sensible distribution to this background 00:11:26.530 --> 00:11:31.080 align:middle line:84% to be studying this system would be the gamma distribution. 00:11:31.080 --> 00:11:35.460 align:middle line:84% Secondly, there are certain, what you've probably 00:11:35.460 --> 00:11:38.880 align:middle line:84% heard as power or scaling laws, that 00:11:38.880 --> 00:11:45.930 align:middle line:84% could be useful in detecting errors in this set of numbers 00:11:45.930 --> 00:11:47.640 align:middle line:84% if they don't have a dimension or they 00:11:47.640 --> 00:11:51.480 align:middle line:84% have a dimension-- things, laws such as Zipf's law or Benford's 00:11:51.480 --> 00:11:53.160 align:middle line:90% law. 00:11:53.160 --> 00:11:58.223 align:middle line:84% For instance, the third digit of every telephone number 00:11:58.223 --> 00:11:59.640 align:middle line:84% in this country, things like that. 00:11:59.640 --> 00:12:02.850 align:middle line:84% There is a certain underlying distribution that could be very 00:12:02.850 --> 00:12:06.270 align:middle line:84% useful in informing you what actually is random and what 00:12:06.270 --> 00:12:06.960 align:middle line:90% actually isn't. 00:12:06.960 --> 00:12:12.120 align:middle line:84% Because if you get a result after some analysis 00:12:12.120 --> 00:12:14.370 align:middle line:84% that you do, whatever it is, you would like, at least, 00:12:14.370 --> 00:12:17.040 align:middle line:84% to be sure that that result is not arising 00:12:17.040 --> 00:12:18.540 align:middle line:90% completely out of chance. 00:12:18.540 --> 00:12:20.160 align:middle line:84% And that is the whole point of this, 00:12:20.160 --> 00:12:25.230 align:middle line:84% that knowing the dimension-ness of the numbers 00:12:25.230 --> 00:12:30.270 align:middle line:84% could help you or guide you in formulating a null hypothesis 00:12:30.270 --> 00:12:31.695 align:middle line:90% for this data set. 00:12:31.695 --> 00:12:34.670 align:middle line:90% 00:12:34.670 --> 00:12:38.500 align:middle line:90% So that's dimensionality. 00:12:38.500 --> 00:12:41.350 align:middle line:84% Why do you-- so let's say you already have 00:12:41.350 --> 00:12:43.030 align:middle line:90% the numbers in front of you. 00:12:43.030 --> 00:12:45.760 align:middle line:84% Why would you want to transform it or rewrite it 00:12:45.760 --> 00:12:47.380 align:middle line:90% in a different format? 00:12:47.380 --> 00:12:49.840 align:middle line:84% There are actually several very good reasons for doing it, 00:12:49.840 --> 00:12:54.070 align:middle line:84% and I'm sure it comes very obvious to a lot of people's 00:12:54.070 --> 00:12:55.270 align:middle line:90% heads. 00:12:55.270 --> 00:13:01.870 align:middle line:84% Number one-- it could simplify mathematical manipulation. 00:13:01.870 --> 00:13:06.550 align:middle line:84% And secondly, rewriting it in a certain way 00:13:06.550 --> 00:13:11.770 align:middle line:84% could uncover certain structures in the data. 00:13:11.770 --> 00:13:13.900 align:middle line:84% We will see examples of these very shortly. 00:13:13.900 --> 00:13:16.550 align:middle line:90% 00:13:16.550 --> 00:13:21.700 align:middle line:84% So number one-- simplifying mathematical manipulation. 00:13:21.700 --> 00:13:28.060 align:middle line:84% It can be argued that any spreadsheet is, essentially, 00:13:28.060 --> 00:13:29.320 align:middle line:90% a matrix. 00:13:29.320 --> 00:13:32.170 align:middle line:84% Of course, the entries of the matrices themselves 00:13:32.170 --> 00:13:33.200 align:middle line:90% are not homogeneous. 00:13:33.200 --> 00:13:34.700 align:middle line:84% They can be very different, and they 00:13:34.700 --> 00:13:37.240 align:middle line:84% could affect what kinds of operations 00:13:37.240 --> 00:13:38.770 align:middle line:84% you can sensibly perform on them. 00:13:38.770 --> 00:13:41.440 align:middle line:84% But essentially, it is a matrix if there 00:13:41.440 --> 00:13:44.050 align:middle line:90% are numbers in these entries. 00:13:44.050 --> 00:13:46.570 align:middle line:84% And as such, if there are matrices 00:13:46.570 --> 00:13:48.970 align:middle line:84% and the entries are homogeneous, they 00:13:48.970 --> 00:13:53.650 align:middle line:84% are subject to formal and very basic linear algebraic roles. 00:13:53.650 --> 00:13:56.770 align:middle line:84% You could do a matrix addition, subtraction, et cetera. 00:13:56.770 --> 00:14:01.420 align:middle line:84% And you could investigate the eigenvalues 00:14:01.420 --> 00:14:04.930 align:middle line:84% that are eigenvectors-- basically, invariant structures 00:14:04.930 --> 00:14:05.680 align:middle line:90% within the data. 00:14:05.680 --> 00:14:09.130 align:middle line:84% And of course, the hope in any scientific endeavor 00:14:09.130 --> 00:14:13.720 align:middle line:84% is that these invariant structures somehow 00:14:13.720 --> 00:14:19.030 align:middle line:84% are related to a physical phenomenon from which 00:14:19.030 --> 00:14:20.420 align:middle line:90% these numbers come from. 00:14:20.420 --> 00:14:23.800 align:middle line:84% There is a hope of all scientists who 00:14:23.800 --> 00:14:29.350 align:middle line:90% are trying to study a system. 00:14:29.350 --> 00:14:30.700 align:middle line:90% And this warning is obvious. 00:14:30.700 --> 00:14:34.300 align:middle line:84% If your matrix-- if you are measuring, say, temperature, 00:14:34.300 --> 00:14:38.350 align:middle line:84% and height, and weight, clearly, you can't just 00:14:38.350 --> 00:14:42.850 align:middle line:84% perform blindly linear algebraic manipulations in this data 00:14:42.850 --> 00:14:46.150 align:middle line:84% and hope to get something sensible out of it. 00:14:46.150 --> 00:14:48.100 align:middle line:90% But that's obvious. 00:14:48.100 --> 00:14:51.820 align:middle line:84% OK, why transform data number two-- second reason. 00:14:51.820 --> 00:14:54.580 align:middle line:90% Revealing intrinsic geometries. 00:14:54.580 --> 00:14:56.590 align:middle line:84% What is meant by intrinsic geometry? 00:14:56.590 --> 00:15:00.310 align:middle line:84% It could be a very unstructured data. 00:15:00.310 --> 00:15:03.790 align:middle line:84% You may not notice upon looking at the numbers themselves 00:15:03.790 --> 00:15:09.680 align:middle line:84% in a table that there is a group of variables, 00:15:09.680 --> 00:15:11.350 align:middle line:90% which are acting a certain way. 00:15:11.350 --> 00:15:15.880 align:middle line:84% But it could turn out that if you write it or transform it, 00:15:15.880 --> 00:15:17.470 align:middle line:90% this feature would stand out. 00:15:17.470 --> 00:15:20.180 align:middle line:84% And we actually will see graphical examples 00:15:20.180 --> 00:15:21.055 align:middle line:90% of when that happens. 00:15:21.055 --> 00:15:26.500 align:middle line:90% 00:15:26.500 --> 00:15:32.122 align:middle line:84% Let's see, so there are actually-- 00:15:32.122 --> 00:15:33.830 align:middle line:84% I'm not sure if you need to know if there 00:15:33.830 --> 00:15:35.990 align:middle line:84% are clues to the existence of these structures. 00:15:35.990 --> 00:15:40.280 align:middle line:84% But oftentimes, it's not obvious, 00:15:40.280 --> 00:15:43.385 align:middle line:84% and you could use some help by transforming the data. 00:15:43.385 --> 00:15:46.730 align:middle line:90% 00:15:46.730 --> 00:15:51.080 align:middle line:84% Now, these internal structures may be explicit or implicit. 00:15:51.080 --> 00:15:52.610 align:middle line:90% In what sense? 00:15:52.610 --> 00:15:54.710 align:middle line:84% The explicit structures are clearly 00:15:54.710 --> 00:15:56.930 align:middle line:84% the notation you give, like gene 1, 00:15:56.930 --> 00:16:00.020 align:middle line:84% gene 2, gene 3, or condition 1, 2, and 3. 00:16:00.020 --> 00:16:03.590 align:middle line:84% These are very obvious things of clinical phenotypes. 00:16:03.590 --> 00:16:09.620 align:middle line:84% The implicit ones are the relation between gene-- 00:16:09.620 --> 00:16:12.290 align:middle line:84% how they interact with one another, 00:16:12.290 --> 00:16:14.870 align:middle line:84% at least as captured by these numbers. 00:16:14.870 --> 00:16:16.700 align:middle line:84% Now, that, you don't know for sure. 00:16:16.700 --> 00:16:18.770 align:middle line:84% You actually have to go in and look for them. 00:16:18.770 --> 00:16:21.320 align:middle line:84% But it's just good to keep in mind 00:16:21.320 --> 00:16:24.320 align:middle line:84% that this aspect-- explicit and implicitness 00:16:24.320 --> 00:16:26.300 align:middle line:90% of these structures. 00:16:26.300 --> 00:16:29.190 align:middle line:90% 00:16:29.190 --> 00:16:31.850 align:middle line:90% Well, the whole idea of-- 00:16:31.850 --> 00:16:36.560 align:middle line:84% it's a bit of a jump from data transformation to modeling use. 00:16:36.560 --> 00:16:38.660 align:middle line:84% This is the real-world system, and here 00:16:38.660 --> 00:16:42.530 align:middle line:84% is what exists in our heads, or at least in our computers. 00:16:42.530 --> 00:16:45.170 align:middle line:84% You're trying to understand a system-- 00:16:45.170 --> 00:16:46.610 align:middle line:90% a physical phenomenon, say. 00:16:46.610 --> 00:16:48.830 align:middle line:90% And it's going to be-- 00:16:48.830 --> 00:16:51.200 align:middle line:84% there clearly has to be some physical quantity 00:16:51.200 --> 00:16:52.400 align:middle line:90% one is measuring. 00:16:52.400 --> 00:16:55.460 align:middle line:84% And this physical quantity is subject to some kind 00:16:55.460 --> 00:16:57.410 align:middle line:90% of perturbation. 00:16:57.410 --> 00:16:59.210 align:middle line:90% You make your measurement here. 00:16:59.210 --> 00:17:00.950 align:middle line:84% This is where the data representation 00:17:00.950 --> 00:17:06.260 align:middle line:84% occurs you map it into a set of numbers or symbols, 00:17:06.260 --> 00:17:09.140 align:middle line:90% which there's an algebra on it. 00:17:09.140 --> 00:17:15.470 align:middle line:84% And then any perturbation of this physical system 00:17:15.470 --> 00:17:19.819 align:middle line:84% will manifest itself as numeric fluctuations. 00:17:19.819 --> 00:17:25.160 align:middle line:84% And oftentimes, when you get a lot of data, 00:17:25.160 --> 00:17:28.910 align:middle line:84% it's not clear if you're actually seeing a fluctuation 00:17:28.910 --> 00:17:32.720 align:middle line:84% or you're just seeing, quote, unquote, "noise." 00:17:32.720 --> 00:17:34.910 align:middle line:84% And then you form a model of the system. 00:17:34.910 --> 00:17:36.590 align:middle line:84% And hopefully, your model somehow 00:17:36.590 --> 00:17:39.200 align:middle line:84% will corroborate with the physical phenomenon. 00:17:39.200 --> 00:17:41.900 align:middle line:84% I mean, these are very obvious ideas of modeling. 00:17:41.900 --> 00:17:45.450 align:middle line:90% 00:17:45.450 --> 00:17:48.810 align:middle line:84% Now, I give an example of uncovering internal structures. 00:17:48.810 --> 00:17:52.830 align:middle line:84% Now, let's say that you have two different patient populations, 00:17:52.830 --> 00:17:57.075 align:middle line:84% X and O. Say the X's are cancer patients, 00:17:57.075 --> 00:18:00.360 align:middle line:90% and the O are normal patients. 00:18:00.360 --> 00:18:02.730 align:middle line:84% And they're being controlled for age, gender, 00:18:02.730 --> 00:18:04.630 align:middle line:90% et cetera, things like that. 00:18:04.630 --> 00:18:07.890 align:middle line:84% And for each patient, you make two gene measurements-- 00:18:07.890 --> 00:18:11.050 align:middle line:84% RNA measurements, protein, doesn't really matter. 00:18:11.050 --> 00:18:12.390 align:middle line:90% This is just an illustration. 00:18:12.390 --> 00:18:16.040 align:middle line:84% And let's call the measurements G1 and G2. 00:18:16.040 --> 00:18:19.320 align:middle line:84% OK, so each patient, you measure two of these quantities. 00:18:19.320 --> 00:18:22.950 align:middle line:84% Now, what let's graphically represent this data. 00:18:22.950 --> 00:18:25.830 align:middle line:84% After all, we're talking about data representation, right? 00:18:25.830 --> 00:18:29.910 align:middle line:84% And let's suppose that the measurements came out this way. 00:18:29.910 --> 00:18:33.120 align:middle line:90% 00:18:33.120 --> 00:18:37.240 align:middle line:84% Now, what is the point here that I'm trying to make? 00:18:37.240 --> 00:18:44.460 align:middle line:84% Well, if you were simply using measurement 1 00:18:44.460 --> 00:18:47.220 align:middle line:84% and trying to use measurement 1, somehow, 00:18:47.220 --> 00:18:51.120 align:middle line:84% to discriminate between the crosses and the zeros, 00:18:51.120 --> 00:18:52.650 align:middle line:90% it would not work right. 00:18:52.650 --> 00:18:54.540 align:middle line:90% Do you see why it does not work? 00:18:54.540 --> 00:18:59.130 align:middle line:84% Basically, you simply project the axis and the zeros 00:18:59.130 --> 00:19:03.840 align:middle line:90% onto the G1 axis, here. 00:19:03.840 --> 00:19:08.040 align:middle line:84% If you project it, you notice that there is just 00:19:08.040 --> 00:19:12.750 align:middle line:84% a alternation of crosses and X's, right? 00:19:12.750 --> 00:19:16.500 align:middle line:84% So G1 by itself certainly does not discriminate the cancer 00:19:16.500 --> 00:19:17.820 align:middle line:90% from the normal. 00:19:17.820 --> 00:19:20.880 align:middle line:90% The same goes for quantity G2. 00:19:20.880 --> 00:19:24.300 align:middle line:84% If you project this onto the G2 axis, 00:19:24.300 --> 00:19:28.560 align:middle line:84% you don't see that G2 actually segregates the two 00:19:28.560 --> 00:19:32.140 align:middle line:90% different population samples. 00:19:32.140 --> 00:19:35.550 align:middle line:84% However, now, when you perform something 00:19:35.550 --> 00:19:39.060 align:middle line:84% called principal component analysis, which we will briefly 00:19:39.060 --> 00:19:41.970 align:middle line:84% touch on, but it is very standard in any textbook, 00:19:41.970 --> 00:19:45.690 align:middle line:84% basically, on multivariate analysis, 00:19:45.690 --> 00:19:48.420 align:middle line:90% the data is simply rotated. 00:19:48.420 --> 00:19:50.070 align:middle line:90% It's an Fn transformation. 00:19:50.070 --> 00:19:52.150 align:middle line:90% It is rotation and translation. 00:19:52.150 --> 00:19:54.480 align:middle line:84% And now, in this new coordinates, 00:19:54.480 --> 00:19:58.110 align:middle line:90% principal component 1 and 2-- 00:19:58.110 --> 00:20:00.300 align:middle line:90% this is how it looks. 00:20:00.300 --> 00:20:01.230 align:middle line:90% It's the same picture. 00:20:01.230 --> 00:20:02.400 align:middle line:90% And what do you see? 00:20:02.400 --> 00:20:03.540 align:middle line:90% It's the same thing. 00:20:03.540 --> 00:20:04.800 align:middle line:90% You're just rotating. 00:20:04.800 --> 00:20:08.580 align:middle line:84% But the simple act of rotation itself is highly useful. 00:20:08.580 --> 00:20:09.420 align:middle line:90% Why? 00:20:09.420 --> 00:20:14.670 align:middle line:84% Well, you notice, then, that while PC1 does not 00:20:14.670 --> 00:20:19.030 align:middle line:84% distinguish the two populations, PC2 certainly does. 00:20:19.030 --> 00:20:23.010 align:middle line:84% And the discriminating quantity is G1 minus G2. 00:20:23.010 --> 00:20:27.810 align:middle line:84% If G1 minus G2 is positive or negative, you are either cancer 00:20:27.810 --> 00:20:29.350 align:middle line:90% or you're normal. 00:20:29.350 --> 00:20:30.990 align:middle line:84% This is the beauty of this technique. 00:20:30.990 --> 00:20:32.370 align:middle line:90% And why is this? 00:20:32.370 --> 00:20:34.740 align:middle line:84% So this is a very kindergarten example, 00:20:34.740 --> 00:20:36.810 align:middle line:84% but there is something to be learnt from here. 00:20:36.810 --> 00:20:40.440 align:middle line:84% Because as human beings, we can't visualize beyond three 00:20:40.440 --> 00:20:41.730 align:middle line:90% or four dimensions. 00:20:41.730 --> 00:20:45.870 align:middle line:84% And when you're making multiple measurements and-- 00:20:45.870 --> 00:20:49.770 align:middle line:84% so imagine that you are a person who lives in one dimension, 00:20:49.770 --> 00:20:52.810 align:middle line:84% and you are making these two measurements. 00:20:52.810 --> 00:20:56.220 align:middle line:84% Your vision of this whole process 00:20:56.220 --> 00:21:00.600 align:middle line:84% is only somehow captured in these projections 00:21:00.600 --> 00:21:01.950 align:middle line:90% under G1 and G2. 00:21:01.950 --> 00:21:05.400 align:middle line:84% So in your one-dimensional world, or in mine, 00:21:05.400 --> 00:21:08.070 align:middle line:84% I would not have realized that G1 and G2 actually 00:21:08.070 --> 00:21:09.430 align:middle line:90% does anything. 00:21:09.430 --> 00:21:12.210 align:middle line:84% However, this transformation helps me. 00:21:12.210 --> 00:21:16.170 align:middle line:84% Because then, using this one dimension alone-- remember, 00:21:16.170 --> 00:21:17.760 align:middle line:90% I am a one-dimensional beast-- 00:21:17.760 --> 00:21:21.120 align:middle line:84% I can tell that this quantity here 00:21:21.120 --> 00:21:23.550 align:middle line:84% is a linear combination actually distinguishes one 00:21:23.550 --> 00:21:25.710 align:middle line:90% from the other. 00:21:25.710 --> 00:21:30.030 align:middle line:84% And imagine, now, you have 10,000 variables, not just 00:21:30.030 --> 00:21:30.600 align:middle line:90% three. 00:21:30.600 --> 00:21:36.840 align:middle line:84% And the power of this method immediately comes to the fore. 00:21:36.840 --> 00:21:44.050 align:middle line:84% Another example-- say you have two different-- 00:21:44.050 --> 00:21:46.530 align:middle line:90% so each dot is a gene. 00:21:46.530 --> 00:21:51.330 align:middle line:84% And you have-- let's just suppose that you have two 00:21:51.330 --> 00:21:56.010 align:middle line:84% patients, or two persons, or two animals-- 00:21:56.010 --> 00:21:57.420 align:middle line:90% red and blue animals. 00:21:57.420 --> 00:22:03.060 align:middle line:84% And for each animal, you measure 5,000 genes. 00:22:03.060 --> 00:22:05.370 align:middle line:84% And these 5,000 genes are measured 00:22:05.370 --> 00:22:07.210 align:middle line:90% under three conditions. 00:22:07.210 --> 00:22:12.510 align:middle line:84% So you could get a brown mouse and a white mouse-- 00:22:12.510 --> 00:22:17.040 align:middle line:84% microarray 5,000 protein levels or something-- 00:22:17.040 --> 00:22:18.570 align:middle line:84% under three different conditions, 00:22:18.570 --> 00:22:24.000 align:middle line:84% say, heat shock, starvation, exercise. 00:22:24.000 --> 00:22:25.290 align:middle line:90% And then you plot it. 00:22:25.290 --> 00:22:31.860 align:middle line:90% 00:22:31.860 --> 00:22:33.120 align:middle line:90% I hope you're not colorblind. 00:22:33.120 --> 00:22:37.680 align:middle line:84% But let's suppose that the red and blue are not there. 00:22:37.680 --> 00:22:39.160 align:middle line:90% All you see is one color. 00:22:39.160 --> 00:22:42.270 align:middle line:90% And you did this measurement. 00:22:42.270 --> 00:22:44.790 align:middle line:84% And let's say you have a three-dimensional vision 00:22:44.790 --> 00:22:47.160 align:middle line:90% of this whole process. 00:22:47.160 --> 00:22:50.043 align:middle line:84% Look at the projections here, here, and here-- 00:22:50.043 --> 00:22:51.210 align:middle line:90% two-dimensional projections. 00:22:51.210 --> 00:22:55.980 align:middle line:84% You don't actually see that blue and red are just together, 00:22:55.980 --> 00:22:57.270 align:middle line:90% right? 00:22:57.270 --> 00:22:59.880 align:middle line:84% Now, if you do the same principle component 00:22:59.880 --> 00:23:03.480 align:middle line:84% analysis, simply rotating the data, remember, 00:23:03.480 --> 00:23:05.920 align:middle line:90% is simply an Fn transformation. 00:23:05.920 --> 00:23:07.980 align:middle line:84% The first projection, you see nothing. 00:23:07.980 --> 00:23:11.220 align:middle line:84% But this is where the power comes. 00:23:11.220 --> 00:23:14.490 align:middle line:84% Again, you have to somehow remove the green-- 00:23:14.490 --> 00:23:18.900 align:middle line:84% the blue and the red, and you see that clearly. 00:23:18.900 --> 00:23:21.300 align:middle line:84% The two populations reveal themselves. 00:23:21.300 --> 00:23:24.040 align:middle line:84% In a lower dimension, that's the whole point. 00:23:24.040 --> 00:23:27.660 align:middle line:84% So we are cutting down from three to two-- 00:23:27.660 --> 00:23:29.082 align:middle line:90% basically just one dimension. 00:23:29.082 --> 00:23:31.290 align:middle line:84% Clearly, one dimension actually distinguishes the two 00:23:31.290 --> 00:23:32.550 align:middle line:90% different populations. 00:23:32.550 --> 00:23:36.150 align:middle line:84% The power is you could have 10,000 dimensions. 00:23:36.150 --> 00:23:40.330 align:middle line:84% It cuts it down all the way to the first five or 10. 00:23:40.330 --> 00:23:42.417 align:middle line:90% It depends on the system. 00:23:42.417 --> 00:23:43.500 align:middle line:90% It's just a demonstration. 00:23:43.500 --> 00:23:45.600 align:middle line:84% But this is simulated data, by the way. 00:23:45.600 --> 00:23:46.395 align:middle line:90% It's not real. 00:23:46.395 --> 00:23:50.550 align:middle line:90% 00:23:50.550 --> 00:23:52.320 align:middle line:84% Data transformation example two-- 00:23:52.320 --> 00:23:55.200 align:middle line:84% I'm not quite sure now why I give 00:23:55.200 --> 00:23:58.410 align:middle line:84% you all this principal component analysis examples. 00:23:58.410 --> 00:24:01.920 align:middle line:90% But this is actually real data-- 00:24:01.920 --> 00:24:04.830 align:middle line:84% pancreatic development time series. 00:24:04.830 --> 00:24:12.150 align:middle line:84% And at-- I think it was 11 time points versus something 00:24:12.150 --> 00:24:13.770 align:middle line:90% like 10,000 genes, I believe. 00:24:13.770 --> 00:24:16.210 align:middle line:90% I'm not quite sure. 00:24:16.210 --> 00:24:17.670 align:middle line:90% So it's a matrix. 00:24:17.670 --> 00:24:21.180 align:middle line:84% Now, there are two ways to look at the system. 00:24:21.180 --> 00:24:27.180 align:middle line:84% It's basically, say, 10,000 genes by 11 conditions. 00:24:27.180 --> 00:24:37.180 align:middle line:84% You can look at it as time points in a gene space-- 00:24:37.180 --> 00:24:38.710 align:middle line:90% 10,000 dimensions. 00:24:38.710 --> 00:24:41.650 align:middle line:84% That's one vision of this experiment. 00:24:41.650 --> 00:24:44.170 align:middle line:84% The other vision is the transpose vision, 00:24:44.170 --> 00:24:48.040 align:middle line:84% which is that the objects, the graphical objects number 00:24:48.040 --> 00:24:51.730 align:middle line:84% are genes, and they live in 11-dimensional sample 00:24:51.730 --> 00:24:54.527 align:middle line:90% space or temporal space. 00:24:54.527 --> 00:24:55.360 align:middle line:90% I hope you see that. 00:24:55.360 --> 00:24:58.690 align:middle line:84% So there are two ways to look at this study. 00:24:58.690 --> 00:25:02.440 align:middle line:84% And the two ways are, when you do further analysis on them, 00:25:02.440 --> 00:25:06.550 align:middle line:84% will actually bring out different aspects 00:25:06.550 --> 00:25:11.200 align:middle line:90% of the experiment. 00:25:11.200 --> 00:25:19.090 align:middle line:84% So you can look at the system sample-wise, meaning that dots, 00:25:19.090 --> 00:25:24.010 align:middle line:84% genes in time-space, or you can look at it gene-wise-- 00:25:24.010 --> 00:25:26.500 align:middle line:84% actually, their names could have been switched-- 00:25:26.500 --> 00:25:31.900 align:middle line:84% which is each point is a time, is a whole pancreas, 00:25:31.900 --> 00:25:35.050 align:middle line:84% and the space it's sitting on is actually genes. 00:25:35.050 --> 00:25:36.910 align:middle line:90% So it's like 10,000 dimensions. 00:25:36.910 --> 00:25:39.040 align:middle line:90% And CLT is, basically-- 00:25:39.040 --> 00:25:40.780 align:middle line:90% Central Limit Theorem-- scaling. 00:25:40.780 --> 00:25:43.060 align:middle line:84% It's a fancy term for saying, you 00:25:43.060 --> 00:25:45.917 align:middle line:84% normalize the data to mean 0 variance 1. 00:25:45.917 --> 00:25:47.500 align:middle line:84% There are reasons that actually do it, 00:25:47.500 --> 00:25:48.940 align:middle line:90% but we come to that later. 00:25:48.940 --> 00:25:51.660 align:middle line:90% 00:25:51.660 --> 00:25:56.400 align:middle line:84% So the vision of each dot being a gene sitting in time-space-- 00:25:56.400 --> 00:25:59.570 align:middle line:84% I'm only showing you 3 times 11, OK? 00:25:59.570 --> 00:26:01.370 align:middle line:90% Well, when you do that-- 00:26:01.370 --> 00:26:04.100 align:middle line:84% when you don't do anything at all, you just plot it, 00:26:04.100 --> 00:26:07.670 align:middle line:90% you get this cloud. 00:26:07.670 --> 00:26:10.550 align:middle line:84% It's not very informative, really. 00:26:10.550 --> 00:26:12.860 align:middle line:84% Maybe there is some information, but I didn't go deeper 00:26:12.860 --> 00:26:16.700 align:middle line:90% into see what these things did. 00:26:16.700 --> 00:26:20.030 align:middle line:84% Now, when you do a principal component analysis 00:26:20.030 --> 00:26:24.620 align:middle line:84% of the time axis and re-plot this, 00:26:24.620 --> 00:26:27.130 align:middle line:84% what you see is a circular object. 00:26:27.130 --> 00:26:30.170 align:middle line:84% And the reason it's actually circular or looks like an egg 00:26:30.170 --> 00:26:32.630 align:middle line:90% is because of the scaling. 00:26:32.630 --> 00:26:36.110 align:middle line:84% This scaling, if you know linear algebra, 00:26:36.110 --> 00:26:38.720 align:middle line:84% actually maps everything to the unit hypersphere, 00:26:38.720 --> 00:26:42.110 align:middle line:84% which is why it's not a surprise you get an egg-shape like this. 00:26:42.110 --> 00:26:45.810 align:middle line:84% But why is this more informative than the previous one, 00:26:45.810 --> 00:26:47.630 align:middle line:90% the previous slide? 00:26:47.630 --> 00:26:48.720 align:middle line:90% There is a reason. 00:26:48.720 --> 00:26:53.150 align:middle line:84% So I will claim, actually, that the density 00:26:53.150 --> 00:26:55.700 align:middle line:84% of the-- oh, and the first principal component 00:26:55.700 --> 00:27:00.000 align:middle line:90% captures 45% of the variance. 00:27:00.000 --> 00:27:02.550 align:middle line:84% The second principal component captures 15%. 00:27:02.550 --> 00:27:04.460 align:middle line:84% So the idea of principal component analysis 00:27:04.460 --> 00:27:08.043 align:middle line:90% is that as you-- 00:27:08.043 --> 00:27:10.460 align:middle line:84% the first principal component captures the greatest amount 00:27:10.460 --> 00:27:11.010 align:middle line:90% of variance. 00:27:11.010 --> 00:27:13.580 align:middle line:84% The second captures the second greatest, third, et cetera. 00:27:13.580 --> 00:27:15.455 align:middle line:84% They are all orthogonal, one to the other. 00:27:15.455 --> 00:27:19.100 align:middle line:90% 00:27:19.100 --> 00:27:23.240 align:middle line:84% So the utility of doing this is now you can, I claim, 00:27:23.240 --> 00:27:27.470 align:middle line:84% that you can actually represent 10,000 different profiles 00:27:27.470 --> 00:27:30.500 align:middle line:84% in a convenient egg-shape like this. 00:27:30.500 --> 00:27:32.000 align:middle line:90% How so? 00:27:32.000 --> 00:27:36.240 align:middle line:84% Well, the first component captures a lot 00:27:36.240 --> 00:27:37.690 align:middle line:90% of variance in the system. 00:27:37.690 --> 00:27:42.180 align:middle line:84% And so what is the profile of a gene? 00:27:42.180 --> 00:27:45.690 align:middle line:84% If you pick any gene from here, say, and you plot its profile, 00:27:45.690 --> 00:27:46.950 align:middle line:90% how does it look like? 00:27:46.950 --> 00:27:50.520 align:middle line:84% It turns out that the profile looks like that. 00:27:50.520 --> 00:27:53.820 align:middle line:84% So in English, essentially, it is 00:27:53.820 --> 00:27:58.290 align:middle line:84% a gene of a protein which is highly-expressed early on 00:27:58.290 --> 00:28:01.110 align:middle line:90% and it goes down later on. 00:28:01.110 --> 00:28:02.730 align:middle line:84% What about if you pick something here 00:28:02.730 --> 00:28:06.030 align:middle line:84% from the complete opposite end of PC1? 00:28:06.030 --> 00:28:08.250 align:middle line:90% That's the shape that you get. 00:28:08.250 --> 00:28:10.060 align:middle line:90% It goes the opposite direction. 00:28:10.060 --> 00:28:14.820 align:middle line:84% In fact, if you picked samples going all the way from here 00:28:14.820 --> 00:28:18.390 align:middle line:84% to there, you will notice that there is a gradual shifting 00:28:18.390 --> 00:28:19.710 align:middle line:90% of this shape to that. 00:28:19.710 --> 00:28:23.400 align:middle line:90% It morphs one into the other. 00:28:23.400 --> 00:28:26.250 align:middle line:84% And so this is an example of something picked out of the 90 00:28:26.250 --> 00:28:29.160 align:middle line:90% degrees, counting from 12:00. 00:28:29.160 --> 00:28:30.383 align:middle line:90% That's how it looks like. 00:28:30.383 --> 00:28:32.550 align:middle line:84% So it's a very convenient way to display everything. 00:28:32.550 --> 00:28:35.700 align:middle line:84% And another utility is a density of dots. 00:28:35.700 --> 00:28:38.610 align:middle line:84% You notice that there is a huge absence of anything 00:28:38.610 --> 00:28:39.570 align:middle line:90% here, right? 00:28:39.570 --> 00:28:41.100 align:middle line:84% So basically, you could state-- you 00:28:41.100 --> 00:28:47.010 align:middle line:84% could claim that there is a family of profiles 00:28:47.010 --> 00:28:51.030 align:middle line:84% that is actually missing from all these genes. 00:28:51.030 --> 00:28:54.330 align:middle line:84% No genes at all express a theoretical profile here. 00:28:54.330 --> 00:28:57.580 align:middle line:84% I think it is something which looks like that 00:28:57.580 --> 00:29:00.390 align:middle line:90% but with some variation. 00:29:00.390 --> 00:29:02.290 align:middle line:90% But there is a density here. 00:29:02.290 --> 00:29:04.800 align:middle line:84% I'm not sure how that looks like. 00:29:04.800 --> 00:29:08.130 align:middle line:84% So it's a convenient way to display everything 00:29:08.130 --> 00:29:09.210 align:middle line:90% as opposed to what? 00:29:09.210 --> 00:29:14.860 align:middle line:84% As opposed to seeing 10,000 of these things. 00:29:14.860 --> 00:29:16.180 align:middle line:90% So it's like a dictionary. 00:29:16.180 --> 00:29:19.390 align:middle line:90% 00:29:19.390 --> 00:29:23.180 align:middle line:84% So that was looking at genes sitting in time-space. 00:29:23.180 --> 00:29:25.390 align:middle line:84% The other transpose way of looking 00:29:25.390 --> 00:29:27.790 align:middle line:84% at the same system, the same data 00:29:27.790 --> 00:29:31.150 align:middle line:84% is to view it as samples, time points. 00:29:31.150 --> 00:29:34.420 align:middle line:84% I've labeled them 1 through-- oh, actually, it's 13-- 00:29:34.420 --> 00:29:38.650 align:middle line:84% 1 through 13 sitting in a genomic space 00:29:38.650 --> 00:29:41.200 align:middle line:90% of 10,000 dimensions. 00:29:41.200 --> 00:29:44.800 align:middle line:84% If you just picked any three genes at random 00:29:44.800 --> 00:29:48.280 align:middle line:90% and plotted the samples-- 00:29:48.280 --> 00:29:51.550 align:middle line:84% the numbers 1 through 13 are consecutive. 00:29:51.550 --> 00:29:53.500 align:middle line:84% I didn't just randomly assign them. 00:29:53.500 --> 00:29:55.010 align:middle line:90% They're consecutive with time. 00:29:55.010 --> 00:29:58.030 align:middle line:84% So time 13 is going to be greater 00:29:58.030 --> 00:30:02.950 align:middle line:84% than time 12, greater than time 11, et cetera, et cetera. 00:30:02.950 --> 00:30:06.948 align:middle line:84% So you've picked any two genes, and what do you notice? 00:30:06.948 --> 00:30:07.990 align:middle line:90% Well, you notice nothing. 00:30:07.990 --> 00:30:11.110 align:middle line:90% It's just a mess. 00:30:11.110 --> 00:30:13.845 align:middle line:84% You pick any three, and I'm just picking three, for example. 00:30:13.845 --> 00:30:15.220 align:middle line:84% I mean, you'd be very lucky if we 00:30:15.220 --> 00:30:17.770 align:middle line:84% pick a set of three that actually 00:30:17.770 --> 00:30:19.900 align:middle line:84% reveals some beautiful structure in here. 00:30:19.900 --> 00:30:24.370 align:middle line:84% But then you have to wonder, what is our noise here? 00:30:24.370 --> 00:30:26.110 align:middle line:84% That goes back to understanding, what 00:30:26.110 --> 00:30:29.330 align:middle line:84% is the underlying null hypothesis of the system? 00:30:29.330 --> 00:30:33.740 align:middle line:84% So now you perform principal component analysis. 00:30:33.740 --> 00:30:36.750 align:middle line:90% What happens? 00:30:36.750 --> 00:30:41.880 align:middle line:84% The first principal component, second, and third. 00:30:41.880 --> 00:30:46.440 align:middle line:84% The most salient thing that jumps out at you 00:30:46.440 --> 00:30:48.780 align:middle line:90% is that PC1 looks-- 00:30:48.780 --> 00:30:55.140 align:middle line:84% looks-- to be correlated with time. 00:30:55.140 --> 00:31:00.040 align:middle line:84% You get 1, 2, 3, 4, 5, 6, 7-- well, 12 is an anomaly. 00:31:00.040 --> 00:31:01.740 align:middle line:84% I don't know why it landed there-- 00:31:01.740 --> 00:31:03.840 align:middle line:90% and 13. 00:31:03.840 --> 00:31:08.940 align:middle line:84% So I'm not sure what the other principal components mean. 00:31:08.940 --> 00:31:11.970 align:middle line:84% Maybe there is some biologic import to them. 00:31:11.970 --> 00:31:13.140 align:middle line:90% It's unclear to me. 00:31:13.140 --> 00:31:18.150 align:middle line:84% But certainly, one captures the progression of time, 00:31:18.150 --> 00:31:19.080 align:middle line:90% it looks to me. 00:31:19.080 --> 00:31:27.480 align:middle line:84% And you are sort of immune from the possibility 00:31:27.480 --> 00:31:29.160 align:middle line:84% that this is actually due to noise. 00:31:29.160 --> 00:31:29.820 align:middle line:90% Why? 00:31:29.820 --> 00:31:32.790 align:middle line:84% Because a principal component, now, it's 00:31:32.790 --> 00:31:34.103 align:middle line:90% not just a single gene or two. 00:31:34.103 --> 00:31:35.520 align:middle line:84% It's actually a linear combination 00:31:35.520 --> 00:31:37.440 align:middle line:90% of something like 10,000. 00:31:37.440 --> 00:31:39.570 align:middle line:84% So that's the power of the method. 00:31:39.570 --> 00:31:41.460 align:middle line:84% Had you just picked any tree randomly 00:31:41.460 --> 00:31:44.237 align:middle line:84% and found a configuration like this, 00:31:44.237 --> 00:31:46.320 align:middle line:84% you have to wonder at the randomness of this thing 00:31:46.320 --> 00:31:46.820 align:middle line:90% happening. 00:31:46.820 --> 00:31:49.500 align:middle line:84% But this is a linear combination of all of them. 00:31:49.500 --> 00:31:51.600 align:middle line:84% This is the power of this methodology. 00:31:51.600 --> 00:31:55.320 align:middle line:90% 00:31:55.320 --> 00:31:58.800 align:middle line:84% I give you another example, but this is very kindergarten. 00:31:58.800 --> 00:32:02.310 align:middle line:84% So Fourier decomposition-- Fourier 00:32:02.310 --> 00:32:08.310 align:middle line:84% transforms-- that is another way of transforming data to reveal 00:32:08.310 --> 00:32:09.720 align:middle line:90% structures within the data. 00:32:09.720 --> 00:32:12.690 align:middle line:84% And the point of doing Fourier analysis 00:32:12.690 --> 00:32:16.560 align:middle line:90% is, you want to reveal-- 00:32:16.560 --> 00:32:20.430 align:middle line:84% the objects that you're looking for are basically frequencies. 00:32:20.430 --> 00:32:23.760 align:middle line:90% So let's say that you have a-- 00:32:23.760 --> 00:32:25.420 align:middle line:84% is not real-world data, of course. 00:32:25.420 --> 00:32:27.900 align:middle line:84% It was completely cooked out of a machine. 00:32:27.900 --> 00:32:36.330 align:middle line:84% If you have this red sinusoid here of period 1, clearly, 00:32:36.330 --> 00:32:39.120 align:middle line:84% the frequency is just one frequency. 00:32:39.120 --> 00:32:41.010 align:middle line:84% I don't know if sequence is something 00:32:41.010 --> 00:32:43.380 align:middle line:90% 1 over 2 pi or whatnot. 00:32:43.380 --> 00:32:46.740 align:middle line:84% When you apply Fourier transforms on it-- 00:32:46.740 --> 00:32:50.370 align:middle line:84% well, discrete, fast Fourier transform-- 00:32:50.370 --> 00:32:54.730 align:middle line:84% you will find you enter the realm of complex numbers, 00:32:54.730 --> 00:32:55.230 align:middle line:90% actually. 00:32:55.230 --> 00:33:01.350 align:middle line:84% But suffice to know that you get a point in frequency space. 00:33:01.350 --> 00:33:03.180 align:middle line:90% That's the point. 00:33:03.180 --> 00:33:05.280 align:middle line:84% And to give you some bearing, let's 00:33:05.280 --> 00:33:10.458 align:middle line:84% take another waveform, which is twice the frequency. 00:33:10.458 --> 00:33:12.000 align:middle line:84% The frequency shouldn't be surprising 00:33:12.000 --> 00:33:13.230 align:middle line:90% that it's actually twice. 00:33:13.230 --> 00:33:15.660 align:middle line:90% It's 5, 2.5. 00:33:15.660 --> 00:33:19.830 align:middle line:84% And now you take yet another waveform which is even faster. 00:33:19.830 --> 00:33:22.500 align:middle line:90% 00:33:22.500 --> 00:33:26.280 align:middle line:84% So the mapping-- so this is the entire waveform maps 00:33:26.280 --> 00:33:29.176 align:middle line:84% to just one point, one point, one point. 00:33:29.176 --> 00:33:31.980 align:middle line:84% So the object of interest here are frequencies. 00:33:31.980 --> 00:33:33.780 align:middle line:90% It's not localized in time. 00:33:33.780 --> 00:33:37.800 align:middle line:84% For localization, the are transformation techniques 00:33:37.800 --> 00:33:39.430 align:middle line:90% such as wavelets. 00:33:39.430 --> 00:33:42.080 align:middle line:90% 00:33:42.080 --> 00:33:45.980 align:middle line:84% Now, of course, the world does not give you data 00:33:45.980 --> 00:33:49.010 align:middle line:84% so nicely in this uniform, band-limited, 00:33:49.010 --> 00:33:51.050 align:middle line:90% three signals like this, right? 00:33:51.050 --> 00:33:57.320 align:middle line:84% So let's suppose that the world makes it more complicated, 00:33:57.320 --> 00:33:59.180 align:middle line:90% adds up these three sinusoids. 00:33:59.180 --> 00:34:03.360 align:middle line:84% When you add up these three sinusoids, what happens? 00:34:03.360 --> 00:34:06.080 align:middle line:90% This is what you would see. 00:34:06.080 --> 00:34:08.090 align:middle line:84% Now, let's say the world presents you with this. 00:34:08.090 --> 00:34:11.502 align:middle line:90% 00:34:11.502 --> 00:34:12.710 align:middle line:90% There are many things can do. 00:34:12.710 --> 00:34:14.752 align:middle line:84% You can actually do principal component analysis, 00:34:14.752 --> 00:34:17.120 align:middle line:84% but the question is, what are you trying to look for? 00:34:17.120 --> 00:34:20.840 align:middle line:84% If you are looking for the predominant frequencies 00:34:20.840 --> 00:34:23.630 align:middle line:84% embedded in this waveform, the most natural thing to do 00:34:23.630 --> 00:34:25.380 align:middle line:90% is Fourier analysis. 00:34:25.380 --> 00:34:26.969 align:middle line:84% And when you do Fourier analysis, 00:34:26.969 --> 00:34:30.020 align:middle line:84% it should not shock you that the answer, when 00:34:30.020 --> 00:34:33.973 align:middle line:84% you map it to frequency space, is three dots. 00:34:33.973 --> 00:34:35.765 align:middle line:84% Happens to be the same three dots up there. 00:34:35.765 --> 00:34:38.270 align:middle line:90% 00:34:38.270 --> 00:34:41.219 align:middle line:90% So that's the beauty of it. 00:34:41.219 --> 00:34:43.760 align:middle line:84% And there are applications, very real applications, 00:34:43.760 --> 00:34:50.659 align:middle line:84% actually, when you enter into sequential genomics. 00:34:50.659 --> 00:34:53.030 align:middle line:84% Because the alphabets, A, T, C, G 00:34:53.030 --> 00:34:56.210 align:middle line:84% can easily be mapped into 0, 1, 2, 3. 00:34:56.210 --> 00:34:58.260 align:middle line:90% Of course, the ordering-- 00:34:58.260 --> 00:35:01.310 align:middle line:84% I'm not quite sure if the ordering actually matters 00:35:01.310 --> 00:35:04.470 align:middle line:84% or the ordinality, but I don't think it does. 00:35:04.470 --> 00:35:11.630 align:middle line:84% And if you are interested in repeating structures 00:35:11.630 --> 00:35:14.570 align:middle line:84% in the genome, it could be helpful. 00:35:14.570 --> 00:35:18.930 align:middle line:84% I'm sure people have done this, in fact. 00:35:18.930 --> 00:35:20.880 align:middle line:84% So the summary of data transformation 00:35:20.880 --> 00:35:27.840 align:middle line:84% is, basically, somebody gives you a vector x, data x. 00:35:27.840 --> 00:35:34.350 align:middle line:84% And you simply rewrite it into a different form 00:35:34.350 --> 00:35:41.270 align:middle line:84% based on a set of basis elements that 00:35:41.270 --> 00:35:42.890 align:middle line:84% are different from the original ones. 00:35:42.890 --> 00:35:43.910 align:middle line:90% How do I say it? 00:35:43.910 --> 00:35:45.980 align:middle line:84% Typically, when somebody gives you data, 00:35:45.980 --> 00:35:49.430 align:middle line:84% there's going to be real numbers, let's say a matrix. 00:35:49.430 --> 00:35:53.540 align:middle line:84% And the standard basis is the basis 00:35:53.540 --> 00:35:58.010 align:middle line:84% they're thinking of, that 10000100, et cetera. 00:35:58.010 --> 00:36:00.380 align:middle line:84% What you can do is actually transform it 00:36:00.380 --> 00:36:03.680 align:middle line:84% by principal component analysis, or Fourier transforms, or even 00:36:03.680 --> 00:36:04.550 align:middle line:90% wavelets. 00:36:04.550 --> 00:36:08.300 align:middle line:84% All of these techniques are simply 00:36:08.300 --> 00:36:11.330 align:middle line:84% names to describe these basis elements. 00:36:11.330 --> 00:36:14.450 align:middle line:84% They are actual new representation 00:36:14.450 --> 00:36:16.050 align:middle line:90% forms are come out. 00:36:16.050 --> 00:36:20.090 align:middle line:90% But in these A subjects-- 00:36:20.090 --> 00:36:23.105 align:middle line:84% you can actually read about this in several textbooks 00:36:23.105 --> 00:36:25.820 align:middle line:84% that say, do the heart and pattern 00:36:25.820 --> 00:36:33.500 align:middle line:84% recognition, which I shall provide in leaflet handouts. 00:36:33.500 --> 00:36:37.220 align:middle line:84% And clearly, not all these transformation techniques 00:36:37.220 --> 00:36:37.760 align:middle line:90% are equal. 00:36:37.760 --> 00:36:41.180 align:middle line:84% They are going to reveal for you very different things, 00:36:41.180 --> 00:36:43.550 align:middle line:84% very different internal structures in the data. 00:36:43.550 --> 00:36:44.945 align:middle line:90% That should be obvious. 00:36:44.945 --> 00:36:47.660 align:middle line:90% 00:36:47.660 --> 00:36:50.900 align:middle line:84% And I claim that there is almost always 00:36:50.900 --> 00:36:56.090 align:middle line:84% a geometric interpretation of any given data set. 00:36:56.090 --> 00:36:58.880 align:middle line:84% Secondary users would be denoising. 00:36:58.880 --> 00:37:00.890 align:middle line:84% And feature reduction we have seen, actually, 00:37:00.890 --> 00:37:01.880 align:middle line:90% in the case of PCA. 00:37:01.880 --> 00:37:06.260 align:middle line:84% Denoising is, say, Fourier transforms, 00:37:06.260 --> 00:37:10.280 align:middle line:84% you could-- if you believe that noise are higher frequencies, 00:37:10.280 --> 00:37:12.230 align:middle line:84% you could band limit or you could 00:37:12.230 --> 00:37:17.055 align:middle line:84% model all the dots which appear much higher in that frequency 00:37:17.055 --> 00:37:17.555 align:middle line:90% space. 00:37:17.555 --> 00:37:20.150 align:middle line:90% 00:37:20.150 --> 00:37:24.080 align:middle line:84% No, I don't think I'll talk about this. 00:37:24.080 --> 00:37:28.460 align:middle line:84% Do we actually have time for the next part? 00:37:28.460 --> 00:37:31.130 align:middle line:90% We do? 00:37:31.130 --> 00:37:33.800 align:middle line:84% Now that we are done with data representation, 00:37:33.800 --> 00:37:37.700 align:middle line:84% I'm going to try to cover some common terms that 00:37:37.700 --> 00:37:41.840 align:middle line:84% occurs over and over again in the area of microarray analysis 00:37:41.840 --> 00:37:43.100 align:middle line:90% or genomics. 00:37:43.100 --> 00:37:45.620 align:middle line:84% You hear it all the time, but you wonder what they mean. 00:37:45.620 --> 00:37:47.660 align:middle line:84% I sometimes wonder what they mean, 00:37:47.660 --> 00:37:51.260 align:middle line:84% but this is my understanding, at least, of what may mean. 00:37:51.260 --> 00:37:53.160 align:middle line:84% A very important thing I believe, 00:37:53.160 --> 00:37:56.960 align:middle line:84% as do a lot of people, I'm sure, is that nature makes no leaps, 00:37:56.960 --> 00:38:00.260 align:middle line:84% that physical phenomenon, at least at the microscopic level, 00:38:00.260 --> 00:38:01.370 align:middle line:90% what you observe-- 00:38:01.370 --> 00:38:04.130 align:middle line:84% microscopic atoms bumping around-- 00:38:04.130 --> 00:38:08.570 align:middle line:84% at least at a microscopic level, it cannot occur abruptly. 00:38:08.570 --> 00:38:10.980 align:middle line:84% There has to be a continuity to these processes. 00:38:10.980 --> 00:38:13.790 align:middle line:84% And this is a very important guiding principle, at least, 00:38:13.790 --> 00:38:17.780 align:middle line:90% in definition of noise. 00:38:17.780 --> 00:38:19.210 align:middle line:90% So I'll give an example. 00:38:19.210 --> 00:38:20.960 align:middle line:84% I don't know why I called it example four. 00:38:20.960 --> 00:38:24.680 align:middle line:84% I make 100 separate measurements of the room temperature 00:38:24.680 --> 00:38:27.650 align:middle line:84% in this room in a 1-minute interval 00:38:27.650 --> 00:38:30.080 align:middle line:90% at different locations. 00:38:30.080 --> 00:38:32.990 align:middle line:84% Depending on the accuracy of the device I'm using, 00:38:32.990 --> 00:38:35.060 align:middle line:84% it is not very likely that all these measurements 00:38:35.060 --> 00:38:37.650 align:middle line:90% are going to be the same. 00:38:37.650 --> 00:38:40.190 align:middle line:90% So the question is-- 00:38:40.190 --> 00:38:41.708 align:middle line:84% and it is an ill-posed question-- 00:38:41.708 --> 00:38:43.250 align:middle line:84% what is the temperature in this room? 00:38:43.250 --> 00:38:46.040 align:middle line:84% On average, or what is the temperature distribution 00:38:46.040 --> 00:38:47.300 align:middle line:90% in this room? 00:38:47.300 --> 00:38:48.860 align:middle line:90% Questions like this. 00:38:48.860 --> 00:38:53.330 align:middle line:84% Now, this is my working definition of noise. 00:38:53.330 --> 00:38:57.110 align:middle line:84% And I'm sure that there are going to be-- 00:38:57.110 --> 00:38:58.700 align:middle line:90% it can be argued. 00:38:58.700 --> 00:39:02.420 align:middle line:84% In a narrow sense, noise is any measurable divergence 00:39:02.420 --> 00:39:03.470 align:middle line:90% from axiom 1-- 00:39:03.470 --> 00:39:07.490 align:middle line:84% this idea-- or more generally, any applicable axiom 00:39:07.490 --> 00:39:11.880 align:middle line:90% in a studied system. 00:39:11.880 --> 00:39:13.370 align:middle line:90% So if you believe that-- 00:39:13.370 --> 00:39:17.450 align:middle line:84% if you are very, very sure that the room temperature-- 00:39:17.450 --> 00:39:21.020 align:middle line:84% example, if you believe that the room temperature in this room 00:39:21.020 --> 00:39:26.000 align:middle line:84% cannot be so different from me from here to there, 00:39:26.000 --> 00:39:28.270 align:middle line:90% then you make 100 measurements. 00:39:28.270 --> 00:39:31.640 align:middle line:90% 00:39:31.640 --> 00:39:34.220 align:middle line:84% And you believe that there is an idealized temperature 00:39:34.220 --> 00:39:37.880 align:middle line:84% or some belief that there should be a static quantity. 00:39:37.880 --> 00:39:42.950 align:middle line:84% Any fluctuation, any fluctuating observations 00:39:42.950 --> 00:39:46.900 align:middle line:84% you made away from this idealized temperature-- which 00:39:46.900 --> 00:39:47.900 align:middle line:90% you don't know, anyways. 00:39:47.900 --> 00:39:50.360 align:middle line:84% The best you can do is estimate with the average-- 00:39:50.360 --> 00:39:52.820 align:middle line:90% any fluctuation is noise. 00:39:52.820 --> 00:39:54.800 align:middle line:90% So it's practical. 00:39:54.800 --> 00:39:57.210 align:middle line:84% I'm not sure if it is useful at all. 00:39:57.210 --> 00:40:00.410 align:middle line:84% And in ideal situations, math theorems 00:40:00.410 --> 00:40:03.740 align:middle line:84% will apply-- things like the central limit theorem 00:40:03.740 --> 00:40:05.120 align:middle line:90% and law of large numbers. 00:40:05.120 --> 00:40:09.380 align:middle line:84% They are very, very robust if you have a lot of-- 00:40:09.380 --> 00:40:12.050 align:middle line:84% the question is, or the problem is 00:40:12.050 --> 00:40:15.350 align:middle line:84% that you need to have a lot of observations in order for these 00:40:15.350 --> 00:40:17.090 align:middle line:90% to kick in and help you. 00:40:17.090 --> 00:40:21.920 align:middle line:90% 00:40:21.920 --> 00:40:25.850 align:middle line:84% Now, what is a replicate and repeated measurement? 00:40:25.850 --> 00:40:30.800 align:middle line:84% And what I'm going to say here is not going to be new to you-- 00:40:30.800 --> 00:40:33.140 align:middle line:90% how people define replicate. 00:40:33.140 --> 00:40:35.990 align:middle line:90% It, in a way, also depends on-- 00:40:35.990 --> 00:40:39.350 align:middle line:84% in fact, replicates and reproducibility 00:40:39.350 --> 00:40:40.400 align:middle line:90% go hand in hand. 00:40:40.400 --> 00:40:46.610 align:middle line:90% 00:40:46.610 --> 00:40:51.210 align:middle line:84% I'll give you three examples of a replicate measurement. 00:40:51.210 --> 00:40:55.130 align:middle line:84% When you talk about replicate, it always involves two things. 00:40:55.130 --> 00:40:57.840 align:middle line:84% You have something, and you need something to compare it with. 00:40:57.840 --> 00:41:04.370 align:middle line:84% So let's say that you want to do replicate assays of mice 00:41:04.370 --> 00:41:11.240 align:middle line:84% pancreas-- normal mice pancreas, whole pancreas, RNA analysis. 00:41:11.240 --> 00:41:16.070 align:middle line:84% And they had been controlled for weight, for gender, et cetera. 00:41:16.070 --> 00:41:19.430 align:middle line:84% Now, there are three different situations, 00:41:19.430 --> 00:41:22.460 align:middle line:84% or three different ways of defining replicates, right? 00:41:22.460 --> 00:41:25.370 align:middle line:84% Number one-- you take the pancreas from each of these 00:41:25.370 --> 00:41:25.890 align:middle line:90% mice-- 00:41:25.890 --> 00:41:28.400 align:middle line:90% they are, say, the same litter-- 00:41:28.400 --> 00:41:32.090 align:middle line:90% and hybridize it, et cetera. 00:41:32.090 --> 00:41:35.120 align:middle line:84% This could be a definition one of a replicate. 00:41:35.120 --> 00:41:38.480 align:middle line:84% The other is you take the pancreas from one mouse 00:41:38.480 --> 00:41:41.240 align:middle line:84% and you split it and hybridize the other two. 00:41:41.240 --> 00:41:43.250 align:middle line:90% That's the other one. 00:41:43.250 --> 00:41:46.100 align:middle line:84% Notice, in this case, the biological variation. 00:41:46.100 --> 00:41:47.360 align:middle line:90% There is none. 00:41:47.360 --> 00:41:48.380 align:middle line:90% The call comes to one. 00:41:48.380 --> 00:41:50.070 align:middle line:84% Here, clearly, a biological variation 00:41:50.070 --> 00:41:51.690 align:middle line:90% is going to be very important. 00:41:51.690 --> 00:41:58.670 align:middle line:84% The third one is to somehow homogenize 00:41:58.670 --> 00:42:01.940 align:middle line:84% the biological variation at the very top level. 00:42:01.940 --> 00:42:05.750 align:middle line:84% You just mix them all up and split them. 00:42:05.750 --> 00:42:08.870 align:middle line:84% And there are arguments for using this way, this way, 00:42:08.870 --> 00:42:09.920 align:middle line:90% this way-- 00:42:09.920 --> 00:42:12.170 align:middle line:90% what you want to control. 00:42:12.170 --> 00:42:17.060 align:middle line:84% It can also be argued that you can remove biological variation 00:42:17.060 --> 00:42:18.980 align:middle line:84% but only at a later stage when you have 00:42:18.980 --> 00:42:23.150 align:middle line:84% the numbers, whereas, you are pulling them here, 00:42:23.150 --> 00:42:25.820 align:middle line:90% you can pull them later on. 00:42:25.820 --> 00:42:30.290 align:middle line:84% There is no better or worse definition of a replicate. 00:42:30.290 --> 00:42:32.930 align:middle line:84% But one should be aware, when you are reading papers, 00:42:32.930 --> 00:42:36.136 align:middle line:84% you should be aware, at least, what do they mean by replicate? 00:42:36.136 --> 00:42:39.750 align:middle line:90% 00:42:39.750 --> 00:42:46.200 align:middle line:84% Because in a way, you will also notice that replicate-- 00:42:46.200 --> 00:42:52.230 align:middle line:84% the notion of a replicate will actually guide or affect 00:42:52.230 --> 00:42:54.870 align:middle line:90% how you define noise. 00:42:54.870 --> 00:42:57.630 align:middle line:84% Because if you believe that these two mice should give you 00:42:57.630 --> 00:43:02.190 align:middle line:84% identical reading, then any deviation 00:43:02.190 --> 00:43:06.240 align:middle line:84% of this mouse from that one is going to be noise. 00:43:06.240 --> 00:43:12.060 align:middle line:84% And that deviations-- it seems very obvious things. 00:43:12.060 --> 00:43:15.420 align:middle line:90% Now, yeah, actually, here it is. 00:43:15.420 --> 00:43:17.940 align:middle line:84% The definition of replicate will have implication 00:43:17.940 --> 00:43:20.040 align:middle line:90% on how you define noise. 00:43:20.040 --> 00:43:22.980 align:middle line:84% And there is-- we talked about this-- biological versus 00:43:22.980 --> 00:43:25.020 align:middle line:90% measurement variation. 00:43:25.020 --> 00:43:28.530 align:middle line:84% This being biological variation, this being measurable 00:43:28.530 --> 00:43:29.160 align:middle line:90% variation. 00:43:29.160 --> 00:43:33.945 align:middle line:84% And arguably, this could be measurement variation, too. 00:43:33.945 --> 00:43:37.750 align:middle line:90% 00:43:37.750 --> 00:43:40.270 align:middle line:84% However, if you're too restrictive in your definition 00:43:40.270 --> 00:43:45.640 align:middle line:84% of replicate, it could actually hinder the generalizability 00:43:45.640 --> 00:43:47.680 align:middle line:90% of your study. 00:43:47.680 --> 00:43:51.190 align:middle line:84% Your study is only applicable to mice 00:43:51.190 --> 00:43:56.170 align:middle line:84% with a certain genetic background, very, very 00:43:56.170 --> 00:43:58.300 align:middle line:90% restricted, et cetera. 00:43:58.300 --> 00:44:01.210 align:middle line:84% One has to consider these things. 00:44:01.210 --> 00:44:04.450 align:middle line:84% But despite taking all precautions, 00:44:04.450 --> 00:44:07.780 align:middle line:84% it is not very likely that your, quote, unquote, 00:44:07.780 --> 00:44:11.770 align:middle line:84% "replicate assay" will, down the line, 00:44:11.770 --> 00:44:16.030 align:middle line:84% give you numerically identical results. 00:44:16.030 --> 00:44:19.060 align:middle line:84% And there is this old saying, I think from the Greek alchemist, 00:44:19.060 --> 00:44:21.940 align:middle line:84% that you never step into the same river twice. 00:44:21.940 --> 00:44:23.005 align:middle line:90% It's very true here. 00:44:23.005 --> 00:44:26.510 align:middle line:90% 00:44:26.510 --> 00:44:31.870 align:middle line:84% And as a result, people often will 00:44:31.870 --> 00:44:35.140 align:middle line:84% try to massage the data later on after the test 00:44:35.140 --> 00:44:37.210 align:middle line:90% tube comes the numbers-- 00:44:37.210 --> 00:44:40.660 align:middle line:84% well, it has to go through certain scanners and machines. 00:44:40.660 --> 00:44:44.590 align:middle line:84% But when the numbers come out, then they massage it-- 00:44:44.590 --> 00:44:47.860 align:middle line:84% they normalize, in other words-- to somehow account 00:44:47.860 --> 00:44:54.880 align:middle line:84% for biological variation or measurement variation. 00:44:54.880 --> 00:44:59.620 align:middle line:84% There are arguments for and against normalization. 00:44:59.620 --> 00:45:06.760 align:middle line:84% And there is no blanket principle whether to do this-- 00:45:06.760 --> 00:45:07.880 align:middle line:90% to normalize or not. 00:45:07.880 --> 00:45:11.110 align:middle line:84% It depends on your experiment design 00:45:11.110 --> 00:45:18.010 align:middle line:84% OK, I give an example of that pancreatic development data-- 00:45:18.010 --> 00:45:22.930 align:middle line:84% embryonic day 12, 14, 18, postnatal day 2, 00:45:22.930 --> 00:45:24.670 align:middle line:90% and the adult pancreas. 00:45:24.670 --> 00:45:29.590 align:middle line:84% And I'm simply plotting that 10,000 genes against itself, 00:45:29.590 --> 00:45:31.270 align:middle line:90% here-- 00:45:31.270 --> 00:45:33.520 align:middle line:90% embryonic day 12 against itself. 00:45:33.520 --> 00:45:35.890 align:middle line:90% And E12 against its aliquot. 00:45:35.890 --> 00:45:38.260 align:middle line:90% What is aliquot? 00:45:38.260 --> 00:45:42.130 align:middle line:84% I think, actually, aliquot, in this case, was this. 00:45:42.130 --> 00:45:43.720 align:middle line:90% It's a measurement variation. 00:45:43.720 --> 00:45:46.240 align:middle line:90% 00:45:46.240 --> 00:45:49.200 align:middle line:84% Though, in a way, it's not very wise in the case of-- clearly, 00:45:49.200 --> 00:45:51.820 align:middle line:84% you cannot remove the whole pancreas from the same mouse 00:45:51.820 --> 00:45:52.930 align:middle line:90% twice. 00:45:52.930 --> 00:45:55.180 align:middle line:84% So it may have been wiser to have 00:45:55.180 --> 00:45:56.710 align:middle line:84% done this, come to think of it, now, 00:45:56.710 --> 00:45:58.750 align:middle line:84% to account for biological variation. 00:45:58.750 --> 00:46:12.410 align:middle line:84% Because in this case, then, it was measurement variation. 00:46:12.410 --> 00:46:17.180 align:middle line:84% Then, clearly, there are two confounding factors here. 00:46:17.180 --> 00:46:19.670 align:middle line:84% Number one is measurement variation, number two, 00:46:19.670 --> 00:46:20.618 align:middle line:90% biological variation. 00:46:20.618 --> 00:46:22.160 align:middle line:84% So it would have been nice, in a way, 00:46:22.160 --> 00:46:24.960 align:middle line:84% if we had biological variation captured here. 00:46:24.960 --> 00:46:29.120 align:middle line:84% So E12, you see it start to spread like a comet. 00:46:29.120 --> 00:46:31.970 align:middle line:84% And as you progress with development, 00:46:31.970 --> 00:46:35.250 align:middle line:84% you can see that it spreads even worse. 00:46:35.250 --> 00:46:36.740 align:middle line:90% So it looks less and less alike. 00:46:36.740 --> 00:46:38.690 align:middle line:90% The most alike is here. 00:46:38.690 --> 00:46:40.400 align:middle line:84% They should align up right-- you're 00:46:40.400 --> 00:46:43.940 align:middle line:84% supposed to be taking the same reading of the same thing. 00:46:43.940 --> 00:46:46.032 align:middle line:84% Of course, by the time it reaches adulthood, 00:46:46.032 --> 00:46:46.865 align:middle line:90% anything can happen. 00:46:46.865 --> 00:46:51.170 align:middle line:90% 00:46:51.170 --> 00:46:54.160 align:middle line:84% We don't know-- what are your priority 00:46:54.160 --> 00:46:57.370 align:middle line:84% assumptions about the system guides you 00:46:57.370 --> 00:46:59.380 align:middle line:90% on whether or not to normalize. 00:46:59.380 --> 00:47:03.430 align:middle line:84% If you believe-- if you have done 10 whole microarrays 00:47:03.430 --> 00:47:07.090 align:middle line:84% for some system and you believe that your system should 00:47:07.090 --> 00:47:10.810 align:middle line:84% be remaining consistent despite the biological variation, 00:47:10.810 --> 00:47:19.750 align:middle line:84% et cetera, then you could claim that the average of these array 00:47:19.750 --> 00:47:23.800 align:middle line:84% readings have to be the same, or the variance 00:47:23.800 --> 00:47:25.960 align:middle line:90% should be a certain quantity. 00:47:25.960 --> 00:47:30.850 align:middle line:84% But that's actually putting your own assumptions 00:47:30.850 --> 00:47:35.560 align:middle line:84% of how the system behaves into this actual phenomenon that's 00:47:35.560 --> 00:47:37.320 align:middle line:90% unfolding. 00:47:37.320 --> 00:47:39.190 align:middle line:90% So one has to be careful. 00:47:39.190 --> 00:47:42.940 align:middle line:84% The common normalization technique that people use 00:47:42.940 --> 00:47:46.060 align:middle line:84% are the one which we have actually talked about, 00:47:46.060 --> 00:47:50.410 align:middle line:84% central limit theorem scaling, meaning that this 00:47:50.410 --> 00:47:52.000 align:middle line:90% is a vector, actually-- 00:47:52.000 --> 00:47:53.680 align:middle line:84% vector x against the reference r. 00:47:53.680 --> 00:47:57.190 align:middle line:84% So you simply subtract the average 00:47:57.190 --> 00:48:00.760 align:middle line:84% from each component of the vector x 00:48:00.760 --> 00:48:03.410 align:middle line:84% and you divide it by the standard deviation. 00:48:03.410 --> 00:48:06.400 align:middle line:84% The end result, actually, is a quantity which 00:48:06.400 --> 00:48:09.670 align:middle line:90% has mean 0 and variance 1. 00:48:09.670 --> 00:48:13.000 align:middle line:90% 00:48:13.000 --> 00:48:14.802 align:middle line:84% Some people have reason to do this. 00:48:14.802 --> 00:48:16.510 align:middle line:84% Some people don't have a reason to do it, 00:48:16.510 --> 00:48:19.060 align:middle line:90% but they do it anyways. 00:48:19.060 --> 00:48:21.160 align:middle line:84% You just have to be very careful. 00:48:21.160 --> 00:48:24.670 align:middle line:84% So what happens when-- well we just saw what happens. 00:48:24.670 --> 00:48:26.800 align:middle line:84% By so doing, you have mapped, actually, 00:48:26.800 --> 00:48:31.390 align:middle line:84% this entire vector into an element of the unit 00:48:31.390 --> 00:48:33.730 align:middle line:90% hypersphere, it turns out. 00:48:33.730 --> 00:48:35.110 align:middle line:90% So you do lose information. 00:48:35.110 --> 00:48:41.530 align:middle line:84% Maybe, you lose all notion of absoluteness, 00:48:41.530 --> 00:48:44.230 align:middle line:90% absolute intensities. 00:48:44.230 --> 00:48:47.980 align:middle line:84% The second common method of normalization 00:48:47.980 --> 00:48:52.750 align:middle line:84% is you have a reference data set and you 00:48:52.750 --> 00:48:56.320 align:middle line:84% regress against that reference data set. 00:48:56.320 --> 00:48:57.530 align:middle line:90% What do I mean? 00:48:57.530 --> 00:49:01.660 align:middle line:84% Well, if you-- back to this thing. 00:49:01.660 --> 00:49:07.360 align:middle line:84% If you take the E12 to be the reference data set 00:49:07.360 --> 00:49:11.980 align:middle line:84% and you're trying to regress everything to it 00:49:11.980 --> 00:49:18.280 align:middle line:84% or normalize everything to it, then, for example, the E12-- 00:49:18.280 --> 00:49:22.900 align:middle line:84% second aliquot against the first E12. 00:49:22.900 --> 00:49:27.610 align:middle line:84% In an ideal world, you have to get scatter that's clear. 00:49:27.610 --> 00:49:32.350 align:middle line:84% However, even through the scatter, 00:49:32.350 --> 00:49:35.080 align:middle line:84% some people believe that the linear regression 00:49:35.080 --> 00:49:41.510 align:middle line:84% line has to be of slope 1 and going through the origin. 00:49:41.510 --> 00:49:44.770 align:middle line:90% If it isn't, make it so. 00:49:44.770 --> 00:49:48.280 align:middle line:84% So it's a linear transformation of the second aliquot by, 00:49:48.280 --> 00:49:51.040 align:middle line:84% basically, you subtract and you divide. 00:49:51.040 --> 00:49:54.200 align:middle line:84% It's just a translation of the data 00:49:54.200 --> 00:49:56.920 align:middle line:84% so that the regression of one against the other 00:49:56.920 --> 00:50:02.560 align:middle line:84% is now going through the origin and has slope 1. 00:50:02.560 --> 00:50:07.760 align:middle line:84% You do it for the entire time series, in this case, 00:50:07.760 --> 00:50:09.880 align:middle line:90% for example. 00:50:09.880 --> 00:50:11.180 align:middle line:90% There are reasons to do it. 00:50:11.180 --> 00:50:13.210 align:middle line:90% There are reasons not to do it. 00:50:13.210 --> 00:50:15.580 align:middle line:84% What actually happens after you do it? 00:50:15.580 --> 00:50:18.130 align:middle line:84% Well, clearly then, all the samples 00:50:18.130 --> 00:50:22.750 align:middle line:84% are now going to have regression intercept 0, slope 00:50:22.750 --> 00:50:24.310 align:middle line:90% 1 against the reference. 00:50:24.310 --> 00:50:26.410 align:middle line:84% A second thing that comes out of this 00:50:26.410 --> 00:50:31.660 align:middle line:84% is all the vectors, the newly-normalized vectors, 00:50:31.660 --> 00:50:33.820 align:middle line:90% will have the same average. 00:50:33.820 --> 00:50:35.350 align:middle line:90% You can show that for yourself. 00:50:35.350 --> 00:50:36.610 align:middle line:90% You can do the arithmetic. 00:50:36.610 --> 00:50:38.140 align:middle line:84% You'll see that come out for free. 00:50:38.140 --> 00:50:41.920 align:middle line:90% 00:50:41.920 --> 00:50:45.940 align:middle line:84% Then, there is a notion of a fold 00:50:45.940 --> 00:50:48.520 align:middle line:84% that you hear all the time which is very natural when 00:50:48.520 --> 00:50:50.350 align:middle line:90% it comes to PCR and blots. 00:50:50.350 --> 00:50:53.050 align:middle line:84% But the question is, does a fold actually 00:50:53.050 --> 00:50:58.690 align:middle line:84% make sense in the context of Affymetrix chips? 00:50:58.690 --> 00:51:01.210 align:middle line:84% For example, you often encounter this. 00:51:01.210 --> 00:51:08.440 align:middle line:84% You have a sample population, A, and you have three readings 00:51:08.440 --> 00:51:12.310 align:middle line:84% for that sample population A versus B, 00:51:12.310 --> 00:51:14.350 align:middle line:84% which has another three readings. 00:51:14.350 --> 00:51:19.480 align:middle line:84% And people will ask, what is the fold change from here to there? 00:51:19.480 --> 00:51:23.470 align:middle line:84% And notice that there is a negative here, too. 00:51:23.470 --> 00:51:29.320 align:middle line:84% Yes, many ways have been proposed to solve this problem. 00:51:29.320 --> 00:51:33.070 align:middle line:84% You could take the arithmetic average here then 00:51:33.070 --> 00:51:35.570 align:middle line:84% divide it by the arithmetic average there. 00:51:35.570 --> 00:51:37.540 align:middle line:84% There's a negative there, though-- 00:51:37.540 --> 00:51:38.230 align:middle line:90% negative 1. 00:51:38.230 --> 00:51:43.360 align:middle line:84% So does the fault actually make sense in its setting? 00:51:43.360 --> 00:51:48.250 align:middle line:84% Alternatively, some people have used the geometric average 00:51:48.250 --> 00:51:51.940 align:middle line:84% rather than the arithmetic average. 00:51:51.940 --> 00:51:53.410 align:middle line:90% Or they have logged it, somehow. 00:51:53.410 --> 00:51:59.200 align:middle line:84% But logs only-- well, posed on a set of positive numbers-- 00:51:59.200 --> 00:52:03.280 align:middle line:90% none, not even zero. 00:52:03.280 --> 00:52:06.160 align:middle line:84% One argument against using fold actually, 00:52:06.160 --> 00:52:08.230 align:middle line:90% is that it is not stable. 00:52:08.230 --> 00:52:12.820 align:middle line:84% It is incredibly not stable and highly-sensitive 00:52:12.820 --> 00:52:13.960 align:middle line:90% to its denominator. 00:52:13.960 --> 00:52:20.080 align:middle line:84% For instance, 20 over 10, 50 over 25, and 200 over 100 00:52:20.080 --> 00:52:22.480 align:middle line:90% are all equal to 2, right? 00:52:22.480 --> 00:52:26.440 align:middle line:84% Let's actually perturb them by the same quantity, epsilon. 00:52:26.440 --> 00:52:28.090 align:middle line:90% This is a perturbation epsilon. 00:52:28.090 --> 00:52:31.430 align:middle line:90% Can be positive or negative. 00:52:31.430 --> 00:52:36.550 align:middle line:84% Now, 20 over 10-- when you perturb it by epsilon, 00:52:36.550 --> 00:52:42.520 align:middle line:84% it swings all the way from 3 fold to something so small-- 00:52:42.520 --> 00:52:45.010 align:middle line:90% 1.8 folds there. 00:52:45.010 --> 00:52:47.960 align:middle line:84% But this quantity here is more stable. 00:52:47.960 --> 00:52:50.740 align:middle line:84% So does it mean that folds actually only makes sense 00:52:50.740 --> 00:52:56.110 align:middle line:84% if the absolute numbers involved or the absolute averages 00:52:56.110 --> 00:52:59.387 align:middle line:84% involved are large, so it's much more flatter here? 00:52:59.387 --> 00:53:00.595 align:middle line:90% You can say it's more robust. 00:53:00.595 --> 00:53:04.158 align:middle line:90% 00:53:04.158 --> 00:53:05.700 align:middle line:84% So these are things that one actually 00:53:05.700 --> 00:53:09.600 align:middle line:84% should be aware of when doing analysis 00:53:09.600 --> 00:53:14.460 align:middle line:84% or when reading the papers and what they mean by folds. 00:53:14.460 --> 00:53:18.840 align:middle line:84% Again, you recall that we have covered, actually, all of this. 00:53:18.840 --> 00:53:22.080 align:middle line:84% The study design-- prototypical study designs 00:53:22.080 --> 00:53:26.610 align:middle line:84% on data representation, background, and the last slide, 00:53:26.610 --> 00:53:28.750 align:middle line:84% essentially, is just miscellaneous. 00:53:28.750 --> 00:53:32.250 align:middle line:84% So our discussion so far, if you notice, 00:53:32.250 --> 00:53:37.590 align:middle line:90% does not require biology. 00:53:37.590 --> 00:53:40.428 align:middle line:84% It's just makes nominal reference to biology. 00:53:40.428 --> 00:53:42.720 align:middle line:84% The approaches are very general, if you think about it. 00:53:42.720 --> 00:53:45.330 align:middle line:84% It would apply in any data set, whether or not 00:53:45.330 --> 00:53:47.790 align:middle line:90% it came from biology. 00:53:47.790 --> 00:53:51.960 align:middle line:84% But one should be aware that the math or the statistics 00:53:51.960 --> 00:53:54.780 align:middle line:84% will only provide a tool for biological discovery. 00:53:54.780 --> 00:53:58.020 align:middle line:84% And the key thing we are all here to do 00:53:58.020 --> 00:54:00.990 align:middle line:90% is to understand the biology. 00:54:00.990 --> 00:54:05.040 align:middle line:84% And because that, actually, will dictate for you 00:54:05.040 --> 00:54:09.570 align:middle line:84% the experiment design, the appropriate measure 00:54:09.570 --> 00:54:15.810 align:middle line:84% or similarity space to formulate your problem, 00:54:15.810 --> 00:54:19.050 align:middle line:84% and, also, in reading and making sense of what your model gives 00:54:19.050 --> 00:54:21.900 align:middle line:90% you after you have done this. 00:54:21.900 --> 00:54:25.260 align:middle line:84% And a very, very important thing is, no study 00:54:25.260 --> 00:54:26.880 align:middle line:90% is ever hypothesis-free. 00:54:26.880 --> 00:54:29.650 align:middle line:84% You are going to read things into your study, 00:54:29.650 --> 00:54:30.600 align:middle line:90% no matter what. 00:54:30.600 --> 00:54:33.570 align:middle line:84% It is best to be very explicit in the beginning what 00:54:33.570 --> 00:54:34.860 align:middle line:90% your hypotheses are. 00:54:34.860 --> 00:54:38.730 align:middle line:84% And a lot of studies, I noticed, have completely 00:54:38.730 --> 00:54:40.860 align:middle line:84% deluded themselves into thinking that they 00:54:40.860 --> 00:54:43.740 align:middle line:84% are unsupervised or hypothesis-free. 00:54:43.740 --> 00:54:46.680 align:middle line:84% There is some hypothesis, I'm sorry to say. 00:54:46.680 --> 00:54:48.540 align:middle line:84% Even principal component analysis 00:54:48.540 --> 00:54:50.940 align:middle line:84% has been claimed to be unsupervised. 00:54:50.940 --> 00:54:52.440 align:middle line:90% That's not true. 00:54:52.440 --> 00:54:54.030 align:middle line:84% Well, it is unsupervised, in the sense 00:54:54.030 --> 00:54:55.530 align:middle line:90% that you just let things fall. 00:54:55.530 --> 00:54:58.890 align:middle line:84% But the underlying assumption is that the measure of similarity 00:54:58.890 --> 00:55:02.010 align:middle line:90% there is Euclidean distance. 00:55:02.010 --> 00:55:05.730 align:middle line:90% That matters, actually. 00:55:05.730 --> 00:55:12.260 align:middle line:84% And I end early with a quote from 00:55:12.260 --> 00:55:16.080 align:middle line:84% An evolutionary by a biologist, a French one, 00:55:16.080 --> 00:55:19.230 align:middle line:84% from the 18th century that the discoveries that one 00:55:19.230 --> 00:55:22.980 align:middle line:84% can make with a microscope amount to very little. 00:55:22.980 --> 00:55:26.820 align:middle line:84% So you can see with your mind's eyes and without the microscope 00:55:26.820 --> 00:55:29.220 align:middle line:84% the real existence of these little beings. 00:55:29.220 --> 00:55:32.190 align:middle line:84% I think he was referring to microbes 00:55:32.190 --> 00:55:35.740 align:middle line:90% or very small features. 00:55:35.740 --> 00:55:37.520 align:middle line:90% Thank you.