1 00:00:04,210 --> 00:00:09,670 Random variables can be transformed through a function, suppose that we have a random variable X and 2 00:00:09,670 --> 00:00:11,020 it's associated PDF. 3 00:00:11,350 --> 00:00:18,160 It is possible to apply a mathematical function to the BBF to find the transform PDF for the transform 4 00:00:18,160 --> 00:00:18,940 random variable. 5 00:00:19,780 --> 00:00:27,340 So if we have a random variable X and we have the associated PDF, if X, it is possible to apply a 6 00:00:27,340 --> 00:00:35,740 mathematical function so Y easily equals G of X to the PDF to find the PDF of the transform variable 7 00:00:35,860 --> 00:00:38,440 Y, which is the fly. 8 00:00:39,880 --> 00:00:46,070 So let's assume the transfer functions are monotonic so they can be monotonically increasing or decreasing. 9 00:00:46,090 --> 00:00:51,820 It doesn't matter as long as this transformation function, this why of G of X is monotonic. 10 00:00:52,370 --> 00:00:54,220 So we have the transformation function. 11 00:00:54,370 --> 00:00:56,170 Why is equal G of X.. 12 00:00:56,950 --> 00:01:02,590 Now we want to find the inverse mapping, so we want to find the function, the inverse G that maps 13 00:01:02,590 --> 00:01:05,500 from the Y variable to back to the X variable. 14 00:01:05,920 --> 00:01:10,780 And we're going to rename this inverse mapping the inverse G into H. 15 00:01:12,020 --> 00:01:17,870 We then can express the transformation of the PDF using this function here, so we won't go through 16 00:01:17,870 --> 00:01:24,290 the derivation of this function, but this is the function that will transform the PDF effects into 17 00:01:24,290 --> 00:01:31,720 the PDF if y so this takes the random variable X and works out the random variable Y and the associated 18 00:01:31,720 --> 00:01:32,540 the PDF of it. 19 00:01:32,970 --> 00:01:38,120 So you can say it's pretty much the derivative of the inverse function, the absolute value of it. 20 00:01:38,600 --> 00:01:44,510 Thom's pi the PDF for the X variable with the function substituted into it. 21 00:01:47,250 --> 00:01:52,350 So using this relationship here, we then can look into the example of a linear transformation of a 22 00:01:52,350 --> 00:01:54,690 Gaussian PDF and calculate the result. 23 00:01:56,210 --> 00:02:01,760 So let's suppose we have a random variable X and we know that it is the normal distribution or Gaussian 24 00:02:01,760 --> 00:02:06,990 distribution and it has a mean of X bar and a variant of Sigma X squared. 25 00:02:07,790 --> 00:02:14,770 We know from the previous video that this here is a function of the PDF for this normal distribution. 26 00:02:15,290 --> 00:02:18,140 So we have the main here and the variance here. 27 00:02:19,220 --> 00:02:23,420 So now suppose that we have the transformation and we want to do a linear transformation. 28 00:02:23,420 --> 00:02:27,300 So we have A X plus B equals Y. 29 00:02:27,350 --> 00:02:32,300 So this is going to be a linear transformation function that we're going to apply to this random variable. 30 00:02:33,050 --> 00:02:40,190 And in this of course, this coefficient A and B have to be real numbers and A cannot be equal to zero. 31 00:02:40,220 --> 00:02:42,000 Otherwise, it's not a transformation. 32 00:02:42,920 --> 00:02:47,270 So the first thing we want to do is to work out the inverse G mapping function. 33 00:02:47,780 --> 00:02:52,350 So that's just going to be Y minus B divided by A. 34 00:02:52,400 --> 00:02:56,860 So this is going to be our investment in function and we're going to define it to be hech. 35 00:02:58,910 --> 00:03:03,230 Now we know that we need to get the derivative of its derivative have to exist to be able to do this 36 00:03:03,230 --> 00:03:09,320 mapping and the derivative of the hedge function is simply just one over a that's fairly straightforward 37 00:03:09,320 --> 00:03:11,810 from first order of differentiation. 38 00:03:13,410 --> 00:03:18,540 So using the relationship that we presented on the previous slide, we can now fill in all this other 39 00:03:18,540 --> 00:03:19,230 information. 40 00:03:19,800 --> 00:03:25,530 So first we know that the derivative is just one Ivereigh, so we can fill that in here. 41 00:03:27,090 --> 00:03:33,290 Next, we want to work out the second time here so the PDF for X with the function substituted in. 42 00:03:33,570 --> 00:03:40,110 So what we can do is we can take our hedge function substituted in for X into the X PDF. 43 00:03:41,220 --> 00:03:47,760 Once we've done that substitution we can then take the whole function and fill it in for this term here. 44 00:03:48,390 --> 00:03:51,750 So if we do this, we end up with this equation here. 45 00:03:51,780 --> 00:03:59,120 So this is the transformation and this gives us the PDF for the Y random variable after we do the transformation. 46 00:03:59,820 --> 00:04:04,380 So you can see that this function here looks very similar to this function over here. 47 00:04:04,950 --> 00:04:08,100 So in fact, they're both Gaussian distributions. 48 00:04:08,430 --> 00:04:14,730 The only difference you can see is that we've changed the meaning of the Gaussian distribution and we've 49 00:04:14,730 --> 00:04:17,460 also changed the variance of the Gaussian distribution. 50 00:04:18,390 --> 00:04:23,840 So equating the two parameters, we can work out the new meaning of this random variable y. 51 00:04:23,850 --> 00:04:29,910 It's just a time to previous mean plus B, so this is basically just doing the transformation on the 52 00:04:29,910 --> 00:04:30,410 main. 53 00:04:30,450 --> 00:04:31,730 So it's fairly straightforward. 54 00:04:32,580 --> 00:04:37,720 Also, we can see the variance here is just being scaled by the scale factor, eh. 55 00:04:38,010 --> 00:04:43,380 So the new variance for the Y pdf is just going to be a squared time. 56 00:04:43,390 --> 00:04:44,130 The variance. 57 00:04:47,000 --> 00:04:52,550 So we can see that the linear transformation of a Gaussian PDF is just another Gaussian PDF with the 58 00:04:52,550 --> 00:04:54,440 mean and variance transformed. 59 00:04:54,950 --> 00:05:03,110 So here's the original PDF of of main exposure and variance Sigma X, we apply the mapping function 60 00:05:03,110 --> 00:05:09,920 of A, X plus B, so a linear transformation to get the new wife the random variable. 61 00:05:10,550 --> 00:05:16,520 All we have to do is we just have to transform the main using this equation here and transform the variance 62 00:05:16,850 --> 00:05:18,140 using this equation here. 63 00:05:19,520 --> 00:05:23,480 So this is a major take away from a Bayesian probabilistic estimation. 64 00:05:23,480 --> 00:05:27,620 A linear transformation of Gaussian PDF is just another Gaussian PDAF. 65 00:05:28,220 --> 00:05:34,010 If we can express the PDF as a Gaussian, then we can dramatically simplify the mathematics involved 66 00:05:34,010 --> 00:05:39,620 in estimation when we're working with probabilities in particular, we won't we will not have to carry 67 00:05:39,620 --> 00:05:46,190 out integrations or payday's to a probability which may or may not have close form solutions. 68 00:05:46,490 --> 00:05:53,330 So this is why when we're doing probabilistic data fusion, that ideally we'd like to use Gaussian distributions 69 00:05:53,330 --> 00:05:57,140 because we can apply these simple transformations and find the results. 70 00:05:59,040 --> 00:06:05,040 So let's suppose we have a random vector X and this random vector is pulled from a multidimensional 71 00:06:05,040 --> 00:06:12,990 Gaussian distribution with a of our exposure and covariance of C of X, let's suppose that we want to 72 00:06:12,990 --> 00:06:20,040 transform this Gaussian distribution using the linear transform of X plus B, so A, here is going to 73 00:06:20,040 --> 00:06:26,430 be a transformation matrix multiplied by our random vector plus an offset vector. 74 00:06:26,850 --> 00:06:28,810 So this function here, we're going to call it G. 75 00:06:28,980 --> 00:06:34,440 This is the transformation that we want to apply to this Gaussian distribution up here. 76 00:06:36,590 --> 00:06:42,260 So the first step is to work out our inversed mapping function, so g inverse, and that's just going 77 00:06:42,260 --> 00:06:49,850 to be our inverse, a Thom's Y minus our inverse A times B, so this gives us our inverse mapping function 78 00:06:50,220 --> 00:06:55,880 and then we can differentiate this mapping function with respect to Y just to end up with our inverse. 79 00:06:55,880 --> 00:07:02,480 A Now we can use the relationship that we've covered in the past, which allows us to perform a mathematical 80 00:07:02,480 --> 00:07:06,510 transformation of a probability density function from one to the other. 81 00:07:06,530 --> 00:07:09,440 So it allows us to apply a function to the density function. 82 00:07:10,440 --> 00:07:15,420 So filling in the terms here, we know the probability density function is going to be our equation 83 00:07:15,420 --> 00:07:22,380 for our gorshin multidimensional distribution, we have a derivative of our investment function G. 84 00:07:22,870 --> 00:07:26,420 So if we fill all this in, we end up with this equation here. 85 00:07:26,430 --> 00:07:33,840 So we find out this is a probability of the multidimensional Gaussian distribution after is undergone 86 00:07:33,840 --> 00:07:36,330 our transformation of X plus B.. 87 00:07:37,080 --> 00:07:40,080 So now inspecting these terms here, we can have a look. 88 00:07:40,080 --> 00:07:48,210 You can say now this is a function of why we know our Y bar is now just going to be X bar plus B, we 89 00:07:48,210 --> 00:07:54,090 also know that the covariance of this relationship, of this multidimensional Gaussian distribution 90 00:07:54,330 --> 00:07:55,980 is now going to be this term here. 91 00:07:56,460 --> 00:08:03,090 So this new covariance of the covariance of Y is just going to be a times OK, variance, matrix times 92 00:08:03,110 --> 00:08:04,440 array transpose. 93 00:08:05,040 --> 00:08:11,010 So by looking at these terms, we can see that the linear transform of a multidimensional Gaussian distribution 94 00:08:11,250 --> 00:08:17,520 is just again a linear transform of the main and covariance parameters. 95 00:08:18,810 --> 00:08:25,440 So the linear transform of Gaussian pdf is just another Gaussian UPDF with the mean and variance transformed. 96 00:08:26,160 --> 00:08:31,440 So if we have this random variable vector X here and it comes from a multidimensional Gaussian distribution 97 00:08:31,440 --> 00:08:38,670 described by these parameters here, and we want to apply a linear transformation of X plus B, we end 98 00:08:38,670 --> 00:08:41,780 up with this Gaussian distribution described by these parameters here. 99 00:08:41,790 --> 00:08:42,120 So. 100 00:08:43,140 --> 00:08:48,120 So we just transform the main and we just transform the covariance using this relationship. 101 00:08:50,040 --> 00:08:56,550 So this is pretty important, it tells us if Sea X represents the uncertainty covariance, then it can 102 00:08:56,550 --> 00:09:01,590 be transformed into another frame using the linear transformation Y is equal to X.. 103 00:09:01,980 --> 00:09:07,650 So if we have the covariance in frame X and we want to transform it to frame Y, and we know the transformation 104 00:09:07,650 --> 00:09:11,750 matrix, we can transform the covariance using this relationship here. 105 00:09:12,030 --> 00:09:17,410 So covariance and why is this going to be a times covariance in X times transpose? 106 00:09:18,060 --> 00:09:23,150 So we're going to use this transformation relationship quite a lot later on inside our computer. 107 00:09:23,490 --> 00:09:27,570 So this is an important relationship and an important outcome of this process here. 108 00:09:28,380 --> 00:09:33,870 So if we have a random variable vector X and it undergoes a transformation and we know the original 109 00:09:33,870 --> 00:09:39,210 covariance before the transformation, we can work out the uncertainty after the transformation so we 110 00:09:39,210 --> 00:09:42,560 can transform the uncertainty from one frame to other. 111 00:09:42,690 --> 00:09:45,870 Just like that, we can transform a vector from one frame to another.