1 00:00:03,850 --> 00:00:09,760 In this video, we're going to look at how we can use the basic concepts of probability inside an estimation 2 00:00:09,760 --> 00:00:15,850 problem, so we're going to represent the current state to be estimated as a random variable vector 3 00:00:15,850 --> 00:00:17,710 with a probability distribution. 4 00:00:18,580 --> 00:00:22,450 The distribution main is going to be the best estimate of the current vector. 5 00:00:22,480 --> 00:00:28,720 So the solution to the minimum main squared error, while the distribution covariance is the estimate 6 00:00:28,720 --> 00:00:30,900 of the uncertainty around the current state. 7 00:00:31,150 --> 00:00:34,630 And this is, for example, the covariance of the estimation error. 8 00:00:37,070 --> 00:00:42,500 We can also represent the measurements or observations as a random variable vector as well, with a 9 00:00:42,530 --> 00:00:44,300 given probability density function. 10 00:00:45,880 --> 00:00:51,820 The distribution main is then the measurement value, while the distribution covariance is the estimation 11 00:00:51,820 --> 00:00:53,590 of the uncertainty of the measurement. 12 00:00:55,240 --> 00:01:00,690 We can also use the dynamic system to estimate changes in the state domain and the uncertainty. 13 00:01:00,700 --> 00:01:04,040 So the covariance between the measurements or updates with time. 14 00:01:05,530 --> 00:01:10,510 So, for example, we could try to estimate the position of a car and we can represent the estimated 15 00:01:10,510 --> 00:01:11,710 position for the car. 16 00:01:11,740 --> 00:01:14,240 So this is the car position here. 17 00:01:14,260 --> 00:01:15,880 This is the estimated position. 18 00:01:16,240 --> 00:01:20,220 We can represent the position estimate as a normal distribution. 19 00:01:20,350 --> 00:01:21,900 So we have a main one. 20 00:01:21,910 --> 00:01:28,670 So the current main four timestep one is this estimate and we have a given uncertainty of the measurement. 21 00:01:28,690 --> 00:01:30,240 So this is our sigma squared one. 22 00:01:30,250 --> 00:01:35,800 So the uncertainty at this point of time and we can represent it as a normal or Gaussian distribution. 23 00:01:37,100 --> 00:01:43,240 So we know that the car moves with time so we can use the dynamic system to predict the estimate forward. 24 00:01:43,540 --> 00:01:47,370 So basically using the dynamic system, we can predict the estimate forward. 25 00:01:47,380 --> 00:01:53,300 So we move this position estimate through the dynamic system to get this estimate over here. 26 00:01:53,320 --> 00:01:56,160 So now we have a new position estimate for time, too. 27 00:01:56,530 --> 00:02:01,710 It has a new main, has a new variance, and in this case, the variance here. 28 00:02:01,720 --> 00:02:06,190 So Sigma squared two is going to be greater than our original position. 29 00:02:06,190 --> 00:02:13,030 Estimate Sigma squared one, because as time goes, we get less certain with the position of the vehicle. 30 00:02:13,480 --> 00:02:18,050 So now you can see that the estimate of position in here, which is the true position, is here. 31 00:02:18,730 --> 00:02:23,950 So there's a bit more error now because we've had to predict forward and we haven't constrained the 32 00:02:23,950 --> 00:02:26,700 system with any extra information, however. 33 00:02:26,890 --> 00:02:31,930 Now, let's say we get a bit of additional information so we get a sense of measurement, so we get 34 00:02:31,930 --> 00:02:33,890 a sense of mission such as from GPS. 35 00:02:33,910 --> 00:02:35,860 So this is going to measure the position of the car. 36 00:02:35,920 --> 00:02:39,970 So the measurement is going to be basically the main position of the measurement. 37 00:02:40,000 --> 00:02:45,700 So this is going to be our Munim and the measurement itself is going to have some level of uncertainty 38 00:02:45,700 --> 00:02:46,080 to it. 39 00:02:46,090 --> 00:02:48,860 And again, we model this as a normal distribution. 40 00:02:48,880 --> 00:02:53,000 So this is going to have our sigma squared m sorry, the uncertainty inside the measurement. 41 00:02:53,830 --> 00:02:59,170 Now we can take our sense of measurement and our predicted state and we can combine them together to 42 00:02:59,170 --> 00:03:00,160 get a better estimate. 43 00:03:01,060 --> 00:03:05,210 So once we do that, once we use the measurement in, we're going to get this new estimate. 44 00:03:05,230 --> 00:03:11,860 So this is our updated position estimate for the car and see now if the main is well and truly near 45 00:03:11,860 --> 00:03:16,330 the position of the car and it's going to have a new main and it's going to have a new uncertainty. 46 00:03:16,330 --> 00:03:17,710 So new sigma squared. 47 00:03:18,370 --> 00:03:23,950 Now, what we can see is that once we used in the measurement, we can see the new uncertainty for the 48 00:03:23,950 --> 00:03:25,600 updated position for time. 49 00:03:25,600 --> 00:03:31,940 Step two is going to be smaller than the original position four times up to without the additional information. 50 00:03:32,890 --> 00:03:37,530 So this is a quick example of how we can use probability in our estimation problem. 51 00:03:37,540 --> 00:03:42,460 We can represent the things that we want to estimate as a probabilistic distribution, and then we can 52 00:03:42,460 --> 00:03:48,430 use rules of probability to fuse in different estimates and then use Bayes Theorem to update our estimates 53 00:03:48,430 --> 00:03:51,400 to get a better estimate using all the available information. 54 00:03:51,880 --> 00:03:55,790 So this is one path away of how we can use Bayesian estimation. 55 00:03:55,810 --> 00:04:01,950 So mixing the concepts of probability and estimation theory together to form a powerful framework for 56 00:04:01,960 --> 00:04:02,800 estimation.