1 00:00:03,660 --> 00:00:08,610 The normal or Gaussian distribution is one of the most important probability density functions when 2 00:00:08,610 --> 00:00:14,490 it comes to data fusion, the distribution is a very natural way of expressing an estimated value and 3 00:00:14,490 --> 00:00:16,950 the associated uncertainty of the estimate. 4 00:00:17,460 --> 00:00:22,560 The whole distribution is continuous, which makes it a mathematically nice function to operate with. 5 00:00:23,250 --> 00:00:27,510 The shape of the distribution is shown here along with its function. 6 00:00:28,140 --> 00:00:31,620 The complete distribution is fully described by two parameters. 7 00:00:31,620 --> 00:00:36,740 It is described by the mean and the variance, and it's usually notated up here. 8 00:00:36,750 --> 00:00:38,610 So we have the capital N for the normal. 9 00:00:38,610 --> 00:00:41,520 Distribution has the main and it has a variance. 10 00:00:42,210 --> 00:00:47,910 Now the fact that it's the food distribution is described by only these two families to get this complete 11 00:00:47,910 --> 00:00:52,070 shape makes it a very compact way of describing the distribution. 12 00:00:53,190 --> 00:00:59,430 As usual, the complete area under the curve under the distribution has to equal one for it to be a 13 00:00:59,460 --> 00:01:00,790 valid PDF. 14 00:01:01,680 --> 00:01:07,740 This is why there's a normalization factor out of this distribution as the variance of the distribution 15 00:01:07,740 --> 00:01:08,420 shrinks. 16 00:01:08,430 --> 00:01:14,640 So as the edges get closer, the likelihood or peak value of the distribution has to increase to maintain 17 00:01:14,640 --> 00:01:16,350 the unit area under the curve. 18 00:01:16,950 --> 00:01:21,900 The majority of the random values produced by the distribution are clustered around the main or the 19 00:01:21,900 --> 00:01:27,720 pache value, and they become less probable as they move further out either side of the main. 20 00:01:28,380 --> 00:01:34,710 As the main position shifts, the take shifts as variance shrinks, regrows, the spread of the distribution 21 00:01:34,710 --> 00:01:39,480 shrinks or growers are calling a very useful property of the Gaussian distribution. 22 00:01:39,480 --> 00:01:44,940 Is the three sigma rule the probability that the random value is within one segment of the main. 23 00:01:45,240 --> 00:01:49,770 So if the value lies in this area, here is sixty eight percent. 24 00:01:50,190 --> 00:01:57,660 So if we integrate the PDF between around the main so the main minus one sigma all the way up to the 25 00:01:57,660 --> 00:02:02,970 main plus one sigma, if we integrate the probability density function we get point six eight. 26 00:02:03,600 --> 00:02:05,120 So sixty 68 percent. 27 00:02:05,820 --> 00:02:10,650 Now if we do the same thing for two sigma, so if we integrate the area under the curve between these 28 00:02:10,650 --> 00:02:15,960 two bands here, we then get a probability of point nine, five or 95 percent. 29 00:02:16,440 --> 00:02:21,360 And then again if we do it with three sigma, so we integrate the complete area between these two elements 30 00:02:21,360 --> 00:02:25,950 here we end up with a probability of point nine, nine or 99 percent. 31 00:02:26,850 --> 00:02:31,620 This gives us a likely bound on the random number that might be produced by the random variable. 32 00:02:31,890 --> 00:02:37,200 It conversely, also gives us a measure of how likely any value might be consistent with the random 33 00:02:37,200 --> 00:02:38,550 variable or distribution. 34 00:02:39,060 --> 00:02:44,040 We call it the Three Sigma rule, because pretty much any value produced by the distribution is most 35 00:02:44,040 --> 00:02:48,210 likely to be contained within plus or minus three sigma of the main value. 36 00:02:48,360 --> 00:02:51,930 And we can be 99 percent confident that this value is. 37 00:02:53,160 --> 00:02:58,710 One of the most important properties of the Gaussian distribution is the fact that any linear transformation 38 00:02:58,710 --> 00:03:02,580 of the Gaussian distribution is another Gaussian distribution. 39 00:03:03,150 --> 00:03:08,400 This will be explored later on in depth as it is a key property for many of the estimation processes 40 00:03:08,400 --> 00:03:09,000 later on. 41 00:03:09,570 --> 00:03:15,630 But basically, this allows the computationally expensive operations involving convolutions or integrations 42 00:03:15,630 --> 00:03:22,050 of the probability density functions to be simplified down to a transformation of the main and variance 43 00:03:22,050 --> 00:03:27,190 parameters themselves, rather than carrying out the complete transformation of the Paideia function. 44 00:03:27,960 --> 00:03:29,970 And again, we will look at this in more detail. 45 00:03:29,970 --> 00:03:33,120 But this is a very important concept of the Gaussian distribution. 46 00:03:34,280 --> 00:03:39,120 So now let's have a look at an estimation example to highlight the usefulness of this distribution. 47 00:03:39,890 --> 00:03:42,670 Imagine that we're trying to estimate a position of a car. 48 00:03:43,070 --> 00:03:48,710 So the estimation process gives us a Gaussian distribution for the estimate of the car described by 49 00:03:48,710 --> 00:03:50,270 a mean and variance. 50 00:03:50,300 --> 00:03:56,270 So the distribution for the position estimate looks something like this and it can be described using 51 00:03:56,270 --> 00:03:57,130 these numbers here. 52 00:03:57,140 --> 00:03:59,250 So imagine we have an estimate of the position. 53 00:03:59,780 --> 00:04:06,230 It has a main value of one hundred and twenty five meters and a variance of four square meters squared. 54 00:04:07,340 --> 00:04:09,450 So we know the most likely position of the car. 55 00:04:09,470 --> 00:04:11,270 And we also know how good the estimate is. 56 00:04:11,540 --> 00:04:17,780 So using the three sigma rule, we know that the composition has to be within plus or minus three sigma 57 00:04:17,780 --> 00:04:22,260 of the main and we can be ninety nine point eight percent sure of this confidence. 58 00:04:23,210 --> 00:04:25,510 This gives us a bounce on a position estimate. 59 00:04:25,880 --> 00:04:31,880 So therefore the estimated position must be one hundred and twenty five meters, plus or minus three 60 00:04:31,880 --> 00:04:33,240 sigma or 12 meters. 61 00:04:33,770 --> 00:04:38,420 So this is a very useful way of expressing an estimate position and it's uncertainty. 62 00:04:40,220 --> 00:04:44,930 So hopefully you can see why the Gaussian distribution is a very nice way of expressing an estimated 63 00:04:44,930 --> 00:04:50,180 value and its associated uncertainty, and this is going to be very useful later on when we look in 64 00:04:50,180 --> 00:04:52,490 depth at different estimation processes.