The normal or Gaussian distribution is one of the most important probability density functions when
it comes to data fusion, the distribution is a very natural way of expressing an estimated value and
the associated uncertainty of the estimate.
The whole distribution is continuous, which makes it a mathematically nice function to operate with.
The shape of the distribution is shown here along with its function.
The complete distribution is fully described by two parameters.
It is described by the mean and the variance, and it's usually notated up here.
So we have the capital N for the normal.
Distribution has the main and it has a variance.
Now the fact that it's the food distribution is described by only these two families to get this complete
shape makes it a very compact way of describing the distribution.
As usual, the complete area under the curve under the distribution has to equal one for it to be a
valid PDF.
This is why there's a normalization factor out of this distribution as the variance of the distribution
shrinks.
So as the edges get closer, the likelihood or peak value of the distribution has to increase to maintain
the unit area under the curve.
The majority of the random values produced by the distribution are clustered around the main or the
pache value, and they become less probable as they move further out either side of the main.
As the main position shifts, the take shifts as variance shrinks, regrows, the spread of the distribution
shrinks or growers are calling a very useful property of the Gaussian distribution.
Is the three sigma rule the probability that the random value is within one segment of the main.
So if the value lies in this area, here is sixty eight percent.
So if we integrate the PDF between around the main so the main minus one sigma all the way up to the
main plus one sigma, if we integrate the probability density function we get point six eight.
So sixty 68 percent.
Now if we do the same thing for two sigma, so if we integrate the area under the curve between these
two bands here, we then get a probability of point nine, five or 95 percent.
And then again if we do it with three sigma, so we integrate the complete area between these two elements
here we end up with a probability of point nine, nine or 99 percent.
This gives us a likely bound on the random number that might be produced by the random variable.
It conversely, also gives us a measure of how likely any value might be consistent with the random
variable or distribution.
We call it the Three Sigma rule, because pretty much any value produced by the distribution is most
likely to be contained within plus or minus three sigma of the main value.
And we can be 99 percent confident that this value is.
One of the most important properties of the Gaussian distribution is the fact that any linear transformation
of the Gaussian distribution is another Gaussian distribution.
This will be explored later on in depth as it is a key property for many of the estimation processes
later on.
But basically, this allows the computationally expensive operations involving convolutions or integrations
of the probability density functions to be simplified down to a transformation of the main and variance
parameters themselves, rather than carrying out the complete transformation of the Paideia function.
And again, we will look at this in more detail.
But this is a very important concept of the Gaussian distribution.
So now let's have a look at an estimation example to highlight the usefulness of this distribution.
Imagine that we're trying to estimate a position of a car.
So the estimation process gives us a Gaussian distribution for the estimate of the car described by
a mean and variance.
So the distribution for the position estimate looks something like this and it can be described using
these numbers here.
So imagine we have an estimate of the position.
It has a main value of one hundred and twenty five meters and a variance of four square meters squared.
So we know the most likely position of the car.
And we also know how good the estimate is.
So using the three sigma rule, we know that the composition has to be within plus or minus three sigma
of the main and we can be ninety nine point eight percent sure of this confidence.
This gives us a bounce on a position estimate.
So therefore the estimated position must be one hundred and twenty five meters, plus or minus three
sigma or 12 meters.
So this is a very useful way of expressing an estimate position and it's uncertainty.
So hopefully you can see why the Gaussian distribution is a very nice way of expressing an estimated
value and its associated uncertainty, and this is going to be very useful later on when we look in
depth at different estimation processes.