We’re going to spend a good deal of time on the following general form of the relationship between our parameters and our measurements:

    \begin{equation*} \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0} \end{equation*}

But we need an example before we lose the meaning of what we’re actually talking about..

Example 2: Our first functional model
(of a distance)

Let’s consider an extension of the example where we considered measurements made to estimate the width of a room. In Example 1 our measurement l_i was a direct observation of the distance d_{AB} between unknown stations A and B.

But in this example, I want you to imagine that instead of wanting to know the distance between the two points (e.g. the width of the room), we want to know the coordinates of the stations in a simple planar 2D coordinate system as shown below. We’re still going to measure l_i, but this time the parameter of interest is different.

Sketch

From Example 2

This example

Situation

The desired parameter vector \mathbf{x} is made up by the coordinates of the stations A and B. In other words, E_AN_A, E_B, and N_B will become part of \mathbf{x}.

And the measurements l_i of the distance d_{AB} will become part of the measurement vector \mathbf{l}_{true}.

Required

I want you to find the relationship between the parameters and the measurements.

Solution

The model that relates the true distance to the actual (unknown) coordinates of A and B is given by the Pythagorean Theorem as follows:

    \begin{equation*} l_{true} = d_{AB} = \sqrt{(E_B-E_A)^2 + (N_B-N_A)^2} \end{equation*}

which gives us an expression of our observables (parameters) in terms of our true observations.

And now you know how to parametrize the given situation.

Pretty simple, I know. But I wanted to have that example in hand before we went any deeper with \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0}, which we’re ready to do now.

The general form of this functional model

Let’s go back and clarify what we wrote before.

    \begin{equation*} \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0} \end{equation*}

is nothing more than a vector function, which means a set of r equations like this:

    \begin{equation*} \begin{bmatrix} f_1(\mathbf{x},\mathbf{l}_{true}) = 0 \\ f_2(\mathbf{x},\mathbf{l}_{true}) = 0 \\ \vdots \\ f_r(\mathbf{x},\mathbf{l}_{true}) = 0 \end{bmatrix} \end{equation*}

And we just saw the general form of these for this example, which can be rearranged as follows:

    \begin{equation*} l_{true} - \sqrt{(E_B-E_A)^2 + (N_B-N_A)^2} = 0 \end{equation*}

which, incidentally, can be written in the following general form for each row in the vector function:

    \begin{equation*} l_{true} - f_j(x) = 0 \end{equation*}

or as follows:

    \begin{equation*} \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) =\mathbf{0} \end{equation*}

Later we will see that the ability to separate the model in this example into these separate terms means that it belongs to a certain class of functional models.

But whatever form one writes it in, it still represents nothing more than the basic geometry or physics relating the parameters to the observed quantities.

The general functional model

Generally speaking, we are going to deal throughout this course with situations (problems to solve) where:

we need to estimate u parameters (or unknowns)

from n observations

with a functional math model containing r equations.

And usually, things will be designed so you have redundancy, meaning that there will be more observations than are strictly necessary to solve the problem, i.e.:

n \geq r \geq u

We will call ru the degrees of freedom, i.e. the statistical degrees of freedom is the number of equations minus the number of unknown parameters.

And, therefore, to express it formally, the general functional model for geomatics networks is:

    \begin{equation*} \boxed{\mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0}} \end{equation*}

where:

\mathbf{F} is a vector function of r equations

\mathbf{x} is the parameter vector of size u unknowns

\mathbf{l}_{true} is the observation vector of size n

I’ve been deliberately super explicit with my subscripts throughout this mini course so far, but it’s worth noting that this basic mathematical model applies to the true values of the parameters and (what the) observed quantities (would be if error free).

Now let’s take a look at some more examples of functions \mathbf{F}(\mathbf{x},\mathbf{l}_{true}) = \mathbf{0}} to make sure we understand, and to help us generalize the different kinds of such models.

Example 3: Our second functional model
(of an azimuth)

In this example, I want you to imagine that we want to know the coordinates of two points in a simple planar 2D coordinate system as shown below. But instead of measuring the distance between them, we’re going to measure the azimuth of point B measured from point A, as shown below.

Sketch

Situation

The desired parameter vector \mathbf{x} is again made up by the coordinates of the stations A and B. In other words, E_AN_A, E_B, and N_B will become part of \mathbf{x}.

And the measurements \alpha of the azimuth will become part of the measurement vector \mathbf{l}_{true}.

(Note that a more rigorous treatment of this azimuth example can be found here.)

Required

Again, I want you to find the relationship between the parameters and the measurements.

Solution

The model that relates the true azimuth to the actual (unknown) coordinates of A and B is given by the following:

    \begin{equation*} tan(\alpha_{AB})=\dfrac{E_B-E_A}{N_B-N_A} \end{equation*}

or:

    \begin{equation*} \alpha_{AB} - tan^{-1}(\dfrac{E_B-E_A}{N_B-N_A}) = 0 \end{equation*}

which, again, can be written in the following general form for each row in the vector function:

    \begin{equation*} l_{true} - f_j(x) = 0 \end{equation*}

or as follows:

    \begin{equation*} \mathbf{l}_{true} - \mathbf{F}(\mathbf{x}) =\mathbf{0} \end{equation*}

Let’s do two other (different) examples before drawing some conclusions.

Example 4: Our third functional model
(of a best fit line)

In this example, I want you to consider the challenge of estimating the slope, m, and vertical-intercept, b, of the line that best fits through some known points in our 2D planar coordinate system.

This is shown below and could arise, for example, in the case where you need to fit a “road vector” to a set of map coordinates derived from GPS.

Sketch

Situation

This time the desired parameter vector \mathbf{x} is made up by the quantities m and b.

And the measurements are made up by the coordinates of the stations 1234 and 5. In other words, E_1N_1, E_2, N_2, E_3, N_3, E_4, N_4, E_5, N_5 will become part of the measurement vector \mathbf{l}_{true}.

(Note that a more rigorous treatment of this azimuth example can be found here.)

Required

Again, I want you to find the relationship between the parameters and the measurements.

Solution

We know from the general equation of a straight line y = mx + b that the following is true in this case:

    \begin{equation*} N_i = mE_i + b \end{equation*}

or:

    \begin{equation*} N_i - mE_i - b = 0 \end{equation*}

This time, it’s important to notice that our functional model can’t be simplified beyond the following:

    \begin{equation*} f_j(x,l_{true}) = 0 \end{equation*}

or as follows in terms of the vector function:

    \begin{equation*} \mathbf{F}(\mathbf{x}, \mathbf{l}_{true}) =\mathbf{0} \end{equation*}

because we’re not able to separate the parameters from the observed quantities.

Example 5: Our fourth functional model
(of the internal angles of a triangle)

In this example, assume we want to know the internal angles of a triangle from observed angles, again on a planar 2D surface. This is shown below.

Sketch

Situation

This time the angles \alpha_A\alpha_B\alpha_C become part of the measurement vector \mathbf{l}_{true} and there are no separate parameters to be related.

Required

Again, I want you to find the relationship between the parameters and the measurements.

Solution

We know from the geometric principles of a triangle that:

    \begin{equation*} \alpha_A + \alpha_B + \alpha_C = 180^o \end{equation*}

or:

    \begin{equation*} \alpha_A + \alpha_B + \alpha_C - 180^o = 0 \end{equation*}

This time, this can be reduced to the following general form for each row of the vector function:

    \begin{equation*} f_j(l_{true}) = 0 \end{equation*}

or as follows:

    \begin{equation*} \mathbf{F}(\mathbf{l}_{true}) =\mathbf{0} \end{equation*}

If you’ve been watching carefully, you’ll recognize that this gives rise to a third general form. We’ll explore all three in the next lesson.