Example 3.2.7

Suppose we work for a company that records its quarterly revenue, in millions of dollars, as:

Quarter Revenue
1 10.3
2 13.1
3 7.5
4 8.2
Table 3.2.8 Quarterly revenue

Rather than using a table to record the data, we could display it in a graph or write it as a vector in \(\real^4\text{:}\)

\begin{equation*} \xvec=\fourvec{10.3}{13.1}{7.5}{8.2}\text{.} \end{equation*}

Let's now consider a new basis \(\bcal\) for \(\real^4\) using vectors

\begin{equation*} \vvec_1=\fourvec{1}{1}{1}{1}, \vvec_2=\fourvec{1}{1}{-1}{-1}, \vvec_3=\fourvec{1}{-1}{0}{0}, \vvec_4=\fourvec{0}{0}{1}{-1}\text{.} \end{equation*}

We may view these basis elements graphically, as in FigureĀ 3.2.9

Figure 3.2.9 A representation of the basis elements of \(\bcal\text{.}\)

As we wish to convert our revenue vectors into the coordinates given by \(\bcal\text{,}\) we form the matrices:

\begin{equation*} C_{\bcal} = \left[\begin{array}{rrrr} 1 \amp 1 \amp 1 \amp 0 \\ 1 \amp 1 \amp -1 \amp 0 \\ 1 \amp -1 \amp 0 \amp 1 \\ 1 \amp -1 \amp 0 \amp -1 \\ \end{array}\right], C_{\bcal}^{-1} = \left[\begin{array}{rrrr} \frac14 \amp \frac14 \amp \frac14 \amp \frac14 \\ \frac14 \amp \frac14 \amp -\frac14 \amp -\frac14 \\ \frac12 \amp -\frac12 \amp 0 \amp 0 \\ 0 \amp 0 \amp \frac12 \amp -\frac12 \\ \end{array}\right] \end{equation*}

and compute

\begin{equation*} \coords{\xvec}{\bcal} = C_{\bcal}^{-1} \xvec = C_{\bcal}^{-1} \fourvec{10.3}{13.1}{7.5}{8.2} = \fourvec{9.775}{1.925}{-1.400}{-0.350}\text{.} \end{equation*}

This means that our revenue vector is

\begin{equation*} \xvec = 9.775 \vvec_1 + 1.925 \vvec_2 - 1.400\vvec_3 - 0.350 \vvec_4\text{.} \end{equation*}

We will think about what these coordinates mean by adding the basis vectors together one at a time.

The first coordinate gives us the average revenue over the year: \(9.775\vvec_1\text{.}\)

Adding in the second component shows how the averages in the first and second halves of year differ from the annual average: \(9.775\vvec_1 + 1.925\vvec_2\text{.}\)

The third and fourth components break down the behavior in the first and second halves of the year into quarters:

\begin{equation*} \begin{aligned} \xvec = \amp 9.775 \vvec_1 + 1.925 \vvec_2 \\ \amp - 1.400\vvec_3 - 0.350 \vvec_4\text{.} \end{aligned} \end{equation*}

If we write \(\coords{\xvec}{\bcal} = \fourvec{c_1}{c_2}{c_3}{c_4}\text{,}\) we see that the coefficient \(c_1\) measures the average revenue over the year, \(c_2\) measures the deviation from the annual average in the first and second halves of the year, and \(c_3\) measures how the revenue in the first and second quarter differs from the average in the first half of the year. In this way, the coefficients provide a view of the revenue over different time scales, from an annual summary to a finer view of quarterly behavior.

This basis is sometimes called a Haar wavelet basis, and the change of basis is known as a Haar wavelet transform. In the next section, we will see how this basis provides a useful way to store digital images.