Multivariate data

We will be measuring several variables on two different kinds of clam shells - white (left) and calico (right). Although both of these are bivalve shells, and have a roughly similar morphology, we will use multivariate techniques to see how they differ across several morphological measurements at once.

Everyone should measure 2 or 3 shells of each type until all are measured. Make sure you measure each variable on the shells you select - one missing variable forces us to drop the entire shell from the data set! Please measure carefully.

The measurements we will use will be:

Weight - weigh the shells on the balance.

Major axis - maximum distance measured along the longest dimension, with a calipers, to the nearest 0.01 mm.

Minor axis - maximum distance measured roughly perpendicular to the major axis. Make sure the shell sits flat along the jaws of the calipers.

Height - measured from top to bottom. Be careful that both sides of the open part of the shell rest on the jaws of the calipers, and not in the gap at the back.

Surface length - use a tape measure to to measure the surface distance along the major axis. Record this to the nearest 0.5 mm (between the 1 mm hash marks).

Depth - measure the thickness of a metal ruler and zero the calipers on it. Then, lay the shell on the ruler, and measure the depth from the top of the ruler to the inside of the shell at its deepest point using the bottom of the caliper as a depth gauge.

Hoofed mammals with even numbers of toes are placed in the Order Artiodactyla, and Artiodactyls with permanent horns are placed in the Family Bovidae. Bovids are predominantly grazers (i.e. they eat grass), and their teeth are adapted to this diet. Their cheek teeth (i.e. molars and pre-molars) are adapted for grinding up grass. We won't be working with cheek teeth today.

Bovids have incisors that they use to clip off grass to eat, but only on their lower jaw (as you can see in the first picture with both the upper and lower jaw visible). They press the grass against a thick, tough dental pad on their upper jaw, and then clip off the grass against their incisors.

We have 100 teeth from an Indian buffalo species, like the one shown in these pictures. The incisors are expected to vary in size for a couple of reasons: a) individual variation, and b) variation attributable to position in the mouth. As you can see in these two pictures, the teeth in the middle at the front of the mouth are bigger than the ones towards the back of the mouth. The pictures also suggest we will see variation in shape, as the tooth shape also appears to change with position.

NASA and the US Geological Survey jointly operate the LandSat remote sensing program, which uses satellites to record reflectance of electromagnetic radiation from the earth's surface. At least one satellite has been in orbit continuously since 1972, and the orbital paths allow each satellite to record the same location about once per month. We will be using an image from the LandSat 5 satellite taken in April of 2011 that covers most of San Diego County.

If you are familiar with how a digital camera works, visible light is recorded in three different bands (red, green, and blue), and these three primary colors of light are combined on the page, or on a computer screen, to make all the colors we see. Digital images are an array of square pixels, each of which have a digital number recorded that indicates the amount of light that was recorded in the band. Digital numbers can have values between 0 to 255, with 0 indicating no light recorded in that band, and 255 being the maximum amount of light that the sensor can record in that band.

For a typical color image, a single pixel on the screen will thus have a DN for each of the three recorded color bands, reported as an RGB code. For example, an RGB code of 255,0,0 would have a maximum amount of red, and no green or blue, which will display as bright red on the screen. An RGB value of 255,255,255 is white, an RBG of 0,0,0 is black, and any set of numbers that are the same is a shade of gray (150,150,150 would be a medium gray, 250,250,250 would be a very light gray).

LandSat satellites record all three of the visible light bands, but they also include additional bands that are not visible to the naked eye. The LandSat 5 sensor that recorded the image we will use had 7 bands, the first three of which were visible blue, green, and red, and the remaining four of which were infrared (band 4 and 5 are near infrared, band 6 is mid infrared, and band 7 is thermal infrared).

The LandSat data can be used to make color composites that either look natural (if they use only the three visible bands), or that don't look natural (if one or more of the infrared bands is assigned to the R, G, or B channel on the computer monitor). Assigning the bands that record visible red, visible green, and visible blue light to the R,G, and B channels on the computer monitor gives you a "true color" composite, like the one on the right. You can see that different types of land cover have different colors - this is because different cover types absorb more light in some bands and reflect more in others, which gives us a different mix of R, G, and B digital numbers for different cover types.

But, we have more than the three visible bands to work with. If we assign the first near-infrared band (band 4) to the red color channel on the computer screen, we get a "color infrared" image. Visible blue and visible green are still assigned to the blue and green channels, respectively, but just by substituting infrared for red we can see certain differences more clearly than before, and we can also see some differences in the vegetation that wasn't apparent. It turns out that actively growing vegetation reflects more strongly in the infrared bands than does living but less active vegetation, and even more strongly than does dormant or dead vegetation.