Exercise 2b - random numbers, color, and sound on a computer

Random numbers on a computer

If there is one thing that a computer is not it's spontaneous. Computers are unable to do something truly unexpected. This may sound wrong to you, since from a user's perspective computers can do things we don't expect, and even programmers are surprised by what their programs do at times. But, what has actually happened when we are surprised by what a computer does is that the computer executed its instructions exactly as it was told to do, but the instructions we gave it worked differently than we expected or intended. In our ignorance of what is happening under the hood we may be surprised, but the computer will always do exactly what it is told. That is, a computer is deterministic - given a set of inputs, it will always give the same output (unless it breaks, then all bets are off!).

Randomness is true unpredictability. With a random variable we receive different outputs for the same input, and we can only predict what the output will be with a probability statement. Another word for random is stochastic, which is the opposite of deterministic - for a given input a stochastic process gives different outputs that can only be predicted as a probability.

This is a problem for us as scientists, because we use random numbers a lot. When we assign our experimental subjects to different treatment groups we are supposed to do so randomly, and the most reliable way to do that is to use random numbers - closing your eyes and picking isn't really random, and can lead to biases in experiments. We will use random numbers later in class when we learn how to do things like randomization testing, and when we build our simulation model of genetic drift.

Because computers are deterministic, they are not capable of producing truly random numbers. They are capable of producing pseudo-random numbers, which are sequences of numbers that do not have a sequential pattern, generated using an input value (the "seed") applied to an algorithm (which is what computer scientists call a procedure). If we use the same initial input value the algorithm will perfectly reproduce the same set of sequentially unpatterned set of values. Since they are unpatterned we can treat them as though they are randomly generated - provided that we don't use the same seed number twice we get different, unpatterned numbers every time. In other words, we can simulate random numbers on a computer so well with pseudo-random numbers that they are indistinguishable from the real thing.

1. Set up the pseudo-random number generator. Start Excel with a new file. In cell A1 enter the word "Seed", then in B1 enter the number 1.

In cell B2 enter the formula =mod(48271*B1, 2^31-1) and hit ENTER. You'll see the number 48271. The mod() command is the modulo function, which returns the remainder left when dividing the first number by the second. The first number is 48271 multiplied by the seed in cell B1, and the second number is 2³¹-1 (exponents are denoted in Excel formulas with the carat symbol, SHIFT+6).

Now, grab the fill handle and extend downward to about the 20th row - this will copy and paste the formula to cells B3 to B20. If you look at B3 you'll see the number displayed is 182605794, but if you select the cell the formula that is producing this number shows in the formula bar as =MOD(48271*B2,2^31-1) - the formula is the same as in B2, but since we copied the formula down one row it updated the reference so that it points to B2 instead of B1. The numbers in the rows are now un-patterned, pseudo-random numbers.

2. Make random uniform probabilities. Enter the word "My uniform" in cell C1, and in cell C2 enter the formula =b2/(2^31-1) - this divides the modulo value in cell b2 by the big number we used in the modulo function, which is also the largest remainder possible. Since the smallest possible remainder is 0, and the largest is 2³¹-1, this operation gives us numbers that have to fall between 0 and 1, and we can interpret them as probabilities. The remainders in column B are equally likely to fall anywhere between 0 and 2^31, so these probabilities are equally likely to fall anywhere between 0 and 1, which means they are random numbers from a uniform probability distribution. Extend this cell down to the 20th row as well, and you will have a set of uniform (pseudo-) random numbers.

These are the kind of random numbers Excel generates for you with the rand() function - in the cell below the last of your random uniform numbers enter =rand() and you'll see a number between 0 and 1, but this one changes each time you re-calculate the worksheet because it is not based on an unchanging, constant seed like the ones you just produced are.

3. Generate random numbers from a normal distribution. Enter the word "My normal" in cell D1. Then in cell D2 enter the formula =norminv(c2,0,1) - this function takes the uniform distribution probabilities in C2 and translates them into a value drawn from a normal distribution with a mean of 0 and a standard deviation of 1. Extend these down to the 20th row and you'll have a set of (pseudo-) random numbers drawn from the normal distribution. We could get random normal numbers with any mean and standard deviation we want by changing the arguments in norminv() - for example, to get numbers with a mean of 100 and a standard deviation of 10 we would change the function in D2 to =norminv(c2,100,10), and then copy/paste the the rest of the rows.

Now that you have your formulas set up, change the seed in cell B1 to 2 - you'll see the pseudo-random numbers all change to a different set. But, if you change the seed back to 1 again they all go back to exactly the same values you had originally. Since the numbers are unpatterned we can use them as though they're random numbers in our work, and as long as we use a different seed each time we'll get numbers that are indistinguishable from truly random values, even though they were generated deterministically.

How does Excel avoid producing exactly the same set of random numbers every time? Each time you use its built-in random number function (rand()) it uses system time as its seed. Remember from lecture that system time is the number of ticks that have elapsed since the epoch, which is never the same twice, so the random numbers are never the same. We can simulate this by entering =now() into cell B1 - this will enter the date and time in B1, which is then used as the seed for the random numbers generated below it. We'll learn more about how Excel is able to interpret the date and time as a single number in a couple of weeks - for now note that each time you re-calculate the worksheet (hit F9 to do this manually) a new time stamp is entered into B1, and this causes a new set of random numbers to be generated. Since it will never be now again, this means that the random numbers we're generating will never be the same twice.

Answer the questions about random number generation on your worksheet.

The RGB color model

Color is usually represented on a computer using the RGB color model, which mixes together the intensity of red, green, and blue light to generate millions of possible colors. When you mix pigments the three primary colors are red, yellow, and blue, which can be mixed to form all the other pigment colors. On a computer we see color on a monitor, which is mixing together different colors of light - a computer monitor has three light emitting diodes in each pixel, one for each primary color, and the intensity of light emitted by each diode sets the color of that pixel.

What about printers?

Mixing color on a monitor involves mixing different colors of light, but printing involves mixing pigments (ink or toner). The color model computers use for printing is the CMYK model.

Like the RGB model the CMYK model mixes primary colors to get all the colors it requires. The letters stand for cyan, magenta, yellow, and black. The three colors (C, M, and Y) are mixed to get the hue (color) that is desired, and black is used to set the lightness or darkness of the color (its value). On the printed page (assuming white paper), the RGB code for white is 255, 255, 255, but for CMYK would be 0, 0, 0, 0 - the lack of all pigment is white on the printed page. Black for RGB is 0, 0, 0, but is 0, 0, 0, 1 for CMYK (the CMYK numbers are proportions of a total, so 1 means 100% coverage with black). There is a one to one match between RGB and CMYK colors, so computers are able to translate between the RGB and CMYK models to make the printed page look like the image on the computer screen.

Since the number of possible combinations of red, green, and blue light levels exceeds the number that the human eye can discriminate, 24 bit RGB color is called "true" color.

1. Translate colors between colors and RGB codes. Refer to the color picker to the left - you can either specify a color by entering R, G, or B values to see what color results, or you can click on the colored square to pick a color, and see which R, G, B values result.

To begin the R, G, and B channels are all set to 0 - since there is no red, green, or blue to display the color is black.

Try the following:

Set Red to 255 - the box below the number will turn bright red, because 255 is the maximum red intensity possible. Since green and blue are still 0 the square is still red - an RGB value of 255,0,0 is bright red.
Set Green to 255, and red back to 0 - the box is now bright green.
Set Blue to 255, and green back to 0 - the box is bright blue.
Set red to 255 and green to 255, but leave blue as 0 - you should see bright yellow. Mixing light is different from mixing pigment - the combination of red and green pigment is a muddy brown color, but red and green light mix to make yellow.
Now click on the square and select a color - you should see that the Red, Green, and Blue numbers update for the color you selected, and the intensity of each of the primary colors is shown below the numbers.

Using these three colors of light it is possible to display all the colors we could ever need, given the limitations of the human eye.

Calling 24 bit RGB color "true color" only makes sense given the limitations of the human eye. The RGB model uses only wavelengths of visible light, where "visible" is defined by what humans can see. However, some animals can see wavelengths that we cannot. Some animals can see into the ultraviolet (wavelengths shorter than violet light) and infrared (wavelengths longer than red light), and the RGB model can't represent those. It wouldn't make much sense to use a color model that extended into wavelengths that we can't see, because...we wouldn't be able to see them!

What we can do, however, is to use the RGB model to make false color images - for example, if we had measurements of the amount of infrared radiation reflected from a scene we could assign the intensity of infrared radiation to be displayed by the red channel on our computer screens, which would show us something like this:

By assigning a wavelength band that we can't see to one of the color channels that we can see, we're able to visualize characteristics of the scene that would be invisible otherwise. It should be fairly obvious from where the brightest reds are located in this image that growing vegetation reflects a lot of infrared light compared to buildings and water - we can make use of this fact to help us identify where the vegetation is growing well, and where it is dormant.

Many insects can see into the ultraviolet part of the spectrum, and flowers evolve patterns that attract them that are only visible to those wavelengths. The picture on the left shows a dandelion as we see it, on the right side, and how it appears to an insect on the left side. Measuring the amount of UV radiation reflected off the flower and assigning it to the red channel shows that the middle of the flower has a pattern that we can't see, but insects can.

So, while the "true color" designation of 24 bit RGB color is only true relative to human vision, we can take advantage of the RGB color model to make invisible wavelengths visible to us by assigning them to visible color channels on the computer.

Digitizing sound waves

1. Evaluate how many different levels of sound intensity would be possible at different bit depths. If you remember from lecture, sound is converted to a digital representation by sampling the sound wave form at discrete time points. The two factors that determine how well the sound wave is represented are...

The graph that shows the closest match between the blue bars (indicating the digital recordings of sound intensity) and the curve (indicating the actual sound wave) occurs with high sampling rates and high bit depth. Note that even the best digital sound recording doesn't perfectly match the analog sound wave, but it can get close enough that the human ear can't hear the difference.

On your worksheet answer the questions about how bit depth affects fidelity of digital sound recordings.

File storage and lossy compression

Digital files can get really big, and although storage media capacities keep increasing we keep finding ways to fill them up. Finding ways to store information with fewer bits allows us to fit more data onto our hard drives, CD-ROM's, and flash drives.

When you are looking at pictures on a web page, or watching video, or listening to music, as long as everything looks and sounds okay to you there would be no reason to worry about whether the images or audio were exactly as they were recorded. If the recorded data had been changed in some way to save storage space, as long as the results didn't look or sound different it wouldn't matter.

When you ware doing scientific work, though, digital data in image, sound, or video files is used to measure the properties of a system being studied. A change in data that isn't intentional is a problem, and the fact that the change may not be noticeable to the researcher actually makes the problem worse, as the data may be used as though it's accurate when it isn't anymore.

Using lossy file formats is a way of corrupting your digital data unintentionally, and possibly without knowing that you did so. Today you will evaluate how use of lossy file formats can affect digital sound data and digital image data. In general, use of lossy file formats in scientific work is a bad idea, and should only be done when a researcher knows how the data will be altered, and is certain that the change will not affect his or her work.

Lossy compression methods commonly used for sound files throw out sounds that aren't audible to the human ear. This is fine for audio recordings that are meant for entertainment, but if we are recording animal sounds to study how they communicate we have to be aware that animals can hear different ranges of sound than we can, and throwing out data because we can't hear it is problematic.

To see how lossy compression affects sound files, we are going to use a program called Audacity to analyze a recording of a European Blackbird. European Blackbirds are not closely related to blackbirds we have in North America, they are actually much more closely related to the American Robin. If you know birds, the picture of a European Blackbird to the left looks just like a black robin.

We will start with a recording of a blackbird song stored in an uncompressed WAV file, export it to a lossy compression format (called Ogg Vorbis), and then compare the compressed file to the original to see how lossy compression changes the data in the file.

1. Start Audacity from Cougar Apps. You can open a copy of the uncompressed WAV file by selecting "File" → "Open", clicking on the "This PC" icon in the "Select one or more files" window, and then navigating to "Public on Viking (P:)" → "Biology" → "kristanw" → "Biol365" → "Ex2", and then open the file "blackbird_w.wav".

When the file opens the sound is displayed as a waveform graph, which has time on the x-axis, and loudness (in decibels) on the y-axis. If you are able to hear it when you play the sound, you'll hear that the blackbird has a lilting, flute-like song that covers a wide range of different pitches - this will make it a good test case.

2. Before we try out some lossy compression, we will test the method we will use to compare the original sound file to the compressed one. We will make a copy of the original uncompressed WAV file, invert it, and then mix the original with the inverted copy. An example of a tiny clip from the sound file is shown to the left, with the original on top and the inverted original on the bottom - inverting the sound file flips the wave vertically around 0, so that positive values become negative, and negative become positive - if you look at the discrete sampled sound intensities on the two they are mirror images of one another.

Mixing these two sound waves together means to add them, and if two waves are exact inverses of one another they will cancel each other out. The result will be a flat line at 0, which is silence. Since the second is an exact inverse of the other this is what we'll get - a flat line at 0.

First we'll make an exact copy of the song by selecting the track (by clicking the "Select" button in the controls to the left of the waveform graph), then duplicating it using "Edit" → "Duplicate" - you will now see a second copy of the original in a second "track". below the first.

Next, select just the duplicate track by clicking its "Select" button. When a track is selected it lightens and gets a light yellow border around it, so the second track should be lighter than the first.

Now, select "Effect" → "Invert". If you watch the waveform graph as you invert it you'll see a subtle change in the wave form, but otherwise you probably won't be able to tell that it's different. Don't worry if you don't see the change - you can double-check that it happened by checking the "Edit" menu, and if it gives you the option to undo the invert then you're good to go.

Now to combine the tracks hold down the CTRL key while you hit the A key to select both of the tracks, and then use "Tracks" → "Mix" → "Mix and render to new track". The two tracks will be combined into one, and you'll see a flat line. If you select this new mixed track and play it you won't hear anything. This is what happens when you mix two tracks are inverses of one another - they cancel each other out exactly, and the result is silence.

Before you move on to the next step you can close the second, duplicate track by clicking the little x next to the name of the track in the properties box on the left side of the track (next to "blackbird_w" in the screen shot above), and do the same for the mixed flat line track. You should just have the original track left open before going on.

3. Now select "File" → "Export" → "Export as OGG". In the "Export Audio" window that pops up:

You may not have heard of Ogg Vorbis before - it is a lossy sound file format intended for the same uses as mp3 files, primarily for sound files meant for human ears. We would have used mp3 since it is so common, but there is a bug in the mp3 encoder used by Audacity that inserts a few milliseconds of silence at the beginning of each file, and this causes the sound waves to misalign when we try to compare them. However, any lossy sound file format, including mp3, would change the sound data and give us similar results to what we will get with Ogg Vorbis.

4. Import the lossy version of the file as a second track. Select "File" → "Import" → "Audio", and go find the Ogg Vorbis file you just exported, select it, and import it.

Audacity will make a new track for this file, add it below the original, and select it. The wave forms don't obviously look different, and if you select the lossy one and play it you won't be notice any obvious differences.

You should now have a mixed track that isn't a flat line. Anywhere that isn't flat is sound data that was changed when the file was saved to a lossy format.

If you try to play the mixed track you may not hear much, it's a fairly subtle amount of difference. But, it's not zero difference, and for scientific work it is a bad idea to allow a choice of file format to change your data.

Some image file formats use raw, uncompressed image data (e.g. TIFF), but there are also lossless (e.g. PNG), and lossy compression (e.g. JPEG) formats to choose from.

Just like with sound file compression, lossy compression of images discards image data strategically, so that the image still looks okay when viewed at its intended magnification. However, when images are being used as the basis for data collection these changes in the image corrupt the data.

We need to use an image editing program to see how compression changes an image file, so we will use the Gnu Image Manipulation Program (GIMP). The GIMP is an open source program that is similar to PhotoShop.

1. Start the GIMP from Cougar Apps, and open the original TIFF file on the P: drive. You will find it in the same folder as the blackbird song - it's called csusm.tif.

When you open it you'll see it's a high resolution aerial photo of part of the CSUSM campus taken in 2012 (before the USU was built, although you can see that construction had started).

2. Hold down the SHIFT key and hit + to zoom in. When you've zoomed in far enough to see individual pixels stop. This image has a resolution if 0.6 feet (7.2 inches on a side), so each pixel is equivalent to 0.36 square feet on the ground.

Select "Windows" → "Dockable Dialogs" → "Pointer" to bring up the pointer window. With the pointer window visible, move your pointer around the image and you'll see that the Red, Green, and Blue values change as you move from pixel to pixel. Even though some pixels are pretty similar in color, you will have a hard time finding RGB values that are identical for adjacent pixels.

You can zoom back out so that you can see the whole scene now by hitting the "-" key (to the left of +).

3. Now it's time to export the file as a lossy file format, jpeg. We're going to use jpeg because it is very common, and sufficiently good at retaining a good looking image that many digital cameras use it as their native format. The jpeg format uses a combination of lossy and lossless compression techniques, and the amount of compression is configurable as a "quality" setting - higher quality means less lossy compression, and if a quality setting of 100% is used then only lossless compression is used. However, even digital cameras will typically use a quality setting less than 100% by default, and some degree of lossy compression is typically used when the image is originally acquired.

Choose "File" → "Export as", and in the "Export Image" window give the file the name "csusm.jpg" and save it to your OneDrive. Using the jpg extension will cause the GIMP to export as a jpeg file, so we don't need to select that as an option. We will accept the default of 90%, so don't change this setting.

When you click "Export" an "Export image as JPEG" window will pop up so that you can set the quality. The default is 90%, which is considered very high (70% is considered high quality for images used on web pages). Click "Export".

4. Import the jpeg file you just created as a new layer by selecting "File" → "Open as layers". Navigate to the csusm.jpg, select and open it.

This will add csusm.jpg as a second layer in the same image as csusm.tif, rather than opening the jpeg file as a separate image. If you look at the "Layers" tab in the lower right section of the GIMP window you'll see that you now have a layer for csusm.jpg above the original csusm.tif layer.

You can turn the csusm.jpg layer on and off by clicking on the eye icon to the left of it, but you probably won't notice big differences unless you zoom in to pixel level, and even at a very high magnification the differences are very slight - using 90% quality results in barely perceptible changes in the appearance of the image.

But just because our eyes aren't seeing differences doesn't mean they aren't there - we'll confirm that saving as jpeg changed the image data in the next step.

5. To do the comparison of csusm.tif and csusm.jpg we will take advantage of a built-in function that is used to make animations from multiple images.

Animation and video takes up a lot of file space because they use multiple images that are played in quick succession to give the illusion of movement. If each frame in the animation is a complete image then the animation file size will be the size of a single frame multiplied by the number of frames - since video tends to use 30 frames per second or more to produce smooth playback the number of individual frames becomes huge for even a few minutes of video. Animation and video files get big quickly if they are not compressed.

However, not very much typically changes from one frame to the next when each is only 1/30 of a second apart in time. Sequential frames in an animation thus usually have a lot of pixels that are identical, and only a few pixels that actually change between frames. If the animation file contains a full image for the first frame, and then saves only the differences between the first frame and the frames that follow, pixels that don't change only have to be saved once, which can save a lot of space. We will use a tool that makes these differenced images, but instead of making an animation from them we'll use the differenced images to see how jpeg changes the data from the original tif image.

Select "Filters" → "Animation" → "Optimize (difference)". This will make a new image in a new tab with two layers, the first being a complete copy of the original lossless tif image, and the second being pixels in the jpeg that are different from the original tif. Pixels that are the same between the tif and jpeg images are transparent in the second layer, so turning the original on and off by clicking the eye icon next to csusm.tif(100ms) will show where the differences are - transparent pixels were the same between the tif and jpeg images, and any that are not transparent are at least a little different between the tif and jpeg layers.

Now when you turn csusm.tif(100ms) off you will see red through the transparent pixels - red pixels are those that did not change when you saved the original tif image as a jpeg with 90% quality. Any pixel that is not red is at least a little different from the tif image.

Since most of the image isn't red, there were at least small changes to most of the pixels in the image when it was saved as a jpeg. The differences are mostly very slight, but just like with the audio file, we wouldn't want our choice of file format to change our data at all, so non-zero changes are a problem.

In general, the best way to avoid corruption to your digital data is to use either uncompressed file formats, or file formats that use lossless compression.

That's it! Upload your completed Excel file to the course web site, and complete your worksheet to turn in next week.

	2 bit (only 3 intensities recorded)	4 bit (16 intensities recorded)	8 bit (65536 intensities recorded)
Bit depth = the number of bits used to record sound intensity. The more bits used the larger the number of discrete levels that can be recorded. The bars can get closer to the continuous wave if there are more different levels possible along the y-axis

	8 bit with low sampling rate	8 bit with moderate sampling rate	8 bit with high sampling rate
Sampling rate - the amount of time between samples of the sound wave. The shorter the time between measurements of sound intensity the narrower the bars, and the more continuous they become

Exercise 2b - randomness, color, and sound

Random numbers on a computer

The RGB color model

Digitizing sound waves

File storage and lossy compression