120 Statistics Scatter Plots and Smoothing
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An Example Car Stopping Distances
An experiment was conducted to measure how the stopping distance of a car depends on its speed. The experiment used a random selection of cars and a variety of speeds. The measurements are contained in the R data set "cars," which can be loaded with the command: data(cars)
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Car Stopping Distances Imperial Units
mph 4 4 7 7 8 9 10 10 10 11 11 12 12 ft 2 10 4 22 16 10 18 26 34 17 28 14 20 mph 12 12 13 13 13 13 14 14 14 14 15 15 15 ft 24 28 26 34 34 46 26 36 60 80 20 26 54 mph 16 16 17 17 17 18 18 18 18 19 19 19 20 ft 32 40 32 40 50 42 56 76 84 36 46 68 32 mph ft 20 48 20 52 20 56 20 64 22 66 23 54 24 70 24 92 24 93 24 120 25 85
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Car Stopping Distances Metric Units
kph m 6.4 0.6 6.4 3.0 11.3 1.2 11.3 6.7 12.9 4.9 14.5 3.0 16.1 5.5 16.1 7.9 16.1 10.4 17.7 5.2 17.7 8.5 19.3 4.3 19.3 6.1 kph 19.3 19.3 20.9 20.9 20.9 20.9 22.5 22.5 22.5 22.5 24.1 24.1 24.1 m 7.3 8.5 7.9 10.4 10.4 14.0 7.9 11.0 18.3 24.4 6.1 7.9 16.5 kph 25.7 25.7 27.4 27.4 27.4 29.0 29.0 29.0 29.0 30.6 30.6 30.6 32.2 m 9.8 12.2 9.8 12.2 15.2 12.8 17.1 23.2 25.6 11.0 14.0 20.7 9.8 kph 32.2 32.2 32.2 32.2 35.4 37.0 38.6 38.6 38.6 38.6 40.2 m 14.6 15.8 17.1 19.5 20.1 16.5 21.3 28.0 28.3 36.6 25.9
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Question
Why would anyone collect this kind of data?
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Graphical Investigation
We are going to use the value to investigate the relationship between speed and stopping distance. The best way to investigate the relationship between two related variables is to simply plot the pairs of values. The basic plot is produced with plot. > data(cars) > attach(cars) > plot(speed, dist) Using default labels is fine for exploratory work, but not for publication.
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Comments
There is a general trend for stopping distance to increase with speed. There is evidence that the variability in the stopping distances also increases with speed. It is difficult to be more precise about the form of the relationship by just looking at the scatter of points.
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Scatterplot Smoothing
One way to try to uncover the nature of the relationship is to add a line which conveys the basic trend in the plot. This can be done using a technique known as scatterplot smoothing. R has a smoothing procedure called L OWESS which can be used to add the trend line. L OWESS is a relatively complicated procedure, but it is easy to use. plot(speed, dist) lines(lowess(speed, dist))
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Conclusions
The "smooth" confirms that stopping distance increases with speed, but it gives us more detail. The relationship is not of the form y = a + bx but has an unknown mathematical form. If we are just interested in determining the stopping distance we can expect for a given speed this doesn't matter. We can just read the answer off the graph.
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Turning a Smooth into a Function
It is useful to have a a computational procedure for "reading off the results" from the lowess curve. This can be done by fitting a spline curve through the points returned by lowess. > z = lowess(speed, dist) > u = !duplicated(z$x) > f = splinefun(z$x[u], z$y[u]) The function f can now be used to do the lookup of values on the curve. > f(10:12) [1] 21.28031 24.12928 27.11955
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Mathematical Modelling
While the curve obtained by the L OWESS lets us read off the kind of stopping distance we can expect for a given speed, it does not help understand why the relationship is the way it is. It is possible to use the data to try to fit a well defined mathematical curve to the data points. This suffers from the same difficulty. It is much better to try to understand the mechanism which produced the data.
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Conservation of Energy
A moving car has kinetic energy associated with it. The kinetic energy is dissipated as work is done against friction during breaking. When the car comes to rest the Kinetic energy dissipated equals work done.
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Equations from Physics
Thanks Isaac to Newton (and others) we know the following. Kinetic Energy = 1 mv2 2 where m is the mass of the car and v is the car speed. Work Done = F d where F is the frictional force and d is the distance travelled. When the car comes to a halt, all the kinetic energy has been dissipated as work done against the frictional force.
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Conservation of Energy
Because energy is conserved, we can equate right-hand side of the previous equations. F d = 1 mv2 2 Ignoring constants, this says that d v2 or d v.
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Using Plots
We can check whether these are really the underlying relationships with scatterplots. Either plot distance against speed-squared or plot the square-root of distance against speed.
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Producing the Plots
> plot(speed^2, dist, main = "Car Stopping Distances", xlab = "Speed-squared (MPH^2)", ylab = "Stopping Distance (Feet)") > lines(lowess(speed^2, dist)) > plot(speed, sqrt(dist), main = "Car Stopping Distances", xlab = "Speed (MPH)", ylab = "Square Root Stopping Distance (Feet)") > lines(lowess(speed, sqrt(dist)))
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Car Stopping Distances
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Speed-squared (MPH^2)
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Car Stopping Distances
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Conclusions
Both the plots indicate that there is close to a straight line relationship between speed-squared and distance. From a statistical point-of-view, the second plot is preferable because the scatter of points about the line is independent of speed. (I.e. it is possible to compare apples with apples). The straight line of best fit to the plot of square-root distance versus speed is: d = 1.28 + 0.32 v Dropping the intercept, the best fit is: d = 0.4 v
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How Lowess Works
It is worth spending a little time to see how lowess works. We'll consider how to get an estimate of the lowess curve at just one location in a scatter plot. We will compute the value of the lowess curve at the 6th point in the following plot. The lowess procedure does this for every point in the plot.
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Step 1 : Find the neighbours of the point.
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