MATLAB雙語教學視頻第17講

MATLAB雙語教學視頻第18講

Summarizing Data

In this section...

“Overview” on page 5-10

“Measures of Location” on page 5-10

“Measures of Scale” on page 5-11

“Shape of a Distribution” on page 5-11Overview

Many MATLAB functions enable you to summarize the overall location, scale,

and shape of a data sample.

One of the advantages of working in MATLAB is that functions operate on

entire arrays of data, not just on single scalar values. The functions are said

to be vectorized. Vectorization allows for both efficient problem formulation,

using array-based data, and efficient computation, using vectorized statistical

functions.

Note This section continues the data analysis from “Preprocessing Data”

on page 5-3.Measures of Location

Summarize the location of a data sample by finding a “typical” value.

Common measures of location or “central tendency” are computed by the

functions mean, median, and mode:

load count.dat

x1 = mean(count)

x1 =

32.0000 46.5417 65.5833

x2 = median(count)

x2 =

23.5000 36.0000 39.0000

x3 = mode(count)

x3 =

11 9 9

Like all of its statistical functions, the MATLAB functions above summarize

data across observations (rows) while preserving variables (columns). The

functions compute the location of the data at each of the three intersections

in a single call.Measures of Scale

There are many ways to measure the scale or “dispersion” of a data sample.

The MATLAB functions max, min, std, and var compute some common

measures:

dx1 = max(count)-min(count)

dx1 =

107 136 250

dx2 = std(count)

dx2 =

25.3703 41.4057 68.0281

dx3 = var(count)

dx3 =

1.0e+003 *

0.6437 1.7144 4.6278

Like all of its statistical functions, the MATLAB functions above summarize

data across observations (rows) while preserving variables (columns). The

functions compute the scale of the data at each of the three intersections

in a single call.Shape of a Distribution

The shape of a distribution is harder to summarize than its location or

scale. The MATLAB hist function plots a histogram that provides a visual

summary:

figure

hist(count)

legend('Intersection 1',...

'Intersection 2',...

'Intersection 3')

Parametric models give analytic summaries of distribution shapes.

Exponential distributions, with parameter mu given by the data mean, are a

good choice for the traffic data:

c1 = count(:,1); % Data at intersection 1

[bin_counts,bin_locations] = hist(c1);

bin_width = bin_locations(2) - bin_locations(1);

hist_area = (bin_width)*(sum(bin_counts));

figure

hist(c1)

hold on

mu1 = mean(c1);

exp_pdf = @(t)(1/mu1)*exp(-t/mu1); % Integrates

% to 1

t = 0:150;

y = exp_pdf(t);

plot(t,(hist_area)*y,'r','LineWidth',2)

legend('Distribution','Exponential Fit')

are beyond the scope of this Getting Started guide. Statistics Toolbox

software provides functions for computing maximum likelihood estimates

of distribution parameters.

See “Descriptive Statistics” in the MATLAB Data Analysis documentation for

more information on summarizing data samples.

Visualizing Data

In this section...

“Overview” on page 5-14

“2-D Scatter Plots” on page 5-14

“3-D Scatter Plots” on page 5-16

“Scatter Plot Arrays” on page 5-18

“Exploring Data in Graphs” on page 5-19Overview

You can use many MATLAB graph types for visualizing data patterns and

trends. Scatter plots, described in this section, help to visualize relationships

among the traffic data at different intersections. Data exploration tools let

you query and interact with individual data points on graphs.

Note This section continues the data analysis from “Summarizing Data”

on page 5-10.2-D Scatter Plots

A 2-D scatter plot, created with the scatter function, shows the relationship

between the traffic volume at the first two intersections:

load count.dat

c1 = count(:,1); % Data at intersection 1

c2 = count(:,2); % Data at intersection 2

figure

scatter(c1,c2,'filled')

xlabel('Intersection 1')

ylabel('Intersection 2')

The covariance, computed by the cov function measures the strength of the

linear relationship between the two variables (how tightly the data lies along

a least-squares line through the scatter):

C12 = cov([c1 c2])

C12 =

1.0e+003 *

0.6437 0.9802

0.9802 1.7144

The results are displayed in a symmetric square matrix, with the covariance

of the ith and jth variables in the (i, j)th position. The ith diagonal element

is the variance of the ith variable.

Covariances have the disadvantage of depending on the units used to measure

the individual variables. You can divide a covariance by the standard

deviations of the variables to normalize values between +1 and –1. The

corrcoef function computes correlation coefficients:

R12 = corrcoef([c1 c2])

R12 =

1.0000 0.9331

0.9331 1.0000

r12 = R12(1,2) % Correlation coefficient

r12 =

0.9331

r12sq = r12^2 % Coefficient of determination

r12sq =

0.8707

Because it is normalized, the value of the correlation coefficient is readily

comparable to values for other pairs of intersections. Its square, the coefficient

of determination, is the variance about the least-squares line divided by

the variance about the mean. Thus, it is the proportion of variation in the

response (in this case, the traffic volume at intersection 2) that is eliminated

or statistically explained by a least-squares line through the scatter.

3-D Scatter Plots

A 3-D scatter plot, created with the scatter3 function, shows the relationship

between the traffic volume at all three intersections. Use the variables c1,

c2, and c3 that you created in the previous step:

figure

scatter3(c1,c2,c3,'filled')

xlabel('Intersection 1')

ylabel('Intersection 2')

zlabel('Intersection 3')

Measure the strength of the linear relationship among the variables in the

3-D scatter by computing eigenvalues of the covariance matrix with the eig

function:

vars = eig(cov([c1 c2 c3]))

vars =

1.0e+003 *

0.0442

0.1118

6.8300

explained = max(vars)/sum(vars)

explained =

0.9777

The eigenvalues are the variances along the principal components of the data.

The variable explained measures the proportion of variation explained by the

first principal component, along the axis of the data. Unlike the coefficient

of determination for 2-D scatters, this measure distinguishes predictor and

response variables.Scatter Plot Arrays

Use the plotmatrix function to make comparisons of the relationships

between multiple pairs of intersections:

figure

plotmatrix(count)

The plot in the (i, j)th position of the array is a scatter with the i th variable

on the vertical axis and the jth variable on the horizontal axis. The plot in the

ith diagonal position is a histogram of the ith variable.

For more information on statistical visualization, see “Plotting Data” and

“Interactive Data Exploration” in the MATLAB Data Analysis documentation.Exploring Data? in Graphs

Using your mouse, you can pick observations on almost any MATLAB graph

with two tools from the figure toolbar:

? Data Cursor

? Data Brushing

These tools each place you in exploratory modes in which you can select data

points on graphs to identify their values and create workspace variables to

contain specific observations. When you use data brushing, you can also copy,

remove or replace the selected observations.

For example, make a scatter plot of the first and third columns of count:

load count.dat

scatter(count(:,1),count(:,3))

Select the Data Cursor Tool and click the right-most data point. A datatip

displaying the point’s x and y value is placed there.

Datatips display x-, y-, and z- (for 3-D plots) coordinates by default. You

can drag a datatip from one data point to another to see new values or add

additional datatips by right-clicking a datatip and using the context menu.

You can also customize the text that datatips display using MATLAB code.

For more information, see the datacursormode function and “Interacting with

Graphed Data” in the MATLAB Data Analysis documentation.

Data brushing is a related feature that lets you highlight one or more

observations on a graph by clicking or dragging. To enter data brushing

mode, click the left side of the Data Brushing tool on the figure toolbar.

Clicking the arrow on the right side of the tool icon drops down a color palette

for selecting the color with which to brush observations. This figure shows

the same scatter plot as the previous figure, but with all observations beyond

one standard deviation of the mean (as identified using the Tools > Data

Statistics GUI) brushed in red.

After you brush data observations, you can perform the following operations

on them:

? Delete them.

? Replace them with constant values.

? Replace them with NaN values.

? Drag or copy, and paste them to the Command Window.

? Save them as workspace variables.

For example, use the Data Brush context menu or the

Tools > Brushing > Create new variable option to create a new

variable called count13high.

A new variable in the workspace results:

count13high

count13high =

61 186

75 180

114 257

For more information, see the MATLAB brush function and “Marking Up

Graphs with Data Brushing” in the MATLAB Data Analysis documentation.

Linked plots, or data linking, is a feature closely related to data brushing. A

plot is said to be linked when it has a live connection to the workspace data it

depicts. The copies of variables stored in a plot object’s XData, YData, (and,

where appropriate, ZData), automatically updated whenever the workspace

variables to which they are linked change or are deleted. This causes the

graphs on which they appear to update automatically.

Linking plots to variables lets you track specific observations through

different presentations of them. When you brush data points in linked plots,

brushing one graph highlights the same observations in every graph that is

linked to the same variables.

Data linking establishes immediate, two-way communication between

figures and workspace variables, in the same way that the Variable Editor

communicates with workspace variables. You create links by activating the

Data Linking tool on a figure’s toolbar. Activating this tool causes the

Linked Plot information bar, displayed in the next figure, to appear at the top

of the plot (possibly obscuring its title). You can dismiss the bar (shown in

the following figure) without unlinking the plot; it does not print and is not

saved with the figure.

The following two graphs depict scatter plot displays of linked data after

brushing some observations on the left graph. The common variable, count

carries the brush marks to the right figure. Even though the right graph

is not in data brushing mode, it displays brush marks because it is linked

to its variables.

figure

scatter(count(:,1),count(:,2))

xlabel ('count(:,1)')

ylabel ('count(:,2)')

figure

scatter(count(:,3),count(:,2))

xlabel ('count(:,3)')

ylabel ('count(:,2)')

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