Exploratory Data Analysis_16 doc

Given the definition of the lag plot, Yiversus Yi-1, a good candidate model is a model of the form Fit Output A linear fit of this model generated the following output.. 4-Plot ofResidua

I STAT EXP(STAT) SD(STAT) Z

STATISTIC = NUMBER OF RUNS TOTAL

OF LENGTH I OR MORE

I STAT EXP(STAT) SD(STAT) Z

LENGTH OF THE LONGEST RUN UP = 10 LENGTH OF THE LONGEST RUN DOWN = 7 LENGTH OF THE LONGEST RUN UP OR DOWN = 10

NUMBER OF POSITIVE DIFFERENCES = 258 NUMBER OF NEGATIVE DIFFERENCES = 241 NUMBER OF ZERO DIFFERENCES = 0

Values in the column labeled "Z" greater than 1.96 or less than -1.96 are statistically significant at the 5% level Numerous values in this column are much larger than +/-1.96,

so we conclude that the data are not random.

Distributional

Assumptions

Since the quantitative tests show that the assumptions of randomness and constant location and scale are not met, the distributional measures will not be meaningful.

Therefore these quantitative tests are omitted.

1.4.2.3.2 Test Underlying Assumptions

1 Exploratory Data Analysis

1.4 EDA Case Studies

Since the underlying assumptions did not hold, we need to develop a better model.

The lag plot showed a distinct linear pattern Given the definition of the lag plot, Yiversus Yi-1, a good candidate model is a model of the form

Fit

Output

A linear fit of this model generated the following output.

LEAST SQUARES MULTILINEAR FIT SAMPLE SIZE N = 499 NUMBER OF VARIABLES = 1

NO REPLICATION CASE

PARAMETER ESTIMATES (APPROX ST DEV.) TVALUE

1 A0 0.501650E-01 (0.2417E-01) 2.075

2 A1 YIM1 0.987087 (0.6313E-02) 156.4

RESIDUAL STANDARD DEVIATION = 0.2931194 RESIDUAL DEGREES OF FREEDOM = 497

The slope parameter, A1, has a t value of 156.4 which is statistically significant Also, the residual standard deviation is 0.29 This can be compared to the standard deviation

shown in the summary table , which is 2.08 That is, the fit to the autoregressive model has reduced the variability by a factor of 7.

1.4.2.3.3 Develop A Better Model

1 Exploratory Data Analysis

1.4 EDA Case Studies

1.4.2.3.4 Validate New Model

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4-Plot of

Residuals

Interpretation The assumptions are addressed by the graphics shown above:

The run sequence plot (upper left) indicates no significant shifts

in location or scale over time.

1.4.2.3.4 Validate New Model

in that we knew how the data were constructed, it is common and desirable to use scientific and engineering knowledge of the process that generated the data in formulating and testing models for the data Quite often, several competing models will produce nearly equivalent mathematical results In this case, selecting the model that best

approximates the scientific understanding of the process is a reasonable choice.

1.4.2.3.4 Validate New Model

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Time Series

Model

This model is an example of a time series model More extensive discussion of time series is given in the Process Monitoring chapter.1.4.2.3.4 Validate New Model

1 Exploratory Data Analysis

1.4 EDA Case Studies

Click on the links below to start Dataplot and run this case

study yourself Each step may use results from previous steps,

so please be patient Wait until the software verifies that the

current step is complete before clicking on the next step.

The links in this column will connect you with more detailed information about each analysis step from the case study description.

1 Invoke Dataplot and read data

1 Read in the data

4 Detect drift in variation by

dividing the data into quarters and

computing Levene's test for equal

1 Based on the 4-plot, there are shifts

in location and scale and the data are not random

2 The summary statistics table displays 25+ statistics

3 The linear fit indicates drift in location since the slope parameter

3 Generate the randomness plots.

1 Generate an autocorrelation plot

2 Generate a spectral plot

1 The autocorrelation plot shows significant autocorrelation at lag 1

2 The spectral plot shows a single dominant low frequency peak

4 Fit Yi = A0 + A1*Yi-1 + Ei

and validate

1 Generate the fit

2 Plot fitted line with original data

3 Generate a 4-plot of the residuals

from the fit

4 Generate a uniform probability plot

of the residuals

1 The residual standard deviation from the fit is 0.29 (compared to the standard deviation of 2.08 from the original data)

2 The plot of the predicted values with the original data indicates a good fit

3 The 4-plot indicates that the assumptions

of constant location and scale are valid The lag plot indicates that the data are random However, the histogram and normal probability plot indicate that the uniform disribution might be a better model for the residuals than the normal

distribution

4 The uniform probability plot verifies that the residuals can be fit by a uniform distribution

1.4.2.3.5 Work This Example Yourself

1 Exploratory Data Analysis

1.4 EDA Case Studies

1 Exploratory Data Analysis

1.4 EDA Case Studies

1.4.2 Case Studies

1.4.2.4 Josephson Junction Cryothermometry

1.4.2.4.1 Background and Data

Generation This data set was collected by Bob Soulen of NIST in October, 1971 as

a sequence of observations collected equi-spaced in time from a volt meter to ascertain the process temperature in a Josephson junction cryothermometry (low temperature) experiment The response variable

is voltage counts.

Motivation The motivation for studying this data set is to illustrate the case where

there is discreteness in the measurements, but the underlying assumptions hold In this case, the discreteness is due to the data being integers.

This file can be read by Dataplot with the following commands:

SKIP 25 SET READ FORMAT 5F5.0 SERIAL READ SOULEN.DAT Y SET READ FORMAT

1 Exploratory Data Analysis

1.4 EDA Case Studies

1.4.2 Case Studies

1.4.2.4 Josephson Junction Cryothermometry

1.4.2.4.2 Graphical Output and

Interpretation

Goal The goal of this analysis is threefold:

Determine if the univariate model:

is appropriate and valid.

Determine if the confidence interval

is appropriate and valid where s is the standard deviation of the

original data.

3

1.4.2.4.2 Graphical Output and Interpretation

4-Plot of

Data

Interpretation The assumptions are addressed by the graphics shown above:

The run sequence plot (upper left) indicates that the data do not have any significant shifts in location or scale over time.

3

The normal probability plot (lower right) is difficult to interpret due to the fact that there are only a few distinct values with many repeats.

4

The integer data with only a few distinct values and many repeats accounts for the discrete appearance of several of the plots (e.g., the lag plot and the normal probability plot) In this case, the nature of the data makes the normal probability plot difficult to interpret, especially since each number is repeated many times However, the histogram indicates that a normal distribution should provide an adequate model for the data.

From the above plots, we conclude that the underlying assumptions are valid and the data can be reasonably approximated with a normal

distribution Therefore, the commonly used uncertainty standard is valid and appropriate The numerical values for this model are given in1.4.2.4.2 Graphical Output and Interpretation

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the Quantitative Output and Interpretation section.

1 Exploratory Data Analysis

1.4 EDA Case Studies

1.4.2 Case Studies

1.4.2.4 Josephson Junction Cryothermometry

1.4.2.4.3 Quantitative Output and Interpretation

Summary

Statistics As a first step in the analysis, a table of summary statistics is computed from the data. The following table, generated by Dataplot , shows a typical set of statistics.

SUMMARY

NUMBER OF OBSERVATIONS = 700

***********************************************************************

* LOCATION MEASURES * DISPERSION MEASURES

* ***********************************************************************

* MIDRANGE = 0.2898500E+04 * RANGE = 0.7000000E+01

* RANDOMNESS MEASURES * DISTRIBUTIONAL MEASURES

* ***********************************************************************

* AUTOCO COEF = 0.3148023E+00 * ST 3RD MOM = 0.6580265E-02

* = * TUK -.5 PPCC = 0.7935873E+00

*

* = * CAUCHY PPCC = 0.4231319E+00

* ***********************************************************************

Location One way to quantify a change in location over time is to fit a straight line to the data set

using the index variable X = 1, 2, , N, with N denoting the number of observations If there is no significant drift in the location, the slope parameter should be zero For this data set, Dataplot generates the following output:

LEAST SQUARES MULTILINEAR FIT SAMPLE SIZE N = 700 NUMBER OF VARIABLES = 1

NO REPLICATION CASE

PARAMETER ESTIMATES (APPROX ST DEV.) TVALUE

1 A0 2898.19 (0.9745E-01) 0.2974E+05

2 A1 X 0.107075E-02 (0.2409E-03) 4.445

RESIDUAL STANDARD DEVIATION = 1.287802 RESIDUAL DEGREES OF FREEDOM = 698

The slope parameter, A1, has a t value of 2.1 which is statistically significant (the critical value is 1.98) However, the value of the slope is 0.0011 Given that the slope is nearly zero, the assumption of constant location is not seriously violated even though it is (just barely) statistically significant.

Variation One simple way to detect a change in variation is with a Bartlett test after dividing the

data set into several equal-sized intervals However, the Bartlett test is not robust for non-normality Since the nature of the data (a few distinct points repeated many times) makes the normality assumption questionable, we use the alternative Levene test In partiuclar, we use the Levene test based on the median rather the mean The choice of the number of intervals is somewhat arbitrary, although values of 4 or 8 are reasonable.

Dataplot generated the following output for the Levene test.

LEVENE F-TEST FOR SHIFT IN VARIATION (ASSUMPTION: NORMALITY)

1 STATISTICS NUMBER OF OBSERVATIONS = 700 NUMBER OF GROUPS = 4 LEVENE F TEST STATISTIC = 1.432365

FOR LEVENE TEST STATISTIC

99.9 % POINT = 5.482234

76.79006 % Point: 1.432365

3 CONCLUSION (AT THE 5% LEVEL):

THERE IS NO SHIFT IN VARIATION

THUS THE GROUPS ARE HOMOGENEOUS WITH RESPECT TO VARIATION

Since the Levene test statistic value of 1.43 is less than the 95% critical value of 2.67, we conclude that the standard deviations are not significantly different in the 4 intervals.

Randomness

There are many ways in which data can be non-random However, most common forms

of non-randomness can be detected with a few simple tests The lag plot in the previous section is a simple graphical technique.

Another check is an autocorrelation plot that shows the autocorrelations for various lags Confidence bands can be plotted at the 95% and 99% confidence levels Points outside this band indicate statistically significant values (lag 0 is always 1) Dataplot generated the following autocorrelation plot.

The lag 1 autocorrelation, which is generally the one of most interest, is 0.31 The critical values at the 5% level of significance are -0.087 and 0.087 This indicates that the lag 1 autocorrelation is statistically significant, so there is some evidence for non-randomness.

A common test for randomness is the runs test

RUNS UP

STATISTIC = NUMBER OF RUNS UP

OF LENGTH EXACTLY I

I STAT EXP(STAT) SD(STAT) Z

1 102.0 145.8750 12.1665 -3.61

1.4.2.4.3 Quantitative Output and Interpretation

STATISTIC = NUMBER OF RUNS UP

OF LENGTH I OR MORE

I STAT EXP(STAT) SD(STAT) Z

RUNS DOWN

STATISTIC = NUMBER OF RUNS DOWN

OF LENGTH EXACTLY I

I STAT EXP(STAT) SD(STAT) Z

STATISTIC = NUMBER OF RUNS DOWN

OF LENGTH I OR MORE

9 0.0 0.0002 0.0132 -0.01

10 0.0 0.0000 0.0040 0.00

RUNS TOTAL = RUNS UP + RUNS DOWN

STATISTIC = NUMBER OF RUNS TOTAL

OF LENGTH EXACTLY I

I STAT EXP(STAT) SD(STAT) Z

STATISTIC = NUMBER OF RUNS TOTAL

OF LENGTH I OR MORE

I STAT EXP(STAT) SD(STAT) Z

LENGTH OF THE LONGEST RUN UP = 7 LENGTH OF THE LONGEST RUN DOWN = 6 LENGTH OF THE LONGEST RUN UP OR DOWN = 7

NUMBER OF POSITIVE DIFFERENCES = 262 NUMBER OF NEGATIVE DIFFERENCES = 258 NUMBER OF ZERO DIFFERENCES = 179

Values in the column labeled "Z" greater than 1.96 or less than -1.96 are statistically significant at the 5% level The runs test indicates some mild non-randomness.

Although the runs test and lag 1 autocorrelation indicate some mild non-randomness, it is

not sufficient to reject the Yi = C + Ei model At least part of the non-randomness can be explained by the discrete nature of the data.

1.4.2.4.3 Quantitative Output and Interpretation

Chi-square and Kolmogorov-Smirnov goodness-of-fit tests are alternative methods for assessing distributional adequacy The Wilk-Shapiro and Anderson-Darling tests can be used to test for normality Dataplot generates the following output for the

Anderson-Darling normality test.

ANDERSON-DARLING 1-SAMPLE TEST THAT THE DATA CAME FROM A NORMAL DISTRIBUTION

1 STATISTICS:

NUMBER OF OBSERVATIONS = 700 MEAN = 2898.562 STANDARD DEVIATION = 1.304969

ANDERSON-DARLING TEST STATISTIC VALUE = 16.76349 ADJUSTED TEST STATISTIC VALUE = 16.85843

2 CRITICAL VALUES:

90 % POINT = 0.6560000

95 % POINT = 0.7870000 97.5 % POINT = 0.9180000

99 % POINT = 1.092000

3 CONCLUSION (AT THE 5% LEVEL):

THE DATA DO NOT COME FROM A NORMAL DISTRIBUTION

The Anderson-Darling test rejects the normality assumption because the test statistic, 16.76, is greater than the 99% critical value 1.092.

Although the data are not strictly normal, the violation of the normality assumption is not

severe enough to conclude that the Yi = C + Ei model is unreasonable At least part of the non-normality can be explained by the discrete nature of the data.

1 STATISTICS:

NUMBER OF OBSERVATIONS = 700 MINIMUM = 2895.000 MEAN = 2898.562 MAXIMUM = 2902.000 STANDARD DEVIATION = 1.304969

GRUBBS TEST STATISTIC = 2.729201

1.4.2.4.3 Quantitative Output and Interpretation

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