Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
18.104.22.168. Beam Deflections
The goal of this analysis is threefold:
|4-Plot of Data|
The assumptions are addressed by the graphics shown above:
We need to develop a better model. Non-random data can frequently be modeled using time series mehtodology. Specifically, the circular pattern in the lag plot indicates that a sinusoidal model might be appropriate. The sinusoidal model will be developed in the next section.
|Individual Plots||The plots can be generated individually for more detail. In this case, only the run sequence plot and the lag plot are drawn since the distributional plots are not meaningful.|
|Run Sequence Plot|
We have drawn some lines and boxes on the plot to better isolate the outliers. The following data points appear to be outliers based on the lag plot.
INDEX Y(i-1) Y(i) 158 -506.00 300.00 157 300.00 201.00 3 -15.00 -35.00 5 115.00 141.00That is, the third, fifth, 157th, and 158th points appear to be outliers.
When the lag plot indicates significant non-randomness, it can be
helpful to follow up with a an
This autocorrelation plot shows a distinct cyclic pattern. As with the lag plot, this suggests a sinusoidal model.
Another useful plot for non-random data is the
This spectral plot shows a single dominant peak at a frequency of 0.3. This frequency of 0.3 will be used in fitting the sinusoidal model in the next section.
|Quantitative Results||Although the lag plot, autocorrelation plot, and spectral plot clearly show the violation of the randomness assumption, we supplement the graphical output with some quantitative measures.|
As a first step in the analysis, summary statistics are
computed from the data.
Sample size = 200 Mean = -177.4350 Median = -162.0000 Minimum = -579.0000 Maximum = 300.0000 Range = 879.0000 Stan. Dev. = 277.3322
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.
Coefficient Estimate Stan. Error t-Value A0 -178.175 39.47 -4.514 A1 0.7366E-02 0.34 0.022 Residual Standard Deviation = 278.0313 Residual Degrees of Freedom = 198The slope parameter, A1, has a t value of 0.022 which is statistically not significant. This indicates that the slope can in fact be considered zero.
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
the non-randomness of this data does not allows us to assume normality,
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.
H0: σ12 = σ22 = σ32 = σ42 Ha: At least one σi2 is not equal to the others. Test statistic: W = 0.09378 Degrees of freedom: k - 1 = 3 Sample size: N = 200 Significance level: α = 0.05 Critical value: Fα,k-1,N-k = 2.651 Critical region: Reject H0 if W > 2.651In this case, the Levene test indicates that the variances are not significantly different in the four intervals since the test statistic value, 0.9378, is less than the critical value of 2.651.
A runs test
is used to check for randomness
H0: the sequence was produced in a random manner Ha: the sequence was not produced in a random manner Test statistic: Z = 2.6938 Significance level: α = 0.05 Critical value: Z1-α/2 = 1.96 Critical region: Reject H0 if |Z| > 1.96The absolute value of the test statistic is larger than the critical value at the 5 % significance level, so we conclude that the data are not random.
|Distributional Assumptions||Since the quantitative tests show that the assumptions of constant scale and non-randomness are not met, the distributional measures will not be meaningful. Therefore these quantitative tests are omitted.|