Diagnostics Post Processing

Analyze the response surface quality.

Analyze Response Surface Quality

Analyze the accuracy of the Fit using the diagnostics accessed in the Diagnostics post processing tab.

From the Post Processing step, click the Diagnostics tab.
In the work area, select the output response to analyze.
Click the tabs, below the output responses, to change the diagnostics used to analyze the selected output response.
- Detailed Diagnostics displays diagnostic information for the Input matrix, Cross-Validation matrix, and Testing Matrix.
- Regression Terms displays the confidence intervals which consist of an upper and lower bound on the coefficients of the regression equation.
  Bounds represent the confidence that the true value of the coefficient lies within the bounds, based on the given sample.
  
  Change the confidence value from the % Confidence settings. A higher confidence value will result in wider bounds; a 95% confidence interval is typically used.
- Regression Equation displays the complete formula for the predictive model as a function of the input variables.
  Note: Only available for Least Squares Regression.
- ANOVA estimates the error variance and determines the relative importance of various factors.
  Often used to identify which variables are explaining the variance in the data. This is done by examining the resulting increase in the unexplained error when variables are removed.
  
  Note: Only available for Least Squares Regression.

Configure the Diagnostics tab display settings by clicking

(located in the top, right corner of the work area). For more information about these settings, refer to Diagnostic Tab Settings.

Diagnostic Tab Settings

Settings to configure the diagnostics displayed in the Diagnostic post processing tab.

Access settings from the menu that displays when you click

(located in the top, right corner of the pane that displays the different types of diagnostics).

% Confidence: Change the confidence value.
Note: Only available for Regression Terms diagnostics.

Diagnostic Definitions

Definitions used to describe diagnostic concepts.

For a given set of $n$ input values, denoted as $y_{i}$ , the Fit predictions at the same points are denoted as ${\bar{y}}_{i}$ . The mean of the input values is expressed $\bar{y}$ . For a Least Squares Regression, $p$ is the number of unknown coefficients in the regression.

The following values are defined as follows:

Total Sum of Squares: $S S_{t o t} = \sum_{i = 1}^{n} {[y_{i} - \bar{y}]}^{2}$
Explained Sum of Squares: $S S_{\exp} = \sum_{i = 1}^{n} {[{\bar{y}}_{i} - \bar{y}]}^{2}$
Residual Sum of Squares: $S S_{e e r} = \sum_{i = 1}^{n} {[y_{i} - {\bar{y}}_{i}]}^{2}$
Average Absolute Error: $\frac{1}{n} \sum_{i = 1}^{n} [a b s (y_{i} - {\bar{y}}_{i})]$
Standard Deviation: $\sqrt{(\frac{1}{n} \sum_{i = 1}^{n} {[y_{i} - {\bar{y}}_{i}]}^{2})}$

Detailed Diagnostic

Data displayed in the Detailed Diagnostic tab of the Diagnostics post process tool.

Input Matrix

The Input Matrix column shows the diagnostic information using only the input matrix. For methods which go through the data points, such as HyperKriging or Radial Basis Functions, input matrix diagnostics are not useful.

Cross-Validation Matrix

The Cross-Validation Matrix column shows the diagnostic information using a k-fold scheme, which means input data is broken into k groups. For each group, the group's data is used as a validation set for a new approximate model using only the other k-1 group's data. This allows for diagnostic information without the need of a testing matrix.

Testing Matrix

The Testing Matrix column compares the approximate model, which was built using the input matrix, against a separate set of user supplied points. Using a Testing matrix is the best method to get accurate diagnostic information.

Criterion

R-Sqaure

Commonly called the coefficient of determination, is a measure of how well the Fit can reproduce known data points. Graphically, this can be visualized by scatter plotting the known values versus the predicted values. If the model perfectly predicts the known values, R-Square will have its maximum possible value of 1.0, and the scatter points will lie on a perfect diagonal line, as shown in the Figure 1.

More typically, the Fit introduces modeling error, and the scatter points will deviate from the straight diagonal line, as shown in the Figure 2.

The value of R-Square decreases as errors increase and the scatter plot deviates more from a straight line. The main interpretation of R-Square is that it represents the proportion of variance within the data which is explained by the Fit. For example, if R-Square = 0.84, then 84% of the variance in the data is predictable by the Fit. The higher the value of R-Square, the better the quality of the Fit. In practice, a value above 0.92 is often very good and a value lower than 0.7 necessitates investigation using other metrics. If R-Square is 1.0, you should be skeptical of this result unless the data was expected to be perfectly predicted by the Fit. There are some cases in which R-Square can be negative. A negative R-Square value indicates that using the raw mean would be a better predictor than the Fit itself; the Fit is very poor quality.

In the work area, these numbers are presented with a spark line to indicate the relative value of the number (values typically vary between 0 and 1). Values are color coded based on the following:

Red: When R² is less than 0.65 (R² < 0.65) it is displayed red, which indicates the value is not good.
Green: When R² is between 0.8 and 0.995 (0.8 < R² < 0.995) it is displayed green, which indicates the value is good.
Black: Indicates that you should apply judgment when determining whether the value is or is not good.

R-Sqaure is defined as:

R^{2} = 1 - \frac{\sum_{i = 1}^{n} {[y_{i} - {\bar{y}}_{i}]}^{2}}{\sum_{i = 1}^{n} {[y_{i} - \bar{y}]}^{2}}

R-Sqaure Adjusted

Due to its formulation, adding a variable to the model will always increase R-Square. R-Square Adjusted is a modification of R-Square that adjusts for the explanatory terms in the model. Unlike R-Square, R-Square Adjusted increases only if the new term improves the model more than would be expected by chance. The adjusted R-Square can be negative, and will always be less than or equal to R-Square. If R-Square and R-Square Adjusted differ dramatically, it indicates that non-significant terms may have been included in the model.

R-Sqaure Adjusted is defined as:

R^{2} a d j u s t e d = 1 - \frac{n - 1}{n - p - 1} (1 - R^{2})

In the work area, these numbers are presented with a spark line to indicate the relative value of the number (values typically vary between 0 and 1). Values are color coded based on the following:

Red: When R² adjusted is less than 0.65 (R² adjusted < 0.65) it is displayed red, which indicates the value is not good.
Green: When R² adjusted is between 0.8 and 0.995 (0.8 < R² adjusted < 0.995) it is displayed green, which indicates the value is good.
Black: Indicates that you should apply judgment when determining whether the value is or is not good.

Multiple R

The multiple correlation coefficient between actual and predicted values, and in most cases it is the square root of R-Square. It is an indication of the relationship between two variables.

Note: Only available for Least Squares Regression.

Relative Average Absolute Error

The ratio of the average absolute error to the standard deviation. A low ratio is more desirable as it indicates that the variance in the Fit's predicted value are dominated by the actual variance in the data and not by modeling error.

Relative Average Absolute Error is defined as:

\frac{\frac{1}{n} \sum_{i = 1}^{n} [a b s (y_{i} - {\bar{y}}_{i})]}{\sqrt{(\frac{1}{n} \sum_{i = 1}^{n} {[y_{i} - \bar{y}]}^{2})}}

Maximum Absolute Error

The maximum difference, in absolute value, between the observed and predicted values. For the input and validation matrices, this value can also be observed in the Residuals tab.

Maximum Absolute Error is defined as:

\max (a b s (y_{i} - {\bar{y}}_{i}))

Root Mean Square Error

A measure of weighted average error. A higher quality Fit will have a lower value.

Root Mean Square Error is defined as:

\sqrt{\frac{\sum_{i = 1}^{n} {[y_{i} - {\bar{y}}_{i}]}^{2}}{n}}

Number of Samples

The number of data points used in the diagnostic computations.

Regression Terms

Data displayed in the Regression Terms tab of the Diagnostics post process tool.

t-value is defined as:

t_{j} = \frac{β_{j}}{\sqrt{σ^{2} c_{j j}}}

where β_j is the corresponding regression coefficient (the Values column) and SE is the standard error. The standard error is defined as:

S E = \sqrt{σ^{2} c_{j j}}

and

σ^{2} = \frac{\sum_{i = 1}^{n} {[y_{i} - {\bar{y}}_{i}]}^{2}}{n - p}

where c_jj is the diagonal coefficient of the information matrix used during the regression calculation.

p-values are computed using the standard error and t-value to perform a student’s t-test. The p-value indicates the statistical probability that the quantity in the Value column could have resulted from a random sample and that the real value of the coefficient is actually zero (the null hypothesis). A low value, typically less than 0.05, leads to a rejection of the null-hypothesis, meaning the term is statistically significant.

ANOVA

Data displayed in the ANOVA (Analysis of Variance) tab of the Diagnostics post process tool.

Degrees of Freedom

Number of terms in the regression associated with the variable. All degrees of freedom not associated with a variable are retained in the Error assessment. More degrees of freedom associated with the error increases the statistical certainty of the results: the p-values. Higher order terms have more degrees of freedom; for example a second order polynomial will have two degrees of freedom for a variable: one for both the linear and quadratic terms.

Sum of Squares

For each variable, the quantity shown is the increase in unexplained variance if the variable were to be removed from the regression. A variable which has a small value is less critical in explaining the data variance than a variable which has a larger value.

The row Error, represents the variance not explained by the model, which is SS_err.

The row Total, which is SS_tot, will generally not equal to the sum of the others rows.

Mean Squares

The ratio between unexplained error increase and degrees of freedom, computed as the Sum of Squares divided by the associated degrees of freedom.

Mean Squares Percent

Interpreted as the relative contribution of the variables to the Fit quality, computed as the ratio of the Mean Square to the summed total of the Mean Squares. A variable with a higher percentage is more critical to explaining the variance in the given data than a variable with a lower percentage.

F-value

The quotient of the mean squares from the variable to the mean squares from the error. This is a relative measure of the variable’s explanatory variance to overall unexplained variance.

p-value

The result of an F-test on the corresponding F-value. The p-value indicates the statistical probability that the same pattern of relative variable importance could have resulted from a random sample and that the variable actually has no effect at all (the null hypothesis). A low value, typically less than 0.05, leads to a rejection of the null-hypothesis, meaning the variable is statistically significant.