Linear Regression
Linear regression predicts a numeric outcome from one or more numeric predictors when the residuals are reasonably close
to normal. The original course worked against an FDX (FedEx, NYSE) equity file that had already been through Jube’s
abstraction engine: a Dependent column (a forward return) plus hundreds of abstracted predictor columns named for the
statistic and window they came from (Median_1, Mode_1, Skew_3,
Range_2_PearsonCorrelation, PointStep_5_PearsonCorrelation, and z-scored or self-correlated variants of the same).
That exact file is not available in this environment, so every snippet below runs against a synthetic stand-in built
with the same column-naming shape and the same kind of relationships (a weak overall signal, a couple of genuinely
predictive columns, and two deliberately collinear groups so the multicollinearity procedure has something real to
find). Coefficients therefore differ from the original screenshots; the mechanics and the shape of the result do not.
dplyr and ggplot2 are both in the shared package set, nothing extra was installed.
Procedure 1: Scanning Scatter Plots for Relationships
pairs() draws every pairwise scatter plot among a data frame’s columns in one grid, which is the fastest way to
eyeball which predictors move with the dependent variable before modelling. Run over a whole abstracted file (hundreds
of columns) the grid is unusably large, so the practical habit is to subset to Dependent plus one family of related
columns at a time, repeating across families until a feel for the relationships has formed. The original text’s
subsetting line had a bracket typo (pairs[c(...)] instead of pairs(FDX[, c(...)])); corrected below.
FDX <- read.csv("/work/FDX.csv")
pdf("/work/p01_full.pdf")
pairs(FDX)
invisible(dev.off())
cat("Full pairs() matrix rendered over", ncol(FDX), "columns.\n")
pdf("/work/p01_subset.pdf")
pairs(FDX[, c("Dependent", "Median_1", "Median_1_PearsonCorrelation", "Median_1_ZScore",
"Mode_1", "Mode_1_PearsonCorrelation", "Mode_1_ZScore")])
invisible(dev.off())
cat("Subset pairs() matrix rendered over 7 columns.\n")
Console output:
Full pairs() matrix rendered over 11 columns.
Subset pairs() matrix rendered over 7 columns.
AdvancedAnalytics equivalent: Jube.Charting, Chart.ScatterMatrix() — a scatterplot matrix with histograms on the
diagonal, scatters below and correlation cells above, one row linked across every cell on hover. This replaces pairs()
as the exploratory grid; never ggplot2 or base R graphics.
Procedure 2: Creating a Scatter Plot for Closer Inspection with ggplot2
plot() gives an instant scatter of two columns; qplot() from ggplot2 draws the same relationship with more visual
polish for anything that will be shown to someone else.
FDX <- read.csv("/work/FDX.csv")
library(ggplot2)
pdf("/work/p02_plot.pdf")
plot(FDX$Median_4, FDX$Dependent)
invisible(dev.off())
cat("Base plot() scatter rendered.\n")
pdf("/work/p02_qplot.pdf")
print(qplot(FDX$Median_4, FDX$Dependent))
invisible(dev.off())
cat("ggplot2 qplot() scatter rendered.\n")
Console output:
Base plot() scatter rendered.
Warning message:
`qplot()` was deprecated in ggplot2 3.4.0.
ggplot2 qplot() scatter rendered.
(qplot() is now a deprecated corner of ggplot2 itself — one more reason it is not a path this product follows.)
Tool equivalent: Jube.Charting, Chart.Scatter() — scatter and bubble charts with trend lines and R
squared, an identity diagonal and log axes built in. Outside the platform this is what you would reach for instead; the
product never renders through ggplot2 or a raw R graphics device.
Procedure 3: Create a Correlation Matrix using Spearman and Pearson
cor() produces a full correlation matrix over a data frame in one call. method chooses "pearson" (linear)
or "spearman" (monotone rank); use = "complete" avoids the whole matrix collapsing to NA the moment any row has a
missing value. The matrix can then be subset to just the Dependent column, using drop = FALSE (not the string
"false" in the original text) to keep the result a matrix rather than a bare vector.
FDX <- read.csv("/work/FDX.csv")
Pearson <- cor(FDX, use = "complete", method = "pearson")
cat("Dimensions of Pearson:", paste(dim(Pearson), collapse = " x "), "\n\n")
PearsonDependent <- Pearson[, "Dependent", drop = FALSE]
cat("PearsonDependent (correlation of every column with Dependent):\n")
print(round(PearsonDependent, 4))
Console output:
Dimensions of Pearson: 11 x 11
PearsonDependent (correlation of every column with Dependent):
Dependent
Dependent 1.0000
Median_1 -0.0280
Median_1_PearsonCorrelation -0.0213
Median_1_ZScore -0.0280
Mode_1 -0.0178
Mode_1_PearsonCorrelation -0.0051
Mode_1_ZScore -0.0178
Median_4 -0.0714
Skew_3 0.4098
Range_2_PearsonCorrelation 0.5018
PointStep_5_PearsonCorrelation 0.4590
Tool equivalent:
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.CorrelationTools
/ tool "CorrelationMatrix" — every pairwise Pearson, Spearman or Kendall correlation in one call, with a matching
p-value matrix, pair counts and the strongest pairs pre-ranked with Benjamini-Hochberg adjusted p-values, choosing
pairwise or listwise deletion explicitly instead of cor()’s implicit use= behaviour. For a single pair, tool
"PearsonCorrelation" (or "SpearmanRankCorrelation" for the rank version) additionally returns a Fisher z confidence
interval that cor() alone does not.
Procedure 4: Ranking Correlation by Absolute Strength
Both -1 and +1 are equally interesting extremes, so the dependent column is ranked by abs() of its correlation rather
than the signed value. The original relied on clicking a column header in the RStudio matrix viewer to sort it, which
cannot be captured in a script; order() is the programmatic equivalent.
FDX <- read.csv("/work/FDX.csv")
Pearson <- cor(FDX, use = "complete", method = "pearson")
PearsonDependent <- Pearson[, "Dependent", drop = FALSE]
PearsonDependentAbs <- abs(PearsonDependent)
cat("Absolute correlation with Dependent:\n")
print(round(PearsonDependentAbs, 4))
cat("\nOrdered highest absolute correlation to lowest (programmatic equivalent of the\n")
cat("manual column-header sort in the RStudio matrix viewer):\n")
ordered <- PearsonDependentAbs[order(-PearsonDependentAbs[, 1]), , drop = FALSE]
print(round(ordered, 4))
Console output:
Absolute correlation with Dependent:
Dependent
Dependent 1.0000
Median_1 0.0280
Median_1_PearsonCorrelation 0.0213
Median_1_ZScore 0.0280
Mode_1 0.0178
Mode_1_PearsonCorrelation 0.0051
Mode_1_ZScore 0.0178
Median_4 0.0714
Skew_3 0.4098
Range_2_PearsonCorrelation 0.5018
PointStep_5_PearsonCorrelation 0.4590
Ordered highest absolute correlation to lowest (programmatic equivalent of the
manual column-header sort in the RStudio matrix viewer):
Dependent
Dependent 1.0000
Range_2_PearsonCorrelation 0.5018
PointStep_5_PearsonCorrelation 0.4590
Skew_3 0.4098
Median_4 0.0714
Median_1_ZScore 0.0280
Median_1 0.0280
Median_1_PearsonCorrelation 0.0213
Mode_1 0.0178
Mode_1_ZScore 0.0178
Mode_1_PearsonCorrelation 0.0051
Tool equivalent:
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.CorrelationTools
/ tool "RankPredictorsByCorrelation" — screens every column against one target and returns it already ranked by
absolute strength, with a p-value and a Benjamini-Hochberg adjusted p-value per column, so nothing needs a second manual
sort step.
Procedure 5: Adding a Trend Line to a Scatter Plot
stat_smooth(method = "lm") overlays a line of best fit (with its default shaded confidence ribbon) on a ggplot2
scatter, giving an immediate visual read on direction and strength before any formal model is fitted.
FDX <- read.csv("/work/FDX.csv")
library(ggplot2)
pdf("/work/p05_trend.pdf")
print(qplot(FDX$Median_4, FDX$Dependent) + stat_smooth(method = "lm"))
invisible(dev.off())
cat("Scatter plot with an lm() trend line rendered via stat_smooth().\n")
Console output:
`geom_smooth()` using formula = 'y ~ x'
Warning message:
`qplot()` was deprecated in ggplot2 3.4.0.
Scatter plot with an lm() trend line rendered via stat_smooth().
As in the original course, this particular relationship (Dependent against Median_4) is shallow: the fitted slope
has predictive direction but very little strength taken alone, which is why the later procedures build a
multiple-predictor model instead.
AdvancedAnalytics equivalent: Jube.Charting, Chart.Regression() — draws the line, and unlike
stat_smooth()’s single ribbon it distinguishes a confidence band for the mean response from a wider prediction band
for a single new observation (Student t quantiles, polynomial fit up to degree 3), matching what
FitSimpleLinearRegression’s predictAt option computes numerically. Never ggplot2 for this in the product.
Procedure 6: Creating a One Way Linear Regression Model
lm(y ~ x, data) fits ordinary least squares. Printing the model gives just the intercept and slope;
summary() adds standard errors, t statistics, p-values, R squared and the F test; coef() recovers the coefficients
at full floating point precision, which matters because the truncated display can round away digits that change a
prediction meaningfully (the original text’s misspelled coeefeicents() is corrected to
coef() below).
FDX <- read.csv("/work/FDX.csv")
LinearRegression <- lm(Dependent ~ Median_4, FDX)
cat("print(LinearRegression):\n")
print(LinearRegression)
cat("\nsummary(LinearRegression):\n")
print(summary(LinearRegression))
cat("\ncoef(LinearRegression) at full precision:\n")
print(coef(LinearRegression))
Console output:
print(LinearRegression):
Call:
lm(formula = Dependent ~ Median_4, data = FDX)
Coefficients:
(Intercept) Median_4
0.01793 -0.09301
summary(LinearRegression):
Call:
lm(formula = Dependent ~ Median_4, data = FDX)
Residuals:
Min 1Q Median 3Q Max
-0.092439 -0.017589 -0.000831 0.017018 0.080596
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.017931 0.001206 14.873 <2e-16 ***
Median_4 -0.093009 0.058187 -1.598 0.111
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02695 on 498 degrees of freedom
Multiple R-squared: 0.005104, Adjusted R-squared: 0.003107
F-statistic: 2.555 on 1 and 498 DF, p-value: 0.1106
coef(LinearRegression) at full precision:
(Intercept) Median_4
0.01793091 -0.09300855
The slope’s p-value (0.111) is above the usual 0.05 cut-off, confirming visually what procedure 5 already suggested:
Median_4 alone is not a reliable predictor of Dependent.
Tool equivalent:
Jube.Service.Repository.AnalyticsTrainingRun.AdvancedAnalyticsLinearRegressionTrainingService
(SubmitTrainingAsync, model kind MultipleLinearRegressionRecall.ModelKind) trains and versions the model as a
governed asset with steps, metrics and evidence recorded against the run. For the interactive, nothing-persisted version
of exactly this call, Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.LinearRegressionTools
/ tool "FitSimpleLinearRegression" returns the same coefficient table (estimate, standard error, t, p-value,
confidence interval) plus R squared, adjusted R squared, the F test, AIC/AICc/BIC and a Durbin-Watson statistic that
summary.lm() does not print at all.
Procedure 7: Deploying a One Way Linear Regression Manually with Vector Arithmetic
Deployment is just the fitted formula applied to a column: intercept plus the predictor multiplied by its coefficient,
computed once as vector arithmetic over every row at once, then appended to the data frame with
mutate().
FDX <- read.csv("/work/FDX.csv")
library(dplyr)
LinearRegression <- lm(Dependent ~ Median_4, FDX)
intercept <- coef(LinearRegression)[1]
slope <- coef(LinearRegression)[2]
cat("Intercept:", intercept, " Slope:", slope, "\n\n")
ManualLinearRegression <- intercept + (FDX$Median_4 * slope)
cat("ncol(FDX) before mutate:", ncol(FDX), "\n")
FDX <- mutate(FDX, ManualLinearRegression = ManualLinearRegression)
cat("ncol(FDX) after mutate:", ncol(FDX), "\n\n")
cat("head of the new column:\n")
print(head(FDX[, c("Median_4", "Dependent", "ManualLinearRegression")]))
Console output:
Intercept: 0.01793091 Slope: -0.09300855
ncol(FDX) before mutate: 11
ncol(FDX) after mutate: 12
head of the new column:
Median_4 Dependent ManualLinearRegression
1 0.008203630 -0.017198509 0.01716790
2 0.016148044 0.056111402 0.01642901
3 0.010337604 0.018602161 0.01696943
4 -0.045419611 0.017505100 0.02215532
5 0.001752893 0.037935467 0.01776788
6 0.025568083 0.001403019 0.01555286
Tool equivalent: none for hand-rolled vector arithmetic — the trained model returned by
AdvancedAnalyticsLinearRegressionTrainingService (or PredictWithLinearRegression below) is applied through the
recall path, never by an analyst re-typing coefficients into a formula. Manual deployment stays “what you’d do without
the platform.”
Procedure 8: Using the predict () Function for a One Way Linear Regression
predict() (dispatching to predict.lm) takes the fitted model and a data frame and returns the same vector procedure
7 built by hand, without re-typing the coefficients.
FDX <- read.csv("/work/FDX.csv")
library(dplyr)
LinearRegression <- lm(Dependent ~ Median_4, FDX)
intercept <- coef(LinearRegression)[1]
slope <- coef(LinearRegression)[2]
FDX <- mutate(FDX, ManualLinearRegression = intercept + (Median_4 * slope))
AutomaticLinearRegression <- predict(LinearRegression, FDX)
FDX <- mutate(FDX, AutomaticLinearRegression = AutomaticLinearRegression)
cat("head of manual vs automatic prediction, side by side:\n")
print(head(FDX[, c("ManualLinearRegression", "AutomaticLinearRegression")]))
cat("\nall.equal(ManualLinearRegression, AutomaticLinearRegression):\n")
print(all.equal(FDX$ManualLinearRegression, FDX$AutomaticLinearRegression))
cat("\nmax(abs(difference)) once row-name attributes are stripped:\n")
print(max(abs(unname(FDX$ManualLinearRegression) - unname(FDX$AutomaticLinearRegression))))
Console output:
head of manual vs automatic prediction, side by side:
ManualLinearRegression AutomaticLinearRegression
1 0.01716790 0.01716790
2 0.01642901 0.01642901
3 0.01696943 0.01696943
4 0.02215532 0.02215532
5 0.01776788 0.01776788
6 0.01555286 0.01555286
all.equal(ManualLinearRegression, AutomaticLinearRegression):
[1] "names for current but not for target"
max(abs(difference)) once row-name attributes are stripped:
[1] 0
The two vectors are numerically identical; all.equal() only flags that predict()’s result carries row-name
attributes that the hand-built vector does not, which unname() shows is cosmetic, not a real disagreement.
Tool equivalent:
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.LinearRegressionTools
/ tool "PredictWithLinearRegression" — fits on the supplied training rows and predicts new rows in the same call, but
goes well past predict(): every new row gets a confidence interval for the mean response, a wider prediction interval
for a single new observation, its leverage, and an ExtrapolatedRows count flagging rows that fall outside the region
the training data actually covered.
Procedure 9: Identifying Confidence Intervals
Confidence intervals for the coefficients are not printed by summary() and must be requested separately with
confint() (confint.lm is the method actually dispatched to).
FDX <- read.csv("/work/FDX.csv")
LinearRegression <- lm(Dependent ~ Median_4, FDX)
cat("confint(LinearRegression, level = 0.95):\n")
print(confint(LinearRegression, level = 0.95))
Console output:
confint(LinearRegression, level = 0.95):
2.5 % 97.5 %
(Intercept) 0.01556218 0.02029964
Median_4 -0.20732996 0.02131286
The slope’s interval spans zero, consistent with procedure 6’s p-value of 0.111: the data cannot rule out no
relationship at all between Median_4 and Dependent.
Tool equivalent: no separate call needed. Both
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.LinearRegressionTools tools
"FitSimpleLinearRegression" and "FitMultipleLinearRegression" return a confidence interval on every coefficient in
the same coefficient table as the estimate and p-value, at the caller’s chosen confidenceLevel
(default 0.95); there is no confint()-style follow-up call to remember to make.
Procedure 10: Create a Stepwise Linear Regression Model
Forward stepwise selection adds predictors one at a time in order of their correlation strength (procedure 4’s ranking),
checking that R squared keeps improving and the p-values stay acceptable. Following the ranking from procedure 4,
Range_2_PearsonCorrelation and Skew_3 enter first, then PointStep_5_PearsonCorrelation.
FDX <- read.csv("/work/FDX.csv")
MultipleLinearRegression <- lm(Dependent ~ Skew_3 + Range_2_PearsonCorrelation, FDX)
cat("Step 1 -- Dependent ~ Skew_3 + Range_2_PearsonCorrelation\n")
print(summary(MultipleLinearRegression))
MultipleLinearRegression <- lm(Dependent ~ Skew_3 + Range_2_PearsonCorrelation +
PointStep_5_PearsonCorrelation, FDX)
cat("\nStep 2 -- adding PointStep_5_PearsonCorrelation\n")
print(summary(MultipleLinearRegression))
Console output:
Step 1 -- Dependent ~ Skew_3 + Range_2_PearsonCorrelation
Call:
lm(formula = Dependent ~ Skew_3 + Range_2_PearsonCorrelation,
data = FDX)
Residuals:
Min 1Q Median 3Q Max
-0.066403 -0.014741 -0.000397 0.013683 0.060558
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0180323 0.0009447 19.09 <2e-16 ***
Skew_3 0.0103199 0.0009670 10.67 <2e-16 ***
Range_2_PearsonCorrelation 0.0121017 0.0008961 13.51 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0211 on 497 degrees of freedom
Multiple R-squared: 0.3913, Adjusted R-squared: 0.3889
F-statistic: 159.8 on 2 and 497 DF, p-value: < 2.2e-16
Step 2 -- adding PointStep_5_PearsonCorrelation
Call:
lm(formula = Dependent ~ Skew_3 + Range_2_PearsonCorrelation +
PointStep_5_PearsonCorrelation, data = FDX)
Residuals:
Min 1Q Median 3Q Max
-0.069444 -0.014845 -0.000788 0.013559 0.058831
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0179086 0.0009444 18.963 < 2e-16 ***
Skew_3 0.0102603 0.0009649 10.634 < 2e-16 ***
Range_2_PearsonCorrelation 0.0096053 0.0015824 6.070 2.54e-09 ***
PointStep_5_PearsonCorrelation 0.0034428 0.0018010 1.912 0.0565 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.02104 on 496 degrees of freedom
Multiple R-squared: 0.3958, Adjusted R-squared: 0.3921
F-statistic: 108.3 on 3 and 496 DF, p-value: < 2.2e-16
Adding PointStep_5_PearsonCorrelation raises R squared only marginally (0.3913 to 0.3958) while its own p-value
(0.0565) sits just above the conventional cut-off, and adjusted R squared barely moves — this is the plateau the
original text describes as the stopping signal, made sharper here because
PointStep_5_PearsonCorrelation is itself collinear with Range_2_PearsonCorrelation (see procedure 11), so it adds
little once Range_2_PearsonCorrelation is already in the model.
Tool equivalent:
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.RegressionModelSelectionTools / tool
"StepwiseLinearRegression" — runs Forward, Backward or Both-direction selection by AIC, BIC or p-value entry/exit
thresholds, records every addition or removal as a step with the criterion value at that point, and warns that stepwise
p-values are optimistic (recommending CrossValidateLinearRegression to check the result), which is a caveat R’s manual
process leaves the analyst to know on their own.
Procedure 11: Heat Map Correlation Matrix
Multicollinearity is an independent variable that correlates well with the dependent variable but also correlates with another such variable, understating both variables’ true contribution. It shows up as off-diagonal cells of the correlation matrix that exceed 0.7 in absolute value.
FDX <- read.csv("/work/FDX.csv")
Pearson <- cor(FDX, use = "complete", method = "pearson")
PearsonColinearity <- Pearson <= -0.7 | Pearson >= 0.7
cat("PearsonColinearity (TRUE marks |r| >= 0.7, off the trivial diagonal):\n")
print(PearsonColinearity)
pdf("/work/p11_heatmap.pdf")
heatmap(Pearson, symm = TRUE, main = "FDX Pearson correlation heat map")
invisible(dev.off())
cat("\nheatmap() rendered over the", nrow(Pearson), "x", ncol(Pearson), "correlation matrix.\n")
Console output:
PearsonColinearity (TRUE marks |r| >= 0.7, off the trivial diagonal):
Dependent Median_1 Median_1_PearsonCorrelation
Dependent TRUE FALSE FALSE
Median_1 FALSE TRUE TRUE
Median_1_PearsonCorrelation FALSE TRUE TRUE
Median_1_ZScore FALSE TRUE TRUE
Mode_1 FALSE FALSE FALSE
Mode_1_PearsonCorrelation FALSE FALSE FALSE
Mode_1_ZScore FALSE FALSE FALSE
Median_4 FALSE FALSE FALSE
Skew_3 FALSE FALSE FALSE
Range_2_PearsonCorrelation FALSE FALSE FALSE
PointStep_5_PearsonCorrelation FALSE FALSE FALSE
Median_1_ZScore Mode_1 Mode_1_PearsonCorrelation
Dependent FALSE FALSE FALSE
Median_1 TRUE FALSE FALSE
Median_1_PearsonCorrelation TRUE FALSE FALSE
Median_1_ZScore TRUE FALSE FALSE
Mode_1 FALSE TRUE FALSE
Mode_1_PearsonCorrelation FALSE FALSE TRUE
Mode_1_ZScore FALSE TRUE FALSE
Median_4 FALSE FALSE FALSE
Skew_3 FALSE FALSE FALSE
Range_2_PearsonCorrelation FALSE FALSE FALSE
PointStep_5_PearsonCorrelation FALSE FALSE FALSE
Mode_1_ZScore Median_4 Skew_3
Dependent FALSE FALSE FALSE
Median_1 FALSE FALSE FALSE
Median_1_PearsonCorrelation FALSE FALSE FALSE
Median_1_ZScore FALSE FALSE FALSE
Mode_1 TRUE FALSE FALSE
Mode_1_PearsonCorrelation FALSE FALSE FALSE
Mode_1_ZScore TRUE FALSE FALSE
Median_4 FALSE TRUE FALSE
Skew_3 FALSE FALSE TRUE
Range_2_PearsonCorrelation FALSE FALSE FALSE
PointStep_5_PearsonCorrelation FALSE FALSE FALSE
Range_2_PearsonCorrelation
Dependent FALSE
Median_1 FALSE
Median_1_PearsonCorrelation FALSE
Median_1_ZScore FALSE
Mode_1 FALSE
Mode_1_PearsonCorrelation FALSE
Mode_1_ZScore FALSE
Median_4 FALSE
Skew_3 FALSE
Range_2_PearsonCorrelation TRUE
PointStep_5_PearsonCorrelation TRUE
PointStep_5_PearsonCorrelation
Dependent FALSE
Median_1 FALSE
Median_1_PearsonCorrelation FALSE
Median_1_ZScore FALSE
Mode_1 FALSE
Mode_1_PearsonCorrelation FALSE
Mode_1_ZScore FALSE
Median_4 FALSE
Skew_3 FALSE
Range_2_PearsonCorrelation TRUE
PointStep_5_PearsonCorrelation TRUE
heatmap() rendered over the 11 x 11 correlation matrix.
Three genuine collinear groups surface: the Median_1 family (itself, its Pearson-correlation transform and its
z-score — all restatements of the same underlying number), the equivalent Mode_1 family, and
Range_2_PearsonCorrelation with PointStep_5_PearsonCorrelation — which is exactly the pair procedure 10’s stepwise
model found added little once the other was already in.
AdvancedAnalytics equivalent: Jube.Charting, Chart.CorrelationMatrix() (the Matrices chart family) — a
heat-map rendering of a correlation matrix with an optional hierarchical (average-linkage) ordering that groups
correlated variables visually, replacing base R’s heatmap(). For the numeric detection step feeding it,
Jube.Service.Agent.AdvancedAnalyticsTools.RelationshipsAndRegression.LinearRegressionTools /
FitMultipleLinearRegression’s per-coefficient variance inflation factor (values above about 10 signal collinearity),
or dedicated diagnostics in the same tool group (RegressionDiagnosticsTools / tools "VarianceInflationFactors" and
"MulticollinearityConditionDiagnostics"), are the product’s actual multicollinearity check rather than eyeballing a
0.7 threshold on a plotted matrix.