A3 Error Metrics for Predictive Models

Description

A package for the generation of accurate, accessible, and adaptable error metrics for developing high quality predictions and inferences. The name A3 (pronounced "A-Cubed") comes from the combination of the first letters of these three primary adjectives.

Details

The overarching purpose of the outputs and tools in this package are to make the accurate assessment of model errors more accessible to a wider audience. Furthermore, a standardized set of reporting features are provided by this package which create consistent outputs for virtually any predictive model. This makes it straightforward to compare, for instance, a linear regression model to more exotic techniques such as Random forests or Support vector machines.

The standard outputs for each model fit provided by the A3 package include:

• Average Slope: Equivalent to a linear regression coefficient.

• Cross Validated R^2: Robust calculation of R^2 (percent of squared error explained by the model compared to the null model) values adjusting for over-fitting.

• p Values: Robust calculation of p-values requiring no parametric assumptions other than independence between observations (which may be violated if compensated for).

The primary functions that will be used are a3 for arbitrary modeling functions and a3.lm for linear models. This package also includes print.A3 and plot.A3 for outputting the A3 results.

Author(s)

Scott Fortmann-Roe scottfr@berkeley.edu http://Scott.Fortmann-Roe.com

A3 Results for Arbitrary Model

Description

This function calculates the A3 results for an arbitrary model construction algorithm (e.g. Linear Regressions, Support Vector Machines or Random Forests). For linear regression models, you may use the a3.lm convenience function.

Usage

a3(formula, data, model.fn, model.args = list(), ...)

Arguments

Value

S3 A3 object; see a3.base for details

References

Scott Fortmann-Roe (2015). Consistent and Clear Reporting of Results from Diverse Modeling Techniques: The A3 Method. Journal of Statistical Software, 66(7), 1-23. <http://www.jstatsoft.org/v66/i07/>

Examples

 ## Standard linear regression results:

 summary(lm(rating ~ ., attitude))

 ## A3 Results for a Linear Regression model:

 # In practice, p.acc should be <= 0.01 in order
 # to obtain finer grained p values.

 a3(rating ~ ., attitude, lm, p.acc = 0.1)


 ## A3 Results for a Random Forest model:

 # It is important to include the "+0" in the formula
 # to eliminate the constant term.

 require(randomForest)
 a3(rating ~ .+0, attitude, randomForest, p.acc = 0.1)

 # Set the ntrees argument of the randomForest function to 100

 a3(rating ~ .+0, attitude, randomForest, p.acc = 0.1, model.args = list(ntree = 100))

 # Speed up the calculation by doing 5-fold cross-validation.
 # This is faster and more conservative (i.e. it should over-estimate error)

 a3(rating ~ .+0, attitude, randomForest, n.folds = 5, p.acc = 0.1)

 # Use Leave One Out Cross Validation. The least biased approach,
 # but, for large data sets, potentially very slow.

 a3(rating ~ .+0, attitude, randomForest, n.folds = 0, p.acc = 0.1)

 ## Use a Support Vector Machine algorithm.

 # Just calculate the slopes and R^2 values, do not calculate p values.

 require(e1071)
 a3(rating ~ .+0, attitude, svm, p.acc = NULL)
 

Base A3 Results Calculation

Description

This function calculates the A3 results. Generally this function is not called directly. It is simpler to use a3 (for arbitrary models) or a3.lm (specifically for linear regressions).

Usage

a3.base(formula, data, model.fn, simulate.fn, n.folds = 10,
  data.generating.fn = replicate(ncol(x), a3.gen.default), p.acc = 0.01,
  features = TRUE, slope.sample = NULL, slope.displacement = 1)

Arguments

Value

S3 A3 object containing:

Stochastic Data Generators

Description

The stochastic data generators generate stochastic noise with (if specified correctly) the same properties as the observed data. By replicating the stochastic properties of the original data, we are able to obtain the exact calculation of p values.

Usage

a3.gen.default(x, n.reps)

Arguments

Details

Generally these will not be called directly but will instead be passed to the data.generating.fn argument of a3.base.

Value

A list of of length n.reps of vectors of stochastic noise. There are a number of different methods of generating noise:

Examples

 # Calculate the A3 results assuming an auto-correlated set of observations.
 # In usage p.acc should be <=0.01 in order to obtain more accurate p values.

 a3.lm(rating ~ ., attitude, p.acc = 0.1,
   data.generating.fn = replicate(ncol(attitude), a3.gen.autocor))
 

 ## A general illustration:

 # Take x as a sample set of observations for a feature
 x <- c(0.349, 1.845, 2.287, 1.921, 0.803, 0.855, 2.368, 3.023, 2.102, 4.648)

 # Generate three stochastic data series with the same autocorrelation properties as x
 rand.x <- a3.gen.autocor(x, 3)

 plot(x, type="l")
 for(i in 1:3) lines(rand.x[[i]], lwd = 0.2)

A3 for Linear Regressions

Description

This convenience function calculates the A3 results specifically for linear regressions. It uses R's glm function and so supports logistic regressions and other link functions using the family argument. For other forms of models you may use the more general a3 function.

Usage

a3.lm(formula, data, family = gaussian, ...)

Arguments

Value

S3 A3 object; see a3.base for details

Examples

 ## Standard linear regression results:

 summary(lm(rating ~ ., attitude))

 ## A3 linear regression results:

 # In practice, p.acc should be <= 0.01 in order
 # to obtain fine grained p values.

 a3.lm(rating ~ ., attitude, p.acc = 0.1)

 # This is equivalent both to:

 a3(rating ~ ., attitude, glm, model.args = list(family = gaussian), p.acc = 0.1)

 # and also to:

 a3(rating ~ ., attitude, lm, p.acc = 0.1)
 

Cross-Validated R^2

Description

Applies cross validation to obtain the cross-validated R^2 for a model: the fraction of the squared error explained by the model compared to the null model (which is defined as the average response). A pseudo R^2 is implemented for classification.

Usage

a3.r2(y, x, simulate.fn, cv.folds)

Arguments

Value

A list comprising of the following elements:

Boston Housing Prices

Description

A dataset containing the prices of houses in the Boston region and a number of features. The dataset and the following description is based on that provided by UCI Machine Learning Repository (http://archive.ics.uci.edu/ml/datasets/Housing).

Usage

data(housing)

Details

• CRIME: Per capita crime rate by town

• ZN: Proportion of residential land zoned for lots over 25,000 sq.ft.

• INDUS: Proportion of non-retail business acres per town

• CHAS: Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)

• NOX: Nitrogen oxides pollutant concentration (parts per 10 million)

• ROOMS: Average number of rooms per dwelling

• AGE: Proportion of owner-occupied units built prior to 1940

• DISTANCE: Weighted distances to five Boston employment centres

• HIGHWAY: Index of accessibility to radial highways

• TAX: Full-value property-tax rate per ten thousand dollar

• PUPIL.TEACHER: Pupil-teacher ratio by town

• MINORITY: 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town

• LSTAT: Percent lower status of the population

• MED.VALUE: Median value of owner-occupied homes in thousands of dollars

References

Frank, A. & Asuncion, A. (2010). UCI Machine Learning Repository [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science.

Harrison, D. and Rubinfeld, D.L. Hedonic prices and the demand for clean air, J. Environ. Economics & Management, vol.5, 81-102, 1978.

Ecosystem Multifunctionality

Description

This dataset relates multifunctionality to a number of different biotic and abiotic features in a global survey of drylands. The dataset was obtained from (http://www.sciencemag.org/content/335/6065/214/suppl/DC1). The dataset contains the features listed below.

Usage

data(multifunctionality)

Details

• ELE: Elevation of the site

• LAT & LONG: Location of the site

• SLO: Site slope

• SAC: Soil sand content

• PCA_C1, PCA_C2, PCA_C3, PCA_C4: Principal components of a set of 21 climatic features

• SR: Species richness

• MUL: Multifunctionality

References

Maestre, F. T., Quero, J. L., Gotelli, N. J., Escudero, A., Ochoa, V., Delgado-Baquerizo, M., et al. (2012). Plant Species Richness and Ecosystem Multifunctionality in Global Drylands. Science, 335(6065), 214-218. doi:10.1126/science.1215442

Plot A3 Results

Description

Plots an 'A3' object results. Displays predicted versus observed values for each observation along with the distribution of slopes measured for each feature.

Usage

## S3 method for class 'A3'
plot(x, ...)

Arguments

Examples

 data(housing)
 res <- a3.lm(MED.VALUE ~ NOX + ROOMS + AGE + HIGHWAY + PUPIL.TEACHER, housing, p.acc = NULL)
 plot(res)
 

Plot Predicted versus Observed

Description

Plots an 'A3' object's values showing the predicted versus observed values for each observation.

Usage

plotPredictions(x, show.equality = TRUE, xlab = "Observed Value",
  ylab = "Predicted Value", main = "Predicted vs Observed", ...)

Arguments

Examples

data(multifunctionality)
 x <- a3.lm(MUL ~ ., multifunctionality, p.acc = NULL, features = FALSE)
 plotPredictions(x)

Plot Distribution of Slopes

Description

Plots an 'A3' object's distribution of slopes for each feature and observation. Uses Kernel Density Estimation to create an estimate of the distribution of slopes for a feature.

Usage

plotSlopes(x, ...)

Arguments

Examples

 require(randomForest)
 data(housing)

 x <- a3(MED.VALUE ~ NOX + PUPIL.TEACHER + ROOMS + AGE + HIGHWAY + 0,
   housing, randomForest, p.acc = NULL, n.folds = 2)

 plotSlopes(x)
 

Print Fit Results

Description

Prints an 'A3' object results table.

Usage

## S3 method for class 'A3'
print(x, ...)

Arguments

Examples

x <- a3.lm(rating ~ ., attitude, p.acc = NULL)
 print(x)

Nicely Formatted Fit Results

Description

Creates a LaTeX table of results. Depends on the xtable package.

Usage

## S3 method for class 'A3'
xtable(x, ...)

Arguments

Examples

x <- a3.lm(rating ~ ., attitude, p.acc = NULL)
 xtable(x)