HurdleDMR.jl

HurdleDMR.jl is a Julia implementation of the Hurdle Distributed Multinomial Regression (HDMR), as described in:

Bryan Kelly, Asaf Manela & Alan Moreira (2021) Text Selection, Journal of Business & Economic Statistics (ungated preprint).

It includes a Julia implementation of the Distributed Multinomial Regression (DMR) model of Taddy (2015).

Setup

Install the HurdleDMR package, switch to Pkg mode (hit ])

pkg> add HurdleDMR

Add parallel workers and make package available to workers

using Distributed
addprocs(4)
import HurdleDMR; @everywhere using HurdleDMR

Setup your data into an n-by-p covars matrix, and a (sparse) n-by-d counts matrix. Here we generate some random data.

using CSV, GLM, DataFrames, Distributions, Random, LinearAlgebra, SparseArrays
n = 100
p = 3
d = 4

Random.seed!(13)
m = 1 .+ rand(Poisson(5),n)
covars = rand(n,p)
ηfn(vi) = exp.([0 + i*sum(vi) for i=1:d])
q = [ηfn(covars[i,:]) for i=1:n]
rmul!.(q,ones(n)./sum.(q))
counts = convert(SparseMatrixCSC{Float64,Int},hcat(broadcast((qi,mi)->rand(Multinomial(mi, qi)),q,m)...)')
covarsdf = DataFrame(covars,[:vy, :v1, :v2])

Distributed Multinomial Regression (DMR)

The Distributed Multinomial Regression (DMR) model of Taddy (2015) is a highly scalable approximation to the Multinomial using distributed (independent, parallel) Poisson regressions, one for each of the d categories (columns) of a large counts matrix, on the covars.

To fit a DMR:

m = dmr(covars, counts)

or with a dataframe and formula

mf = @model(c ~ vy + v1 + v2)
m = fit(DMR, mf, covarsdf, counts)

in either case we can get the coefficients matrix for each variable + intercept as usual with

coef(m)

By default we only return the AICc maximizing coefficients. To also get back the entire regulatrization paths, run

paths = fit(DMRPaths, mf, covarsdf, counts)

we can now select, for example the coefficients that minimize 10-fold CV mse (takes a while)

coef(paths, MinCVKfold{MinCVmse}(10))

HurdleDMR.DCR — Type

Abstract Distributed Counts Regression (DCR) returned object

HurdleDMR.DMR — Type

Abstract DMR returned object

HurdleDMR.DMRCoefs — Type

Relatively light object used to return DMR results when we only care about estimated coefficients.

HurdleDMR.DMRPaths — Type

Relatively heavy object used to return DMR results when we care about the regulatrization paths.

Distributions.ncategories — Method

Number of categories (terms/words/phrases) used for DMR estimation

HurdleDMR.dmr — Method

Shorthand for fit(DMR,covars,counts). See also fit(::DMR)

HurdleDMR.dmrpaths — Method

Shorthand for fit(DMRPaths,covars,counts). See also fit(::DMRPaths)

HurdleDMR.hasintercept — Method

Whether the model includes an intercept in each independent counts (e.g. hurdle) regression

HurdleDMR.ncoefs — Method

Number of coefficient potentially including intercept used for each independent Poisson regression

HurdleDMR.ncovars — Method

Number of covariates used for DMR estimation

StatsAPI.coef — Method

coef(m::DMRCoefs)

Returns the coefficient matrices fitted with DMR using the segment selected during fit (MinAICc by default).

Example:

  m = fit(DMR,covars,counts)
  coef(m)

StatsAPI.fit — Method

fit(DMR,covars,counts; <keyword arguments>)
dmr(covars,counts; <keyword arguments>)

Fit a Distributed Multinomial Regression (DMR) of counts on covars.

DMR fits independent poisson gamma lasso regressions to each column of counts to approximate a multinomial, picks a segement of each path, and returns a coefficient matrix (wrapped in DMRCoefs) representing point estimates for the entire multinomial (includes the intercept if one was included).

Example:

  m = fit(DMR,covars,counts)

Arguments

covars n-by-p matrix of covariates
counts n-by-d matrix of counts (usually sparse)

Keywords

intercept::Bool=false include an intercept in each poisson
parallel::Bool=true parallelize the poisson fits
local_cluster::Bool=true use local_cluster mode that shares memory across parallel workers that is appropriate on a single multicore machine, or remote cluster mode that is more appropriate when distributing across machines for which sharing memory is costly.
verbose::Bool=true
showwarnings::Bool=false
select::SegSelect=MinAICc() which path segment to pick
kwargs... additional keyword arguments passed along to fit(GammaLassoPath,...)

StatsAPI.fit — Method

fit(DMRPaths,covars,counts; <keyword arguments>)
dmrpaths(covars,counts; <keyword arguments>)

Fit a Distributed Multinomial Regression (DMR) of counts on covars, and returns the entire regulatrization paths, which may be useful for plotting or picking coefficients other than the AICc optimal ones. Same arguments as fit(::DMR).

StatsAPI.fit — Method

fit(DMR,@model(c ~ x1*x2),df,counts; <keyword arguments>)

Fits a DMR but takes a model formula and dataframe instead of the covars matrix. See also fit(::DMR).

c must be specified on the lhs to indicate the model for counts.

StatsAPI.nobs — Method

Number of observations used. May be lower than provided after removing all zero obs.

StatsAPI.predict — Method

predict(m,newcovars; <keyword arguments>)

Predict counts using a fitted DMRPaths object and given newcovars.

Example:

  m = fit(DMRPaths,covars,counts)
  newcovars = covars[1:10,:]
  countshat = predict(m, newcovars; select=MinAICc())

Arguments

m::DMRPaths fitted DMRPaths model (DMRCoefs currently not supported)
newcovars n-by-p matrix of covariates of same dimensions used to fit m.

Keywords

select=MinAICc() See coef(::RegularizationPath).
kwargs... additional keyword arguments passed along to predict() for each category j=1..size(counts,2)

Hurdle Distributed Multinomial Regression (HDMR)

For highly sparse counts, as is often the case with text that is selected for various reasons, the Hurdle Distributed Multinomial Regression (HDMR) model of Kelly, Manela, and Moreira (2018), may be superior to the DMR. It approximates a higher dispersion Multinomial using distributed (independent, parallel) Hurdle regressions, one for each of the d categories (columns) of a large counts matrix, on the covars. It allows a potentially different sets of covariates to explain category inclusion ($h=1{c>0}$), and repetition ($c>0$).

Both the model for zeroes and for positive counts are regularized by default, using GammaLassoPath, picking the AICc optimal segment of the regularization path.

HDMR can be fitted:

m = hdmr(covars, counts; inpos=1:2, inzero=1:3)

or with a dataframe and formula

mf = @model(h ~ vy + v1 + v2, c ~ vy + v1)
m = fit(HDMR, mf, covarsdf, counts)

where the h ~ equation is the model for zeros (hurdle crossing) and c ~ is the model for positive counts

in either case we can get the coefficients matrix for each variable + intercept as usual with

coefspos, coefszero = coef(m)

By default we only return the AICc maximizing coefficients. To also get back the entire regularization paths, run

paths = fit(HDMRPaths, mf, covarsdf, counts)

coef(paths, AllSeg())

Syntax:

HurdleDMR.HDMR — Type

Abstract HDMR returned object

HurdleDMR.HDMRCoefs — Type

Relatively light object used to return HDMR results when we only care about estimated coefficients.

HurdleDMR.HDMRPaths — Type

Relatively heavy object used to return HDMR results when we care about the regulatrization paths.

HurdleDMR.hdmr — Method

Shorthand for fit(HDMR,covars,counts). See also fit(::HDMR)

HurdleDMR.hdmrpaths — Method

Shorthand for fit(HDMRPaths,covars,counts). See also fit(::HDMRPaths)

HurdleDMR.ncoefspos — Method

Number of coefficient potentially including intercept used by model for positives

HurdleDMR.ncoefszero — Method

Number of coefficient potentially including intercept used by model for zeros

HurdleDMR.ncovarspos — Method

Number of covariates used for HDMR estimation of positives model

HurdleDMR.ncovarszero — Method

Number of covariates used for HDMR estimation of zeros model

HurdleDMR.posindic — Method

posindic(A)

Returns an array of the same dimensions of indicators for positive entries in A.

HurdleDMR.posindic — Method

Sparse version simply replaces all the non-zero values with ones.

StatsAPI.coef — Function

coef(m::HDMRPaths, select::SegSelect=MinAICc())

Returns all or selected coefficient matrices fitted with HDMR.

Example:

  m = fit(HDMRPaths,covars,counts)
  coefspos, coefszero = coef(m, MinCVKfold{MinCVmse}(5))

StatsAPI.coef — Method

coef(m::HDMRCoefs)

Returns the coefficient matrices fitted with HDMR using the segment selected during fit (MinAICc by default).

Example:

  m = fit(HDMR,covars,counts)
  coefspos, coefszero = coef(m)

StatsAPI.fit — Method

fit(HDMR,covars,counts; <keyword arguments>)
hdmr(covars,counts; <keyword arguments>)

Fit a Hurdle Distributed Multinomial Regression (HDMR) of counts on covars.

HDMR fits independent hurdle lasso regressions to each column of counts to approximate a multinomial, picks a segement of each path, and returns a coefficient matrix (wrapped in HDMRCoefs) representing point estimates for the entire multinomial (includes the intercept if one was included).

Example:

  m = fit(HDMR,covars,counts)

Arguments

covars n-by-p matrix of covariates
counts n-by-d matrix of counts (usually sparse)

Keywords

inpos=1:p indices of covars columns included in model for positive counts
inzero=1:p indices of covars columns included in model for zero counts
intercept::Bool=false include a intercepts in each hurdle regression
parallel::Bool=true parallelize the poisson fits
local_cluster::Bool=true use local_cluster mode that shares memory across parallel workers that is appropriate on a single multicore machine, or remote cluster mode that is more appropriate when distributing across machines for which sharing memory is costly.
verbose::Bool=true
showwarnings::Bool=false
select::SegSelect=MinAICc() path segment selection criterion
kwargs... additional keyword arguments passed along to fit(Hurdle,...)

StatsAPI.fit — Method

fit(HDMRPaths,covars,counts; <keyword arguments>)
hdmrpaths(covars,counts; <keyword arguments>)

Fit a Hurdle Distributed Multinomial Regression (HDMR) of counts on covars, and returns the entire regulatrization paths, which may be useful for plotting or picking coefficients other than the AICc optimal ones. Same arguments as fit(::HDMR).

StatsAPI.fit — Method

fit(HDMR,@model(h ~ x1 + x2, c ~ x1),df,counts; <keyword arguments>)

Fits a HDMR but takes a model formula and dataframe instead of the covars matrix. See also fit(::HDMR).

h and c on the lhs indicate the model for zeros and positives, respectively.

StatsAPI.predict — Method

predict(m,newcovars; <keyword arguments>)

Predict counts using a fitted HDMRPaths object and given newcovars.

Example:

  m = fit(HDMRPaths,covars,counts)
  newcovars = covars[1:10,:]
  countshat = predict(m, newcovars; select=MinAICc())

Arguments

m::HDMRPaths fitted DMRPaths model (HDMRCoefs currently not supported)
newcovars n-by-p matrix of covariates of same dimensions used to fit m.

Keywords

select=MinAICc() See coef(::RegularizationPath).
kwargs... additional keyword arguments passed along to predict() for each category j=1..size(counts,2)

Sufficient reduction projection

A sufficient reduction projection summarizes the counts, much like a sufficient statistic, and is useful for reducing the d dimensional counts in a potentially much lower dimension matrix z.

To get a sufficient reduction projection in direction of vy for the above example

z = srproj(m,counts,1,1)

Here, the first column is the SR projection from the model for positive counts, the second is the the SR projection from the model for hurdle crossing (zeros), and the third is the total count for each observation.

Syntax:

HurdleDMR.srproj — Function

srproj calculates the MNIR Sufficient Reduction projection from text counts on to the attribute dimensions of interest (covars in mnlm). In particular, for counts C, with row sums m, and mnlm coefficients φj corresponding to attribute j, zj = C'φj/m is the SR projection in the direction of j. The MNIR paper explains how V=[v1 ... vK], your original covariates/attributes, are independent of text counts C given SR projections Z=[z1 ... z_K]. dir == nothing returns projections in all directions.

HurdleDMR.srproj — Method

srproj for hurdle dmr takes two coefficent matrices coefspos, coefszero, and a two specific directions and returns an n-by-4 matrix Z = [zpos zzero m l]. dirpos = 0 omits positive counts projections and dirzero = 0 omits zero counts projections. Setting any of these to nothing will return projections in all directions.

HurdleDMR.srproj — Method

srproj for hurdle dmr takes two coefficent matrices coefspos, coefszero, and a two specific directions and returns an n-by-4 matrix Z = [zpos zzero m l]. dirpos = 0 omits positive counts projections and dirzero = 0 omits zero counts projections. Setting any of these to nothing will return projections in all directions.

HurdleDMR.srproj — Method

srproj calculates the MNIR Sufficient Reduction projection from text counts on to the attribute dimensions of interest (covars in mnlm). In particular, for counts C, with row sums m, and mnlm coefficients φj corresponding to attribute j, zj = C'φj/m is the SR projection in the direction of j. The MNIR paper explains how V=[v1 ... vK], your original covariates/attributes, are independent of text counts C given SR projections Z=[z1 ... z_K]. dir == nothing returns projections in all directions.

HurdleDMR.srproj — Method

Like srproj but efficiently interates over a sparse counts matrix, and only projects in a single direction (dir).

HurdleDMR.srprojX — Method

Builds the design matrix X for predicting covar in direction projdir hdmr version Assumes that covars include all variables for both positives and zeros models and indicates which variables are where with the index arrays inpos and inzero. inz=[1,2] if both zpos and zzero are included inz=[2] if zpos is dropped due to collinearity

HurdleDMR.srprojX — Method

Builds the design matrix X for predicting covar in direction projdir dmr version inz=[1] and testrank=false always for dmr, so variables are ignored and only here for convinence of unified calling function

Counts Inverse Regression (CIR)

Counts inverse regression allows us to predict a covariate with the counts and other covariates. Here we use hdmr for the backward regression and another model for the forward regression. This can be accomplished with a single command, by fitting a CIR{HDMR,FM} where the forward model is FM <: RegressionModel.

cir = fit(CIR{HDMR,LinearModel},mf,covarsdf,counts,:vy; nocounts=true)

where the nocounts=true means we also fit a benchmark model without counts.

we can get the forward and backward model coefficients with

coefbwd(cir)
coeffwd(cir)

The fitted model can be used to predict vy with new data

yhat = predict(cir, covarsdf[1:10,:], counts[1:10,:])

We can also predict only with the other covariates, which in this case is just a linear regression

yhat_nocounts = predict(cir, covarsdf[1:10,:], counts[1:10,:]; nocounts=true)

Syntax:

HurdleDMR.CIR — Type

Counts inverse regression (CIR) model supports both multinomial and hurdle inverse regressions and holds both the inverse and forward regression model estimates

HurdleDMR.coefbwd — Method

Returns coefficients for backward model for counts as function of covariates

HurdleDMR.coeffwd — Method

Returns coefficients of forward regression model. Set nocounts=true to get coefficients for the benchmark model without counts.

StatsAPI.fit — Method

fit(CIR{DMR,FM},m,df,counts,projdir[,fmargs...]; <keyword arguments>)

Version of fit(CIR{DMR,FM}...) that takes a @model() and a dataframe instead of a covars matrix, and a projdir::Symbol specifies the dependent variable. See also fit(CIR...).

Example:

  m = fit(CIR{DMR,LinearModel}, @model(c~x1+x2), df, counts, :x1; nocounts=true)

where c~ is the model for counts. x1 (projdir) is the variable to predict. We can then predict with a dataframe as well

  yhat = predict(m, df, counts)
  yhatnc = predict(m, df, counts; nocounts=true)

StatsAPI.fit — Method

fit(CIR{HDMR,FM},m,df,counts,projdir[,fmargs...]; <keyword arguments>)

Version of fit(CIR{HDMR,FM}...) that takes a @model() and a dataframe instead of a covars matrix, and a projdir::Symbol specifies the dependent variable. See also fit(CIR...).

Example:

  m = fit(CIR{HDMR,LinearModel}, @model(h~x1+x2, c~x1), df, counts, :x1; nocounts=true)

where h~ is the model for zeros, c~ is the model for positives. x1 (projdir) is the variable to predict. We can then predict with a dataframe as well

  yhat = predict(m, df, counts)
  yhatnc = predict(m, df, counts; nocounts=true)

StatsAPI.fit — Method

fit(::CIR{BM,FM},covars,counts,projdir[,fmargs...]; <keyword arguments>)

Fit a Counts Inverse Regression (CIR) of covars[:,projdir] ~ counts + covars[:,~projdir].

CIR involves three steps:

Fit a backward regression model BM<:DCR: counts ~ covars
Calculate an sufficient reduction projection in direction projdir
Fit a forward regression model FM<:RegressionModel:

covars[:,projdir] ~ srproj(counts) + covars[:,~projdir]

Example:

  m = fit(CIR{DMR,LinearModel}, covars, counts, 1; nocounts=true)
  yhat = predict(m, covars, counts)
  yhatnc = predict(m, covars, counts; nocounts=true)

Arguments

covars n-by-p matrix of covariates
counts n-by-d matrix of counts (usually sparse)
projdir index of covars column used as dependent variable in forward model
fmargs... optional arguments passed along to the forward regression model

Keywords

nocounts::Bool=false whether to also fit a benchmark model without counts
bmkwargs... keyword arguments passed along to the backward regression model

StatsAPI.predict — Method

Predict using a fitted Counts inverse regression (CIR) given new covars dataframe and counts. See also predict(::CIR).

StatsAPI.predict — Method

Predict using a fitted Counts inverse regression (CIR) given new covars and counts.

Keywords

Set nocounts=true to predict using a benchmark model without counts.

Hurdle

This package also provides a regularized Hurdle model (Mullahy, 1986) that can be fit using a fast coordinate decent algorithm, or simply by running two fit(GeneralizedLinearModel,...) regressions, one for each of its two parts.

Syntax:

HurdleDMR.Hurdle — Type

Hurdle returned object

HurdleDMR.MinCVKfold — Type

Selects the RegularizationPath segment according to CVSegSelect with k-fold cross-validation

StatsAPI.coef — Method

coef(m::Hurdle; select=MinAICc())

Returns a selected segment of the coefficient matrices of the fitted the TwoPartModel.

Example:

  m = fit(Hurdle,GeneralizedLinearModel,X,y; Xpos=Xpos)
  coefspos, coefszero = coef(m)

Keywords

kwargs... are passed along to two coef() calls on the two model parts.

StatsAPI.coef — Method

Selects the RegularizationPath segment coefficients according to S with k-fold cross-validation

StatsAPI.coef — Method

Selects the RegularizationPath segment coefficients according to S with k-fold cross-validation

StatsAPI.fit — Method

fit(Hurdle,M,f,df; fpos=Xpos, <keyword arguments>)

Takes dataframe and two formulas, one for each model part. Otherwise same arguments as fit(::Hurdle)

Example

  fit(Hurdle,GeneralizedLinearModel,@formula(y ~ x1*x2), df; fpos=@formula(y ~ x1*x2+x3))

StatsAPI.fit — Method

fit(Hurdle,M,X,y; Xpos=Xpos, <keyword arguments>)

Fit a Hurdle (Mullahy, 1986) of count vector y on X with potentially another covariates matrix Xpos used to model positive counts.

Example with GLM:

  m = fit(Hurdle,GeneralizedLinearModel,X,y; Xpos=Xpos)
  yhat = predict(m, X; Xpos=Xpos)

Example with Lasso regularization:

  m = fit(Hurdle,GammaLassoPath,X,y; Xpos=Xpos)
  yhat = predict(m, X; Xpos=Xpos, select=MinAICc())

Arguments

M::RegressionModel
counts n-by-d matrix of counts (usually sparse)
dzero::UnivariateDistribution = Binomial() distribution for zeros model
dpos::UnivariateDistribution = PositivePoisson() distribution for positives model
lzero::Link=canonicallink(dzero) link function for zeros model
lpos::Link=canonicallink(dpos) link function for positives model

Keywords

Xpos::Union{AbstractMatrix{T},Nothing} = nothing covariates matrix for positives model or nothing to use X for both parts
dofit::Bool = true fit the model or just construct its shell
wts::V = ones(y) observation weights
offsetzero::AbstractVector = similar(y, 0) offsets for zeros model
offsetpos::AbstractVector = similar(y, 0) offsets for positives model
offset::AbstractVector = similar(y, 0) offsets for both model parts
verbose::Bool=true
showwarnings::Bool=false
fitargs... additional keyword arguments passed along to fit(M,...)

StatsAPI.predict — Method

predict(m,X; Xpos=Xpos, <keyword arguments>)

Predict using a fitted TwoPartModel given new X (and potentially Xpos).

Example with GLM:

  m = fit(Hurdle,GeneralizedLinearModel,X,y; Xpos=Xpos)
  yhat = predict(m, X; Xpos=Xpos)

Example with Lasso regularization:

  m = fit(Hurdle,GammaLassoPath,X,y; Xpos=Xpos)
  yhat = predict(m, X; Xpos=Xpos, select=MinAICc())

Arguments

m::Hurdle fitted Hurdle model
X n-by-p matrix of covariates of same dimensions used to fit m.

Keywords

Xpos::Union{AbstractMatrix{T},Nothing} = nothing covariates matrix for positives model or nothing to use X for both parts
kwargs... additional keyword arguments passed along to predict() for each of the two model parts.

Positive Poisson

This package also implements the PositivePoisson distribution and the GLM necessary methods to facilitate fit with fit(::GeneralizedLinearModel.

Syntax:

HurdleDMR.PositivePoisson — Type

PositivePoisson(λ)

A PositivePoisson distribution (aka zero-truncated Poisson, ZTP) descibes the number of independent events occurring within a unit time interval, given the average rate of occurrence λ and, importantly, given that the number is not zero.

$P(X = k) = \frac{λ^k}{k!(1-e^{-λ})} e^{-λ}, \quad \text{ for } k = 1,2,\ldots.$

PositivePoisson()        # PositivePoisson distribution with rate parameter 1
PositivePoisson(lambda)       # PositivePoisson distribution with rate parameter lambda

params(d)        # Get the parameters, i.e. (λ,)
mean(d)          # Get the mean arrival rate, i.e. λ

External links:

PositivePoisson distribution on Wikipedia

API / Index