milr: Multiple-Instance Logistic Regression with Lasso Penalty

1 Introduction

Multiple-instance learning (MIL) is used to model the class labels which are associated with bags of observations instead of the individual observations. This technique has been widely used in solving many different real-world problems. In the early stage of the MIL application, Dietterich et al. (1997) studied the drug-activity prediction problem. A molecule is classified as a good drug if it is able to bind strongly to a binding site on the target molecule. The problem is: one molecule can adopt multiple shapes called the conformations and only one or a few conformations can bind the target molecule well. They described a molecule by a bag of its many possible conformations whose binding strength remains unknown. An important application of MIL is the image and text categorization, such as in Maron and Ratan (1998; Andrews et al. 2003; Zhang et al. 2007; Zhou et al. 2009; Li et al. 2011; Kotzias et al. 2015), to name a few. An image (bag) possessing at least one particular pattern (instance) is categorized into one class; otherwise, it is categorized into another class. For example, Maron and Ratan (1998) treated the natural scene images as bags, and, each bag is categorized as the scene of waterfall if at least one of its subimages is the waterfall. Whereas, Zhou et al. (2009) studied the categorization of collections (bags) of posts (instances) from different newsgroups corpus. A collection is a positive bag if it contains 3% posts from a target corpus category and the remaining 97% posts, as well as all posts in the negative bags, belong to the other corpus categories. MIL is also used in medical researches. The UCSB breast cancer study (Kandemir et al. 2014) is such a case. Patients (bags) were diagnosed as having or not having cancer by doctors; however, the computer, initially, had no knowledge of which patterns (instances) were associated with the disease. Furthermore, in manufacturing processes (Chen et al. 2016), a product (bag) is defective as long as one or more of its components (instances) are defective. In practice, at the initial stage, we only know that a product is defective, and we have no idea which component is responsible for the defect.

Several approaches have been offered to analyze datasets with multiple instances, e.g., Maron (1998; Zhang and Goldman 2002; Xu and Frank 2004; Ray and Craven 2005). From our point of view, the statuses of these components are missing variables, and thus, the Expectation-Maximization (EM) algorithm (Dempster et al. 1977) can play a role in multiple-instance learning. By now the toolboxes or libraries available for implementing MIL methods are developed by other computer softwares. For example, Yang (2008) and Tax and Cheplygina (2016) are implemented in MATLAB software, but neither of them carries the methods based on logistic regression model. Settles et al. (2008) provided the Java codes including the method introduced in Ray and Craven (2005). Thus, for R users, we are first to develop a MIL-related package based on logistic regression modelling which is called multiple-instance logistic regression (MILR). In this package, we first apply the logistic regression defined in Ray and Craven (2005) and Xu and Frank (2004), and then, we use the EM algorithm to obtain maximum likelihood estimates of the regression coefficients. In addition, the popular lasso penalty (Tibshirani 1996) is applied to the likelihood function so that parameter estimation and variable selection can be performed simultaneously. This feature is especially desirable when the number of covariates is relatively large.

To fix ideas, we firstly define the notations and introduce the construction of the likelihood function. Suppose that the dataset consists of $n$ bags and that there are $m_i$ instances in the $i$th bag for $i=1,\dots ,n$. Let $Z_i$ denote the status of the $i$th bag, and let $Y_{ij}$ be the status of the $j$th instance in the $i$th bag along with $x_{ij}\in {\mathrm{\Re }}^p$ as the corresponding covariates. We assume that the $Y_{ij}$ follow independent Bernoulli distributions with defect rates of $p_{ij}$, where $p_{ij}=g({\beta }_0+x_{ij}^T\beta )$ and $g(x)=1/(1+e^{-x})$. We also assume that the $Z_i$ follow independent Bernoulli distributions with defect rates of ${\pi }_i$. Therefore, the bag-level likelihood function is

$$\begin{array}{}\text{(1)}& L({\beta }_0,\beta )=\prod _{i=1}^n{\pi }_i^{z_i}{(1-{\pi }_i)}^{1-z_i}.\end{array}$$

To associate the bag-level defect rate ${\pi }_i$ with the instance-level defect rates $p_{ij}$, several methods have been proposed. The bag-level status is defined as $Z_i=I(\sum _{j=1}^{m_i}{Y}_{ij}>0)$. If the independence assumption among the $Y_{ij}$ holds, the bag-level defect rate is ${\pi }_i=1-\prod _{j=1}^{m_i}(1-p_{ij})$. On the other hand, if the independence assumption might not be held, Xu and Frank (2004) and Ray and Craven (2005) proposed the softmax function to associate ${\pi }_i$ to $p_{ij}$, as follows:

$$\begin{array}{}\text{(2)}& s_i\left(\alpha \right)=\sum _{j=1}^{m_i}{p}_{ij}\exp \left\{\alpha p_{ij}\right\}/\sum _{j=1}^{m_i}\exp \left\{\alpha p_{ij}\right\},\end{array}$$

where $\alpha$ is a pre-specified nonnegative value. Xu and Frank (2004) used $\alpha =0$, therein modeling ${\pi }_i$ by taking the average of $p_{ij}$, $j=1,\dots ,m_i$, whereas Ray and Craven (2005) suggested $\alpha =3$. We observe that the likelihood ((1)) applying neither the ${\pi }_i$ function nor the $s_i(\alpha )$ function results in effective estimators.

Below, we begin by establishing the E-steps and M-steps required for the EM algorithm and then attach the lasso penalty for the estimation and feature selection. Several computation strategies applied are the same as those addressed in Friedman et al. (2010). Finally, we demonstrate the functions provided in the milr package via simulations and on a real dataset.

2 The multiple-instance logistic regression

EM algorithm

If the instance-level statuses, $y_{ij}$, are observable, the complete data likelihood is $$\prod _{i=1}^n\prod _{j=1}^{m_i}{p}_{ij}^{y_{ij}}{q}_{ij}^{1-y_{ij}},$$ where $q_{ij}=1-p_{ij}$. An ordinary approach, such as the Newton method, can be used to solve this maximal likelihood estimate (MLE). However, considering multiple-instance data, we can only observe the statuses of the bags, $Z_i=I(\sum _{j=1}^{m_j}{Y}_{ij}>0)$, and not the statuses of the instances $Y_{ij}$. As a result, we apply the EM algorithm to obtain the MLEs of the parameters by treating the instance-level labels as the missing data.

In the E-step, two conditional distributions of the missing data given the bag-level statuses $Z_i$ are $$Pr(Y_{i1}=0,\dots ,Y_{i{m}_i}=0\mid Z_i=0)=1$$ and $$Pr(Y_{ij}=y_{ij},j=1,\dots ,m_i\mid Z_i=1)=\frac{\prod _{j=1}^{m_i}{p}_{ij}^{y_{ij}}{q}_{ij}^{1-y_{ij}}\times I(\sum _{j=1}^{m_i}{y}_{ij}>0)}{1-\prod _{l=1}^{m_i}{q}_{il}}.$$ Thus, the conditional expectations are

$$E(Y_{ij}\mid Z_i=0)=0\text{ and }E(Y_{ij}\mid Z_i=1)=\frac{p_{ij}}{1-\prod _{l=1}^{m_i}{q}_{il}}\equiv {\gamma }_{ij}.$$

The $Q$ function at step $t$ is $Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)=\sum _{i=1}^n{Q}_i({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)$, where $Q_i$ is the conditional expectation of the complete log-likelihood for the $i$th bag given $Z_i$, which is defined as

$$\begin{array}{rl}{Q}_i({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)& =E(\sum _{j=1}^{m_i}{y}_{ij}\log \left(p_{ij}\right)+(1-y_{ij})\log \left(q_{ij}\right)|Z_i=z_i,{\beta }_0^t,{\beta }^t)\\ & =\sum _{j=1}^{m_i}{z}_i{\gamma }_{ij}^t({\beta }_0+x_{ij}^T\beta )-\log (1+e^{{\beta }_0+x_{ij}^T\beta }).\end{array}$$

Note that all the $p_{ij}$, $q_{ij}$, and ${\gamma }_{ij}$ are functions of ${\beta }_0$ and $\beta$, and thus, we define these functions by substituting ${\beta }_0$ and $\beta$ by their current estimates ${\beta }_0^t$ and ${\beta }^t$ to obtain $p_{ij}^t$, $q_{ij}^t$, and ${\gamma }_{ij}^t$, respectively.

In the M-step, we maximize this $Q$ function with respect to $({\beta }_0,\beta )$. Since the maximization of the nonlinear $Q$ function is computationally expensive, following Friedman et al. (2010), the quadratic approximation to $Q$ is applied. Taking the second-order Taylor expansion about ${\beta }_0^t$ and ${\beta }^t$, we have $Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)=Q_Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)+C+R_2({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)$, where $C$ is a constant in terms of ${\beta }_0$ and $\beta$, $R_2({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)$ is the remainder term of the expansion and $$Q_Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)=-\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^{m_i}{w}_{ij}^t{[u_{ij}^t-{\beta }_0-x_{ij}^T\beta ]}^2,$$ where $u_{ij}^t={\beta }_0+x_{ij}^T{\beta }^t+(z_i{\gamma }_{ij}^t-p_{ij}^t)/\left(p_{ij}^t{q}_{ij}^t\right)$ and $w_{ij}^t=p_{ij}^t{q}_{ij}^t$. In the milr package, instead of maximizing $Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)$, we maximize its quadratic approximation, $Q_Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)$. Since the objective function is quadratic, the roots of $\partial Q_Q/\partial {\beta }_0$ and $\partial Q_Q/\partial \beta$ have closed-form representations.

Variable selection with lasso penalty

We adopt the lasso method (Tibshirani 1996) to identify active features in this MILR framework. The key is to add the $L_1$ penalty into the objective function in the M-step so that the EM algorithm is capable of performing estimation and variable selection simultaneously. To this end, we rewrite the objective function as

$$\begin{array}{}\text{(3)}& \underset{{\beta }_0,\beta }{min}\{-Q_Q({\beta }_0,\beta \mid {\beta }_0^t,{\beta }^t)+\lambda \sum _{k=1}^p\left|{\beta }_k\right|\}.\end{array}$$

Note that the intercept term ${\beta }_0$ is always kept in the model; thus, we do not place a penalty on ${\beta }_0$. In addition, $\lambda$ is the tuning parameter, and we will introduce how to determine this parameter later. We applied the shooting algorithm (Fu 1998; Chen et al. 2016) to update $({\beta }_0^t,{\beta }^t)$.

3 Implementation

The milr package contains a data generator, DGP, which is used to generate the multiple-instance data for the simulation studies, and two estimation approaches, milr and softmax, which are the main tools for modeling the multiple-instance data. In this section, we introduce the usage and default setups of these functions.

Data generator

The function DGP is the generator for the multiple-instance-type data under the MILR framework.

To use the DGP function, the user needs to specify an integer n as the number of bags, a vector m of length $n$ as the number of instances in each bag, and a vector beta of length $p$, with the desired number of covariates, and the regression coefficients, $\beta$, as in DGP(n, m, beta). Note that one can set m as an integer for generating the data with an equal instance size m for each bag. Thus, the total number of observations is $N=\sum _{i=1}^n{m}_i$. The DGP simulates the labels of bags through the following steps:

1. Generate $p$ mutually independent covariates of length $N$ from the standard normal distribution as an $N\times p$ matrix, $X$.

2. Generate the binary response, $Y_{ij}$, for the $j$th instance of the $i$th bag from the Bernoulli distribution with $$p_{ij}=1/(1+\exp \{-x_{ij}^T\beta \})$$ where $x_{ij}$ is the $p$-component vector in the row of $X$ representing the $j$th instance of the $i$th bag.

3. Calculate the observed response for the $i$th bag by $Z_i=I(\sum _{j=1}^{m_i}{Y}_{ij}>0)$.

4. Return the indices of the bags, the covariate matrix $X$ and the bag-level statuses $Z$.

The milr and softmax apporaches

In the milr package, we provide two approaches to model the multiple-instance data: the proposed milr (Chen et al. 2016) and the softmax approach (Xu and Frank 2004). To implement these two approaches, we assume that the number of observations and covariates are $N$ and $p$, respectively. The input data for both milr and softmax are separated into three parts: the bag-level statuses, y, as a vector of length $N$; the $N\times p$ design matrix, x; and bag, the vector of indices of length $N$, representing the indices of the bag to which each instance belongs.

  milr(y, x, bag, lambda, numLambda, lambdaCriterion, nfold, maxit)
  softmax(y, x, bag, alpha, ...)

For the milr function, specifying lambda in different ways controls whether and how the lasso penalty participates in parameter estimation. The default value of lambda is $0$. With this value, the ordinary MLE is applied, i.e., no penalty term is considered. This is the suggested choice when the number of covariates $p$ is small. When $p$ is large or when variable selection is desired, users can specify a $\lambda$ vector of length $\kappa$; otherwise, by letting lambda = -1, the program automatically provides a $\lambda$ vector of length $\kappa =\mathtt{\text{numLambda}}$ as the tuning set. Following Friedman et al. (2010), the theoretical maximal value of $\lambda$ in ((3)) is

$$\begin{array}{}\text{(4)}& {\lambda }_{max}={\left[\prod _{i=1}^n(m_i-1)\right]}^{\frac{1}{2}}{\left[\prod _{i=1}^n{m}_i^{1-2z_i}\right]}^{\frac{1}{2}}.\end{array}$$

The automatically specified sequence of $\lambda$ values ranges from ${\lambda }_{min}={\lambda }_{max}/1000$ to ${\lambda }_{max}$ in ascending order.

The default setting for choosing the optimal $\lambda$ among these $\lambda$ values is the Bayesian information criterion (BIC), $-2\log (likelihood)+p^{\ast }\times \log (n)$, where $p^{\ast }$ is the number of nonzero regression coefficients. Alternatively, the user can use the options lambdaCriterion = "deviance" and nfold = K with an integer K to obtain the best $\lambda$ that minimizes the predictive deviance through “bag-wise” $K$-fold cross validation. The last option, maxit, indicates the maximal number of iterations of the EM algorithm; its default value is $500$.

For the softmax function, the option alpha is a nonnegative real number for the $\alpha$ value in ((2)). The maximum likelihood estimators of the regression coefficients are obtained by the generic function optim. Note that no variable selection approach is implemented for this method.

Two generic accessory functions, coef and fitted, can be used to extract the regression coefficients and the fitted bag-level labels returned by milr and softmax. We also provide the significance test based on Wald’s test for the milr estimations without the lasso penalty through the summary function. In addition, to predict the bag-level statuses for the new data set, the predict function can be used by assigning three items: object is the fitted model obtained by milr or softmax, newdata is the covariate matrix, and bag_newdata is the bag indices of the new dataset. Finally, the MIL model can be used to predict the bag-level labels and the instances-level labels. The option type in fitted and predicted functions controls the type of output labels. The default option is type = "bag" which results the bag-level prediction. Otherwise, by setting type = "instance", the instances-level labels will be presented.

  fitted(object, type)
  predict(object, newdata, bag_newdata, type)

4 Examples

We illustrate the usage of the milr package via simulated and real examples.

Estimation and variable selection

We demonstrate how to apply the milr function for model estimation and variable selection. We simulate data with $n=50$ bags, each containing $m=3$ instances and regression coefficients $\beta =(-2,-1,1,2,0.5,0,0,0,0,0)$. Specifically, the first four covariates are important.

library(magrittr)
library(milr)
set.seed(99)
# set the size of dataset
numOfBag <- 50
numOfInstsInBag <- 3
# set true coefficients: beta_0, beta_1, beta_2, beta_3
trueCoefs <- c(-2, -2, -1, 1, 2, 0.5, 0, 0, 0, 0, 0)
trainData <- DGP(numOfBag, numOfInstsInBag, trueCoefs)
trainData$X %<>% set_colnames(paste0("X", 1:ncol(.)))
tapply(trainData$Z, trainData$ID, function(x) sum(x) > 0) %>% as.numeric

##  [1] 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0
## [36] 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1

Since the number of covariates is small, we then use the milr function to estimate the model parameters with lambda = 0. One can apply summary to produce results including estimates of the regression coefficients and their corresponding standard error, testing statistics and the P-values under Wald’s test. The regression coefficients are returned by the function coef.

# fit milr model
milrFit_EST <- milr(trainData$Z, trainData$X, trainData$ID, lambda = 0)
# call the Wald's test result
summary(milrFit_EST)

## Log-Likelihood: -14.005.

## Estimates:

##           Estimate  Std.Err Z value   Pr(>z)
## intercept -3.28671  1.16695 -2.8165 0.004855 **
## X1        -2.45529  0.92227 -2.6622 0.007762 **
## X2        -1.26351  0.67621 -1.8685 0.061689 .
## X3         0.94016  0.75173  1.2507 0.211054
## X4         3.84173  1.47862  2.5982 0.009372 **
## X5         0.22000  0.66579  0.3304 0.741073
## X6        -1.00740  0.73288 -1.3746 0.169262
## X7        -0.53063  0.59871 -0.8863 0.375463
## X8         0.25334  0.71596  0.3538 0.723451
## X9        -1.92753  0.92437 -2.0852 0.037047 *
## X10        0.12249  0.63054  0.1943 0.845972
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# call the regression coefficients
coef(milrFit_EST)

##  intercept         X1         X2         X3         X4         X5
## -3.2867082 -2.4552903 -1.2635149  0.9401636  3.8417318  0.2199982
##         X6         X7         X8         X9        X10
## -1.0074012 -0.5306309  0.2533409 -1.9275338  0.1224893

The generic function table builds a contingency table of the counts for comparing the true bag-level statuses and the fitted bag-level statuses (obtained by the option type = "bag") and the predict function is used to predict the labels of each bag with corresponding covariate $X$. On the other hand, The fitted and predicted instance-level statuses can also be found by setting type = "instance" in the fitted and predict functions.

fitted(milrFit_EST, type = "bag")

##  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
##  1  1  0  1  1  1  0  1  1  0  1  1  0  0  1  1  0  1  1  1  1  0  0  1  0
## 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
##  0  1  0  1  0  1  1  1  0  0  1  1  0  0  0  1  1  1  0  0  0  1  0  1  1

%#fitted(milrFit_EST, type = "instance") # instance-level fitted labels
table(DATA = tapply(trainData$Z, trainData$ID, function(x) sum(x) > 0) %>% as.numeric,
      FITTED = fitted(milrFit_EST, type = "bag"))

##     FITTED
## DATA  0  1
##    0 18  4
##    1  3 25

# predict for testing data
testData <- DGP(numOfBag, numOfInstsInBag, trueCoefs)
testData$X %<>% set_colnames(paste0("X", 1:ncol(.)))
pred_EST <- predict(milrFit_EST, testData$X, testData$ID, type = "bag")
#predict(milrFit_EST, testData$X, testData$ID,
#        type = "instance") # instance-level prediction
table(DATA = tapply(testData$Z, testData$ID, function(x) sum(x) > 0) %>% as.numeric,
      PRED = pred_EST)

##     PRED
## DATA  0  1
##    0 13  6
##    1  8 23

Next, the $n<p$ cases are considered. We generate a data set with $n=50$ bags, each with 3 instances and $p=100$ covariates. Among these covariates, only the first five of them, $X_1,\dots ,X_5$, are active and their nonzero coefficients are the same as the previous example. First, we manually specify 50 $\lambda$ values manually from 0.01 to 50. The milr function chooses the best tuning parameter which results in the smallest BIC. For this dataset, the chosen model is a constant model.

set.seed(99)
# Set the new coefficient vector (large p)
trueCoefs_Lp <- c(-2, -2, -1, 1, 2, 0.5, rep(0, 95))
# Generate the new training data with large p
trainData_Lp <- DGP(numOfBag, numOfInstsInBag, trueCoefs_Lp)
trainData_Lp$X %<>% set_colnames(paste0("X", 1:ncol(.)))
# variable selection by user-defined tuning set
lambdaSet <- exp(seq(log(0.01), log(50), length = 50))
milrFit_VS <- milr(trainData_Lp$Z, trainData_Lp$X, trainData_Lp$ID,
                   lambda = lambdaSet)
# grep the active factors and their corresponding coefficients
coef(milrFit_VS) %>% .[abs(.) > 0]

##  intercept
## -0.9020893

Second, we try the auto-tuning feature implemented in milr by assigning lambda = -1. The total number of tuning $\lambda$ values is indicated by setting nlambda. The following example shows the result of the best model chosen among 50 $\lambda$ values. The slice lambda shows the auto-tuned $\lambda$ candidates and the slice BIC returns the corresponding value of BIC for every candidate $\lambda$ value. Again, the chosen model is a constant model.

# variable selection using auto-tuning
milrFit_auto_VS <- milr(trainData_Lp$Z, trainData_Lp$X, trainData_Lp$ID,
                        lambda = -1, numLambda = 50)
# the auto-selected lambda values
milrFit_auto_VS$lambda

##  [1]  0.08041559  0.09259014  0.10660786  0.12274780  0.14133125
##  [6]  0.16272815  0.18736444  0.21573056  0.24839117  0.28599645
## [11]  0.32929500  0.37914875  0.43655012  0.50264180  0.57873946
## [16]  0.66635795  0.76724148  0.88339831  1.01714075  1.17113118
## [21]  1.34843505  1.55258192  1.78763568  2.05827549  2.36988893
## [26]  2.72867921  3.14178869  3.61744105  4.16510498  4.79568271
## [31]  5.52172701  6.35769107  7.32021625  8.42846331  9.70449388
## [36] 11.17370961 12.86535784 14.81311383 17.05575111 19.63791336
## [41] 22.61100310 26.03420494 29.97566379 34.51384138 39.73907818
## [46] 45.75539179 52.68254760 60.65844293 69.84185212 80.41558721

# the values of BIC under each lambda value
milrFit_auto_VS$BIC

##  [1] 196.54069 184.90606 161.51628 157.75005 118.76827 118.95214 115.27610
##  [8] 115.55212 115.87195 112.32892 112.75439 113.24620 113.81466 114.48027
## [15] 119.19304 116.18877 121.15644 114.54700 112.02964 110.31200 103.92031
## [22] 101.90760  88.78977  83.57070  82.69910  82.13240  78.00261  74.25620
## [29]  74.19599  78.98716  77.39963  75.33387  78.16814  69.25384  69.25384
## [36]  69.25384  69.25384  69.25384  69.25384  69.25384  69.25384  69.25384
## [43]  69.25384  69.25384  69.25384  69.25384  69.25384  69.25384  69.25384
## [50]  69.25384

# grep the active factors and their corresponding coefficients
coef(milrFit_auto_VS) %>% .[abs(.) > 0]

##  intercept
## -0.9020893

Instead of using BIC, a better way to choose the proper $lambda$ is using the cross validation by setting lambdaCriterion = "deviance". The following example shows the best model chosen by minimizing the predictive deviance via “bag-wise” 10-fold cross validation. The results of the predictive deviance for every candidate $\lambda$ can be found in the slice cv. Twenty-nine covariates were identified including the first four true active covariates, $X_1,\dots ,X_4$.

# variable selection using auto-tuning with cross validation
milrFit_auto_CV <- milr(trainData_Lp$Z, trainData_Lp$X, trainData_Lp$ID,
                        lambda = -1, numLambda = 50,
                        lambdaCriterion = "deviance", nfold = 10)
# the values of predictive deviance under each lambda value
milrFit_auto_CV$cv

##  [1] 10.013948  3.754961  3.132322  2.933881  2.433803  2.346058  2.752407
##  [8]  3.248528  3.858600  4.392568  4.781208  5.249175  5.727995  6.030227
## [15]  6.393522  6.432488  6.379543  6.339838  6.317661  6.329531  5.551296
## [22]  5.222904  5.113070  5.006837  5.078377  5.106067  5.242165  5.579102
## [29]  5.786248  6.178347  6.414204  6.648448  6.659413  6.573462  6.547737
## [36]  6.547737  6.547737  6.547737  6.547737  6.547737  6.547737  6.547737
## [43]  6.547737  6.547737  6.547737  6.547737  6.547737  6.547737  6.547737
## [50]  6.547737

# grep the active factors and their corresponding coefficients
coef(milrFit_auto_CV) %>% .[abs(.) > 0]

##    intercept           X1           X2           X3           X4
## -2.446119887 -0.362833108 -1.479388087  0.541861054  0.535400264
##           X7          X11          X14          X15          X17
##  1.448461978  0.334921736  0.004238594 -0.755908930  0.017708059
##          X18          X25          X26          X30          X32
## -0.586349577 -0.244962971  0.343205919  1.315468844 -0.845118964
##          X33          X37          X48          X58          X61
##  0.370261921 -0.493144745 -0.523001848 -0.044975426  0.208521105
##          X62          X71          X72          X74          X76
##  0.409946699  1.369814722  0.484713157  0.683531448  1.542186462
##          X77          X79          X85          X95         X100
## -0.656669320 -1.685794976 -0.369189815 -0.912145167 -0.135461219

According to another simulation study which is not shown in this paper, in contrast to cross-validation, BIC does not perform well for variable selection in terms of multiple-instance logistic regressions. However, it can be an alternative when performing cross-validation is too time consuming.

Real case study

Hereafter, we denote the proposed method with the lasso penalty by MILR-LASSO for brevity. In the following, we demonstrate the usage of MILR-LASSO and the softmax approach on a real dataset, called MUSK1. The MUSK1 data set consists of 92 molecules (bags) of which 47 are classified as having a musky smell and 45 are classified to be non-musks. The molecules are musky if at least one of their conformers (instances) were responsible for the musky smell. However, knowledge about which conformers are responsible for the musky smell is unknown. There are 166 features that describe the shape, or conformation, of the molecules. The goal is to predict whether a new molecules is musk or non-musk. This dataset is one of the popular benchmark datasets in the field of multiple-instance learning research and one can download the dataset from the following weblink.

dataName <- "MIL-Data-2002-Musk-Corel-Trec9.tgz"
dataUrl <- "http://www.cs.columbia.edu/~andrews/mil/data/"

We use the untar function to decompress the downloaded .tgz file and extract the MUSK1 dataset. Then, with the following data preprocessing, we reassemble the MUSK1 dataset in a "data.frame" format. The first 2 columns of the MUSK1 dataset are the bag indices and the bag-level labels of each observation. Starting with the third column, there are $p=166$ covariates involved in the MUSK1 dataset.

filePath <- file.path(getwd(), dataName)
# Download MIL data sets from the url
download.file(paste0(dataUrl, dataName), filePath)
# Extract MUSK1 data file
untar(filePath, files = "MilData/Musk/musk1norm.svm")
# Read and Preprocess MUSK1
library(reshape2)
tmp <- read.table(file.path(getwd(), "MilData/Musk/musk1norm.svm"),
                  sep = " ", colClasses = "character")
MUSK1 <- colsplit(tmp[,1], ":", names = c("obs", "bag", "label"))[,2:3]
MUSK1 <- cbind(MUSK1, Reduce(cbind,
    lapply(2:ncol(tmp),
        function(i) colsplit(tmp[,i], ":", names = paste0(c("num", "x"), i-1))[,2]
    )))
MUSK1$bag <- MUSK1$bag + 1
MUSK1$label <- (MUSK1$label + 1)/2
MUSK1[,3:ncol(MUSK1)] <- scale(MUSK1[,3:ncol(MUSK1)])
Y <- tapply(MUSK1$label, MUSK1$bag, function(x) sum(x) > 0) %>% as.numeric
nc <- ncol(MUSK1)

To fit an MIL model without variable selection, the milr package provides two functions. The first is the milr function with lambda = 0. The second approach is the softmax function with a specific value of alpha. Here, we apply the approaches that have been introduced in Xu and Frank (2004) and Ray and Craven (2005), called the $s(0)$ (alpha=0) and $s(3)$ (alpha=3) methods, respectively. The optimization method in softmax is chosen as the default settings of the generic function optim, that is, the Nelder-Mead method.

As suggested by one reviewer, it is relevant to compare the computational efficiencies and convergence rates of the milr and softmax functions implemented in this package. Note that, the milr approach is written in C++ and so is the objective function in softmax, and, we only consider their performance affected by their common tuning parameter, maxit, the total number of iterations. For each approach, the total number of iterations are set from 5,000 to 25,000, and, the computation task was performed by a laptop with Intel Core M-5Y71 CPU 1.4 GHz and 8GB RAM. Moreover, the performance in model fitting is assessed based on the classification accuracy. We use the generic function table to produce the contingency tables and calculate the classification accuracy values accordingly.

The left panel of Figure 1 shows the computational cost of each approach along with the increment of the total number of iterations. As expected, the computational cost increases with the number of iterations linearly for both functions. However, the slope for the milr function is much flatter than the slope for the softmax function. A further result of MILR not shown here suggests that, for this dataset, the coefficient estimate of the MILR approach converges between 15,000 and 16,000 iterations. The resulting accuracy of each model is shown in the right panel of Figure 1 which indicates that the MILR approach requires fewer iterations to achieve the best fit.

# set the iterations from 5000 to 25000
itSet <- seq(5000, 25000, 2000)
runtime <- matrix(0, length(itSet), 3)
runacc <- matrix(0, length(itSet), 3)
for (it in 1:length(itSet)) {
    # record the computation time
    runtime[it,1] <- system.time(
        softmaxFit_0 <- softmax(MUSK1$label, MUSK1[,3:nc], MUSK1$bag, alpha = 0,
                                control = list(maxit = itSet[it]))
    )[3]
    runtime[it,2] <- system.time(
        softmaxFit_3 <- softmax(MUSK1$label, MUSK1[,3:nc], MUSK1$bag, alpha = 3,
                                control = list(maxit = itSet[it]))
    )[3]
    runtime[it,3] <- system.time(
      # use a very small lambda so that milr can do the estimation
      # without evaluating the Hessian matrix
        milrFit <- milr(MUSK1$label, MUSK1[,3:nc], MUSK1$bag, lambda = 1e-7,
                        maxit = itSet[it])
    )[3]
    # calculate the accuracy
    tmp <- table(DATA = Y, FIT_s0 = fitted(softmaxFit_0, type = "bag"))
    runacc[it,1] <- sum(diag(tmp))/sum(tmp)
    tmp <- table(DATA = Y, FIT_s3 = fitted(softmaxFit_3, type = "bag"))
    runacc[it,2] <- sum(diag(tmp))/sum(tmp)
    tmp <- table(DATA = Y, FIT_MILR = fitted(milrFit, type = "bag"))
    runacc[it,3] <- sum(diag(tmp))/sum(tmp)
}

Figure 1: Computational efficiency of softmax methods and milr approach.

For variable selection, we apply the MILR-LASSO approach. First, the tuning parameter set is chosen automatically by setting $\lambda =-1$, and the best $\lambda$ value is obtained by minimizing the predictive deviance with 10-fold cross validation among nlambda = 100 candidates. In total it costs about 130 seconds to choose the optimal $\lambda$ value and there are 19 active covariates detected by the MILR-LASSO approach. Using these active covariates, the reduced MILR model performs 89.13% classification accuracy.

# MILR-LASSO
milrSV <- milr(MUSK1$label, MUSK1[,3:nc], MUSK1$bag,
               lambda = -1, numLambda = 100,
               lambdaCriterion = "deviance", maxit = 16000)
sv_ind <- which(coef(milrSV)[-1] != 0) + 2
# show the detected active covariates
names(MUSK1)[sv_ind]

##  [1] "V31"  "V36"  "V37"  "V76"  "V83"  "V105" "V106" "V108" "V109" "V116"
## [11] "V118" "V124" "V126" "V129" "V132" "V136" "V147" "V162" "V163"

# use a very small lambda so that milr can do the estimation
# without evaluating the Hessian matrix
milrREFit <- milr(MUSK1$label, MUSK1[,sv_ind], MUSK1$bag,
                  lambda = 1e-7, maxit = 16000)
table(DATA = Y, FIT_MILR = fitted(milrREFit, type = "bag"))

##     FIT_MILR
## DATA  0  1
##    0 39  6
##    1  4 43

Following the discussion above, we use 10-fold cross validation and compare the prediction accuracy among four MIL models which are $s(0)$, $s(3)$, the MILR model with all covariates, and, the MILR model fitted by the selected covariates via MILR-LASSO. The resulting prediction accuracies are 83.70%, 77.17%, 75.00% and 81.52%, respectively.

predY <- matrix(0, length(Y), 4); colnames(predY) <- c("s0", "s3", "milr", "milr_sv")
set.seed(99)
folds <- 10; foldSize <- floor(length(Y)/folds)
foldBag <- c(rep(1:folds, foldSize), sample(1:folds, length(Y) - folds*foldSize))
foldBag <- sample(foldBag, length(foldBag))
foldIns <- rep(foldBag, table(MUSK1$bag))
for (i in 1:folds) {
  # prepare training and testing sets
  ind <- which(foldIns == i)
  training <- MUSK1[-ind,]; testing <- MUSK1[ind,]
  # train models
  fit_s0 <- softmax(training$label, training[,3:nc], training$bag,
                    alpha = 0, control = list(maxit = 25000))
  fit_s3 <- softmax(training$label, training[,3:nc], training$bag,
                    alpha = 3, control = list(maxit = 25000))
  # milr, use a very small lambda so that milr do the estimation
  #       without evaluating the Hessian matrix
  fit_milr <- milr(training$label, training[,3:nc], training$bag,
                   lambda = 1e-7, maxit = 16000)
  fit_milr_sv <- milr(training$label, training[,sv_ind], training$bag,
                      lambda = 1e-7, maxit = 16000)
  # store the predicted labels
  ind2 <- which(foldBag == i)
  # predict function returns bag response in default
  predY[ind2,1] <- predict(fit_s0, as.matrix(testing[,3:nc]), testing$bag)
  predY[ind2,2] <- predict(fit_s3, as.matrix(testing[,3:nc]), testing$bag)
  predY[ind2,3] <- predict(fit_milr, as.matrix(testing[,3:nc]), testing$bag)
  predY[ind2,4] <- predict(fit_milr_sv, as.matrix(testing[,sv_ind]), testing$bag)
}

table(DATA = Y, PRED_s0 = predY[,1])

##     PRED_s0
## DATA  0  1
##    0 36  9
##    1  6 41

table(DATA = Y, PRED_s3 = predY[,2])

##     PRED_s3
## DATA  0  1
##    0 28 17
##    1  4 43

table(DATA = Y, PRED_MILR = predY[,3])

##     PRED_MILR
## DATA  0  1
##    0 32 13
##    1 10 37

table(DATA = Y, PRED_MILR_SV = predY[,4])

##     PRED_MILR_SV
## DATA  0  1
##    0 35 10
##    1  7 40

5 Summary

This article introduces the usage of the R package milr for analyzing multiple-instance data under the framework of logistic regression. In particular, the package contains two approaches: summarizing the mean responses within each bag using the softmax function (Xu and Frank 2004; Ray and Craven 2005) and treating the instance-level statuses as hidden information as well as applying the EM algorithm for estimation (Chen et al. 2016). In addition, to estimate the MILR model, a lasso-type variable selection technique is incorporated into the latter approach. The limitations of the developed approaches are as follows. First, we ignore the potential dependency among instance statuses within one bag. Random effects can be incorporated into the proposed logistic regression to represent the dependency. Second, according to our preliminary simulation study, not shown in this paper, the maximum likelihood estimator might be biased when the number of instances in a bag is large, say, $m_i=100$ or more. Bias reduction methods, such as Firth (1993) and Quenouille (1956), can be applied to alleviate this bias. These attempts are deferred to our future work.