This study deals with the missing answers prediction problem. We address this problem using coupled analysis of ImageCLEF2014 dataset by representing it as a heterogeneous data, i.e., dataset in the form of matrices. We propose to use an approach based on probabilistic interpretation of tensor factorization models, i.e., Generalized Coupled Tensor Factorization, which can simultaneously fit a large class of matrix/tensor models to higher-order matrices/tensors with common latent factors using different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorization gives high prediction performance.
In this study, the task is to predict user expressed (UsE) features by using the
computer generated (CoG) features. UsE features are some predefined questions
related to liver, vessels, lesions and CoG features are the features which are
derived from image itself. Our aim is to fill in a pre-prepared form and
provide computer aided automatic annotation of liver CT volumes by using image
features [
We consider, for answer prediction, both KL-divergence and Euclidean
distancebased cost functions as well as coupled matrix factorization models [
GCTF framework is a generalization of the Probabilistic Latent Tensor
factorization (PLTF) [
X(v0)
X^ (v0) = X Y Z (v );
v0
where some of the factors are possibly fixed. Here, we define v as the set of all
indices in a model, v0 as the set of visible indices, v as the set of indices in
Z , and v = v v as the set of all indices not in Z and = 1; :::K as the
factor index. Since the product Q Z (v ) is collapsed over a set of indices, the
factorization is latent. The approximation problem is cast as an optimization
problem where we minimize the divergence d(X; X^ ), where d is a divergence
between the observed data X and model prediction X^ . In applications, d is
typically taken as Euclidean (EUC), Kullback-Leibler (KL) or Itakura-Saito (IS)
[
Here, we have two-dimensional datasets so we use nonnegative variant of the matrix factorization model. The matrix factorization model can be defined in the PLTF notation as follows. Given a matrix X, its factorization model is defined as:
X(i; j)
X^ (i; j) =
X Z1(i; r)Z2(j; r) r where the index sets V = fi; j; k; rg, V0 = fi; jg, V1 = fi; rg and V2 = fj; rg.
The update equation for non-negative generalized matrix/tensor factorization is expressed as:
Z
Z
(M
(M
where is the Hadamard product (element-wise product), M is a 0 1 mask
array with M (v0) = 1 (M (v0) = 0) if X(v0) is observed (missing). Here p
determines the cost function, i.e., p = f0; 1; 2g correspond to the -divergence
[
X A(v0) Y Z 0 (v 0 ) v 06= (A) is an object, the same size of Z , obtained simply by multiplying all factors other than the one being updated with an object of the order of the data. Hence the key observation is that the function is just computing a tensor product and collapses this product over indices not appearing in Z , which is algebraically equivalent to computing a marginal sum.
This update rule can be used iteratively for all non-negative Z and converges to a local minimum provided we start from some non-negative initial values. For updating Z , we need to compute the function twice for arguments A = M X^ p X and A = M X^ 1 p. It is easy to verify that update equations for the KL-NMF (non-negative matrix factorization) problem (for p = 1) are obtained as a special case of (3).
The Generalized Coupled Tensor Factorization (GCTF) [
X (v0; )
X^ (v0; ) = X Y Z (v )R ;
v0; R ; =
iofthXerwainsde Z connected : where = 1; :::j j and R is a coupling matrix that is defined as in (6). Note that, distinct from PLTF model, there are multiple visible index sets (v0; ) in the GCTF model, each specifying the attributes of the observed tensor X .
The inference, i.e., estimation of the shared latent factors Z , can be achieved
via iterative optimization (see [
Z
P
R ; P
R ; ; (M ; (M (5) (6) (7) (8)
3.1
Experimental Setting In this section, we address the missing answers (UsE features) prediction task on user expressed (UsE) features. UsE features are provided by a radiologist and represented by a 73 6 cell array. In our experiments, we used one column of that data which stands for annotation’s values. For the experiments, we choose and separate the questions into two groups. In the first group, we used the question set:
S1 = f3; 5; 7; 9; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28g and in the second group we used the question set:
S2 = f29; 30; 31; 38; 39; 40; 41; 44; 48; 60; 66; 68; 71g. The questions in the first
group have answers between f0 3g and the questions in the second group have
binary values f0 1g. ImageCLEF2014 (liver CT annotation task) [
X1 ques1 X2 ques2 ! ! e l p m a s
Z1 f eature x x e r u t a e f e r u t a e f
Z2 ques1 Z3 ques2
To predict the UsE features, we use Computer generated (CoG) features. These features are image-based features and they are generated by an interactive segmentation software. These features are stored in a 60 4 cell array. Likewise UsE features, we used feature’s value in our experiments. Here, we generate sample-features matrix Z1 of size 60 447. In matrix Z1, an entry Z1(i; j) indicates the value of the feature j for sample i.
Finally, we generate question-feature matrices Z2 of size 447 21 and Z3 of size 447 13 in order to complete the missing answers in X1 and X2 respectively.
The first 50 rows of X1 and X2 are observed and used for training while the rows from 51 to 60 are not observed and used for test. In order to fill in the missing entries in X1 and X2, we applied the coupled approach to a CPstyle tensor factorization model by analysing the tensor X1 with X2. They have samples mode in common and this gives us the following model: X^1(i; k) = Z1(i; j)Z2(j; k)
X^2(i; p) = Z1(i; j)Z3(j; p) (See Figure 1 for the visualization of the general structure of the model.)
After solving this model and obtain the values for latent factors, we construct the estimated matrices for X1 and X2. Here, our goal is to predict ordinal values for answers on a scale of f1; : : : Rg. We can reduce this prediction task to a set of binary threshold problems, namely, predicting r 1, r 2, . . . , r R. We try 1000 threshold values between 0 1, compute the accuracy of prediction for each value and choose the threshold value that gives the best accuracy performance. Threshold selection is not an easy task when we have different types of questions. For example, in the datasets there are several questions that their answers could have different values from all other questions. So, we have to find a unique value for that specific question. In this study, we ignore the questions with varied answers. That is why we have chosen these two sets of questions: to group the questions that have same type of answers is easier for this stage. 3.2
Results Evaluation: Completeness and accuracy of the prediction are the two main evaluation criteria of this study. The predicted annotations are compared to the manual annotations of the test dataset. To measure the completeness, the number of predicted features are divided by total number of features and to measure the accuracy, the number of correct predicted features divided by total number of predicted features. Finally, the total score is given in terms of the ratio of correct answers in separate groups.
In the previous run we have used KL-divergence to estimate the factors and non-optimal threshold values to relate the real-valued predictions to the discrete predictions. We just choose a one threshold value manually, then predict all answers by using that value. At the end, we obtained the results given in Table 2.
Then, we used the thresholding method that we described in the
Experimental Setting section. It determines the optimum value and we rerun the
experiments by using this method. Here, we used both Euclidean distance and
KL-divergence. Afterwards, we obtained the results with higher accuracy given
in Table 3. We can clearly see that, our new method outperforms the
previous one and the selection of the threshold significantly affects the prediction
accuracy.
In this study, we have studied answer prediction problem using coupled
analysis of imageCLEF2014 data [