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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

We formulate a multi-matrices factorization model (MMF) for the missing sensor data estimation problem. The estimation problem is adequately transformed into a matrix completion one. With MMF, an _{1}, _{2}, … , _{t}, where the entry, _{i,j}_{j}_{(1)}, _{(2)}, … , _{(}_{t}_{)}, and a probabilistic ^{d}^{×}^{t}

In this work, we study the following missing sensor data imputation problem: Let the matrix, ^{n}^{×}^{t}_{1}, _{2}, … _{n}_{1} < _{2} < … < _{t}_{i,j}_{i}_{j}_{i}_{j}_{i,j}

Many efforts have been devoted to the missing sensor data imputation problem. Typical examples include

Matrix factorization (MF), as a

We in this paper, we propose a multi-matrices factorization model (MMF), which can be outlined as follows. For a matrix, _{j}_{(1)}, _{(2)}, …, _{(t)} ∈ ℝ^{d}^{×}^{n}^{d}^{×}^{t}_{(}_{i}_{)} is referred to as the _{(}_{i}_{)}_{,j}_{j}_{i}_{j}_{j}_{(1)}, _{(2)}, …, _{(}_{t}_{)} (single sub-indexes of matrix mean columns) and _{i,j}

The remainder of the paper is organized as follows: Section 2 summarizes the notations used in the paper. Section 3 studies the related work on matrix factorization. In Section 4, we present our multi-matrices factorization model. The algorithm to solve the proposed model is outlined in Section 5. Section 6 is devoted to the experimental evaluations. Finally, our conclusions are presented in Section 7, followed by a presentation of future work, acknowledgments and a list of references.

For a vector _{1}_{2}_{n}^{n}_{0}, ‖_{1} and ‖_{2} to denote its 0-Norm, 1-Norm and 2-Norm, respectively, as follows:

For a matrix, ^{n}^{×}^{m}

The essence of the ^{T}V^{T}V^{T}V^{T}V^{2}) [

It is notable that for

As a transductive model, Model (2) has many nice mathematical properties, such as the generalization error bound [

In this section, we elaborate on our multi-matrices factorization (MMF) approach. Given the sensor data matrix, _{i,j}_{i}_{j}_{(1)}, _{(2)}, …, _{(}_{t}_{)} ∈ ℝ^{d}^{×}^{n}^{d}^{×}^{t}_{(}_{j}_{)} is regarded to be composed of the spatial features of all areas at _{j}_{j}_{(}_{j}_{)},_{i}, which corresponds to the spatial feature value of _{i}_{j}_{j}_{j}

Taking advantage of the knowledge of the probability graph model, we assume that the dependent structure of the data in _{(1)}, _{(2)}, …, _{(}_{t}_{)},

Columns of _{(}_{j}_{)} (1 ≤

_{(1),}_{i}

_{(}_{j}_{),}_{i} (1 ≤ _{U}_{U}

Moreover, for _{(j),}_{i}_{(j-1),}_{i}_{U}

The columns of _{V}_{V}

We also assume that, for

The (

Now, given _{U}_{V}_{R}_{U}_{V}_{(1)}, _{(2)}, …, _{(}_{t}_{)}}; below, we find a solution to the following equation:
_{U}_{V}_{R}_{U}_{V}_{U}_{V}_{R}_{U}_{V}

As a supplement, we have the following comments on Model (6):

_{(1)} and

_{j}_{−1} and _{j}_{j}_{j}_{−1}| → 0), then for any area, _{i}_{i,j}_{−1} and _{i,j}_{i,j}_{i,j}_{−1}| → 0). This can been enforced by tuning the parameters, γ and λ (see the following elaboration).

First of all, since for any ^{n}_{2} ≤ ‖_{1}, we have:
_{j}_{j}_{−1}| → 0, we can simply take _{j}_{j}_{−1}‖_{2} → 0.

On the other side, when |_{j}_{j}_{−1}| → ∞, as is well known, the values in _{j}_{j}_{−1} are regarded as being independent. In this case, we can take _{j}_{j−1}.

_{1} _{1} terms in _{2} terms (equivalently, use the Gaussian distribution instead of the Laplace distribution in Assumptions (III.) and (IV.)), e.g., replacing ‖_{j}_{j}_{−1}‖_{1} with
_{j}_{j}_{−1}‖_{2} can still be bounded via tuning the regularization parameters, _{1} norm here is two-fold: Firstly, as shown above, the ℓ_{1} terms can lead to the bounded difference norm, ‖_{j}_{j}_{−1}‖_{2}, and hence, the proposed model accommodates the ability to characterize the dynamics of the sensor data; secondly, according to the recent emerging works on compressed sensing [_{1} norm is similar to that of the ℓ_{0} norm. In terms of our model, this result indicates that the ℓ_{1} terms can restrict not only the magnitudes of the dynamics happening to the features, but also the number of features that changed in adjacent time points. In other words, with ℓ_{1} norms, our model gains more expressibility.

Below, we present the algorithm to solve Model (6). We denote:
_{(}_{j}_{),}_{i}_{j}

First, we introduce the signum function, _{1}, _{2}, …_{n}^{n}_{1}), _{2}), …, _{n}

Then, we calculate the partial subgradient of _{(}_{j}_{)}_{,i}

For _{j}_{,}_{i}_{,1} = _{(}_{j}_{),}_{i}_{(}_{j}_{−1),}_{i}

For _{j}_{,}_{i}_{,2} = –_{j}_{+1,}_{i}_{,1}.

Let _{1,}_{i}_{,1} = _{t,i,2}

For

For

Similarly, we calculate the partial subgradient of _{j}

For 2 ≤ _{j,}_{1} = _{j}_{j}_{−1}).

For 1 ≤ _{j}_{,2} = – G_{j+1,1}.

Let _{1,1} = _{t}_{,2} = 0, and we have:

Finally, with the results above, we present the solution algorithm in Algorithm 1.

In this section, we evaluate our approach through two large-sized datasets and compare the results with two state-of-the-art algorithms in terms of parametric sensitivity, convergence and missing data recovery performance. The following paragraphs describe the set-up, evaluation methodology and the results obtained. To simplify the parameter tuning, we set

Three state-of-the-art algorithms are selected for comparison to the proposed MMP model. The first one is the _{i,j}_{i,j}

The second algorithm is the probabilistic principle components analysis model (PPCA) [_{o}_{o}_{m}

The third algorithm is the probabilistic matrix factorization model (PMF) [_{i,j}

_{1},

_{2},

_{3}and

_{4}; regularization parameters,

_{(1),1},

_{(1),2}, …,

_{(1)}

_{,n}

_{1}∼

_{j}

_{j}

_{−1}+

_{(}

_{j}

_{),}

_{i}

_{(}

_{j}

_{-1)}

_{,i}

_{2}= inf;

_{2}–

_{1}| >

_{2}=

_{1};

_{(j),i}

_{j}s

_{1};

These three algorithms, as well as the proposed MMF, are employed to perform missing imputations for the incomplete matrix,

The testing protocol adopted here is the

Similar to many other missing imputation problems [_{1}, _{2}, … _{n}_{1}, _{2}, … _{n}_{i}_{i}

To conduct a synthetic validation of the studied approaches, we randomly draw a 100 × 10, 000 matrix _{1}, _{2}, …, _{n}_{i,j}_{i}_{j}_{i}_{,}_{j}_{i}_{j}_{−1} to _{j}_{i}_{j}_{−1} to _{j}

_{i}

_{,1}∼

_{i,j}

_{i,j}

_{i,j}

_{−1}+

_{i,j}R

_{i,j}

_{−1};

We first evaluate the sensitivity of the proposed algorithm to the regularization parameters, ^{n}^{n}

In ^{7} = 128), the RMSE still remains stable. A similar result also appears in the experiments on the parameter,

The second experiment we conduct is to study the prediction ability of the proposed algorithm, as well as that of the comparison algorithms. In the _{1} = _{2} = _{3} = _{4} = 0.01. All results are summarized in

As shown in

When compared with PMF, our algorithm also performs better in most of the settings: PMF achieves lower RMSEs than MMF only in two cases, in which

We also exam the convergence speed of the proposed algorithm. In the missing recovery experiments conducted above, for each

To evaluate the feasibility of the proposed approach on real-world applications, in this section, we conduct another experiment on a traffic speed dataset, which was collected in the urban road network of Zhuhai City [_{i,j}_{i,j}

We perform missing imputation on matrix _{1} = _{2} = _{3} = _{4} = 0.5. We summarize all results in

Missing estimation is one of the main concerns in current studies on sensor data-based applications. In this work, we formulate the estimation problem as a matrix completion one and present a multi-matrices factorization model to address it. In our model, each column, _{j}_{(}_{j}_{)}, and a temporal feature vector, _{j}_{j}_{j}

Reviewing the present work, it is notable that the proposed model only incorporates the temporal structure information, while the information on the spatial structure is disregarded, e.g., the data collected in two adjacent areas, _{k}_{l}

This work has been partially supported by National High-tech R&D Program (863 Program) of China under Grant 2012AA12A203.

The authors declare no conflict of interest.

_{1}Norm Factorization in the Presence of Outliers and Missing Data by Alternative Convex Programming

The structure assumptions.

Empirical studies on parameter sensitivity.

Empirical studies on convergence speed.

Recovery errors on the synthetic dataset (mean ± std).

10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | ||
---|---|---|---|---|---|---|---|---|---|---|

knn | 13.41 ± 0.62 | 6.80 ± 0.19 | 4.44 ± 0.06 | 3.12 ± 0.05 | 2.27 ± 0.02 | 1.81 ± 0.09 | 1.72 ± 0.01 | 1.69 ± 0.05 | 1.62 ± 0.06 | |

| ||||||||||

d = 10 | PPCA | 17.09 ± 2.08 | 20.24 ± 1.30 | 22.75 ± 1.32 | 23.96 ± 2.19 | 20.79 ± 1.00 | 16.67 ± 1.34 | 18.68 ± 0.81 | 11.98 ± 0.70 | 5.11 ± 3.10 |

PMF | 3.23 ± 0.23 | 3.33 ± 0.19 | 3.29 ± 0.12 | 3.34 ± 0.07 | 3.29 ± 0.09 | 1.69 ± 0.04 | 1.83 ± 0.02 | 1.81 ± 0.03 | 1.85 ± 0.04 | |

MMF | 3.07 ± 0.07 | 2.21 ± 0.10 | 2.14 ± 0.09 | 1.98 ± 0.06 | 1.93 ± 0.06 | 1.92 ± 0.04 | 1.75 ± 0.03 | 1.84 ± 0.02 | 1.80 ± 0.03 | |

| ||||||||||

d = 30 | PPCA | 19.65±3.01 | 22.48 ± 0.86 | 24.86 ± 1.54 | 23.99 ± 0.61 | 22.67 ± 0.49 | 20.22 ± 1.46 | 17.53 ± 0.70 | 14.23 ± 1.70 | 11.14 ± 1.72 |

PMF | 3.20 ± 0.21 | 3.32 ± 0.13 | 3.35 ± 0.09 | 3.36 ± 0.11 | 3.31 ± 0.07 | 1.78 ± 0.04 | 1.81 ± 0.02 | 1.79 ± 0.02 | 1.86 ± 0.11 | |

MMF | 3.06 ± 0.05 | 2.17 ± 0.08 | 2.07 ± 0.05 | 1.94 ± 0.03 | 1.74 ± 0.03 | 1.62 ± 0.02 | 1.64 ± 0.03 | 1.65 ± 0.01 | 1.69 ± 0.03 |

Recovery errors on the transportation dataset (mean ± std).

10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90% | |
---|---|---|---|---|---|---|---|---|---|

knn | 40.47 ± 0.02 | 31.79 ± 0.01 | 25.41 ± 0.02 | 21.35 ± 0.00 | 18.33 ± 0.00 | 15.89 ± 0.01 | 13.73 ± 0.01 | 11.67 ± 0.00 | 9.45 ± 0.02 |

PPCA | 17.90 ± 0.01 | 17.36 ± 0.01 | 13.00 ± 0.02 | 12.25 ± 0.01 | 11.47 ± 0.01 | 11.31 ± 0.03 | 11.19 ± 0.02 | 11.14 ± 0.04 | 11.16 ± 0.10 |

PMF | 14.41 ± 0.01 | 12.83 ± 0.03 | 12.43 ± 0.01 | 12.33 ± 0.02 | 12.36 ± 0.01 | 12.35 ± 0.01 | 12.13 ± 0.00 | 11.99 ± 0.03 | 11.96 ± 0.02 |

MMF | 11.79 ± 0.02 | 11.51 ± 0.01 | 11.43 ± 0.01 | 11.05 ± 0.02 | 11.05 ± 0.01 | 11.01 ± 0.00 | 10.83 ± 0.01 | 10.69 ± 0.01 | 10.70 ± 0.02 |