Learning from streaming data with unsupervised heterogeneous domain adaptation

Abstract

Many efforts on domain adaptation focus on stationary environments and assume that the target domain samples are available before the learning process. However, real-world applications frequently involve the availability of non-stationary data sequentially. This study develops an unsupervised heterogeneous domain adaptation approach to address non-stationary scenarios where data streams continually feed the learning model. This process employs a fuzzy-based model that has been trained on a different but related domain. Subsequently, a neighborhood-based weight assignment fine-tunes the attraction and repulsion between neighbors based on prior knowledge about their domains and the similarity between class labels. To avoid unnecessary adaptation for each target domain chunk, domain adaptation is triggered only when concept drift is detected. This way, the model gradually adjusts to the evolving data, incorporating the unique characteristics of the new domain. When no drift is detected, existing parameters are reused for feature adaptation. At the end, the source domain is updated by incorporating the drifting data and their predicted labels. The proposed method offers several advantages, including avoidance of excessive alignment, reduction in domain adaptation cost, and a gradual reduction in dependency on the source domain for domain adaptation. To evaluate the method’s performance, experiments were conducted on several tasks extracted from two benchmark datasets, considering different types of concept drift. The experimental results demonstrate that the proposed model significantly improves classification accuracy while reducing computational time.

Appendix I

This section contains some detailed information for computing the metric $\mathcal{D}$ introduced by [17]. For each fuzzy feature vector ${\overline{A} }_{i}\left({a}_{i1}\text{,}{a}_{i2}\text{,}\dots \text{,}{a}_{in}\right)\in F({\mathbb{R}}^{n})$. For each ${\overline{a} }_{ij}\in F({\mathbb{R}})$, its membership function is computed by Eq. (I-1).

$${\mu }_{\mathrm{ij}}(x|{\overline{a} }_{\mathrm{ij}})=\left\{\begin{array}{c}0 \forall x<{a}_{\mathrm{ij}}-{\rho }_{i} \\ 1-\frac{{\Vert x-{a}_{\mathrm{ij}}\Vert }_{1}}{{\rho }_{i}} \forall {\Vert x-{a}_{\mathrm{ij}}\Vert }_{1}\le {\rho }_{i}\text{,} \, x\in R \\ 0 \forall x>{a}_{\mathrm{ij}}+{\rho }_{i}\end{array}\right.$$

(I-1)

where ${a}_{ij}$ indicates the $j$ th feature value of the $i$ th sample and ${\rho }_{i}$ shows the hesitation degree of the $i$ th sample considering a triangular membership function. Utilizing Eq. (I-2), ${\mu }_{\mathrm{ij}}({\varvec{x}}|{\overline{A} }_{i})$ where ${\varvec{x}}=({x}_{1}\text{,}{ x}_{2}\text{,}\dots \text{, }{x}_{n})$ is obtained.

$$ \mu_{{{\text{ij}}}} ({\varvec{x}}|\overline{A}_{i} ) = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if}} \;\; \exists x_{j} {,} x_{j} < a_{{{\text{ij}}}} - \rho_{i} } \\ {1 - \frac{{{\varvec{x}} - a_{{{\text{ij1}}}} }}{{n\rho_{i} }}} & {{\text{if}} \;\;\forall x_{j} {,} x - a_{{{\text{ij1}}}} \le \rho_{i} {,} x \in {\mathbb{R}}^{n} } \\ 0 & {{\text{if}}\;\; \exists x_{j} {,} x_{j} > a_{{{\text{ij}}}} + \rho_{i} } \\ \end{array} } \right. $$

(I-2)

To define the fuzzy relation between two heterogeneous domains (source and target), the following metric is defined to measure the distance between the fuzzy vectors.

$$\begin{aligned} {\mathcal{D}}\left( {\bar{A}_{i} ,\bar{A}_{j} } \right) &= \frac{1}{n}\mathop \smallint \limits_{0}^{1} \sup \left\{ {{\mathcal{D}}_{\lambda } \left( {u,v} \right):~{\mathcal{D}}_{\lambda } \left( {u,v} \right) \in \Omega \left( \lambda \right)} \right\}{\text{d}}\lambda ;\\ &\quad\quad u \in \bar{A}_{i} \left( \lambda \right),~v \in \bar{A}_{j} \left( \lambda \right)\end{aligned} $$

(I-3)

where $\lambda$ is the membership value, and ${\mathcal{D}}_{\lambda }\left(u\text{,}v\right)$ indicates the distance between points $u$ and $v$ in ${\mathbb{R}}^{n}$ with the given $\lambda$. $\Omega \left(\lambda \right)$ is computed by Eq. (I-4).

$$ \begin{gathered} \Omega \left( \lambda \right) = \left\{ {d\left( {u,\overline{A}_{j} \left( \lambda \right)} \right)} \right\} \cup \left\{ {d\left( {v,\overline{A}_{i} \left( \lambda \right)} \right)} \right\} \hfill \\ d\left( {u,\overline{A}_{j} \left( \lambda \right)} \right) = \min \left\{ {d\left( {u,v} \right), v \in \overline{A}_{j} \left( \lambda \right)} \right\} \hfill \\ d\left( {v,\overline{A}_{i} \left( \lambda \right)} \right) = \min \left\{ {d\left( {v,u} \right), u \in \overline{A}_{i} \left( \lambda \right)} \right\} \hfill \\ \end{gathered} $$

(I-4)

where $d\left(v\text{,}u\right)$ is the ${l}_{1}$-norm between two n-dimensional vectors ($u$ and $v$). Note that the supremum operator (sup) in Eq. (I-3) indicates the longest distance between the fuzzy vector of one specific domain and the fuzzy set of another domain. Eq. (I-3) can be rewritten as Eq. (I-5).

$$\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\frac{1}{n}\underset{0}{\overset{1}{\int }}d\left({A}_{i}\text{,}{A}_{j}\right)+\frac{1}{2}\left|d\left(u\text{,}{\overline{A} }_{j}\left(\lambda \right)\right)+d(v\text{,}{\overline{A} }_{i}(\lambda ))\right|\mathrm{d}\lambda $$

(I-5)

The above equation is de-fuzzified regarding Eqs. (I-1) and (I-2) as follows.

$$ \begin{aligned} &{\mathcal{D}}\left( {\overline{A}_{i} ,\overline{A}_{j} } \right) = \frac{1}{n}d\left( {A_{i} ,A_{j} } \right) + \frac{1}{2}\mathop \smallint \limits_{0}^{1} \left| {\left( {1 - \lambda } \right)\rho_{i} - \left( {1 - \lambda } \right)\rho_{j} } \right|{\text{d}}\lambda \\ & = \frac{1}{n}d\left( {A_{i} ,A_{j} } \right) + \frac{1}{2}\rho_{i} - \rho_{j1} \mathop \smallint \limits_{0}^{1} \left( {1 - \lambda } \right){\text{d}}\lambda \\ &= \frac{1}{n}d\left( {A_{i} ,A_{j} } \right) + \frac{1}{4}\rho_{i} - \rho_{j1} \\ \end{aligned} $$

(I-6)

Eq. (I-6) cannot be used directly for computing the fuzzy relation because it does not satisfy two properties of the fuzzy relation, including (1) symmetry, a condition in which $\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=\mathcal{D}\left({\overline{A} }_{j}\text{,}{\overline{A} }_{i}\right)\text{, }\forall {\overline{A} }_{i}\text{, }{\overline{A} }_{j}$ and (2) reflexivity, a condition in which $\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)=1\text{,} \forall {\overline{A} }_{i}$. Thus, the following function is employed.

$${R}_{\mathcal{D}}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)={e}^{-\frac{\mathcal{D}\left({\overline{A} }_{i}\text{,}{\overline{A} }_{j}\right)}{2{\sigma }^{2}}}$$

(I-7)