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It depends on what exactly your weights apply to.

###Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\frac{1}{\sum w_i}\mathbf X^\top \mathbf W \mathbf X$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.

It depends on what exactly your weights apply to.

Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\frac{1}{\sum w_i}\mathbf X^\top \mathbf W \mathbf X$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.

It depends on what exactly your weights apply to.

###Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\frac{1}{\sum w_i}\mathbf X^\top \mathbf W \mathbf X$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.

It depends on what exactly your weights apply to.

###Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\frac{1}{\sum w_i}\mathbf X^\top \mathbf W \mathbf X$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.

It depends on what exactly your weights apply to.

###Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\mathbf X^\top \mathbf W \mathbf X/(n-1)$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

This is equivalent to multiplying each row of the appropriately centered data matrix by the corresponding $\sqrt{w_i}$ and proceeding with the standard PCA, because $\mathbf X^\top \mathbf W \mathbf X/(n-1)$ is the covariance matrix of $\mathbf W^{1/2} \mathbf X$.

Note that this is conceptually related to rescaling the variables (e.g. standardizing them), when one multiplies each column of the data matrix by a certain value.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.

It depends on what exactly your weights apply to.

###Row weights

Let $\mathbf{X}$ be the data matrix with variables in columns and $n$ observations $\mathbf x_i$ in rows. If each observation has an associated weight $w_i$, then it is indeed straightforward to incorporate these weights into PCA.

First, one needs to compute the weighted mean $\boldsymbol \mu = \frac{1}{\sum w_i}\sum w_i \mathbf x_i$ and subtract it from the data in order to center it.

Then we compute the weighted covariance matrix $\frac{1}{\sum w_i}\mathbf X^\top \mathbf W \mathbf X$, where $\mathbf W = \operatorname{diag}(w_i)$ is the diagonal matrix of weights, and apply standard PCA to analyze it.

Cell weights

The paper by Tamuz et al., 2013, that you found, considers a more complicated case when different weights $w_{ij}$ are applied to each element of the data matrix. Then indeed there is no analytical solution and one has to use an iterative method. Note that, as acknowledged by the authors, they reinvented the wheel, as such general weights have certainly been considered before, e.g. in Gabriel and Zamir, 1979, Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights. This was also discussed here.

As an additional remark: if the weights $w_{ij}$ vary with both variables and observations, but are symmetric, so that $w_{ij}=w_{ji}$, then analytic solution is possible again, see Koren and Carmel, 2004, Robust Linear Dimensionality Reduction.