Algorithm 1: Private and robust estimator for (m_2) | Algorithm 2: Unbounded DP upper-quantile estimator | Algorithm 3: Additive DP histogram | Algorithm 4: Sparse DP histogram | Algorithm 5: Group-wise additive DP histogram | Algorithm 6: Group-wise sparse DP histogram | References

1. DP eigenvalue estimation | $$(\widetilde{\lambda}_1,\ldots,\widetilde{\lambda}_k) | 2. Clipped mean | 3. Private modified winsorized mean (PMWM) | 3.1. Allocation of pair rows | 3.2. Private quantiles and cutoff bounds | $$\zeta | $$L_\ell^ | \mathcal Q_ | $$L_\ell | \min{b,\max{a,L_\ell^{\mathrm{priv}}}},\qquadU_\ell | 3.3. Winsorized mean and Gaussian perturbation | 3.4. Privacy allocation and tuning choices | $$(\varepsilon_M,\delta_M) | 4. Huber mean | 4.1. Huber loss and score | 4.2. Private scale and Huber thresholds | $$\varepsilon_ | \frac | $$\tau_\ell | 4.3. Noisy gradient descent | 5. Nonnegative scree and PVE | 6. DP scree/PVE plots in dppca | References

1. Spherical Kendall matrix and PC directions | Translation invariance | 2. DP PC directions | 3. DP loading plots in dppca | References

1. Two-dimensional PC scores | 2. Private plotting frame | 2.1. Private coordinate medians | 2.2. Private radial quantile and square frame | 3. Histogram grid and sensitivity | 4. Private histogram estimators | 4.1. Additive Gaussian histogram | 4.2. Stability-based sparse histogram | 5. Privacy budget allocation | 6. Synthetic score sampling | 7. Group-wise private score histograms | 8.1. Display centering | $$c_\ell^ | References

1. What is estimated privately? | 2. Privacy budget and shared estimation | 3. Five DP PCA Visualization | 3.1. DP variable plot (correlation-style coordinates) | $$\operatorname{Cor}(X_{\cdot j},S_{\cdot\ell}) | $$\widetilde{\Sigma}_k | $$\widetilde{\sigma}_{j,k}^2 | (\widetilde | $$\widetilde | $$R_j^ | $$|R_j^{\mathrm{DP}}|_2^2 | \widetilde | 3.2. DP biplot | $$\widetilde{d}_\ell | $$G_{i\ell}^{\mathrm{DP}}(\alpha) | \frac{s_{i\ell}^*-c_\ell^{\mathrm{disp}}}{\widetilde{d}\ell^\alpha},\qquadH{j\ell}^{\mathrm{DP}}(\alpha) | For the two selected components, the rows $G_i^{\mathrm{DP}}(\alpha) | \left(G_{ia}^{\mathrm{DP}}(\alpha),G_{ib}^{\mathrm{DP}}(\alpha)\right)$represent synthetic observations, while $H_j^{\mathrm{DP}}(\alpha) | Let$$r_G | Q_{0.9}\left(\left{\left|G_i^{\mathrm{DP}}(\alpha) | $$r_H | $$\gamma | 4. Summary | References and further reading

1. Load the example data | 2. Set the privacy budget and estimate PC directions | 3. Inspect the result | 4. Plot the result | 4.1. Vector plot | 4.2. Point and heatmap plots | 5. Select PCs and variables | 6. Customize the appearance

What you will make | 1. Load the vowel data | 2. Create the privacy allocation | 3. Fit DP PCA analysis | Main arguments of dp_pca() | 4. Inspect the results | 5. Create all five displays | 6. Reuse and export the results

1. Load the example data | 2. Set the privacy budget and estimate a score histogram | 2.1. Additive and sparse methods | 3. Inspect and plot the result | 3.1. Read the fit and stored estimates | 3.2. Show a private histogram | 3.3. Draw synthetic score points | 4. Customize the plot | 5. Group-wise private score histograms | 5.1. Fit a private histogram for each group | 5.2. Plot a group histogram overlay

1. Load the example data | 2. Set the privacy budget and estimate scree values | 3. Inspect and plot the result | 4. Choose a method and its tuning parameters | Clipped mean | Private modified winsorized mean (PMWM) | Huber mean | 5. Compare multiple methods and account for releases | 6. Reusing Private PC Directions | 7. Customize the plot

Choose a starting point | 1. Install and load the package | 2. Prepare your input data | 3. Explore the Example Datasets | 3.1. Gaussian data | 3.2. Vowel data | 4. Set and Allocate the Privacy Budget | 4.1. Default privacy budget allocation | 4.2. Customize the budget allocation | 4.3. Specify individual component budgets | 5. Next steps

1. Available analyses | 2. Getting started | 3. Data and comparison options | 4. Run the app locally

1. Introduction | 2. Population PCA | Let $\xi\in\mathbb{R}^p$ be a random vector with mean $\mu=E[\xi]$ and covariance matrix $\Sigma | 3. Sample PCA | [\widehat \Sigma | 3.1 Directions | [\widehat \Sigma \widehat v_\ell | 3.2 Eigenvalues | The eigenvalues of $\widehat \Sigma$ represent the sample variances along the corresponding PC directions. In particular,[\widehat\lambda_\ell | [\operatorname{PVE}_\ell | 3.3 Scores | where the $\ell$th PC score of observation $i$ is $s_ | 4. PCA visualizations | 4.1 Loading plot | 4.2 Scree plot | 4.3 Score plot | 4.4 Variable plot (correlation circle) | A variable plot displays the correlations between the original variables and the PC scores. Let (\widehat \sigma_j=\sqrt{\widehat \Sigma_{jj}}) denote the sample standard deviation of variable (j). For (\widehat \sigma>0) and (\widehat\lambda_\ell>0), the sample correlation between variable (j) and the (\ell)th PC scores is[\operatorname{Cor}(X_{\cdot j},S_{\cdot\ell}) | Since $|R_j|_2^2 | 4.5 Biplot | Summary of PCA visualizations | References