Estimate shared PCA quantities once

Summarizes an existing fit without new estimation or random sampling.

Usage

dp_pca(
  X,
  privacy,
  scree,
  score = list(),
  axes = c(1, 2),
  cpp.option = TRUE,
  non_private = FALSE,
  V_dp = NULL,
  group = NULL
)

# S3 method for class 'dp_pca'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

# S3 method for class 'dp_pca'
summary(object, ...)

# S3 method for class 'summary.dp_pca'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

X: A numeric matrix or data frame with observations in rows and variables in columns. Non-numeric data-frame columns are excluded with a warning; numeric values must be finite. At least two observations and two numeric features are required. If directions must be estimated, the pairwise-separation condition of dp_loading() also applies.
privacy: A privacy_control() object allocating budget to loading, scree, and score (equal thirds by default).
scree: Required named list with k = NULL (all components), method = "clipped", control (required), and mono = TRUE. k, when supplied, must be between 2 and the number of numeric features. Use clipped_control(), pmwm_control(), or huber_control() for tuning.
score: Named list specifying method = "sparse" or "add" and bins = c(20, 20) by default. Exactly one histogram method is used.
axes: Two distinct PC indices between 1 and the fitted k (c(1, 2) by default).
cpp.option: Whether to use the compiled spherical-Kendall implementation (TRUE by default) when estimating private directions; ignored if V_dp is supplied.
non_private: Whether to compute and store ordinary PCA reference results (FALSE by default).
V_dp: Optional p by p private PC direction matrix (NULL by default), where p is the number of numeric PCA features. Columns must be orthonormal and rows must follow the feature order in X; matching row names are required when both sets of names exist.
group: Optional group column name in X or vector of group labels (NULL by default). A selected group column is excluded from PCA features, even when numeric; group labels and membership are assumed public.
x: A summary.dp_pca object.
digits: Significant digits used when printing tables.
...: Additional arguments. The summary calculation takes no extra options; print methods pass formatting arguments to the printed tables.
object: A dp_pca() result.

Value

A dp_pca object containing private directions, scree, pve, hist, frame, and optional score_groups. nonprivate contains the optional ordinary PCA reference, or NULL when non_private = FALSE. metadata contains data dimensions, privacy allocations, methods, controls, grouping information and axes. The print method returns the fitted object invisibly.

A summary.dp_pca object. importance contains DP eigenvalues, PVE and cumulative PVE, with a column for each estimated component. budget contains the allocated eps and delta for loading, scree, score frame and score histogram, plus the total. Other fields describe data dimensions, selected axes, methods and whether a non-private reference is stored. Raw scores and loading matrices are not copied into the summary. The print method returns its input invisibly.

Details

A single private PC direction matrix is estimated or supplied and reused for scree and score estimation. Scree values estimate variance along those directions using disjoint paired differences; an odd final observation is unused for scree, while the score histogram uses all rows. PVE is normalized over the retained k components, not necessarily all original variables.

The selected scree method uses the entire scree allocation. Of the score epsilon budget, 35% goes to the private frame and 65% to the histogram; all score delta goes to the histogram. With supplied V_dp, the loading budget records its previous release rather than being reassigned.

Grouped data use one direction estimate and private frame. Their disjoint histograms use parallel composition when group labels and membership are public and fixed. The pooled histogram is post-processed from the grouped histograms.

Input is not automatically centered or standardized. Data-dependent preprocessing needs separate privacy accounting, and sample size, schema, selected components, and tuning settings are assumed public. The loading conditions in dp_loading() also apply. The optional non-private reference uses sample-mean-centered ordinary PCA scores and is not private.

PVE is normalized over the k retained components; a total of zero produces zero PVE. When k is smaller than the number of variables, a cumulative PVE of one does not describe all original variation.

The budget table reports allocated budgets, not cumulative privacy spending. With supplied V_dp, the loading allocation records its prior cost. The optional non-private comparison is excluded from the importance table.

Examples

data(gau, package = "dppca")
set.seed(123)
fit <- dp_pca(
  gau[1:200, ],
  privacy = privacy_control(eps = 5, delta = 1e-4),
  scree = list(k = 2, control = clipped_control(C_clip = 3)),
  score = list(method = "sparse", bins = c(10, 10)),
  non_private = FALSE
)
fit
#> Differentially Private PCA
#> Data: 200 observations x 20 variables
#> Components: 20 PC directions; 2 scree estimates
#> Score axes: PC1, PC2 
#> Methods: scree = clipped ; score = sparse 
#> Total privacy budget: eps = 5 ; delta = 1e-04 
#> Non-private reference: not stored 
summary(fit)
#> DP PCA Summary
#> Data: 200 observations x 20 variables
#> Components: 20 PC directions; 2 scree estimates
#> Score axes: PC1, PC2 
#> Methods: scree = clipped ; score = sparse 
#> 
#> DP component importance:
#>                   PC1    PC2
#> Eigenvalue     0.9521 0.8882
#> PVE            0.5174 0.4826
#> Cumulative PVE 0.5174 1.0000
#> 
#> Proportions are normalized over 2 estimated components.
#> 
#> Allocated privacy budget:
#>                    eps     delta
#> Loading         1.6667 3.333e-05
#> Scree           1.6667 3.333e-05
#> Score frame     0.5833 0.000e+00
#> Score histogram 1.0833 3.333e-05
#> Total           5.0000 1.000e-04
#> 
#> Non-private reference: not stored 
fit$directions
#>             PC1          PC2         PC3         PC4          PC5         PC6
#> V1   0.19413754  0.257797730  0.04284186  0.04243370 -0.180986738 -0.01031595
#> V2   0.11370700 -0.316668014 -0.09719936 -0.04104061 -0.051496659  0.39437698
#> V3   0.03944075 -0.008402996 -0.56728045 -0.18160434  0.263973413 -0.10997240
#> V4  -0.10639338  0.187437548 -0.07491462 -0.03641717  0.127752997 -0.11420546
#> V5  -0.10669240 -0.216073257 -0.08186225  0.31926062  0.136641992  0.05545757
#> V6  -0.32776539  0.177324372 -0.22849591  0.37152230  0.377287804 -0.33032518
#> V7  -0.22221606 -0.277439785 -0.07938759 -0.23823756  0.221347153  0.42131868
#> V8   0.40209813 -0.120061102  0.04825535  0.23888581  0.027810689  0.07064820
#> V9   0.17523776  0.167307252  0.21189517 -0.34755632  0.208195520  0.04333997
#> V10 -0.10742646 -0.232271653 -0.15039194 -0.01458754 -0.407721944 -0.28317202
#> V11 -0.34598251  0.361965776  0.06602439  0.07273690 -0.339745625  0.08800841
#> V12  0.01168949 -0.045508426 -0.12241860  0.45072498 -0.208720218  0.34877728
#> V13 -0.30590591 -0.015398251  0.03620102 -0.15858824  0.304988304  0.23591729
#> V14  0.30581684  0.041770263  0.10862067 -0.26894362  0.003417028 -0.21134391
#> V15  0.21157477  0.101964588 -0.26594436  0.09882708 -0.100652539  0.10144672
#> V16  0.06024710  0.409644962 -0.50981428 -0.24922919 -0.160088582  0.18729867
#> V17  0.22213938  0.209131165 -0.20229657  0.09581129  0.054359938  0.26663251
#> V18 -0.09569474  0.175655433  0.20587776 -0.07235810  0.024401079  0.15433075
#> V19 -0.22197050 -0.335419405 -0.22713018 -0.30324872 -0.367207461 -0.14536093
#> V20 -0.31823010  0.214994429  0.16807229 -0.05691178 -0.179024144  0.22658499
#>             PC7          PC8          PC9        PC10         PC11
#> V1   0.09632647  0.007459308  0.350744312  0.20240800 -0.201319529
#> V2  -0.12377897  0.280058004 -0.040612595  0.19382301  0.215128480
#> V3   0.47880682 -0.303460936 -0.003158392  0.23080015 -0.072262162
#> V4  -0.11416819 -0.081030328 -0.605096536  0.23796653  0.317889232
#> V5  -0.29629347 -0.430493133  0.294142472  0.06977654  0.030503769
#> V6  -0.20321276  0.324189049  0.082609141 -0.07527589  0.113296170
#> V7   0.05518077  0.170793053  0.024183483 -0.15224224  0.306194295
#> V8   0.05514612  0.055491412  0.050065573 -0.06722717  0.047614894
#> V9  -0.13567729 -0.160172535  0.019129946  0.09263512 -0.036758138
#> V10 -0.32172285 -0.173415784  0.037922744 -0.15623127  0.123438781
#> V11  0.16808379  0.127570589  0.289880775  0.39707401  0.365736929
#> V12  0.06171612 -0.418127570 -0.175217132  0.05030340  0.105754115
#> V13 -0.10063612 -0.178302301  0.392304244  0.12230237 -0.093999094
#> V14 -0.18187376 -0.218666906  0.157812928  0.13769438  0.579040921
#> V15  0.24438094  0.260985905  0.151170607 -0.21658568  0.221938230
#> V16 -0.28177015 -0.099159609  0.077838612 -0.41699846  0.002004277
#> V17 -0.44047980  0.182621152 -0.141340211  0.44038140 -0.261584168
#> V18 -0.18467804  0.004382302 -0.003231587 -0.22651652 -0.070646563
#> V19 -0.09939377  0.156126454 -0.008815188  0.25635145 -0.245253860
#> V20  0.15373256 -0.195072878 -0.261575307 -0.12029344 -0.092946029
#>              PC12         PC13         PC14        PC15        PC16
#> V1   0.5668866313 -0.133404971  0.291934422 -0.14715535  0.19143711
#> V2   0.1644770016  0.274694819 -0.408814730 -0.03394769  0.03553233
#> V3   0.0638320743  0.140988741 -0.158401571  0.03276296 -0.16858380
#> V4   0.1488432168 -0.396052426  0.002116551 -0.28906063  0.02440537
#> V5  -0.1027713119  0.010176414  0.169597791 -0.45614443 -0.38369387
#> V6   0.2450786055  0.171084732  0.101245882  0.33574991 -0.04636009
#> V7   0.1068348319 -0.008201181  0.519647191 -0.02103049  0.01197135
#> V8   0.1207780823 -0.135096158 -0.085643501  0.30125016 -0.51703790
#> V9  -0.2295190710  0.172079127  0.363853834  0.23104333 -0.05663727
#> V10 -0.1427755026 -0.166584923 -0.041949359  0.21758481  0.08933684
#> V11 -0.2438775116  0.176901537 -0.054447621 -0.05645084 -0.14579442
#> V12  0.1889509152  0.081269740  0.148423686  0.28233633  0.32575583
#> V13  0.0008207009 -0.489246685 -0.345085124  0.26635183  0.21695517
#> V14  0.2337933205  0.012856402 -0.030903125  0.20054413 -0.07060153
#> V15 -0.1986294157 -0.538298803  0.103939033 -0.04113782 -0.10299726
#> V16  0.0124837317  0.181357302 -0.063132469 -0.07781205  0.02763371
#> V17 -0.2070058692 -0.107501508  0.083746160  0.13315009 -0.08387004
#> V18  0.4005474829 -0.014582246 -0.262801967 -0.22127368 -0.25428692
#> V19  0.2377216740 -0.097660632  0.188553016  0.08245285 -0.23984546
#> V20  0.0801137219 -0.059729093  0.023776348  0.32334395 -0.43014143
#>             PC17          PC18        PC19        PC20
#> V1  -0.051573915 -0.3612887358 -0.05991851  0.16134489
#> V2   0.138171570 -0.2550964360 -0.37908268  0.19570525
#> V3  -0.060787146  0.0195494377  0.12624319  0.28881436
#> V4   0.234888522 -0.2205648977  0.04429999 -0.06978389
#> V5  -0.003649037 -0.0535827789 -0.21890492 -0.02447647
#> V6   0.076629919  0.0003913972 -0.15165340  0.06612594
#> V7  -0.202576186 -0.0832092080  0.31726989  0.06025109
#> V8   0.245725091 -0.2790186075  0.39537545 -0.22477158
#> V9   0.549308047 -0.0390164354 -0.16180677  0.27442248
#> V10 -0.039556598 -0.2793965657  0.21405006  0.50983260
#> V11  0.133064460 -0.0065398413  0.24584230 -0.01004403
#> V12  0.205579269  0.3122324201  0.02830726 -0.02583897
#> V13  0.102272795 -0.0507838755  0.01095784 -0.14769852
#> V14 -0.333010186  0.2588696433 -0.13103962 -0.14076626
#> V15  0.112520781  0.2557827008 -0.34939817  0.21114325
#> V16  0.094282927 -0.1400452836  0.02538594 -0.32195027
#> V17 -0.365833358  0.1322774231  0.13939097  0.12692062
#> V18  0.128056622  0.4383242287  0.25771979  0.42335664
#> V19  0.247419590  0.2822400236 -0.08729676 -0.25314761
#> V20 -0.307327055 -0.2160725816 -0.37440964  0.05447205
fit$nonprivate # NULL by default
#> NULL
fit$metadata$privacy
#> $total
#>   eps delta 
#> 5e+00 1e-04 
#> 
#> $budgets
#> $budgets$loading
#>          eps        delta 
#> 1.666667e+00 3.333333e-05 
#> 
#> $budgets$scree
#>          eps        delta 
#> 1.666667e+00 3.333333e-05 
#> 
#> $budgets$score
#>          eps        delta 
#> 1.666667e+00 3.333333e-05 
#> 
#> 
#> attr(,"class")
#> [1] "dppca_privacy_control"

set.seed(123)
plots <- plot(fit) # DP panels only
#> Warning: The private histogram contains no positive probability mass; no synthetic score points were generated.

plots$private$plots$biplot