KixSrSSKrSSKJrJr SSKrSSKJr SSK J r J r SSK J r SSKJr SSKJrJr SS KJrJrJr SS KJrJrJrJr SS KJrJr SS KJ r S"Sjr!\"S/S/\"\SSSS9/SS/\"\SSSS9\"S15/\S/\"\SSSS9/S/\"\SSSS9/S/S/S\"S15/S. S S9S SSSSSSSSSSS. Sj5r""S S!\ 5r#g)#z" Multi-dimensional Scaling (MDS). N)IntegralReal)effective_n_jobs) BaseEstimator _fit_context)IsotonicRegression) ClassicalMDS)euclidean_distancespairwise_distances) check_arraycheck_random_statecheck_symmetric)HiddenInterval StrOptionsvalidate_params)Paralleldelayed) validate_dataT,ư>Fc [USS9nURSn [U5nS[R"U 5- U-R 5n XS:gn Uc$UR X-S9n U RX45n O2URSnXRS:wa[SX4-5eUn [U 5n [SS 9nSn[U5GHnU(aUnOU R 5nUU S:gnUS:aU nOURU U5n[R"U5nUUU S:g'URX45nU[R"XS- -S - US -R5- 5-nUUR -nS XS:H'UU - nU*nU[R""[%U55[R""[%U554==URSS 9- ss'S U - [R&"UU 5-n [U 5n U R 5UR 5- S -R5S - nUS :a[)SUSUS35 UbHU R 5S -R5nUU- US - - U:aU(a[)SUS35 OUnGM U(a=U R 5S -R5n[R"WUS - - 5nU WWS-4$)a Computes multidimensional scaling using SMACOF algorithm. Parameters ---------- dissimilarities : ndarray of shape (n_samples, n_samples) Pairwise dissimilarities between the points. Must be symmetric. metric : bool, default=True Compute metric or nonmetric SMACOF algorithm. When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as missing values. n_components : int, default=2 Number of dimensions in which to immerse the dissimilarities. If an ``init`` array is provided, this option is overridden and the shape of ``init`` is used to determine the dimensionality of the embedding space. init : ndarray of shape (n_samples, n_components), default=None Starting configuration of the embedding to initialize the algorithm. By default, the algorithm is initialized with a randomly chosen array. max_iter : int, default=300 Maximum number of iterations of the SMACOF algorithm for a single run. verbose : int, default=0 Level of verbosity. eps : float, default=1e-6 The tolerance with respect to stress (normalized by the sum of squared embedding distances) at which to declare convergence. .. versionchanged:: 1.7 The default value for `eps` has changed from 1e-3 to 1e-6, as a result of a bugfix in the computation of the convergence criterion. random_state : int, RandomState instance or None, default=None Determines the random number generator used to initialize the centers. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. normalized_stress : bool, default=False Whether to return normalized stress value (Stress-1) instead of raw stress. .. versionadded:: 1.2 .. versionchanged:: 1.7 Normalized stress is now supported for metric MDS as well. Returns ------- X : ndarray of shape (n_samples, n_components) Coordinates of the points in a ``n_components``-space. stress : float The final value of the stress (sum of squared distance of the disparities and the distances for all constrained points). If `normalized_stress=True`, returns Stress-1. A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good, 0.1 fair, and 0.2 poor [1]_. n_iter : int The number of iterations corresponding to the best stress. References ---------- .. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J. Psychometrika, 29 (1964) .. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis" Kruskal, J. Psychometrika, 29, (1964) .. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.; Groenen P. Springer Series in Statistics (1997) Traise_exceptionrNsizez'init matrix should be of shape (%d, %d)clip) out_of_boundsrgh㈵>)axisg?z Iteration z , stress z.4fz)Convergence criterion reached (iteration z).)rshaper nptriraveluniformreshape ValueErrorr rrange fit_transform zeros_likesqrtsumTarangelendotprint)dissimilaritiesmetric n_componentsinitmax_iterverboseeps random_statenormalized_stress n_samplesdissimilarities_flatdissimilarities_flat_wX distancesir old_stressit disparitiesdistances_flatdistances_flat_wdisparities_flatratioBstresssum_squared_distancess W/var/www/html/dynamic-report/venv/lib/python3.13/site-packages/sklearn/manifold/_mds.py_smacof_singlerMsn&otLO%%a(I%l3L !22oELLN1!2KL |  i&> ? IIy/ 0zz!}  1 %9Y)BCsJEbTLM ]`!*!2a!7 < < >#81#<=> fb1f z array-likebooleanrleftclosedwarnr8r:auto) r3r4r5r6n_initn_jobsr7r8r9r: return_n_iterr;prefer_skip_nested_validation r4r5r6rUrVr7r8r9r:rWr;c P^^^^^^^^ US:Xa[R"S[5 Sn[T5m[ U 5n T S:XaT(+m [ TS5(aE[ R"T5R5mUS:Xd"SU-5 SnSup[U5S:XaG[U5H7n[TTTTTTTU T S 9 unnnU bUU :dM#Un UR5n UnM9 OU R[ R"[ R5RUS 9n[!U[TS- S 5S 9"UUUUUUU U4S jU55n[#U6unnn[ R$"U5nUUn UUn UUnU (aXW4$X4$)aoCompute multidimensional scaling using the SMACOF algorithm. The SMACOF (Scaling by MAjorizing a COmplicated Function) algorithm is a multidimensional scaling algorithm which minimizes an objective function (the *stress*) using a majorization technique. Stress majorization, also known as the Guttman Transform, guarantees a monotone convergence of stress, and is more powerful than traditional techniques such as gradient descent. The SMACOF algorithm for metric MDS can be summarized by the following steps: 1. Set an initial start configuration, randomly or not. 2. Compute the stress 3. Compute the Guttman Transform 4. Iterate 2 and 3 until convergence. The nonmetric algorithm adds a monotonic regression step before computing the stress. Parameters ---------- dissimilarities : array-like of shape (n_samples, n_samples) Pairwise dissimilarities between the points. Must be symmetric. metric : bool, default=True Compute metric or nonmetric SMACOF algorithm. When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as missing values. n_components : int, default=2 Number of dimensions in which to immerse the dissimilarities. If an ``init`` array is provided, this option is overridden and the shape of ``init`` is used to determine the dimensionality of the embedding space. init : array-like of shape (n_samples, n_components), default=None Starting configuration of the embedding to initialize the algorithm. By default, the algorithm is initialized with a randomly chosen array. n_init : int, default=8 Number of times the SMACOF algorithm will be run with different initializations. The final results will be the best output of the runs, determined by the run with the smallest final stress. If ``init`` is provided, this option is overridden and a single run is performed. .. versionchanged:: 1.9 The default value for `n_iter` will change from 8 to 1 in version 1.9. n_jobs : int, default=None The number of jobs to use for the computation. If multiple initializations are used (``n_init``), each run of the algorithm is computed in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. max_iter : int, default=300 Maximum number of iterations of the SMACOF algorithm for a single run. verbose : int, default=0 Level of verbosity. eps : float, default=1e-6 The tolerance with respect to stress (normalized by the sum of squared embedding distances) at which to declare convergence. .. versionchanged:: 1.7 The default value for `eps` has changed from 1e-3 to 1e-6, as a result of a bugfix in the computation of the convergence criterion. random_state : int, RandomState instance or None, default=None Determines the random number generator used to initialize the centers. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. return_n_iter : bool, default=False Whether or not to return the number of iterations. normalized_stress : bool or "auto", default="auto" Whether to return normalized stress value (Stress-1) instead of raw stress. By default, metric MDS returns raw stress while non-metric MDS returns normalized stress. .. versionadded:: 1.2 .. versionchanged:: 1.4 The default value changed from `False` to `"auto"` in version 1.4. .. versionchanged:: 1.7 Normalized stress is now supported for metric MDS as well. Returns ------- X : ndarray of shape (n_samples, n_components) Coordinates of the points in a ``n_components``-space. stress : float The final value of the stress (sum of squared distance of the disparities and the distances for all constrained points). If `normalized_stress=True`, returns Stress-1. A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good, 0.1 fair, and 0.2 poor [1]_. n_iter : int The number of iterations corresponding to the best stress. Returned only if ``return_n_iter`` is set to ``True``. References ---------- .. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J. Psychometrika, 29 (1964) .. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis" Kruskal, J. Psychometrika, 29, (1964) .. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.; Groenen P. Springer Series in Statistics (1997) Examples -------- >>> import numpy as np >>> from sklearn.manifold import smacof >>> from sklearn.metrics import euclidean_distances >>> X = np.array([[0, 1, 2], [1, 0, 3], [2, 3, 0]]) >>> dissimilarities = euclidean_distances(X) >>> Z, stress = smacof( ... dissimilarities, n_components=2, n_init=1, eps=1e-6, random_state=42 ... ) >>> Z.shape (3, 2) >>> np.round(stress, 6).item() 3.2e-05 rSz=The default value of `n_init` will change from 8 to 1 in 1.9.rT __array__rzTExplicit initial positions passed: performing only one init of the MDS instead of %dNNr4r5r6r7r8r9r:r;rr)rVr8c 3Z># UH n[[5"TTTTTT TUTS9 v M" g7f)r_N)rrM) .0seedr3r9r6r7r4r5r;r8s rL smacof..sB G  N #)!!"3 s(+)warningsrS FutureWarningr r hasattrr#asarraycopyrr)rMrandintiinfoint32maxrzipargmin)r3r4r5r6rUrVr7r8r9r:rWr;best_pos best_stressrCposrJn_iter_ best_iterseedsresults positionsn_itersbests```` ``` ` rLsmacofrzsP K  !/2O%l3LF" &Jt[!!zz$$$&{ MMDFLM F&H1$-B#1)!)"3 $ C"f{&:$ 88:#  "$$RXXbhh%7%;%;&$I&#gk12EF G G  G  &)']" 67yy Tl T?DM i//$$rNc^\rSrSr%Sr\"\SSSS9/S/\"\SSSS9\"\"S155/\"S S 15\"\"S155/\"\SSSS9/S /\"\ S SSS9/S\/S /\"SS15\"\"S155/\ \ \"S5/\ S/S\"S15/S. r \ \S'SSSSSSSSSSSSSS. SjjrU4SjrS Sjr\"SS9S Sj5rSrU=r$)!MDSiaMultidimensional scaling. Read more in the :ref:`User Guide `. Parameters ---------- n_components : int, default=2 Number of dimensions in which to immerse the dissimilarities. metric_mds : bool, default=True If ``True``, perform metric MDS; otherwise, perform nonmetric MDS. When ``False`` (i.e. non-metric MDS), dissimilarities with 0 are considered as missing values. .. versionchanged:: 1.8 The parameter `metric` was renamed into `metric_mds`. n_init : int, default=4 Number of times the SMACOF algorithm will be run with different initializations. The final results will be the best output of the runs, determined by the run with the smallest final stress. .. versionchanged:: 1.9 The default value for `n_init` will change from 4 to 1 in version 1.9. init : {'random', 'classical_mds'}, default='random' The initialization approach. If `random`, random initialization is used. If `classical_mds`, then classical MDS is run and used as initialization for MDS (in this case, the value of `n_init` is ignored). .. versionadded:: 1.8 .. versionchanged:: 1.10 The default value for `init` will change to `classical_mds`. max_iter : int, default=300 Maximum number of iterations of the SMACOF algorithm for a single run. verbose : int, default=0 Level of verbosity. eps : float, default=1e-6 The tolerance with respect to stress (normalized by the sum of squared embedding distances) at which to declare convergence. .. versionchanged:: 1.7 The default value for `eps` has changed from 1e-3 to 1e-6, as a result of a bugfix in the computation of the convergence criterion. n_jobs : int, default=None The number of jobs to use for the computation. If multiple initializations are used (``n_init``), each run of the algorithm is computed in parallel. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. random_state : int, RandomState instance or None, default=None Determines the random number generator used to initialize the centers. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. dissimilarity : {'euclidean', 'precomputed'} Dissimilarity measure to use: - 'euclidean': Pairwise Euclidean distances between points in the dataset. - 'precomputed': Pre-computed dissimilarities are passed directly to ``fit`` and ``fit_transform``. .. deprecated:: 1.8 `dissimilarity` was renamed `metric` in 1.8 and will be removed in 1.10. metric : str or callable, default='euclidean' Metric to use for dissimilarity computation. Default is "euclidean". If metric is a string, it must be one of the options allowed by `scipy.spatial.distance.pdist` for its metric parameter, or a metric listed in :func:`sklearn.metrics.pairwise.distance_metrics` If metric is "precomputed", X is assumed to be a distance matrix and must be square during fit. If metric is a callable function, it takes two arrays representing 1D vectors as inputs and must return one value indicating the distance between those vectors. This works for Scipy's metrics, but is less efficient than passing the metric name as a string. .. versionchanged:: 1.8 Prior to 1.8, `metric=True/False` was used to select metric/non-metric MDS, which is now the role of `metric_mds`. The support for ``True`` and ``False`` will be dropped in version 1.10, use `metric_mds` instead. metric_params : dict, default=None Additional keyword arguments for the dissimilarity computation. .. versionadded:: 1.8 normalized_stress : bool or "auto" default="auto" Whether to return normalized stress value (Stress-1) instead of raw stress. By default, metric MDS returns raw stress while non-metric MDS returns normalized stress. .. versionadded:: 1.2 .. versionchanged:: 1.4 The default value changed from `False` to `"auto"` in version 1.4. .. versionchanged:: 1.7 Normalized stress is now supported for metric MDS as well. Attributes ---------- embedding_ : ndarray of shape (n_samples, n_components) Stores the position of the dataset in the embedding space. stress_ : float The final value of the stress (sum of squared distance of the disparities and the distances for all constrained points). If `normalized_stress=True`, returns Stress-1. A value of 0 indicates "perfect" fit, 0.025 excellent, 0.05 good, 0.1 fair, and 0.2 poor [1]_. dissimilarity_matrix_ : ndarray of shape (n_samples, n_samples) Pairwise dissimilarities between the points. Symmetric matrix that: - either uses a custom dissimilarity matrix by setting `dissimilarity` to 'precomputed'; - or constructs a dissimilarity matrix from data using Euclidean distances. n_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 n_iter_ : int The number of iterations corresponding to the best stress. See Also -------- sklearn.decomposition.PCA : Principal component analysis that is a linear dimensionality reduction method. sklearn.decomposition.KernelPCA : Non-linear dimensionality reduction using kernels and PCA. TSNE : T-distributed Stochastic Neighbor Embedding. Isomap : Manifold learning based on Isometric Mapping. LocallyLinearEmbedding : Manifold learning using Locally Linear Embedding. SpectralEmbedding : Spectral embedding for non-linear dimensionality. References ---------- .. [1] "Nonmetric multidimensional scaling: a numerical method" Kruskal, J. Psychometrika, 29 (1964) .. [2] "Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis" Kruskal, J. Psychometrika, 29, (1964) .. [3] "Modern Multidimensional Scaling - Theory and Applications" Borg, I.; Groenen P. Springer Series in Statistics (1997) Examples -------- >>> from sklearn.datasets import load_digits >>> from sklearn.manifold import MDS >>> X, _ = load_digits(return_X_y=True) >>> X.shape (1797, 64) >>> embedding = MDS(n_components=2, n_init=1, init="random") >>> X_transformed = embedding.fit_transform(X[:100]) >>> X_transformed.shape (100, 2) For a more detailed example of usage, see :ref:`sphx_glr_auto_examples_manifold_plot_mds.py`. For a comparison of manifold learning techniques, see :ref:`sphx_glr_auto_examples_manifold_plot_compare_methods.py`. rNrPrQrOrSrandom classical_mdsr8gr: euclidean precomputed deprecatedrT) r5 metric_mdsrUr6r7r8r9rVr: dissimilarityr4 metric_paramsr;_parameter_constraintsTrrr) rrUr6r7r8r9rVr:rr4rr;c XlXlXlXlX lX0lX@lXPlXplX`l Xl Xl Xl g)N) r5rr4rrrUr6r7r9r8rVr:r;)selfr5rrUr6r7r8r9rVr:rr4rr;s rL__init__ MDS.__init__sJ")* *$     (!2rNc>[TU]5nURS:HURS:H-URlU$)Nr)super__sklearn_tags__rr4 input_tagspairwise)rtags __class__s rLrMDS.__sklearn_tags__s?w')$($6$6-$G KK= ($   rNc$URXS9 U$)ah Compute the position of the points in the embedding space. Parameters ---------- X : array-like of shape (n_samples, n_features) or (n_samples, n_samples) Input data. If ``metric=='precomputed'``, the input should be the dissimilarity matrix. y : Ignored Not used, present for API consistency by convention. init : ndarray of shape (n_samples, n_components), default=None Starting configuration of the embedding to initialize the SMACOF algorithm. By default, the algorithm is initialized with a randomly chosen array. Returns ------- self : object Fitted estimator. )r6)r*)rr?yr6s rLfitMDS.fits0 1( rNrXctURS:Xa#[R"S[5 SUlOURUlUR S:Xa#[R"S[5 SUlOUR UlURS:waf[UR[5(dURS:wa [S5e[R"S [5 URUl [UR[5(a_[R"S URS URS 3[5 URS:XaSUl URUl O2URS:XaURUl URUl [X5nUR S UR S:Xa&URS:wa[R"S5 URS:Xa Xl[%UR"SS9UlO6['U4SUR0UR(b UR(O0D6UlUbUnO7UR S:Xa%[+SS9nUR-UR"5nOSn[/UR"URUR0UURUR2UR4UR6UR8UR:SUR<S9 uUlUl Ul!UR>$)a Fit the data from `X`, and returns the embedded coordinates. Parameters ---------- X : array-like of shape (n_samples, n_features) or (n_samples, n_samples) Input data. If ``metric=='precomputed'``, the input should be the dissimilarity matrix. y : Ignored Not used, present for API consistency by convention. init : ndarray of shape (n_samples, n_components), default=None Starting configuration of the embedding to initialize the SMACOF algorithm. By default, the algorithm is initialized with a randomly chosen array. Returns ------- X_new : ndarray of shape (n_samples, n_components) X transformed in the new space. rSzwThe default value of `n_init` will change from 4 to 1 in 1.9. To suppress this warning, provide some value of `n_init`.zThe default value of `init` will change from 'random' to 'classical_mds' in 1.10. To suppress this warning, provide some value of `init`.r}rrzIYou provided both `dissimilarity` and `metric`. Please use only `metric`.z^The `dissimilarity` parameter is deprecated and will be removed in 1.10. Use `metric` instead.zUse metric_mds=z instead of metric=z>. The support for metric={True/False} will be dropped in 1.10.rrrzThe provided input is a square matrix. Note that ``fit`` constructs a dissimilarity matrix from data and will treat rows as samples and columns as features. To use a pre-computed dissimilarity matrix, set ``metric='precomputed'``.Trr4Nr~)r4rZ)"rUrerSrf_n_initr6_initr isinstancer4boolr(_metric _metric_mdsrrr"dissimilarity_matrix_rr rr r*rzr5rVr7r8r9r:r; embedding_stress_rs)rr?rr6 init_arraycmdss rLr*MDS.fit_transforms4 ;;& MML  DL;;DL 99  MM(   "DJDJ    -dkk400T[[K5O %  =! $11 dkk4 ( ( MM!$++.A$++OKK  !!\1* #{{D !!\1#{{ #D  $ " 771: # (E MM0  <<= ()* &)8**D*D &*<*||**.););)G4%%R*D &  J ZZ? *}5D++D,F,FGJJ6<  & &##**<<;;]]LL**"44 7 3t|rN)rrrrrrrr9r6r7r4rrr5rUrsrVr;r:rr8)rr^)__name__ __module__ __qualname____firstlineno____doc__rrrrrstrcallabledictr__annotations__rrrrr*__static_attributes__ __classcell__)rs@rLr|r|sJ{|"(AtFCD k Xq$v 6 :vh' ( X78&VHAU:VWh4?@;sD89"'(  ]3 4 :|n- . &"34'VH)=>'$D03  " 3>65v6vrNr|)TrNrrrNF)$rrenumbersrrnumpyr#joblibr sklearn.baserrsklearn.isotonicrsklearn.manifoldr sklearn.metricsr r sklearn.utilsr r rsklearn.utils._param_validationrrrrsklearn.utils.parallelrrsklearn.utils.validationrrMrzr|rNrLrs:"#4/)CJJ 52    fT(>+!(AtFCDt$Haf=z6(?STT"h4?@;q$v67'(#'VH)=> #'(      Y%#"Y%@L-LrN