In the case where all my data are expressed in the same units, how can I express the principal components in terms of the original scale?
In economics and finance, it is customary to summarize data using principal components. I would like to apply principal components analysis to the yield curve of a given country (that is, the interest rate paid by the country for its debt at 1,2,…,10 years maturity). All the yields are expressed in percentage.
However, the interpretation of the principal components for the rest of the things I have to do with the principal components is crucial and should be carried out in the initial scale of the data.
Rather than using the principal components analysis for the yields, researchers also use the following transformations (p23 of this paper, for example):
PC1 (level) = mean(yields);
PC2 (slope) = yield_10years - yield_1year;[2]][2]
PC3 (curvature) = (-yield_1year + 2*yield_3year - yield_10year);
which are obviously expressed in the original scale. I therefore have some kind of benchmark against which I will compare the principal components I obtain with eigenvalues-eigenvectors decomposition of the covariance matrix. I am mostly interested in the first three principal components (blue is PC1, red is PC2, yellow is PC3).
Here are two pictures depending on the method used:
and
If we focus on the first principal components under the two methods, we can see that their shape is very similar, but their scale is not. For the values themselves, the PC1 obtain with PCA is centered in 0 whereas it is centered in 1.8452 for the empirical formulae.
How can I therefore express the PC obtained with PCA such that they have the same range of values as the PC obtained with the empirical formulae?
For all intents and purposes, here are my data, where the row dimension is time and the column dimension is the maturity of the yields. Col 1 is the 1-year, col 3 is the 3-years and col 9 is the 10-years. I have put the loadings obtained after centering my data and the corresponding scores.
# Data in CSV:
3.0875,2.99,2.9325,3.1275,3.5525,3.8075,4.03,4.3875,4.715
2.9175,2.8625,2.8875,3.1525,3.5825,3.8275,4.0375,4.3775,4.7025
2.7725,2.6875,2.61,2.785,3.1725,3.455,3.685,4.05,4.4125
2.7275,2.6225,2.5325,2.6675,3.0425,3.2925,3.52,3.895,4.2675
2.4225,2.3225,2.2475,2.3675,2.7725,3.0325,3.2725,3.6925,4.1325
2.43,2.3375,2.2825,2.4875,2.9125,3.1925,3.43,3.8225,4.2375
2.4225,2.3225,2.2675,2.4725,2.945,3.23,3.47,3.86,4.27
2.19,2.11,2.04,2.1625,2.5525,2.8275,3.0675,3.47,3.9
2.066,1.9895,1.9475,2.1625,2.5675,2.8525,3.1025,3.5175,3.9475
2.0747,2.0556,2.1225,2.5575,3,3.3075,3.555,3.925,4.3125
2.09,2.104,2.2365,2.68,3.175,3.4525,3.6675,4,4.33
2.04,2.0025,2.02,2.3375,2.79,3.0925,3.335,3.7275,4.13
2.0967,2.1275,2.3071,2.7475,3.2275,3.52,3.7475,4.0925,4.4275
2.0977,2.1675,2.415,2.905,3.4125,3.7025,3.925,4.2525,4.57
2.0525,2.0675,2.2125,2.6425,3.16,3.46,3.6975,4.06,4.4075
2.0379,2.0522,2.1911,2.6,3.0925,3.3975,3.64,4.025,4.395
1.98,1.9525,1.9975,2.355,2.79,3.105,3.365,3.785,4.1975
1.9122,1.8713,1.9017,2.1925,2.53,2.965,3.225,3.654,4.087
2.015,2.035,2.145,2.5175,2.885,3.306,3.552,3.943,4.331
2.045,2.0975,2.2825,2.73,2.995,3.482,3.728,4.103,4.477
2.055,2.1169,2.3275,2.763,2.995,3.528,3.753,4.104,4.441
2.0576,2.1127,2.2925,2.7,2.995,3.433,3.654,4.002,4.341
2.0575,2.0874,2.1963,2.514,2.839,3.219,3.44,3.801,4.161
2.072,2.136,2.2995,2.6045,2.876,3.239,3.441,3.778,4.122
2.093,2.129,2.236,2.48,2.739,3.1,3.308,3.655,4.013
2.1,2.128,2.212,2.421,2.651,2.985,3.18,3.514,3.868
2.0875,2.14,2.275,2.519,2.734,3.021,3.177,3.458,3.767
2.071,2.1195,2.223,2.456,2.654,2.928,3.076,3.337,3.629
2.0782,2.131,2.2745,2.563,2.794,3.073,3.231,3.499,3.79
2.088,2.136,2.268,2.535,2.735,3.026,3.175,3.438,3.733
2.074,2.081,2.123,2.303,2.488,2.781,2.937,3.106,3.546
2.0755,2.0638,2.0647,2.195,2.365,2.649,2.803,2.957,3.393
2.0665,2.0422,1.9983,2.069,2.225,2.496,2.532,2.801,3.231
2.0775,2.091,2.156,2.315,2.463,2.5645,2.696,2.942,3.237
2.0845,2.095,2.131,2.253,2.38,2.48,2.589,2.809,3.092
2.1215,2.1505,2.266,2.439,2.556,2.646,2.737,2.915,3.155
2.1995,2.317,2.498,2.7,2.819,2.929,3.023,3.186,3.4
2.3885,2.526,2.705,2.852,2.938,3.033,3.109,3.254,3.444
2.433,2.571,2.775,2.926,3.001,3.071,3.11,3.202,3.338
2.502,2.6305,2.8385,3.0318,3.148,3.233,3.29,3.394,3.537
2.609,2.7235,2.919,3.11,3.226,3.28,3.334,3.426,3.554
2.7485,2.9225,3.179,3.4095,3.525,3.609,3.665,3.76,3.884
2.7865,2.9475,3.2165,3.48,3.667,3.712,3.787,3.906,4.049
2.8765,3.048,3.288,3.51,3.615,3.702,3.775,3.904,4.023
2.9905,3.1655,3.4355,3.662,3.795,3.863,3.93,4.038,4.17
3.0995,3.26,3.459,3.605,3.7135,3.752,3.808,3.913,4.05
3.2115,3.385,3.56,3.605,3.6505,3.654,3.685,3.7625,3.882
3.361,3.5085,3.655,3.675,3.65,3.693,3.711,3.755,3.839
3.48,3.6165,3.76,3.754,3.797,3.744,3.746,3.773,3.835
3.578,3.67,3.77,3.738,3.756,3.719,3.721,3.778,3.7965
3.6585,3.7875,3.9525,3.998,3.98,4.0075,4.0065,4.0265,4.0805
3.7215,3.8505,4.0175,4.0675,4.0985,4.1085,4.1185,4.1555,4.2205
3.8,3.891,3.991,3.9795,3.9685,3.973,3.9795,4.006,4.0715
3.8785,3.9905,4.112,4.1195,4.1195,4.114,4.117,4.14,4.2035
3.967,4.0725,4.215,4.2235,4.2135,4.2055,4.2085,4.2265,4.2825
4.065,4.1923,4.4023,4.495,4.5075,4.5035,4.5045,4.5095,4.549
4.1145,4.2505,4.4555,4.5775,4.609,4.6255,4.6405,4.6695,4.7225
4.1925,4.3,4.4355,4.507,4.4885,4.507,4.5045,4.5255,4.5715
4.097,4.1563,4.231,4.232,4.284,4.4505,4.4525,4.4815,4.5575
4.0713,4.0713,4.125,4.1675,4.2138,4.37,4.388,4.444,4.547
4.0813,4.1335,4.2365,4.2515,4.2738,4.325,4.343,4.38,4.462
4.0225,4.0395,4.047,3.9845,3.9995,4.061,4.112,4.221,4.383
4.0435,4.1265,4.183,4.099,4.1285,4.211,4.264,4.3575,4.507
3.9738,3.874,3.694,3.5685,3.6215,3.692,3.765,3.936,4.132
3.9515,3.827,3.578,3.342,3.376,3.563,3.649,3.857,4.09
3.9985,3.988,3.8655,3.7165,3.7245,3.74,3.779,3.883,4.07
4.034,4.045,4.028,3.9155,3.915,3.9565,3.988,4.0705,4.233
4.08,4.186,4.349,4.3965,4.3425,4.3435,4.3505,4.3835,4.488
4.2935,4.415,4.629,4.732,4.7275,4.7245,4.6775,4.6355,4.6405
4.341,4.3845,4.398,4.319,4.4185,4.3415,4.3515,4.3555,4.391
4.324,4.31,4.267,4.1065,4.101,4.1345,4.1465,4.1745,4.25
4.099,4.0255,3.862,3.7465,3.8225,3.9,3.958,4.044,4.134
2.901,2.7595,2.595,2.735,3.021,3.289,3.467,3.764,4.086
2.198,2.01,1.904,2.146,2.472,2.652,2.847,3.159,3.506
1.734,1.6345,1.559,1.9285,2.209,2.457,2.625,2.9195,3.2485
1.201,1.163,1.211,1.5625,1.946,2.259,2.513,2.909,3.3225
0.865,0.8045,0.889,1.249,1.682,2.0095,2.2555,2.6305,3.0295
0.686,0.665,0.748,1.111,1.575,1.889,2.159,2.573,2.974
0.72,0.728,0.815,1.1775,1.601,1.939,2.221,2.647,3.048
0.787,0.759,0.8105,1.2115,1.704,2.122,2.457,2.934,3.36
0.596,0.669,0.819,1.3385,1.828,2.212,2.494,2.915,3.3
0.443,0.489,0.665,1.2465,1.756,2.135,2.41,2.817,3.199
0.405,0.443,0.675,1.2805,1.769,2.126,2.39,2.777,3.139
0.407,0.466,0.673,1.228,1.695,2.039,2.317,2.736,3.128
0.458,0.525,0.766,1.316,1.752,2.05,2.298,2.701,3.085
0.426,0.512,0.756,1.2165,1.633,1.924,2.184,2.594,2.993
0.3865,0.476,0.803,1.386,1.798,2.064,2.314,2.736,3.148
0.366,0.443,0.686,1.1365,1.542,1.862,2.13,2.568,2.996
0.365,0.433,0.567,0.9065,1.296,1.644,1.948,2.434,2.901
0.372,0.471,0.633,0.9795,1.339,1.649,1.936,2.401,2.86
0.438,0.542,0.664,0.9115,1.243,1.542,1.822,2.274,2.72
0.3965,0.427,0.5345,0.7465,1.011,1.305,1.581,2.025,2.464
0.492,0.5445,0.637,0.815,1.028,1.301,1.566,2.012,2.456
0.545,0.615,0.709,0.931,1.185,1.45,1.701,2.119,2.549
0.4955,0.526,0.584,0.703,0.875,1.055,1.245,1.579,1.935
0.706,0.759,0.859,1.023,1.198,1.358,1.54,1.868,2.22
0.821,0.8817,0.974,1.155,1.359,1.5665,1.765,2.1045,2.4615
0.702,0.747,0.816,1.002,1.242,1.53,1.801,2.228,2.644
0.6005,0.663,0.7575,1.0165,1.33,1.686,1.977,2.416,2.856
0.871,0.958,1.164,1.564,1.901,2.162,2.398,2.733,3.06
0.848,0.989,1.232,1.6275,1.918,2.1705,2.377,2.701,3.04
1.024,1.184,1.469,1.919,2.245,2.477,2.669,2.975,3.2698
1.223,1.366,1.592,1.951,2.222,2.4422,2.625,2.9143,3.227
1.154,1.284,1.466,1.715,1.94,2.154,2.363,2.668,2.985
1.345,1.43,1.556,1.734,1.942,2.187,2.385,2.708,3.056
1.248,1.287,1.32,1.383,1.557,1.788,2.016,2.377,2.748
0.901,0.871,0.834,0.902,1.112,1.399,1.664,2.064,2.444
0.742,0.666,0.658,0.735,0.911,1.162,1.385,1.745,2.074
0.778,0.688,0.649,0.729,0.934,1.151,1.377,1.753,2.1076
0.4895,0.453,0.449,0.565,0.806,1.096,1.375,1.813,2.227
0.391,0.377,0.377,0.463,0.622,0.86,1.104,1.506,1.912
0.361,0.346,0.338,0.371,0.5,0.716,0.952,1.3795,1.812
0.3465,0.3415,0.345,0.411,0.553,0.752,0.97,1.3748,1.804
0.3605,0.362,0.371,0.461,0.613,0.8135,1.028,1.429,1.848
0.3125,0.295,0.28,0.327,0.462,0.66,0.877,1.291,1.733
0.272,0.25,0.236,0.25,0.326,0.452,0.606,0.891,1.213
0.239,0.228,0.226,0.285,0.402,0.581,0.791,1.169,1.573
0.056,0.038,0.04,0.084,0.191,0.343,0.539,0.943,1.377
0.0685,0.055,0.056,0.089,0.19,0.358,0.552,0.929,1.353
0.0855,0.0715,0.07,0.126,0.244,0.426,0.629,1.0159,1.453
0.0885,0.082,0.085,0.145,0.265,0.436,0.632,1.021,1.466
0.064,0.049,0.0465,0.093,0.194,0.354,0.538,0.912,1.354
0.067,0.054,0.0495,0.091,0.181,0.309,0.472,0.829,1.273
0.1395,0.1675,0.2375,0.379,0.534,0.699,0.87,1.206,1.606
0.0765,0.073,0.079,0.144,0.27,0.426,0.607,0.983,1.445
0.082,0.082,0.092,0.131,0.234,0.371,0.54,0.886,1.324
0.071,0.07,0.078,0.119,0.198,0.304,0.454,0.7739,1.2
0.069,0.061,0.074,0.15,0.3015,0.446,0.627,0.994,1.452
0.11,0.124,0.168,0.281,0.444,0.673,0.896,1.271,1.708
0.1065,0.11,0.142,0.23,0.402,0.624,0.8465,1.227,1.675
0.098,0.118,0.171,0.312,0.5164,0.775,1.015,1.4132,1.861
0.105,0.117,0.153,0.239,0.409,0.635,0.865,1.274,1.739
0.114,0.119,0.141,0.206,0.3435,0.5435,0.755,1.1583,1.644
0.1255,0.112,0.105,0.143,0.277,0.482,0.7085,1.136,1.643
0.158,0.154,0.156,0.24,0.427,0.644,0.889,1.3136,1.809
0.1355,0.116,0.103,0.126,0.253,0.437,0.6435,1.051,1.544
0.1383,0.125,0.123,0.157,0.283,0.46,0.649,1.0275,1.505
0.1632,0.152,0.145,0.186,0.293,0.451,0.6245,0.987,1.45
0.169,0.152,0.139,0.165,0.269,0.413,0.5793,0.9261,1.3729
0.107,0.083,0.068,0.072,0.158,0.282,0.4343,0.774,1.2282
0.056,0.051,0.041,0.054,0.118,0.2209,0.3583,0.685,1.134
0.067,0.067,0.06,0.069,0.126,0.214,0.337,0.63,1.05
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# Loadings in CSV:
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# Score in CSV:
5.3362,0.49337,0.34761,-0.093587,0.0361,0.021815,-0.0019689,-0.0051055,-0.0023761
5.2471,0.62294,0.19208,-0.078685,0.046923,0.0099034,-0.0032757,-0.0091293,-0.0017862
4.3345,0.48094,0.37133,-0.070702,0.017014,0.012158,-0.0076307,0.0046608,-0.00096716
3.9783,0.36768,0.40936,-0.063207,0.034922,0.014489,-0.0029844,-0.0026711,-0.00032123
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-6.5451,-1.0307,-0.16826,-0.10474,-0.016395,-0.0032737,-0.0099426,-0.00077963,-0.002771
-6.5375,-1.025,-0.15641,-0.091921,-0.013035,-0.0081941,-0.012645,0.0011498,0.0030226
-6.2131,-0.77588,-0.14262,-0.066379,-0.0047359,-0.0078144,-0.0085249,-0.0017206,-0.00028491
-6.0702,-0.62721,-0.079915,-0.018927,0.0084094,-0.0064769,-0.0027341,-0.0014177,-0.0056457
-6.0593,-0.64019,-0.092493,-0.01849,0.018371,-0.01933,-0.0057625,-0.0008498,-0.0020997
-5.8557,-0.45112,-0.072703,-0.018038,0.0075411,-0.010497,-0.0098942,-4.1133e-05,-0.001368
-6.0629,-0.63234,-0.10198,-0.030226,0.01225,-0.017907,-0.0070876,0.00061621,-0.0034376
-5.8464,-0.48765,-0.12556,-0.046122,0.0039224,-0.0060125,-0.0069918,-0.00023003,-0.003516
EDIT: I have run the PCA function on unscaled (using the covariance rather than the correlation matrix) and uncentered data and this is the result:
I think my mistake was to think that the PC should be of the same order of magnitude as the original data (I have divided the data by 100 to change the order of magnitude of the PC and original data).
Best Answer
I can answer at least the first part of your questions, which is easy. Principal component scores in the scale of the original variables are computed by using the decomposition of the covariance (and not correlation) matrix. The scores are computed by postmultiplying the data matrix (X) by the unscaled eigenvectors (E), or Scores = XE. Often the eigenvectors are called "loadings" but a better usage for that term is the correlation between the scores and the original data.
But, based on the 2nd part of your question, I think you are asking about location and not about scale (scaling shrinks or blows up stuff while translation rigidly moves stuff around to different locations). Your scores are on the same scale as the original data but they have simply been centered (rigidly translated so the multivariate mean is located at the origin). If you want the uncentered scores, don't center the X or the raw scores. Or if you want to re-center to any location, simply add the vector to each case.
This is R code to do this: