Singular Matrix

1. A is a singular matrix which is not invertible if and only if

• $det(A)=0$
• its matrix inverse don’t exists.
• rows (or columns) have an approximately linear dependence
• one of its eigenvalue is zero.

2. sklearn.model_selection.train_test_split divides arrays or matrices into train part and test part of datas.

import numpy as np
from sklearn.model_selection import train_test_split
X, y = np.arange(100).reshape((25, 4)), range(25)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)
print(X_train, X_test, y_train, y_test)

[[20 21 22 23]
 [ 8  9 10 11]
 [48 49 50 51]
 [60 61 62 63]
 [12 13 14 15]
 [16 17 18 19]
 [80 81 82 83]
 [68 69 70 71]
 [84 85 86 87]
 [72 73 74 75]
 [96 97 98 99]
 [28 29 30 31]
 [40 41 42 43]
 [56 57 58 59]
 [76 77 78 79]
 [24 25 26 27]] [[32 33 34 35]
 [64 65 66 67]
 [ 0  1  2  3]
 [92 93 94 95]
 [44 45 46 47]
 [36 37 38 39]
 [52 53 54 55]
 [ 4  5  6  7]
 [88 89 90 91]] [5, 2, 12, 15, 3, 4, 20, 17, 21, 18, 24, 7, 10, 14, 19, 6] [8, 16, 0, 23, 11, 9, 13, 1, 22]

we create a set of datas formed as matrix ,some columns of it have linear relations with each other, as follows.

import numpy as np
from sklearn.model_selection import train_test_split
x1, y = np.arange(100).reshape((25, 4)), range(25)
x2=(x1[:,1]*2+x1[:,3]*5).transpose()
x = np.column_stack((x1,x2))
X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.33, random_state=42)

Next, we try to deal with a set of datas which is a singular matrix.

import numpy as np
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
from sklearn.linear_model import Ridge  # L2正则化


def is_singular_by_eig(matrix, tol=1e-8):
    """
    通过特征值判断
    有零或接近零特征值的矩阵是奇异矩阵
    """
    eigvals = np.linalg.eigvals(matrix)
    return any(np.abs(eigvals) < tol)



x, y = np.arange(400).reshape((20, 20)), range(20)


print(x)
# 示例
print(is_singular_by_eig(x))  # True


X_train, X_test, y_train, y_test = train_test_split(x, y, test_size=0.33, random_state=42)
#regressor = LinearRegression().fit(X_train, y_train)

# 使用Ridge回归
regressor = Ridge(alpha=1.0)  # alpha是正则化强度
regressor.fit(X_train, y_train)
print("Ridge回归得分:", regressor.score(X_train, y_train))

print(regressor.coef_)
print(regressor.intercept_)
y_pred = regressor.predict(X_test)
print(mean_squared_error(y_test, y_pred))

PS E:\learn\learnpy> & D:/Python312/python.exe e:/learn/learnpy/learn1.py
[[  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17
   18  19]
 [ 20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37
   38  39]
 [ 40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57
   58  59]
 [ 60  61  62  63  64  65  66  67  68  69  70  71  72  73  74  75  76  77
   78  79]
 [ 80  81  82  83  84  85  86  87  88  89  90  91  92  93  94  95  96  97
   98  99]
 [100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117
  118 119]
 [120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137
  138 139]
 [140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157
  158 159]
 [160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177
  178 179]
 [180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197
  198 199]
 [200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217
  218 219]
 [220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237
  238 239]
 [240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257
  258 259]
 [260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277
  278 279]
 [280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297
  298 299]
 [300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317
  318 319]
 [320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337
  338 339]
 [340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
  358 359]
 [360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377
  378 379]
 [380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397
  398 399]]
True
Ridge回归得分: 0.9999999999998942
[0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025
 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025]
-0.4749965178404061
4.4036472951780946e-12

references

1. https://scikit-learn.org/stable