Documentation for Basis Functions
¶
Basis Function for NARMAX models.
Bersntein
¶
Bases: BaseBasisFunction
Build Bersntein basis function.
Generate Bernstein basis functions.
This class constructs a new feature matrix consisting of Bernstein basis functions for a given degree. Bernstein polynomials are useful in numerical analysis, curve fitting, and approximation theory due to their smoothness and the ability to approximate any continuous function on a closed interval.
The Bernstein polynomial of degree \(n\) for a variable \(x\) is defined as:
.. math:: B_{i,n}(x) = \binom{n}{i} x^i (1 - x)^{n - i} \quad \text{for} \quad i = 0, 1, \ldots, n
where \(\binom{n}{i}\) is the binomial coefficient, given by:
.. math:: \binom{n}{i} = \frac{n!}{i! (n - i)!}
Bernstein polynomials form a basis for the space of polynomials of degree at most \(n\). They are particularly useful in approximation theory because they can approximate any continuous function on the interval \([0, 1]\) as \(n\) increases.
Be aware that the number of features in the output array scales significantly with the number of inputs, the maximum lag of the input, and the polynomial degree.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
degree | int(max_degree) | The maximum degree of the polynomial features. | 2 |
Notes
Be aware that the number of features in the output array scales significantly as the number of inputs, the max lag of the input and output.
References
- Blog: this method is based on the content provided by Alex Shtoff in his blog. The content is easy to follow and every user is referred to is blog to check not only the Bersntein method, but also different topics that Alex discuss there! https://alexshtf.github.io/2024/01/21/Bernstein.html
- Wikipedia: Bernstein polynomial https://en.wikipedia.org/wiki/Bernstein_polynomial
Source code in sysidentpy\basis_function\_basis_function.py
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 |
|
Fourier
¶
Bases: BaseBasisFunction
Build Fourier basis function.
Generate a new feature matrix consisting of all Fourier features with respect to the number of harmonics.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
degree | int(max_degree) | The maximum degree of the polynomial features. | 2 |
Notes
Be aware that the number of features in the output array scales significantly as the number of inputs, the max lag of the input and output.
Source code in sysidentpy\basis_function\_basis_function.py
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 |
|
fit(data, max_lag=1, predefined_regressors=None)
¶
Build the Polynomial information matrix.
Each columns of the information matrix represents a candidate regressor. The set of candidate regressors are based on xlag, ylag, and degree defined by the user.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
data | ndarray of floats | The lagged matrix built with respect to each lag and column. | required |
max_lag | int | Target data used on training phase. | 1 |
predefined_regressors | ndarray of int | The index of the selected regressors by the Model Structure Selection algorithm. | None |
Returns:
Type | Description |
---|---|
psi = ndarray of floats | The lagged matrix built in respect with each lag and column. |
Source code in sysidentpy\basis_function\_basis_function.py
transform(data, max_lag=1, predefined_regressors=None)
¶
Build Fourier Basis Functions.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
data | ndarray of floats | The lagged matrix built with respect to each lag and column. | required |
max_lag | int | Maximum lag of list of regressors. | 1 |
predefined_regressors | Optional[ndarray] | Regressors to be filtered in the transformation. | None |
Returns:
Name | Type | Description |
---|---|---|
X_tr | {ndarray, sparse matrix} of shape (n_samples, n_features) | Transformed array. |
Source code in sysidentpy\basis_function\_basis_function.py
Polynomial
¶
Bases: BaseBasisFunction
Build polynomial basis function.
Generate a new feature matrix consisting of all polynomial combinations of the features with degree less than or equal to the specified degree.
where \(p\) is the number of regressors, \(\Theta_i\) are the model parameters, and \(a_i, m, b_i, j\) and \(d_i, l \in \mathbb{N}\) are the exponents of the output, input and noise terms, respectively.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
degree | int(max_degree) | The maximum degree of the polynomial features. | 2 |
Notes
Be aware that the number of features in the output array scales significantly as the number of inputs, the max lag of the input and output, and degree increases. High degrees can cause overfitting.
Source code in sysidentpy\basis_function\_basis_function.py
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 |
|
fit(data, max_lag=1, predefined_regressors=None)
¶
Build the Polynomial information matrix.
Each columns of the information matrix represents a candidate regressor. The set of candidate regressors are based on xlag, ylag, and degree defined by the user.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
data | ndarray of floats | The lagged matrix built with respect to each lag and column. | required |
max_lag | int | Target data used on training phase. | 1 |
predefined_regressors | ndarray of int | The index of the selected regressors by the Model Structure Selection algorithm. | None |
Returns:
Type | Description |
---|---|
psi = ndarray of floats | The lagged matrix built in respect with each lag and column. |
Source code in sysidentpy\basis_function\_basis_function.py
transform(data, max_lag=1, predefined_regressors=None)
¶
Build Polynomial Basis Functions.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
data | ndarray of floats | The lagged matrix built with respect to each lag and column. | required |
max_lag | int | Maximum lag of list of regressors. | 1 |
predefined_regressors | Optional[ndarray] | Regressors to be filtered in the transformation. | None |
Returns:
Name | Type | Description |
---|---|---|
X_tr | {ndarray, sparse matrix} of shape (n_samples, n_features) | Transformed array. |