├── README.md ├── src ├── Makefile ├── cylowess.pyx ├── ewma.py ├── liner.py ├── lowess.py └── poly.py └── test ├── host1.json ├── host2.json ├── imgs ├── emwa_window.png ├── holt_winters_average.png ├── liner.png ├── lowess.png └── poly.png ├── readjson.py ├── test.py └── testcylowess.py /README.md: -------------------------------------------------------------------------------- 1 | regression-algorithm 2 | ==================== 3 | Including some experimental code for the LOWESS, Liner, and Polynomial Regression algorithm. 4 | 5 | 6 | Demo 7 | ---- 8 | 9 | #### 1. LOWESS 10 | ![LOWESS](./test/imgs/lowess.png?raw=true "LOWESS") 11 | 12 | #### 2. Liner Regression 13 | ![Liner Regression](./test/imgs/liner.png?raw=true "Liner Regression") 14 | 15 | #### 3. Polynomial Regression 16 | ![Polynomial Regression](./test/imgs/poly.png?raw=true "Polynomial Regression") 17 | 18 | #### 4. Exponentially-Weighted Moving Average 19 | ![Exponentially-Weighted Moving Average with Window](./test/imgs/emwa_window.png?raw=true "Exponentially-Weighted Moving Average with Window") 20 | ![Holt Winters EWMA](./test/imgs/holt_winters_average.png?raw=true "Holt Winters EWMA") 21 | 22 | Dependency 23 | ---------- 24 | It's based on SciPy 25 | 26 | #### 1. Ubuntu 27 | ```shell 28 | sudo apt-get install python-numpy python-scipy python-matplotlib ipython ipython-notebook python-pandas python-sympy python-nose 29 | ``` 30 | #### 2. Redhat/CentOS 31 | ```shell 32 | sudo yum install numpy scipy python-matplotlib ipython python-pandas sympy python-nose 33 | ``` 34 | 35 | #### 3. Windows/Mac 36 | The easiest way to install the packages of the SciPy stack is to download one of these Python distributions, which includes all the key packages: 37 | 38 | * [Anaconda](http://continuum.io/downloads.html): A free distribution for the SciPy stack. Supports Linux, Windows and Mac. 39 | * [Enthought Canopy](http://www.enthought.com/products/canopy/): The free and commercial versions include the core SciPy stack packages. Supports Linux, Windows and Mac. 40 | * [Python(x,y)](http://code.google.com/p/pythonxy/): A free distribution including the SciPy stack, based around the Spyder IDE. Windows only. 41 | * [WinPython](http://code.google.com/p/winpython/): A free distribution including the SciPy stack. Windows only. 42 | * [Pyzo](http://www.pyzo.org/): A free distribution based on Python 3 (see Note on Python 3) with the IEP editor. Supports Linux and Windows. 43 | * [Algorete Loopy](http://algorete.org/): A free, community oriented distribution for the SciPy stack maintained by researches at Dartmouth College. Loopy supports both Python 2 and 3 on Linux, Windows and Mac OSX. The distribution is derived from Anaconda with additional packages (e.g. Space Physics, BioInformatics). 44 | -------------------------------------------------------------------------------- /src/Makefile: -------------------------------------------------------------------------------- 1 | PYTHON_LIB=/usr/include/python2.6 2 | 3 | all: cylowess.so 4 | 5 | cylowess.c: cylowess.pyx 6 | cython cylowess.pyx 7 | 8 | cylowess.o:cylowess.c 9 | gcc -c -fPIC -I${PYTHON_LIB} cylowess.c 10 | 11 | cylowess.so:cylowess.o 12 | gcc -shared -o cylowess.so cylowess.o 13 | 14 | 15 | .PHONY: clean 16 | 17 | clean: 18 | rm -f cylowess.c *.o *.so 19 | -------------------------------------------------------------------------------- /src/cylowess.pyx: -------------------------------------------------------------------------------- 1 | #cython: boundscheck = False 2 | #cython: wraparound = False 3 | #cython: cdivision = True 4 | 5 | ''' 6 | Univariate lowess function, like in R. 7 | 8 | References 9 | ---------- 10 | Hastie, Tibshirani, Friedman. (2009) The Elements of Statistical Learning: Data 11 | Mining, Inference, and Prediction, Second Edition: Chapter 6. 12 | 13 | Cleveland, W.S. (1979) "Robust Locally Weighted Regression and Smoothing 14 | Scatterplots". Journal of the American Statistical Association 74 (368): 829-836. 15 | ''' 16 | 17 | cimport numpy as np 18 | import numpy as np 19 | from cpython cimport bool 20 | cimport cython 21 | from libc.math cimport fabs, fmax 22 | 23 | 24 | DTYPE = np.double 25 | ctypedef np.double_t DTYPE_t 26 | 27 | def fast_lowess(np.ndarray[DTYPE_t, ndim = 1] exog, 28 | np.ndarray[DTYPE_t, ndim = 1] endog, 29 | double frac = 2.0 / 3.0, 30 | Py_ssize_t iter = 3, 31 | double delta = 0.0): 32 | ''' 33 | LOWESS (Locally Weighted Scatterplot Smoothing) 34 | 35 | A lowess function that outs smoothed estimates of endog 36 | at the given exog values from points (exog, endog) 37 | 38 | Parameters 39 | ---------- 40 | exog: 1-D numpy array 41 | The x-values of the observed points 42 | endog: 1-D numpy array 43 | The y-values of the observed points 44 | frac: float 45 | Between 0 and 1. The fraction of the data used 46 | when estimating each y-value. 47 | iter: int 48 | The number of residual-based reweightings 49 | to perform. 50 | delta: float 51 | Distance within which to use linear-interpolation 52 | instead of weighted regression. 53 | 54 | Returns 55 | ------- 56 | out: numpy array 57 | A numpy array with two columns. The first column 58 | is the sorted x values and the second column the 59 | associated estimated y-values. 60 | 61 | Notes 62 | ----- 63 | This lowess function implements the algorithm given in the 64 | reference below using local linear estimates. 65 | 66 | Suppose the input data has N points. The algorithm works by 67 | estimating the `smooth` y_i by taking the frac*N closest points 68 | to (x_i,y_i) based on their x values and estimating y_i 69 | using a weighted linear regression. The weight for (x_j,y_j) 70 | is tricube function applied to |x_i-x_j|. 71 | 72 | If it > 1, then further weighted local linear regressions 73 | are performed, where the weights are the same as above 74 | times the _lowess_bisquare function of the residuals. Each iteration 75 | takes approximately the same amount of time as the original fit, 76 | so these iterations are expensive. They are most useful when 77 | the noise has extremely heavy tails, such as Cauchy noise. 78 | Noise with less heavy-tails, such as t-distributions with df>2, 79 | are less problematic. The weights downgrade the influence of 80 | points with large residuals. In the extreme case, points whose 81 | residuals are larger than 6 times the median absolute residual 82 | are given weight 0. 83 | 84 | delta can be used to save computations. For each x_i, regressions 85 | are skipped for points closer than delta. The next regression is 86 | fit for the farthest point within delta of x_i and all points in 87 | between are estimated by linearly interpolating between the two 88 | regression fits. 89 | 90 | Judicious choice of delta can cut computation time considerably 91 | for large data (N > 5000). A good choice is delta = 0.01 * 92 | range(exog). 93 | 94 | Some experimentation is likely required to find a good 95 | choice of frac and iter for a particular dataset. 96 | 97 | References 98 | ---------- 99 | Cleveland, W.S. (1979) "Robust Locally Weighted Regression 100 | and Smoothing Scatterplots". Journal of the American Statistical 101 | Association 74 (368): 829-836. 102 | 103 | Examples 104 | -------- 105 | The below allows a comparison between how different the fits from 106 | lowess for different values of frac can be. 107 | 108 | >>> import numpy as np 109 | >>> import statsmodels.api as sm 110 | >>> import cylowess 111 | >>> lowess = cylowess.lowess 112 | >>> x = np.random.uniform(low = -2*np.pi, high = 2*np.pi, size=500) 113 | >>> y = np.sin(x) + np.random.normal(size=len(x)) 114 | >>> z = lowess(x,y) 115 | >>> w = lowess(x,y, frac=1./3) 116 | 117 | This gives a similar comparison for when it is 0 vs not. 118 | 119 | >>> import numpy as np 120 | >>> import scipy.stats as stats 121 | >>> import statsmodels.api as sm 122 | >>> import cylowess 123 | >>> lowess = cylowess.lowess 124 | >>> x = np.random.uniform(low = -2*np.pi, high = 2*np.pi, size=500) 125 | >>> y = np.sin(x) + stats.cauchy.rvs(size=len(x)) 126 | >>> z = lowess(x,y, frac= 1./3, it=0) 127 | >>> w = lowess(x,y, frac=1./3) 128 | 129 | ''' 130 | cdef: 131 | Py_ssize_t n 132 | int k 133 | Py_ssize_t robiter, i, left_end, right_end 134 | int last_fit_i, 135 | np.ndarray[np.int_t, ndim = 1] sort_index 136 | np.ndarray[DTYPE_t, ndim = 1] x, y 137 | np.ndarray[DTYPE_t, ndim = 1] y_fit 138 | np.ndarray[DTYPE_t, ndim = 1] weights 139 | np.ndarray[DTYPE_t, ndim = 1] resid_weights 140 | 141 | 142 | # Inputs should be vectors (1-D arrays) of the 143 | # same length. 144 | if exog.ndim != 1: 145 | raise ValueError('exog must be a vector') 146 | if endog.ndim != 1: 147 | raise ValueError('endog must be a vector') 148 | if endog.shape[0] != exog.shape[0] : 149 | raise ValueError('exog and endog must have same length') 150 | 151 | # Cut out missing values 152 | x = exog[(np.isfinite(exog) & np.isfinite(endog))] 153 | y = endog[(np.isfinite(exog) & np.isfinite(endog))] 154 | 155 | # Sort both inputs according to the ascending order of x values 156 | sort_index = np.argsort(exog) 157 | x = np.array(x[sort_index]) 158 | y = np.array(y[sort_index]) 159 | n = x.shape[0] 160 | 161 | # The number of neighbors in each regression. 162 | k = int(frac * n) 163 | 164 | # frac should be set, so that 2 <= k <= n. 165 | # Conform them instead of throwing error. 166 | if k < 2: 167 | k = 2 168 | if k > n: 169 | k = n 170 | 171 | y_fit = np.zeros(n, dtype = DTYPE) 172 | resid_weights = np.zeros(n, dtype = DTYPE) 173 | 174 | iter += 1 # Add one to it for initial run. 175 | for robiter in xrange(iter): 176 | i = 0 177 | last_fit_i = -1 178 | left_end = 0 179 | right_end = k 180 | y_fit = np.zeros(n, dtype = DTYPE) 181 | 182 | # 'do' Fit y[i]'s 'until' the end of the regression 183 | while True: 184 | # Re-initialize the weights for each point x[i]. 185 | weights = np.zeros(n, dtype = DTYPE) 186 | 187 | # Describe the neighborhood around the current x[i]. 188 | left_end, right_end, radius = update_neighborhood(x, i, n, 189 | left_end, 190 | right_end) 191 | 192 | # Calculate the weights for the regression in this neighborhood. 193 | # Determine if at least some weights are positive, so a regression 194 | # is ok. 195 | reg_ok = calculate_weights(x, weights, resid_weights, i, left_end, 196 | right_end, radius, robiter > 0) 197 | 198 | # If ok, run the regression 199 | calculate_y_fit(x, y, i, y_fit, weights, left_end, right_end, 200 | reg_ok) 201 | 202 | # If we skipped some points (because of how delta was set), go back 203 | # and fit them by linear interpolation. 204 | if last_fit_i < (i - 1): 205 | interpolate_skipped_fits(x, y_fit, i, last_fit_i) 206 | 207 | # Update the last fit counter to indicate we've now fit this point. 208 | # Find the next i for which we'll run a regression. 209 | i, last_fit_i = update_indices(x, y_fit, delta, i, n, last_fit_i) 210 | 211 | if last_fit_i >= n-1: 212 | break 213 | 214 | # Calculate residual weights, but don't bother on the last iteration. 215 | if robiter < iter - 1: 216 | resid_weights = calculate_residual_weights(y, y_fit) 217 | 218 | 219 | #return np.array([x, y_fit]).T 220 | return np.array(y_fit).T 221 | 222 | 223 | def update_neighborhood(np.ndarray[DTYPE_t, ndim = 1] x, 224 | Py_ssize_t i, 225 | Py_ssize_t n, 226 | Py_ssize_t left_end, 227 | Py_ssize_t right_end): 228 | ''' 229 | Find the indices bounding the k-nearest-neighbors of the current 230 | point. 231 | 232 | Parameters 233 | ---------- 234 | x: 1-D numpy array 235 | The input x-values 236 | i: indexing integer 237 | The index of the point currently being fit. 238 | n: indexing integer 239 | The length of the input vectors, x and y. 240 | left_end: indexing integer 241 | The index of the left-most point in the neighborhood 242 | of x[i-1] (the previously-fit point). 243 | right_end: indexing integer 244 | The index of the right-most point in the neighborhood 245 | of x[i-1]. Non-inclusive, s.t. the neighborhood is 246 | x[left_end] <= x < x[right_end]. 247 | radius: float 248 | The radius of the current neighborhood. The larger of 249 | distances between x[i] and its left-most or right-most 250 | neighbor. 251 | 252 | Returns 253 | ------- 254 | left_end: indexing integer 255 | The index of the left-most point in the neighborhood 256 | of x[i] (the current point). 257 | right_end: indexing integer 258 | The index of the right-most point in the neighborhood 259 | of x[i]. Non-inclusive, s.t. the neighborhood is 260 | x[left_end] <= x < x[right_end]. 261 | radius: float 262 | The radius of the current neighborhood. The larger of 263 | distances between x[i] and its left-most or right-most 264 | neighbor. 265 | ''' 266 | 267 | cdef double radius 268 | # A subtle loop. Start from the current neighborhood range: 269 | # [left_end, right_end). Shift both ends rightwards by one 270 | # (so that the neighborhood still contains k points), until 271 | # the current point is in the center (or just to the left of 272 | # the center) of the neighborhood. This neighborhood will 273 | # contain the k-nearest neighbors of x[i]. 274 | # 275 | # Once the right end hits the end of the data, hold the 276 | # neighborhood the same for the remaining x[i]s. 277 | while True: 278 | if right_end < n: 279 | 280 | if (x[i] > (x[left_end] + x[right_end]) / 2.0): 281 | left_end += 1 282 | right_end += 1 283 | else: 284 | break 285 | else: 286 | break 287 | 288 | radius = fmax(x[i] - x[left_end], x[right_end-1] - x[i]) 289 | 290 | return left_end, right_end, radius 291 | 292 | cdef bool calculate_weights(np.ndarray[DTYPE_t, ndim = 1] x, 293 | np.ndarray[DTYPE_t, ndim = 1] weights, 294 | np.ndarray[DTYPE_t, ndim = 1] resid_weights, 295 | Py_ssize_t i, 296 | Py_ssize_t left_end, 297 | Py_ssize_t right_end, 298 | double radius, 299 | bool use_resid_weights): 300 | ''' 301 | 302 | Parameters 303 | ---------- 304 | x: 1-D vector 305 | The input x-values. 306 | weights: 1-D numpy array 307 | The vector of regression weights. 308 | resid_weights: 1-D numpy array 309 | The vector of residual weights from the last iteration. 310 | i: indexing integer 311 | The index of the point currently being fit. 312 | left_end: indexing integer 313 | The index of the left-most point in the neighborhood of 314 | x[i]. 315 | right_end: indexing integer 316 | The index of the right-most point in the neighborhood 317 | of x[i]. Non-inclusive, s.t. the neighborhood is 318 | x[left_end] <= x < x[right_end]. 319 | radius: float 320 | The radius of the current neighborhood. The larger of 321 | distances between x[i] and its left-most or right-most 322 | neighbor. 323 | use_resid_weights: boolean 324 | If True, multiply the x-distance weights by the residual 325 | weights from the last iteration of regressions. Set to 326 | False on the first iteration (since there are no residuals 327 | yet) and True on the subsequent ``robustifying`` iterations. 328 | 329 | 330 | Returns 331 | ------- 332 | reg_ok: boolean 333 | If True, at least some points have positive weight, and the 334 | regression will be run. If False, the regression is skipped 335 | and y_fit[i] is set to equal y[i]. 336 | Also, changes elements of weights in-place. 337 | ''' 338 | 339 | cdef: 340 | np.ndarray[DTYPE_t, ndim = 1] x_j = x[left_end:right_end] 341 | np.ndarray[DTYPE_t, ndim = 1] dist_i_j = np.abs(x_j - x[i]) / radius 342 | bool reg_ok = True 343 | double sum_weights 344 | 345 | # Assign the distance measure to the weights, then apply the tricube 346 | # function to change in-place. 347 | # use_resid_weights will be False on the first iteration, then True 348 | # on the subsequent ones, after some residuals have been calculated. 349 | weights[left_end:right_end] = dist_i_j 350 | if use_resid_weights == False: 351 | tricube(weights[left_end:right_end]) 352 | if use_resid_weights == True: 353 | tricube(weights[left_end:right_end]) 354 | weights[left_end:right_end] = (weights[left_end:right_end] * 355 | resid_weights[left_end:right_end]) 356 | 357 | sum_weights = np.sum(weights[left_end:right_end]) 358 | 359 | if sum_weights <= 0.0: 360 | reg_ok = False 361 | else: 362 | weights[left_end:right_end] = weights[left_end:right_end] / sum_weights 363 | 364 | return reg_ok 365 | 366 | 367 | cdef void calculate_y_fit(np.ndarray[DTYPE_t, ndim = 1] x, 368 | np.ndarray[DTYPE_t, ndim = 1] y, 369 | Py_ssize_t i, 370 | np.ndarray[DTYPE_t, ndim = 1] y_fit, 371 | np.ndarray[DTYPE_t, ndim = 1] weights, 372 | Py_ssize_t left_end, 373 | Py_ssize_t right_end, 374 | bool reg_ok): 375 | ''' 376 | Calculate smoothed/fitted y-value by weighted regression. 377 | 378 | Parameters 379 | ---------- 380 | x: 1-D numpy array 381 | The vector of input x-values. 382 | y: 1-D numpy array 383 | The vector of input y-values. 384 | i: indexing integer 385 | The index of the point currently being fit. 386 | y_fit: 1-D numpy array 387 | The vector of fitted y-values. 388 | weights: 1-D numpy array 389 | The vector of regression weights. 390 | left_end: indexing integer 391 | The index of the left-most point in the neighborhood of 392 | x[i]. 393 | right_end: indexing integers 394 | The index of the right-most point in the neighborhood 395 | of x[i]. Non-inclusive, s.t. the neighborhood is 396 | x[left_end] <= x < x[right_end]. 397 | reg_ok: boolean 398 | If True, at least some points have positive weight, and the 399 | regression will be run. If False, the regression is skipped 400 | and y_fit[i] is set to equal y[i]. 401 | 402 | Returns 403 | ------- 404 | Nothing. Changes y_fit[i] in-place. 405 | 406 | Notes 407 | ----- 408 | No regression function (e.g. lstsq) is called. Instead "projection 409 | vector" p_i_j is calculated, and y_fit[i] = sum(p_i_j * y[j]) = y_fit[i] 410 | for j s.t. x[j] is in the neighborhood of x[i]. p_i_j is a function of 411 | the weights, x[i], and its neighbors. 412 | ''' 413 | 414 | cdef: 415 | double sum_weighted_x = 0, weighted_sqdev_x = 0, p_i_j 416 | 417 | if reg_ok == False: 418 | y_fit[i] = y[i] 419 | else: 420 | for j in xrange(left_end, right_end): 421 | sum_weighted_x += weights[j] * x[j] 422 | for j in xrange(left_end, right_end): 423 | weighted_sqdev_x += weights[j] * (x[j] - sum_weighted_x) ** 2 424 | for j in xrange(left_end, right_end): 425 | p_i_j = weights[j] * (1.0 + (x[i] - sum_weighted_x) * 426 | (x[j] - sum_weighted_x) / weighted_sqdev_x) 427 | y_fit[i] += p_i_j * y[j] 428 | 429 | cdef void interpolate_skipped_fits(np.ndarray[DTYPE_t, ndim = 1] x, 430 | np.ndarray[DTYPE_t, ndim = 1] y_fit, 431 | Py_ssize_t i, 432 | Py_ssize_t last_fit_i): 433 | ''' 434 | Calculate smoothed/fitted y by linear interpolation between the current 435 | and previous y fitted by weighted regression. 436 | Called only if delta > 0. 437 | 438 | Parameters 439 | ---------- 440 | x: 1-D numpy array 441 | The vector of input x-values. 442 | y_fit: 1-D numpy array 443 | The vector of fitted y-values 444 | i: indexing integer 445 | The index of the point currently being fit by weighted 446 | regression. 447 | last_fit_i: indexing integer 448 | The index of the last point fit by weighted regression. 449 | 450 | Returns 451 | ------- 452 | Nothing: changes elements of y_fit in-place. 453 | ''' 454 | 455 | cdef np.ndarray[DTYPE_t, ndim = 1] a 456 | 457 | a = x[(last_fit_i + 1): i] - x[last_fit_i] 458 | a = a / (x[i] - x[last_fit_i]) 459 | y_fit[(last_fit_i + 1): i] = a * y_fit[i] + (1.0 - a) * y_fit[last_fit_i] 460 | 461 | 462 | def update_indices(np.ndarray[DTYPE_t, ndim = 1] x, 463 | np.ndarray[DTYPE_t, ndim = 1] y_fit, 464 | double delta, 465 | Py_ssize_t i, 466 | Py_ssize_t n, 467 | Py_ssize_t last_fit_i): 468 | ''' 469 | Update the counters of the local regression. 470 | 471 | Parameters 472 | ---------- 473 | x: 1-D numpy array 474 | The vector of input x-values. 475 | y_fit: 1-D numpy array 476 | The vector of fitted y-values 477 | delta: float 478 | Indicates the range of x values within which linear 479 | interpolation should be used to estimate y_fit instead 480 | of weighted regression. 481 | i: indexing integer 482 | The index of the current point being fit. 483 | n: indexing integer 484 | The length of the input vectors, x and y. 485 | last_fit_i: indexing integer 486 | The last point at which y_fit was calculated. 487 | 488 | Returns 489 | ------- 490 | i: indexing integer 491 | The next point at which to run a weighted regression. 492 | last_fit_i: indexing integer 493 | The updated last point at which y_fit was calculated 494 | 495 | Notes 496 | ----- 497 | The relationship between the outputs is s.t. x[i+1] > 498 | x[last_fit_i] + delta. 499 | 500 | ''' 501 | cdef: 502 | Py_ssize_t k 503 | double cutpoint 504 | 505 | last_fit_i = i 506 | # For most points within delta of the current point, we skip the 507 | # weighted linear regression (which save much computation of 508 | # weights and fitted points). Instead, we'll jump to the last 509 | # point within delta, fit the weighted regression at that point, 510 | # and linearly interpolate in between. 511 | 512 | # This loop increments until we fall just outside of delta distance, 513 | # copying the results for any repeated x's along the way. 514 | cutpoint = x[last_fit_i] + delta 515 | for k in range(last_fit_i + 1, n): 516 | if x[k] > cutpoint: 517 | break 518 | if x[k] == x[last_fit_i]: 519 | # if tied with previous x-value, just use the already 520 | # fitted y, and update the last-fit counter. 521 | y_fit[k] = y_fit[last_fit_i] 522 | last_fit_i = k 523 | 524 | # i, which indicates the next point to fit the regression at, is 525 | # either one prior to k (since k should be the first point outside 526 | # of delta) or is just incremented + 1 if k = i+1. This insures we 527 | # always step forward. 528 | i = max(k-1, last_fit_i + 1) 529 | 530 | return i, last_fit_i 531 | 532 | 533 | def calculate_residual_weights(np.ndarray[DTYPE_t, ndim = 1] y, 534 | np.ndarray[DTYPE_t, ndim = 1] y_fit): 535 | ''' 536 | Calculate residual weights for the next `robustifying` iteration. 537 | 538 | Parameters 539 | ---------- 540 | y: 1-D numpy array 541 | The vector of actual input y-values. 542 | y_fit: 1-D numpy array 543 | The vector of fitted y-values from the current 544 | iteration. 545 | 546 | Returns 547 | ------- 548 | resid_weights: 1-D numpy array 549 | The vector of residual weights, to be used in the 550 | next iteration of regressions. 551 | ''' 552 | 553 | std_resid = np.abs(y - y_fit) 554 | std_resid /= 6.0 * np.median(std_resid) 555 | 556 | # Some trimming of outlier residuals. 557 | std_resid[std_resid >= 1.0] = 1.0 558 | #std_resid[std_resid >= 0.999] = 1.0 559 | #std_resid[std_resid <= 0.001] = 0.0 560 | 561 | resid_weights = bisquare(std_resid) 562 | 563 | return resid_weights 564 | 565 | 566 | cdef void tricube(np.ndarray[DTYPE_t, ndim = 1] x): 567 | ''' 568 | The tri-cubic function (1 - x**3)**3. Used to weight neighboring 569 | points along the x-axis based on their distance to the current point. 570 | 571 | Parameters 572 | ---------- 573 | x: 1-D numpy array 574 | A vector of neighbors` distances from the current point, 575 | in units of the neighborhood radius. 576 | 577 | Returns 578 | ------- 579 | Nothing. Changes array elements in-place 580 | ''' 581 | 582 | # fast_array_cube is an elementwise, in-place cubed-power 583 | # operator. 584 | fast_array_cube(x) 585 | x[:] = np.negative(x) 586 | x += 1 587 | fast_array_cube(x) 588 | 589 | 590 | cdef void fast_array_cube(np.ndarray[DTYPE_t, ndim = 1] x): 591 | ''' 592 | A fast, elementwise, in-place cube operator. Called by the 593 | tricube function. 594 | 595 | Parameters 596 | ---------- 597 | x: 1-D numpy array 598 | 599 | Returns 600 | ------- 601 | Nothing. Changes array elements in-place. 602 | ''' 603 | 604 | x2 = x*x 605 | x *= x2 606 | 607 | 608 | def bisquare(np.ndarray[DTYPE_t, ndim = 1] x): 609 | ''' 610 | The bi-square function (1 - x**2)**2. 611 | 612 | Used to weight the residuals in the `robustifying` 613 | iterations. Called by the calculate_residual_weights function. 614 | 615 | Parameters 616 | ---------- 617 | x: 1-D numpy array 618 | A vector of absolute regression residuals, in units of 619 | 6 times the median absolute residual. 620 | 621 | Returns 622 | ------- 623 | A 1-D numpy array of residual weights. 624 | ''' 625 | 626 | return (1.0 - x**2)**2 627 | 628 | 629 | 630 | -------------------------------------------------------------------------------- /src/ewma.py: -------------------------------------------------------------------------------- 1 | #coding=utf-8 2 | import numpy as np 3 | 4 | def holt_winters_second_order_ewma( x, span=10, beta=1.0/8.0 ): 5 | n = len(x) 6 | alpha = 2.0 / ( 1 + span ) 7 | s = np.zeros(( n, )) 8 | b = np.zeros(( n, )) 9 | s[0] = x[0] 10 | for i in xrange( 1, n ): 11 | s[i] = alpha * x[i] + ( 1 - alpha )*( s[i-1] + b[i-1] ) 12 | b[i] = beta * ( s[i] - s[i-1] ) + ( 1 - beta ) * b[i-1] 13 | return s 14 | 15 | # exponentially-weighted moving average with windows 16 | def ewma_with_window(x, alpha=7.0/8.0, window=30): 17 | n = len(x) 18 | y = np.zeros((n,)) 19 | y[0] = x[0] 20 | if window<=0 : 21 | for i in xrange(1,n): 22 | y[i] = alpha * y[i-1] + (1 - alpha) * x[i] 23 | return y 24 | 25 | for i in xrange(1, n): 26 | if i yest 32 | 33 | Lowess smoother: Robust locally weighted regression. 34 | The lowess function fits a nonparametric regression curve to a scatterplot. 35 | The arrays x and y contain an equal number of elements; each pair 36 | (x[i], y[i]) defines a data point in the scatterplot. The function returns 37 | the estimated (smooth) values of y. 38 | 39 | The smoothing span is given by f. A larger value for f will result in a 40 | smoother curve. The number of robustifying iterations is given by iter. The 41 | function will run faster with a smaller number of iterations. 42 | 43 | x and y should be numpy float arrays of equal length. The return value is 44 | also a numpy float array of that length. 45 | 46 | e.g. 47 | >>> import numpy 48 | >>> x = numpy.array([4, 4, 7, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 12, 49 | ... 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 50 | ... 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 51 | ... 20, 22, 23, 24, 24, 24, 24, 25], numpy.float) 52 | >>> y = numpy.array([2, 10, 4, 22, 16, 10, 18, 26, 34, 17, 28, 14, 20, 24, 53 | ... 28, 26, 34, 34, 46, 26, 36, 60, 80, 20, 26, 54, 32, 40, 54 | ... 32, 40, 50, 42, 56, 76, 84, 36, 46, 68, 32, 48, 52, 56, 55 | ... 64, 66, 54, 70, 92, 93, 120, 85], numpy.float) 56 | >>> result = lowess(x, y) 57 | >>> len(result) 58 | 50 59 | >>> print("[%0.2f, ..., %0.2f]" % (result[0], result[-1])) 60 | [4.85, ..., 84.98] 61 | """ 62 | n = len(x) 63 | r = int(ceil(f*n)) 64 | h = [np.sort(np.abs(x - x[i]))[r] for i in range(n)] 65 | w = np.clip(np.abs((x[:,None] - x[None,:]) / h), 0.0, 1.0) 66 | w = (1 - w**3)**3 67 | smoothy = np.zeros(n) 68 | delta = np.ones(n) 69 | for iteration in range(iter): 70 | for i in range(n): 71 | weights = delta * w[:,i] 72 | b = np.array([np.sum(weights*y), np.sum(weights*y*x)]) 73 | A = np.array([[np.sum(weights), np.sum(weights*x)], 74 | [np.sum(weights*x), np.sum(weights*x*x)]]) 75 | beta = linalg.solve(A, b) 76 | smoothy[i] = beta[0] + beta[1]*x[i] 77 | 78 | residuals = y - smoothy 79 | s = np.median(np.abs(residuals)) 80 | delta = np.clip(residuals / (6.0 * s), -1, 1) 81 | delta = (1 - delta**2)**2 82 | 83 | return smoothy 84 | 85 | -------------------------------------------------------------------------------- /src/poly.py: -------------------------------------------------------------------------------- 1 | from numpy import * 2 | from numpy.random import normal 3 | 4 | def poly_regression(x, y, d ): 5 | m = polyfit(x,y,d) 6 | return poly1d(m)(x), poly1d(m) 7 | 8 | def _chunks(lst, size): 9 | for i in xrange(0, len(lst), size): 10 | yield lst[i:i+size] 11 | 12 | def _poly_regression(x, y, d, chunks_size, reduce_fn): 13 | ys = [reduce_fn(yy) for yy in _chunks(y, chunks_size)] 14 | xs = [max(xx) for xx in _chunks(x, chunks_size)] 15 | poly_y, poly_fn = poly_regression(xs, ys, d) 16 | poly_y = poly_fn(x) 17 | return poly_y, poly_fn 18 | 19 | def poly_regression_max(x, y, d, chunks_size): 20 | return _poly_regression(x, y, d, chunks_size, max) 21 | 22 | def poly_regression_min(x, y, d, chunks_size): 23 | return _poly_regression(x, y, d, chunks_size, min) 24 | -------------------------------------------------------------------------------- /test/host1.json: -------------------------------------------------------------------------------- 1 | {"aggrType":"","dataType":"detail","resultData":{"t172021094005.cm3":{"0xilovexxx":{"cpu":{"2014-02-17 06:48:20":"0.64","2014-02-17 06:49:20":"0.46","2014-02-17 06:50:20":"0.43","2014-02-17 06:51:20":"0.48","2014-02-17 06:52:20":"0.42","2014-02-17 06:53:20":"0.44","2014-02-17 06:54:20":"0.48","2014-02-17 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-------------------------------------------------------------------------------- https://raw.githubusercontent.com/haoel/regression-algorithm/53f720e057c4d3a9ebd9611e871f38fd718cfdcc/test/imgs/liner.png -------------------------------------------------------------------------------- /test/imgs/lowess.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/haoel/regression-algorithm/53f720e057c4d3a9ebd9611e871f38fd718cfdcc/test/imgs/lowess.png -------------------------------------------------------------------------------- /test/imgs/poly.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/haoel/regression-algorithm/53f720e057c4d3a9ebd9611e871f38fd718cfdcc/test/imgs/poly.png -------------------------------------------------------------------------------- /test/readjson.py: -------------------------------------------------------------------------------- 1 | #coding=utf-8 2 | #import pdb 3 | 4 | 5 | def read_json(filename): 6 | 7 | import json 8 | with open(filename) as json_data: 9 | d = json.load(json_data) 10 | json_data.close() 11 | 12 | #result = d['resultData']["t172021094005.cm3"]["0xilovexxx"] 13 | result = d['resultData'] 14 | for k, v in result.iteritems(): 15 | result = v 16 | break; 17 | for k, v in result.iteritems(): 18 | result = v 19 | break; 20 | 21 | cpu, cpu_time = parse( sorted(result["cpu"].items()) ) 22 | mem, mem_time = parse( sorted(result["mem"].items()) ) 23 | load, load_time = parse( sorted(result["load5"].items())) 24 | 25 | return cpu, cpu_time, mem, mem_time, load, load_time 26 | 27 | 28 | def parse(data): 29 | from datetime import datetime 30 | import time 31 | data_list = [] 32 | time_list = [] 33 | for item in data: 34 | t = time.mktime(datetime.strptime(item[0], "%Y-%m-%d %H:%M:%S").timetuple()) 35 | time_list.append( t ) 36 | data_list.append(float(item[1])) 37 | 38 | return (data_list, time_list) 39 | -------------------------------------------------------------------------------- /test/test.py: -------------------------------------------------------------------------------- 1 | import pdb 2 | 3 | import matplotlib 4 | matplotlib.use('Agg') 5 | 6 | import matplotlib.pyplot as plt 7 | fig = plt.gcf() 8 | fig.set_size_inches(22.5,4.5) 9 | 10 | import pylab as pl 11 | 12 | 13 | def testLowess(x, y): 14 | 15 | from lowess import lowess 16 | 17 | smoothy = [] 18 | f = [] 19 | base = 0.005 20 | for i in range(8): 21 | base = base + i*0.015 22 | f.append(base) 23 | smoothy.append( lowess(x, y, f=base, iter=1) ) 24 | 25 | fig = plt.gcf() 26 | 27 | pl.clf() 28 | pl.plot(x, y, label='Raw Data') 29 | for i in range(len(smoothy)): 30 | pl.plot(x, smoothy[i], label='Smooth('+str(f[i])+')') 31 | pl.legend() 32 | pl.savefig('./imgs/lowess') 33 | print "Successfully created picture file lowess.png" 34 | 35 | 36 | def testLinerRegression(x,y): 37 | 38 | from liner import liner_regress 39 | 40 | line1, _, _ = liner_regress(x, y, type=1) 41 | 42 | line2, _, _ = liner_regress(x, y, type=0) 43 | 44 | pl.clf() 45 | 46 | pl.plot(x,line1,'r-',x, line2, 'g-', x,y,'b-') 47 | 48 | pl.savefig("./imgs/liner") 49 | print "Successfully created picture file liner.png" 50 | 51 | 52 | 53 | def testPolyRegression(x, y): 54 | 55 | from poly import poly_regression 56 | 57 | d = 4 58 | polyline, polyfunc = poly_regression(x, y, d) 59 | print polyfunc 60 | 61 | 62 | pl.clf() 63 | 64 | pl.plot(x, y, label="raw data") 65 | pl.plot(x, polyline, label="polyfit") 66 | 67 | pl.legend() 68 | 69 | pl.savefig("./imgs/poly") 70 | print "Successfully created picture file poly.png" 71 | 72 | def testHoltWintersEWMAverage(x, y): 73 | 74 | from ewma import holt_winters_second_order_ewma 75 | 76 | beta = 1.0/8.0; 77 | span = 10.0 78 | n = len(y) 79 | ave_y = holt_winters_second_order_ewma(y, span, beta) 80 | 81 | pl.clf() 82 | pl.plot(x, y, label="raw data") 83 | pl.plot(x, ave_y, label="HoltWintersEMWA") 84 | pl.legend() 85 | pl.savefig("./imgs/holt_winters_average") 86 | print "Successfully created picture file holt_winters_average.png" 87 | 88 | def testEWMAverage(x, y): 89 | 90 | from ewma import ewma_with_window 91 | 92 | ave_y = ewma_with_window(y, 0.8, 50 ) 93 | pl.clf() 94 | pl.plot(x, y, label="raw data") 95 | pl.plot(x, ave_y, label="EWMAverage") 96 | pl.legend() 97 | pl.savefig("./imgs/emwa_window") 98 | print "Successfully created picture file emwa_window.png" 99 | 100 | if __name__ == '__main__': 101 | #pdb.set_trace() 102 | import numpy as np 103 | from readjson import read_json 104 | cpu, cpu_time, mem, mem_time, load, load_time = read_json("./host2.json") 105 | 106 | 107 | x = np.asarray(cpu_time) 108 | y = cpu 109 | 110 | print "----------------- LOWESS ---------------------" 111 | testLowess(x, y) 112 | 113 | print "------------ Liner Regression ----------------" 114 | testLinerRegression(x, y) 115 | 116 | print "------------- Poly Regression ----------------" 117 | testPolyRegression(x, y) 118 | 119 | print "---- Exponential Weighted Moving Average -----" 120 | testEWMAverage(x, y) 121 | testHoltWintersEWMAverage(x, y) 122 | -------------------------------------------------------------------------------- /test/testcylowess.py: -------------------------------------------------------------------------------- 1 | import pdb 2 | 3 | import matplotlib 4 | matplotlib.use('Agg') 5 | 6 | import matplotlib.pyplot as plt 7 | fig = plt.gcf() 8 | fig.set_size_inches(220.5,4.5) 9 | 10 | import pylab as pl 11 | 12 | 13 | 14 | def lowessProcess(x, y): 15 | 16 | from cylowess import fast_lowess 17 | 18 | smoothy = [] 19 | f = [] 20 | base = 0.005 21 | y = np.asarray(y) 22 | #delta = (x.max() - x.min()) * 0.001 23 | delta = 0 24 | for i in range(8): 25 | base = base + i*0.015 26 | f.append(base) 27 | smoothy.append( fast_lowess( x,y, frac=base, iter=1, delta=delta) ) 28 | return f, smoothy 29 | 30 | 31 | def savefile(x,y, smoothy, f): 32 | fig = plt.gcf() 33 | 34 | pl.clf() 35 | pl.plot(x, y, label='Raw Data') 36 | for i in range(len(smoothy)): 37 | pl.plot(x, smoothy[i], label='Smooth('+str(f[i])+')') 38 | pl.legend() 39 | pl.savefig('./imgs/lowess') 40 | print "Successfully created picture file lowess.png" 41 | 42 | 43 | 44 | 45 | if __name__ == '__main__': 46 | #pdb.set_trace() 47 | import numpy as np 48 | from readjson import read_json 49 | cpu, cpu_time, mem, mem_time, load, load_time = read_json("./host2.json") 50 | 51 | 52 | #x = np.asarray(cpu_time) 53 | y=[] 54 | for i in xrange(30): 55 | y = y+cpu 56 | x = np.arange(0.0, len(y)) 57 | 58 | print "----------------- LOWESS ---------------------" 59 | 60 | def chunks(l, n): 61 | for i in xrange(0, len(l), n): 62 | yield l[i:i+n] 63 | 64 | chunk_size = 1000 65 | x_chunks = list(chunks(x,chunk_size)) 66 | y_chunks = list(chunks(y,chunk_size)) 67 | 68 | lowessy=None 69 | f =[] 70 | for i in xrange(len(x_chunks)): 71 | f, smoothy = lowessProcess(x_chunks[i], y_chunks[i]) 72 | if (lowessy == None): 73 | lowessy = smoothy 74 | else: 75 | #pdb.set_trace() 76 | for i in range(len(smoothy)): 77 | lowessy[i]=np.concatenate([lowessy[i],smoothy[i]]) 78 | 79 | savefile(x, y, lowessy, f) 80 | 81 | 82 | --------------------------------------------------------------------------------