python - 使用 'fixed points' 在 Python 中插入数据

Question

我的时间序列数据本质上是周期性的(而且是正弦曲线的)。我根据周期对数据进行了重新定相，以便所有点都位于 0 和 1 之间。您可以认为这与从 0 到周期 2pi 的正弦波中采样点相同。这是一个典型案例:

我尝试使用各种 scipy.interpolate 函数插入这些数据，例如:

>>> scipy.interpolate.UnivariateSpline(x,y)(numpy.linspace(0, 0.99, 100))
array([ 15.13403109,  15.10173144,  15.07070986,  15.04094629,
        15.01242068,  14.98511296,  14.95900308,  14.93407098,
        14.91029659,  14.88765987,  14.86614074,  14.84571915,
        14.82637504,  14.80808836,  14.79083904,  14.77460702,
        14.75937224,  14.74511465,  14.73181418,  14.71945078,
        14.70800439,  14.69745494,  14.68778239,  14.67896666,
        14.6709877 ,  14.66382545,  14.65745985,  14.65187085,
        14.64703838,  14.64294238,  14.6395628 ,  14.63687957,
        14.63487264,  14.63352194,  14.63280742,  14.63270902,
        14.63320668,  14.63428034,  14.63590994,  14.63807542,
        14.64075672,  14.64393378,  14.64758655,  14.65169496,
        14.65623896,  14.66119848,  14.66655347,  14.67228387,
        14.67836961,  14.68479064,  14.69152691,  14.69855834,
        14.70586488,  14.71342648,  14.72122306,  14.72923458,
        14.73744098,  14.74582219,  14.75435815,  14.76302882,
        14.77181411,  14.78069399,  14.78964838,  14.79865724,
        14.80770049,  14.81675809,  14.82580996,  14.83483606,
        14.84381632,  14.85273069,  14.8615591 ,  14.87028149,
        14.87887781,  14.887328  ,  14.895612  ,  14.90370974,
        14.91160117,  14.91926624,  14.92668487,  14.93383702,
        14.94070261,  14.9472616 ,  14.95349392,  14.95937952,
        14.96489834,  14.9700303 ,  14.97475537,  14.97905347,
        14.98290455,  14.98628855,  14.98918541,  14.99157507,
        14.99343747,  14.99475255,  14.99550026,  14.99566053,
        14.9952133 ,  14.99413852,  14.99241612,  14.99002605])

例如，x 为(请注意某些值重复):

>>> x
array([ 0.   ,  0.01 ,  0.016,  0.018,  0.024,  0.029,  0.034,  0.036,
        0.042,  0.046,  0.048,  0.053,  0.058,  0.062,  0.069,  0.071,
        0.072,  0.079,  0.083,  0.091,  0.096,  0.102,  0.102,  0.106,
        0.108,  0.111,  0.112,  0.112,  0.122,  0.131,  0.135,  0.136,
        0.137,  0.145,  0.164,  0.168,  0.172,  0.174,  0.177,  0.178,
        0.179,  0.197,  0.202,  0.205,  0.206,  0.213,  0.215,  0.222,
        0.229,  0.233,  0.235,  0.239,  0.239,  0.241,  0.248,  0.255,
        0.258,  0.259,  0.262,  0.264,  0.266,  0.267,  0.276,  0.28 ,
        0.281,  0.281,  0.285,  0.289,  0.292,  0.292,  0.294,  0.295,
        0.299,  0.304,  0.306,  0.309,  0.313,  0.317,  0.32 ,  0.32 ,
        0.335,  0.34 ,  0.341,  0.353,  0.357,  0.359,  0.364,  0.368,
        0.369,  0.369,  0.388,  0.39 ,  0.394,  0.396,  0.399,  0.401,
        0.404,  0.406,  0.407,  0.413,  0.415,  0.418,  0.423,  0.43 ,
        0.438,  0.439,  0.443,  0.445,  0.454,  0.455,  0.475,  0.478,
        0.478,  0.48 ,  0.48 ,  0.482,  0.485,  0.486,  0.488,  0.488,
        0.498,  0.498,  0.499,  0.508,  0.514,  0.525,  0.527,  0.531,
        0.535,  0.536,  0.546,  0.547,  0.551,  0.553,  0.556,  0.563,
        0.57 ,  0.579,  0.584,  0.59 ,  0.594,  0.595,  0.596,  0.606,
        0.606,  0.619,  0.628,  0.631,  0.632,  0.633,  0.638,  0.64 ,
        0.649,  0.652,  0.654,  0.655,  0.669,  0.674,  0.684,  0.688,
        0.689,  0.692,  0.697,  0.697,  0.703,  0.703,  0.703,  0.704,
        0.706,  0.715,  0.715,  0.717,  0.72 ,  0.721,  0.73 ,  0.739,
        0.746,  0.75 ,  0.751,  0.752,  0.757,  0.762,  0.766,  0.766,
        0.783,  0.785,  0.787,  0.79 ,  0.791,  0.791,  0.806,  0.809,
        0.81 ,  0.813,  0.815,  0.816,  0.816,  0.818,  0.82 ,  0.823,
        0.839,  0.849,  0.857,  0.859,  0.862,  0.864,  0.868,  0.869,
        0.875,  0.877,  0.887,  0.888,  0.893,  0.896,  0.905,  0.907,
        0.908,  0.925,  0.926,  0.936,  0.947,  0.949,  0.955,  0.957,
        0.962,  0.97 ,  0.972,  0.976,  0.979,  0.984,  0.985,  0.986,
        0.993,  1.   ])

例如，y 是:

>>> y
array([ 15.048,  15.046,  15.046,  15.037,  15.035,  15.048,  15.034,
        15.041,  15.03 ,  15.034,  15.037,  15.04 ,  15.038,  15.028,
        14.998,  14.976,  15.012,  15.007,  14.996,  14.979,  14.922,
        14.876,  14.881,  14.931,  14.912,  14.904,  14.906,  14.897,
        14.871,  14.786,  14.778,  14.78 ,  14.782,  14.788,  14.729,
        14.735,  14.661,  14.722,  14.668,  14.657,  14.715,  14.647,
        14.607,  14.627,  14.607,  14.625,  14.619,  14.592,  14.583,
        14.596,  14.596,  14.595,  14.584,  14.593,  14.601,  14.597,
        14.605,  14.596,  14.61 ,  14.6  ,  14.582,  14.609,  14.606,
        14.619,  14.601,  14.612,  14.619,  14.612,  14.612,  14.618,
        14.619,  14.62 ,  14.62 ,  14.619,  14.633,  14.629,  14.611,
        14.62 ,  14.629,  14.618,  14.645,  14.634,  14.633,  14.644,
        14.647,  14.649,  14.67 ,  14.661,  14.658,  14.67 ,  14.667,
        14.682,  14.676,  14.675,  14.68 ,  14.67 ,  14.673,  14.676,
        14.68 ,  14.654,  14.689,  14.699,  14.694,  14.691,  14.699,
        14.703,  14.683,  14.691,  14.706,  14.703,  14.715,  14.73 ,
        14.727,  14.72 ,  14.729,  14.718,  14.712,  14.721,  14.734,
        14.722,  14.738,  14.724,  14.73 ,  14.729,  14.735,  14.751,
        14.741,  14.752,  14.753,  14.765,  14.758,  14.759,  14.766,
        14.766,  14.774,  14.774,  14.768,  14.775,  14.789,  14.788,
        14.793,  14.787,  14.783,  14.808,  14.789,  14.793,  14.804,
        14.804,  14.793,  14.805,  14.808,  14.811,  14.825,  14.816,
        14.827,  14.827,  14.827,  14.838,  14.83 ,  14.839,  14.848,
        14.844,  14.834,  14.838,  14.845,  14.861,  14.856,  14.847,
        14.853,  14.868,  14.845,  14.857,  14.859,  14.859,  14.868,
        14.853,  14.871,  14.873,  14.875,  14.893,  14.882,  14.883,
        14.884,  14.899,  14.904,  14.907,  14.909,  14.903,  14.909,
        14.909,  14.91 ,  14.911,  14.904,  14.909,  14.933,  14.923,
        14.924,  14.907,  14.928,  14.913,  14.939,  14.944,  14.946,
        14.952,  14.935,  14.946,  14.943,  14.948,  14.952,  14.957,
        14.974,  14.981,  14.967,  14.967,  14.977,  14.987,  14.97 ,
        15.013,  14.98 ,  15.011,  15.004,  15.013,  15.   ,  15.017,
        15.02 ,  15.047,  15.03 ,  15.05 ,  15.029,  15.043,  15.038,
        15.03 ,  15.042,  15.052])

该函数在 0 处的计算结果应与 1 处的值(几乎)相同，因为基础数据是周期性的(就像我们期望对正弦进行插值的函数在 0 处与 2pi 处具有相同的值一样)。然而，它显然具有较长的左偏，并且与 0 附近的数据不太相似。0 和 1 处的值之间的差异大约为 0.144，大于数据集的标准差。

有什么想法吗？我可以在设置固定点时以某种方式进行插值，即边界的开始和结束应该大致相同的规范吗？

Answer 1

splrep/splev 函数对声称支持周期样条，参见per 参数。

它在 UnivariateSpline 中不可用是一个错误。这是一个最小的实现(但最好不要使用它，因为访问 _data 可能不向后兼容):

from scipy.interpolate import UnivariateSpline, splrep

class PeriodicUnivariateSpline(UnivariateSpline):
    def __init__(self, x, y, w=None, bbox=[None]*2, k=3, s=0):
        #_data == x,y,w,xb,xe,k,s,n,t,c,fp,fpint,nrdata,ier
        tck, fp, ier, msg = splrep(x, y, k=k, w=w, xb=bbox[0], xe=bbox[1],
                                   s=s, per=1, full_output=1)
        self._data = (x,y,w,bbox[0],bbox[1],k,s,len(tck[0]),tck[0],tck[1],
                      fp,None,None,ier)
        self._reset_class()