参考链接： Python中的numpy.arcsin
1） 巧用 where函数
where函数是numpy的内置，也是一个非常有用的函数，提供了快速并且灵活的计算功能。
def f_norm_1(data, estimate): residule = 0 for row_index in range(data.shape[0]): for column_index in range(data.shape[1]): if data[row_index][column_index] != 0: residule = (data[row_index][column_index] - estimate[row_index][column_index]) ** 2 return residule
def f_norm_2(data, estimate)
return sum(where(data != 0, (data-estimate) **2, 0))
这两段代码完成同样的功能，计算两个矩阵的差，然后将残差进行平方，注意，因为我需要的是考虑矩阵稀疏性，所以不能用内置的norm，函数1是我用普通的python写的，不太复杂，对于规模10*10的矩阵，计算200次耗时0.15s，函数2使用了where函数和sum函数，这两个函数都是为向量计算优化过的，不仅简介，而且耗时仅0.03s, 快了有五倍，不仅如此，有同学将NumPy和matlab做过比较，NumPy稍快一些，这已经是很让人兴奋的结果。
本篇我们看看NumPy中最为基本的Array操作
>>> from numpy import *
创建一个矩阵
>>> a=array([[1,2,3],[4,5,6]])>>> a.shape(2, 3)
>>> b=arange(15);print b[ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14]>>> b.reshape(3,5)array([[ 0, 1, 2, 3, 4],[ 5, 6, 7, 8, 9],[10, 11, 12, 13, 14]])
可以看到，A是2行3列的矩阵。通过arange方法，可以得到一个1维的数组。然后我们可以通过reshape方法改变它的维度。
>>> c=zeros((4,5));print c[[ 0. 0. 0. 0. 0.][ 0. 0. 0. 0. 0.][ 0. 0. 0. 0. 0.][ 0. 0. 0. 0. 0.]]
>>> d=ones((5,7));print d[[ 1. 1. 1. 1. 1. 1. 1.][ 1. 1. 1. 1. 1. 1. 1.][ 1. 1. 1. 1. 1. 1. 1.][ 1. 1. 1. 1. 1. 1. 1.][ 1. 1. 1. 1. 1. 1. 1.]]
>>> e=add(c,arange(20).reshape(4,5))>>> f=dot(e,d);print f[[ 10. 10. 10. 10. 10. 10. 10.][ 35. 35. 35. 35. 35. 35. 35.][ 60. 60. 60. 60. 60. 60. 60.][ 85. 85. 85. 85. 85. 85. 85.]]
使用zeros可以生成一个零矩阵。同理，用ones可以生成值全部为1的矩阵。我选择了一个4*5的矩阵e，和一个5*7的矩阵d做点乘。最后得到f矩阵。再举一个更加明显的例子：
>>> a=arange(5);print a[0 1 2 3 4]>>> b=arange(5).reshape(5,1);print b[[0][1][2][3][4]]>>> print dot(a,b)[30]
点积的效果更加明显了。
ndarray的几个常用属性：
· shape： 代表一个array的形态，是一个向量还是一个矩阵，抑或是一个更复杂的向量组。
· ndim： 代表这个array的维度
· size： 在array中拥有的元素数量
· itemsize： 这个array中每一个元素所需要占的字节数
· nbytes： 这个array的总字节数（=itemsize*size）
· real： 代表一个array中所有元素的实数部分
· imag： 同理，代表一个array中所有元素的虚数部分
· flat： 将这个array整理成一维的，可以索引的一系列的元素组合。它实际上是通过iterator实现的，我们可以通过for x in array.flat来取得到所有的元素
· T： 矩阵转置，同transpose()方法
一些比较有用的方法：
· tolist()： 将array转化成一个Python中的list对象
· item(*args)： 取得某一位置的元素
· dump(file)： 将这个对象序列化至文件。同cPickle中的dump作用
· dumps()： 将序列化的结果通过字符串加以输出
一些关于Array的形态操作：
· reshape()： 改变array的形态
· resize()： 也是改变array的形态。不同的是，resize是直接修改这个对象的，而reshape则会生成一个新的对象
· transpose()： 这个就是矩阵的转置操作啦
· swapaxes()： 将n个维度中任意两个维度（坐标轴）进行调换
· flatten()： 复制一个一维的array出来
还有一些关于Array的运算操作：
· max()：取得所有元素中的最大值
· min()：取得最小值。还有一点值得说，就是max、min这些函数都可以针对某一坐标轴（具体维度）进行运算，例如array.max(axis=0)，就在0坐标上求最大值
· sum()：求和
· cumsum()：求累计和
· prod()：求所有元素之积
· cumprod()：求累计积
· all()：如果所有元素都为真，那么返回真；否则返回假
· any()：只要有一个元素为真则返回真
· mean()：求平均数
Array高级操作
1. Vectorize函数
def t(x):
return x 3
a1 = scipy.zeros((5,4))
a1
NumPy array, format: long
[[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]
[0 0 0 0]]
s = scipy.vectorize(t)
a2 = s(a1)
a2
NumPy array, format: long
[[3 3 3 3]
[3 3 3 3]
[3 3 3 3]
[3 3 3 3]
[3 3 3 3]]
2. NumPy和SciPy相互转化
import numpy
import scipy
a1 = zeros((4,6))
type(a1)
<type 'scipy.ndarray'>
a2 = numpy.asarray(a1)
type(a2)
<type 'numpy.ndarray'>
a3 = numpy.zeros((3,5))
type(a3)
<type 'numpy.ndarray'>
a4 = scipy.asarray(a3)
type(a4)
<type 'scipy.ndarray'>
NumPy 数学函数
Trigonometric functions¶
sin (x[, out])Trigonometric sine, element-wise.cos (x[, out])Cosine elementwise.tan (x[, out])Compute tangent element-wise.arcsin (x[, out])Inverse sine elementwise.arccos (x[, out])Trigonometric inverse cosine, element-wise.arctan (x[, out])Trigonometric inverse tangent, element-wise.hypot (x1, x2[, out])Given two sides of a right triangle, return its hypotenuse.arctan2 (x1, x2[, out])Elementwise arc tangent of x1/x2choosing the quadrant correctly.degrees (x[, out])Convert angles from radians to degrees. This is the same function as rad2deg but the latter is preferred because of the more descriptive name.radians (x[, out])Convert angles from degrees to radians. This function is the same as deg2rad, which is more descriptive..unwrap (p[, discont, axis])Unwrap by changing deltas between values to 2*pi complement.
Hyperbolic functions¶
sinh (x[, out])Hyperbolic sine, element-wise.cosh (x[, out])Hyperbolic cosine, element-wise.tanh (x[, out])Hyperbolic tangent element-wise.arcsinh (x[, out])Inverse hyperbolic sine elementwise.arccosh (x[, out])Inverse hyperbolic cosine, elementwise.arctanh (x[, out])Inverse hyperbolic tangent elementwise.
Rounding¶
around (a[, decimals, out])Evenly round to the given number of decimals.round_ (a[, decimals, out])Round an array to the given number of decimals.rint (x[, out])Round elements of the array to the nearest integer.fix (x[, y])Round to nearest integer towards zero.floor (x[, out])Return the floor of the input, element-wise.ceil (x[, out])Return the ceiling of the input, element-wise.
Sums, products, differences¶
prod (a[, axis, dtype, out])Return the product of array elements over a given axis.sum (a[, axis, dtype, out])Return the sum of array elements over a given axis.nansum (a[, axis])Return the sum of array elements over a given axis treating Not a Numbers (NaNs) as zero.cumprod (a[, axis, dtype, out])Return the cumulative product of elements along a given axis.cumsum (a[, axis, dtype, out])Return the cumulative sum of the elements along a given axis.diff (a[, n, axis])Calculate the nth order discrete difference along given axis.ediff1d (ary[, to_end, to_begin])The differences between consecutive elements of an array.gradient (f, *varargs)Return the gradient of an N-dimensional array.cross (a, b[, axisa, axisb, axisc, ...])Return the cross product of two (arrays of) vectors.trapz (y[, x, dx, axis])Integrate along the given axis using the composite trapezoidal rule.
Exponents and logarithms¶
exp (x[, out])Calculate the exponential of the elements in the input array.expm1 (x[, out])Return the exponential of the elements in the array minus one.log (x[, out])Natural logarithm, element-wise.log10 (x[, out])Compute the logarithm in base 10 element-wise.log2 (x[, y])Return the base 2 logarithm.log1p (x[, out])log(1 x) in base e, elementwise.
Other special functions¶
i0 (x)Modified Bessel function of the first kind, order 0.sinc (x)Return the sinc function.
Floating point routines¶
signbit (x[, out])Returns element-wise True where signbit is set (less than zero).frexp (x[, out1, out2])Split the number, x, into a normalized fraction (y1) and exponent (y2)ldexp (x1, x2[, out])Compute y = x1 * 2**x2.
Arithmetic operations¶
add (x1, x2[, out])Add arguments element-wise.reciprocal (x[, out])Return element-wise reciprocal.negative (x[, out])Returns an array with the negative of each element of the original array.multiply (x1, x2[, out])Multiply arguments elementwise.divide (x1, x2[, out])Divide arguments element-wise.power (x1, x2[, out])Returns element-wise base array raised to power from second array.subtract (x1, x2[, out])Subtract arguments element-wise.true_divide (x1, x2[, out])Returns an element-wise, true division of the inputs.floor_divide (x1, x2[, out])Return the largest integer smaller or equal to the division of the inputs.fmod (x1, x2[, out])Return the remainder of division.mod (x1, x2[, out])Returns element-wise remainder of division.modf (x[, out1, out2])Return the fractional and integral part of a number.remainder (x1, x2[, out])Returns element-wise remainder of division.
Handling complex numbers¶
angle (z[, deg])Return the angle of the complex argument.real (val)Return the real part of the elements of the array.imag (val)Return the imaginary part of array.conj (x[, out])Return the complex conjugate, element-wise.
Miscellaneous¶
convolve (a, v[, mode])Returns the discrete, linear convolution of two one-dimensional sequences.clip (a, a_min, a_max[, out])Clip (limit) the values in an array.sqrt (x[, out])Return the positive square-root of an array, element-wise.square (x[, out])Return the element-wise square of the input.absolute (x[, out])Calculate the absolute value element-wise.fabs (x[, out])Compute the absolute values elementwise.sign (x[, out])Returns an element-wise indication of the sign of a number.maximum (x1, x2[, out])Element-wise maximum of array elements.minimum (x1, x2[, out])Element-wise minimum of array elements.nan_to_num (x)Replace nan with zero and inf with large numbers.real_if_close (a[, tol])If complex input returns a real array if complex parts are close to zero.interp (x, xp, fp[, left, right])One-dimensional linear interpolation.
