Introduction to scientific computing with Python

Maxime Sangnier

September, 2020

Part 5: Additional topics

Object-oriented programming

This section is under construction. Please, refer to the official documentation.

What is it?

A class is probably the most complex and powerful concept you may find in Python (it obviously exists in many other programming languages). It is a template used to create objects (also called instances of the class in question), which generally have many attributes and methods.

Let us remark that Python is a very flexible programming language, with consequently comes with rules we have to obey to cook in occordance with standard practice:

  • classe names are often capitalized, in order to differentiate them from functions;
  • private attributes are suffixed with _. They are normally only aimed at developpers, not at users;
  • attributes should never be modified directly (or very carefully). They are aimed at defining the object and informing the user. In order to modify an object, you should use methods.

Instanciating a class

We start with a first example, aimed at creating an object from the class np.ndarray.

Topics to be addressed:

  • parameters of constructor
  • methods
In [1]:
'rr'.capitalize()
Out[1]:
'Rr'
In [2]:
import numpy as np
help(np.ndarray)
Help on class ndarray in module numpy:

class ndarray(builtins.object)
 |  ndarray(shape, dtype=float, buffer=None, offset=0,
 |          strides=None, order=None)
 |  
 |  An array object represents a multidimensional, homogeneous array
 |  of fixed-size items.  An associated data-type object describes the
 |  format of each element in the array (its byte-order, how many bytes it
 |  occupies in memory, whether it is an integer, a floating point number,
 |  or something else, etc.)
 |  
 |  Arrays should be constructed using `array`, `zeros` or `empty` (refer
 |  to the See Also section below).  The parameters given here refer to
 |  a low-level method (`ndarray(...)`) for instantiating an array.
 |  
 |  For more information, refer to the `numpy` module and examine the
 |  methods and attributes of an array.
 |  
 |  Parameters
 |  ----------
 |  (for the __new__ method; see Notes below)
 |  
 |  shape : tuple of ints
 |      Shape of created array.
 |  dtype : data-type, optional
 |      Any object that can be interpreted as a numpy data type.
 |  buffer : object exposing buffer interface, optional
 |      Used to fill the array with data.
 |  offset : int, optional
 |      Offset of array data in buffer.
 |  strides : tuple of ints, optional
 |      Strides of data in memory.
 |  order : {'C', 'F'}, optional
 |      Row-major (C-style) or column-major (Fortran-style) order.
 |  
 |  Attributes
 |  ----------
 |  T : ndarray
 |      Transpose of the array.
 |  data : buffer
 |      The array's elements, in memory.
 |  dtype : dtype object
 |      Describes the format of the elements in the array.
 |  flags : dict
 |      Dictionary containing information related to memory use, e.g.,
 |      'C_CONTIGUOUS', 'OWNDATA', 'WRITEABLE', etc.
 |  flat : numpy.flatiter object
 |      Flattened version of the array as an iterator.  The iterator
 |      allows assignments, e.g., ``x.flat = 3`` (See `ndarray.flat` for
 |      assignment examples; TODO).
 |  imag : ndarray
 |      Imaginary part of the array.
 |  real : ndarray
 |      Real part of the array.
 |  size : int
 |      Number of elements in the array.
 |  itemsize : int
 |      The memory use of each array element in bytes.
 |  nbytes : int
 |      The total number of bytes required to store the array data,
 |      i.e., ``itemsize * size``.
 |  ndim : int
 |      The array's number of dimensions.
 |  shape : tuple of ints
 |      Shape of the array.
 |  strides : tuple of ints
 |      The step-size required to move from one element to the next in
 |      memory. For example, a contiguous ``(3, 4)`` array of type
 |      ``int16`` in C-order has strides ``(8, 2)``.  This implies that
 |      to move from element to element in memory requires jumps of 2 bytes.
 |      To move from row-to-row, one needs to jump 8 bytes at a time
 |      (``2 * 4``).
 |  ctypes : ctypes object
 |      Class containing properties of the array needed for interaction
 |      with ctypes.
 |  base : ndarray
 |      If the array is a view into another array, that array is its `base`
 |      (unless that array is also a view).  The `base` array is where the
 |      array data is actually stored.
 |  
 |  See Also
 |  --------
 |  array : Construct an array.
 |  zeros : Create an array, each element of which is zero.
 |  empty : Create an array, but leave its allocated memory unchanged (i.e.,
 |          it contains "garbage").
 |  dtype : Create a data-type.
 |  
 |  Notes
 |  -----
 |  There are two modes of creating an array using ``__new__``:
 |  
 |  1. If `buffer` is None, then only `shape`, `dtype`, and `order`
 |     are used.
 |  2. If `buffer` is an object exposing the buffer interface, then
 |     all keywords are interpreted.
 |  
 |  No ``__init__`` method is needed because the array is fully initialized
 |  after the ``__new__`` method.
 |  
 |  Examples
 |  --------
 |  These examples illustrate the low-level `ndarray` constructor.  Refer
 |  to the `See Also` section above for easier ways of constructing an
 |  ndarray.
 |  
 |  First mode, `buffer` is None:
 |  
 |  >>> np.ndarray(shape=(2,2), dtype=float, order='F')
 |  array([[0.0e+000, 0.0e+000], # random
 |         [     nan, 2.5e-323]])
 |  
 |  Second mode:
 |  
 |  >>> np.ndarray((2,), buffer=np.array([1,2,3]),
 |  ...            offset=np.int_().itemsize,
 |  ...            dtype=int) # offset = 1*itemsize, i.e. skip first element
 |  array([2, 3])
 |  
 |  Methods defined here:
 |  
 |  __abs__(self, /)
 |      abs(self)
 |  
 |  __add__(self, value, /)
 |      Return self+value.
 |  
 |  __and__(self, value, /)
 |      Return self&value.
 |  
 |  __array__(...)
 |      a.__array__(|dtype) -> reference if type unchanged, copy otherwise.
 |      
 |      Returns either a new reference to self if dtype is not given or a new array
 |      of provided data type if dtype is different from the current dtype of the
 |      array.
 |  
 |  __array_function__(...)
 |  
 |  __array_prepare__(...)
 |      a.__array_prepare__(obj) -> Object of same type as ndarray object obj.
 |  
 |  __array_ufunc__(...)
 |  
 |  __array_wrap__(...)
 |      a.__array_wrap__(obj) -> Object of same type as ndarray object a.
 |  
 |  __bool__(self, /)
 |      self != 0
 |  
 |  __complex__(...)
 |  
 |  __contains__(self, key, /)
 |      Return key in self.
 |  
 |  __copy__(...)
 |      a.__copy__()
 |      
 |      Used if :func:`copy.copy` is called on an array. Returns a copy of the array.
 |      
 |      Equivalent to ``a.copy(order='K')``.
 |  
 |  __deepcopy__(...)
 |      a.__deepcopy__(memo, /) -> Deep copy of array.
 |      
 |      Used if :func:`copy.deepcopy` is called on an array.
 |  
 |  __delitem__(self, key, /)
 |      Delete self[key].
 |  
 |  __divmod__(self, value, /)
 |      Return divmod(self, value).
 |  
 |  __eq__(self, value, /)
 |      Return self==value.
 |  
 |  __float__(self, /)
 |      float(self)
 |  
 |  __floordiv__(self, value, /)
 |      Return self//value.
 |  
 |  __format__(...)
 |      default object formatter
 |  
 |  __ge__(self, value, /)
 |      Return self>=value.
 |  
 |  __getitem__(self, key, /)
 |      Return self[key].
 |  
 |  __gt__(self, value, /)
 |      Return self>value.
 |  
 |  __iadd__(self, value, /)
 |      Return self+=value.
 |  
 |  __iand__(self, value, /)
 |      Return self&=value.
 |  
 |  __ifloordiv__(self, value, /)
 |      Return self//=value.
 |  
 |  __ilshift__(self, value, /)
 |      Return self<<=value.
 |  
 |  __imatmul__(self, value, /)
 |      Return [email protected]=value.
 |  
 |  __imod__(self, value, /)
 |      Return self%=value.
 |  
 |  __imul__(self, value, /)
 |      Return self*=value.
 |  
 |  __index__(self, /)
 |      Return self converted to an integer, if self is suitable for use as an index into a list.
 |  
 |  __int__(self, /)
 |      int(self)
 |  
 |  __invert__(self, /)
 |      ~self
 |  
 |  __ior__(self, value, /)
 |      Return self|=value.
 |  
 |  __ipow__(self, value, /)
 |      Return self**=value.
 |  
 |  __irshift__(self, value, /)
 |      Return self>>=value.
 |  
 |  __isub__(self, value, /)
 |      Return self-=value.
 |  
 |  __iter__(self, /)
 |      Implement iter(self).
 |  
 |  __itruediv__(self, value, /)
 |      Return self/=value.
 |  
 |  __ixor__(self, value, /)
 |      Return self^=value.
 |  
 |  __le__(self, value, /)
 |      Return self<=value.
 |  
 |  __len__(self, /)
 |      Return len(self).
 |  
 |  __lshift__(self, value, /)
 |      Return self<<value.
 |  
 |  __lt__(self, value, /)
 |      Return self<value.
 |  
 |  __matmul__(self, value, /)
 |      Return [email protected]
 |  
 |  __mod__(self, value, /)
 |      Return self%value.
 |  
 |  __mul__(self, value, /)
 |      Return self*value.
 |  
 |  __ne__(self, value, /)
 |      Return self!=value.
 |  
 |  __neg__(self, /)
 |      -self
 |  
 |  __new__(*args, **kwargs) from builtins.type
 |      Create and return a new object.  See help(type) for accurate signature.
 |  
 |  __or__(self, value, /)
 |      Return self|value.
 |  
 |  __pos__(self, /)
 |      +self
 |  
 |  __pow__(self, value, mod=None, /)
 |      Return pow(self, value, mod).
 |  
 |  __radd__(self, value, /)
 |      Return value+self.
 |  
 |  __rand__(self, value, /)
 |      Return value&self.
 |  
 |  __rdivmod__(self, value, /)
 |      Return divmod(value, self).
 |  
 |  __reduce__(...)
 |      a.__reduce__()
 |      
 |      For pickling.
 |  
 |  __reduce_ex__(...)
 |      helper for pickle
 |  
 |  __repr__(self, /)
 |      Return repr(self).
 |  
 |  __rfloordiv__(self, value, /)
 |      Return value//self.
 |  
 |  __rlshift__(self, value, /)
 |      Return value<<self.
 |  
 |  __rmatmul__(self, value, /)
 |      Return [email protected]
 |  
 |  __rmod__(self, value, /)
 |      Return value%self.
 |  
 |  __rmul__(self, value, /)
 |      Return value*self.
 |  
 |  __ror__(self, value, /)
 |      Return value|self.
 |  
 |  __rpow__(self, value, mod=None, /)
 |      Return pow(value, self, mod).
 |  
 |  __rrshift__(self, value, /)
 |      Return value>>self.
 |  
 |  __rshift__(self, value, /)
 |      Return self>>value.
 |  
 |  __rsub__(self, value, /)
 |      Return value-self.
 |  
 |  __rtruediv__(self, value, /)
 |      Return value/self.
 |  
 |  __rxor__(self, value, /)
 |      Return value^self.
 |  
 |  __setitem__(self, key, value, /)
 |      Set self[key] to value.
 |  
 |  __setstate__(...)
 |      a.__setstate__(state, /)
 |      
 |      For unpickling.
 |      
 |      The `state` argument must be a sequence that contains the following
 |      elements:
 |      
 |      Parameters
 |      ----------
 |      version : int
 |          optional pickle version. If omitted defaults to 0.
 |      shape : tuple
 |      dtype : data-type
 |      isFortran : bool
 |      rawdata : string or list
 |          a binary string with the data (or a list if 'a' is an object array)
 |  
 |  __sizeof__(...)
 |      __sizeof__() -> int
 |      size of object in memory, in bytes
 |  
 |  __str__(self, /)
 |      Return str(self).
 |  
 |  __sub__(self, value, /)
 |      Return self-value.
 |  
 |  __truediv__(self, value, /)
 |      Return self/value.
 |  
 |  __xor__(self, value, /)
 |      Return self^value.
 |  
 |  all(...)
 |      a.all(axis=None, out=None, keepdims=False)
 |      
 |      Returns True if all elements evaluate to True.
 |      
 |      Refer to `numpy.all` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.all : equivalent function
 |  
 |  any(...)
 |      a.any(axis=None, out=None, keepdims=False)
 |      
 |      Returns True if any of the elements of `a` evaluate to True.
 |      
 |      Refer to `numpy.any` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.any : equivalent function
 |  
 |  argmax(...)
 |      a.argmax(axis=None, out=None)
 |      
 |      Return indices of the maximum values along the given axis.
 |      
 |      Refer to `numpy.argmax` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.argmax : equivalent function
 |  
 |  argmin(...)
 |      a.argmin(axis=None, out=None)
 |      
 |      Return indices of the minimum values along the given axis of `a`.
 |      
 |      Refer to `numpy.argmin` for detailed documentation.
 |      
 |      See Also
 |      --------
 |      numpy.argmin : equivalent function
 |  
 |  argpartition(...)
 |      a.argpartition(kth, axis=-1, kind='introselect', order=None)
 |      
 |      Returns the indices that would partition this array.
 |      
 |      Refer to `numpy.argpartition` for full documentation.
 |      
 |      .. versionadded:: 1.8.0
 |      
 |      See Also
 |      --------
 |      numpy.argpartition : equivalent function
 |  
 |  argsort(...)
 |      a.argsort(axis=-1, kind=None, order=None)
 |      
 |      Returns the indices that would sort this array.
 |      
 |      Refer to `numpy.argsort` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.argsort : equivalent function
 |  
 |  astype(...)
 |      a.astype(dtype, order='K', casting='unsafe', subok=True, copy=True)
 |      
 |      Copy of the array, cast to a specified type.
 |      
 |      Parameters
 |      ----------
 |      dtype : str or dtype
 |          Typecode or data-type to which the array is cast.
 |      order : {'C', 'F', 'A', 'K'}, optional
 |          Controls the memory layout order of the result.
 |          'C' means C order, 'F' means Fortran order, 'A'
 |          means 'F' order if all the arrays are Fortran contiguous,
 |          'C' order otherwise, and 'K' means as close to the
 |          order the array elements appear in memory as possible.
 |          Default is 'K'.
 |      casting : {'no', 'equiv', 'safe', 'same_kind', 'unsafe'}, optional
 |          Controls what kind of data casting may occur. Defaults to 'unsafe'
 |          for backwards compatibility.
 |      
 |            * 'no' means the data types should not be cast at all.
 |            * 'equiv' means only byte-order changes are allowed.
 |            * 'safe' means only casts which can preserve values are allowed.
 |            * 'same_kind' means only safe casts or casts within a kind,
 |              like float64 to float32, are allowed.
 |            * 'unsafe' means any data conversions may be done.
 |      subok : bool, optional
 |          If True, then sub-classes will be passed-through (default), otherwise
 |          the returned array will be forced to be a base-class array.
 |      copy : bool, optional
 |          By default, astype always returns a newly allocated array. If this
 |          is set to false, and the `dtype`, `order`, and `subok`
 |          requirements are satisfied, the input array is returned instead
 |          of a copy.
 |      
 |      Returns
 |      -------
 |      arr_t : ndarray
 |          Unless `copy` is False and the other conditions for returning the input
 |          array are satisfied (see description for `copy` input parameter), `arr_t`
 |          is a new array of the same shape as the input array, with dtype, order
 |          given by `dtype`, `order`.
 |      
 |      Notes
 |      -----
 |      .. versionchanged:: 1.17.0
 |         Casting between a simple data type and a structured one is possible only
 |         for "unsafe" casting.  Casting to multiple fields is allowed, but
 |         casting from multiple fields is not.
 |      
 |      .. versionchanged:: 1.9.0
 |         Casting from numeric to string types in 'safe' casting mode requires
 |         that the string dtype length is long enough to store the max
 |         integer/float value converted.
 |      
 |      Raises
 |      ------
 |      ComplexWarning
 |          When casting from complex to float or int. To avoid this,
 |          one should use ``a.real.astype(t)``.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([1, 2, 2.5])
 |      >>> x
 |      array([1. ,  2. ,  2.5])
 |      
 |      >>> x.astype(int)
 |      array([1, 2, 2])
 |  
 |  byteswap(...)
 |      a.byteswap(inplace=False)
 |      
 |      Swap the bytes of the array elements
 |      
 |      Toggle between low-endian and big-endian data representation by
 |      returning a byteswapped array, optionally swapped in-place.
 |      Arrays of byte-strings are not swapped. The real and imaginary
 |      parts of a complex number are swapped individually.
 |      
 |      Parameters
 |      ----------
 |      inplace : bool, optional
 |          If ``True``, swap bytes in-place, default is ``False``.
 |      
 |      Returns
 |      -------
 |      out : ndarray
 |          The byteswapped array. If `inplace` is ``True``, this is
 |          a view to self.
 |      
 |      Examples
 |      --------
 |      >>> A = np.array([1, 256, 8755], dtype=np.int16)
 |      >>> list(map(hex, A))
 |      ['0x1', '0x100', '0x2233']
 |      >>> A.byteswap(inplace=True)
 |      array([  256,     1, 13090], dtype=int16)
 |      >>> list(map(hex, A))
 |      ['0x100', '0x1', '0x3322']
 |      
 |      Arrays of byte-strings are not swapped
 |      
 |      >>> A = np.array([b'ceg', b'fac'])
 |      >>> A.byteswap()
 |      array([b'ceg', b'fac'], dtype='|S3')
 |      
 |      ``A.newbyteorder().byteswap()`` produces an array with the same values
 |        but different representation in memory
 |      
 |      >>> A = np.array([1, 2, 3])
 |      >>> A.view(np.uint8)
 |      array([1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0,
 |             0, 0], dtype=uint8)
 |      >>> A.newbyteorder().byteswap(inplace=True)
 |      array([1, 2, 3])
 |      >>> A.view(np.uint8)
 |      array([0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0,
 |             0, 3], dtype=uint8)
 |  
 |  choose(...)
 |      a.choose(choices, out=None, mode='raise')
 |      
 |      Use an index array to construct a new array from a set of choices.
 |      
 |      Refer to `numpy.choose` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.choose : equivalent function
 |  
 |  clip(...)
 |      a.clip(min=None, max=None, out=None, **kwargs)
 |      
 |      Return an array whose values are limited to ``[min, max]``.
 |      One of max or min must be given.
 |      
 |      Refer to `numpy.clip` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.clip : equivalent function
 |  
 |  compress(...)
 |      a.compress(condition, axis=None, out=None)
 |      
 |      Return selected slices of this array along given axis.
 |      
 |      Refer to `numpy.compress` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.compress : equivalent function
 |  
 |  conj(...)
 |      a.conj()
 |      
 |      Complex-conjugate all elements.
 |      
 |      Refer to `numpy.conjugate` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.conjugate : equivalent function
 |  
 |  conjugate(...)
 |      a.conjugate()
 |      
 |      Return the complex conjugate, element-wise.
 |      
 |      Refer to `numpy.conjugate` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.conjugate : equivalent function
 |  
 |  copy(...)
 |      a.copy(order='C')
 |      
 |      Return a copy of the array.
 |      
 |      Parameters
 |      ----------
 |      order : {'C', 'F', 'A', 'K'}, optional
 |          Controls the memory layout of the copy. 'C' means C-order,
 |          'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous,
 |          'C' otherwise. 'K' means match the layout of `a` as closely
 |          as possible. (Note that this function and :func:`numpy.copy` are very
 |          similar, but have different default values for their order=
 |          arguments.)
 |      
 |      See also
 |      --------
 |      numpy.copy
 |      numpy.copyto
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([[1,2,3],[4,5,6]], order='F')
 |      
 |      >>> y = x.copy()
 |      
 |      >>> x.fill(0)
 |      
 |      >>> x
 |      array([[0, 0, 0],
 |             [0, 0, 0]])
 |      
 |      >>> y
 |      array([[1, 2, 3],
 |             [4, 5, 6]])
 |      
 |      >>> y.flags['C_CONTIGUOUS']
 |      True
 |  
 |  cumprod(...)
 |      a.cumprod(axis=None, dtype=None, out=None)
 |      
 |      Return the cumulative product of the elements along the given axis.
 |      
 |      Refer to `numpy.cumprod` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.cumprod : equivalent function
 |  
 |  cumsum(...)
 |      a.cumsum(axis=None, dtype=None, out=None)
 |      
 |      Return the cumulative sum of the elements along the given axis.
 |      
 |      Refer to `numpy.cumsum` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.cumsum : equivalent function
 |  
 |  diagonal(...)
 |      a.diagonal(offset=0, axis1=0, axis2=1)
 |      
 |      Return specified diagonals. In NumPy 1.9 the returned array is a
 |      read-only view instead of a copy as in previous NumPy versions.  In
 |      a future version the read-only restriction will be removed.
 |      
 |      Refer to :func:`numpy.diagonal` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.diagonal : equivalent function
 |  
 |  dot(...)
 |      a.dot(b, out=None)
 |      
 |      Dot product of two arrays.
 |      
 |      Refer to `numpy.dot` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.dot : equivalent function
 |      
 |      Examples
 |      --------
 |      >>> a = np.eye(2)
 |      >>> b = np.ones((2, 2)) * 2
 |      >>> a.dot(b)
 |      array([[2.,  2.],
 |             [2.,  2.]])
 |      
 |      This array method can be conveniently chained:
 |      
 |      >>> a.dot(b).dot(b)
 |      array([[8.,  8.],
 |             [8.,  8.]])
 |  
 |  dump(...)
 |      a.dump(file)
 |      
 |      Dump a pickle of the array to the specified file.
 |      The array can be read back with pickle.load or numpy.load.
 |      
 |      Parameters
 |      ----------
 |      file : str or Path
 |          A string naming the dump file.
 |      
 |          .. versionchanged:: 1.17.0
 |              `pathlib.Path` objects are now accepted.
 |  
 |  dumps(...)
 |      a.dumps()
 |      
 |      Returns the pickle of the array as a string.
 |      pickle.loads or numpy.loads will convert the string back to an array.
 |      
 |      Parameters
 |      ----------
 |      None
 |  
 |  fill(...)
 |      a.fill(value)
 |      
 |      Fill the array with a scalar value.
 |      
 |      Parameters
 |      ----------
 |      value : scalar
 |          All elements of `a` will be assigned this value.
 |      
 |      Examples
 |      --------
 |      >>> a = np.array([1, 2])
 |      >>> a.fill(0)
 |      >>> a
 |      array([0, 0])
 |      >>> a = np.empty(2)
 |      >>> a.fill(1)
 |      >>> a
 |      array([1.,  1.])
 |  
 |  flatten(...)
 |      a.flatten(order='C')
 |      
 |      Return a copy of the array collapsed into one dimension.
 |      
 |      Parameters
 |      ----------
 |      order : {'C', 'F', 'A', 'K'}, optional
 |          'C' means to flatten in row-major (C-style) order.
 |          'F' means to flatten in column-major (Fortran-
 |          style) order. 'A' means to flatten in column-major
 |          order if `a` is Fortran *contiguous* in memory,
 |          row-major order otherwise. 'K' means to flatten
 |          `a` in the order the elements occur in memory.
 |          The default is 'C'.
 |      
 |      Returns
 |      -------
 |      y : ndarray
 |          A copy of the input array, flattened to one dimension.
 |      
 |      See Also
 |      --------
 |      ravel : Return a flattened array.
 |      flat : A 1-D flat iterator over the array.
 |      
 |      Examples
 |      --------
 |      >>> a = np.array([[1,2], [3,4]])
 |      >>> a.flatten()
 |      array([1, 2, 3, 4])
 |      >>> a.flatten('F')
 |      array([1, 3, 2, 4])
 |  
 |  getfield(...)
 |      a.getfield(dtype, offset=0)
 |      
 |      Returns a field of the given array as a certain type.
 |      
 |      A field is a view of the array data with a given data-type. The values in
 |      the view are determined by the given type and the offset into the current
 |      array in bytes. The offset needs to be such that the view dtype fits in the
 |      array dtype; for example an array of dtype complex128 has 16-byte elements.
 |      If taking a view with a 32-bit integer (4 bytes), the offset needs to be
 |      between 0 and 12 bytes.
 |      
 |      Parameters
 |      ----------
 |      dtype : str or dtype
 |          The data type of the view. The dtype size of the view can not be larger
 |          than that of the array itself.
 |      offset : int
 |          Number of bytes to skip before beginning the element view.
 |      
 |      Examples
 |      --------
 |      >>> x = np.diag([1.+1.j]*2)
 |      >>> x[1, 1] = 2 + 4.j
 |      >>> x
 |      array([[1.+1.j,  0.+0.j],
 |             [0.+0.j,  2.+4.j]])
 |      >>> x.getfield(np.float64)
 |      array([[1.,  0.],
 |             [0.,  2.]])
 |      
 |      By choosing an offset of 8 bytes we can select the complex part of the
 |      array for our view:
 |      
 |      >>> x.getfield(np.float64, offset=8)
 |      array([[1.,  0.],
 |             [0.,  4.]])
 |  
 |  item(...)
 |      a.item(*args)
 |      
 |      Copy an element of an array to a standard Python scalar and return it.
 |      
 |      Parameters
 |      ----------
 |      \*args : Arguments (variable number and type)
 |      
 |          * none: in this case, the method only works for arrays
 |            with one element (`a.size == 1`), which element is
 |            copied into a standard Python scalar object and returned.
 |      
 |          * int_type: this argument is interpreted as a flat index into
 |            the array, specifying which element to copy and return.
 |      
 |          * tuple of int_types: functions as does a single int_type argument,
 |            except that the argument is interpreted as an nd-index into the
 |            array.
 |      
 |      Returns
 |      -------
 |      z : Standard Python scalar object
 |          A copy of the specified element of the array as a suitable
 |          Python scalar
 |      
 |      Notes
 |      -----
 |      When the data type of `a` is longdouble or clongdouble, item() returns
 |      a scalar array object because there is no available Python scalar that
 |      would not lose information. Void arrays return a buffer object for item(),
 |      unless fields are defined, in which case a tuple is returned.
 |      
 |      `item` is very similar to a[args], except, instead of an array scalar,
 |      a standard Python scalar is returned. This can be useful for speeding up
 |      access to elements of the array and doing arithmetic on elements of the
 |      array using Python's optimized math.
 |      
 |      Examples
 |      --------
 |      >>> np.random.seed(123)
 |      >>> x = np.random.randint(9, size=(3, 3))
 |      >>> x
 |      array([[2, 2, 6],
 |             [1, 3, 6],
 |             [1, 0, 1]])
 |      >>> x.item(3)
 |      1
 |      >>> x.item(7)
 |      0
 |      >>> x.item((0, 1))
 |      2
 |      >>> x.item((2, 2))
 |      1
 |  
 |  itemset(...)
 |      a.itemset(*args)
 |      
 |      Insert scalar into an array (scalar is cast to array's dtype, if possible)
 |      
 |      There must be at least 1 argument, and define the last argument
 |      as *item*.  Then, ``a.itemset(*args)`` is equivalent to but faster
 |      than ``a[args] = item``.  The item should be a scalar value and `args`
 |      must select a single item in the array `a`.
 |      
 |      Parameters
 |      ----------
 |      \*args : Arguments
 |          If one argument: a scalar, only used in case `a` is of size 1.
 |          If two arguments: the last argument is the value to be set
 |          and must be a scalar, the first argument specifies a single array
 |          element location. It is either an int or a tuple.
 |      
 |      Notes
 |      -----
 |      Compared to indexing syntax, `itemset` provides some speed increase
 |      for placing a scalar into a particular location in an `ndarray`,
 |      if you must do this.  However, generally this is discouraged:
 |      among other problems, it complicates the appearance of the code.
 |      Also, when using `itemset` (and `item`) inside a loop, be sure
 |      to assign the methods to a local variable to avoid the attribute
 |      look-up at each loop iteration.
 |      
 |      Examples
 |      --------
 |      >>> np.random.seed(123)
 |      >>> x = np.random.randint(9, size=(3, 3))
 |      >>> x
 |      array([[2, 2, 6],
 |             [1, 3, 6],
 |             [1, 0, 1]])
 |      >>> x.itemset(4, 0)
 |      >>> x.itemset((2, 2), 9)
 |      >>> x
 |      array([[2, 2, 6],
 |             [1, 0, 6],
 |             [1, 0, 9]])
 |  
 |  max(...)
 |      a.max(axis=None, out=None, keepdims=False, initial=<no value>, where=True)
 |      
 |      Return the maximum along a given axis.
 |      
 |      Refer to `numpy.amax` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.amax : equivalent function
 |  
 |  mean(...)
 |      a.mean(axis=None, dtype=None, out=None, keepdims=False)
 |      
 |      Returns the average of the array elements along given axis.
 |      
 |      Refer to `numpy.mean` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.mean : equivalent function
 |  
 |  min(...)
 |      a.min(axis=None, out=None, keepdims=False, initial=<no value>, where=True)
 |      
 |      Return the minimum along a given axis.
 |      
 |      Refer to `numpy.amin` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.amin : equivalent function
 |  
 |  newbyteorder(...)
 |      arr.newbyteorder(new_order='S')
 |      
 |      Return the array with the same data viewed with a different byte order.
 |      
 |      Equivalent to::
 |      
 |          arr.view(arr.dtype.newbytorder(new_order))
 |      
 |      Changes are also made in all fields and sub-arrays of the array data
 |      type.
 |      
 |      
 |      
 |      Parameters
 |      ----------
 |      new_order : string, optional
 |          Byte order to force; a value from the byte order specifications
 |          below. `new_order` codes can be any of:
 |      
 |          * 'S' - swap dtype from current to opposite endian
 |          * {'<', 'L'} - little endian
 |          * {'>', 'B'} - big endian
 |          * {'=', 'N'} - native order
 |          * {'|', 'I'} - ignore (no change to byte order)
 |      
 |          The default value ('S') results in swapping the current
 |          byte order. The code does a case-insensitive check on the first
 |          letter of `new_order` for the alternatives above.  For example,
 |          any of 'B' or 'b' or 'biggish' are valid to specify big-endian.
 |      
 |      
 |      Returns
 |      -------
 |      new_arr : array
 |          New array object with the dtype reflecting given change to the
 |          byte order.
 |  
 |  nonzero(...)
 |      a.nonzero()
 |      
 |      Return the indices of the elements that are non-zero.
 |      
 |      Refer to `numpy.nonzero` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.nonzero : equivalent function
 |  
 |  partition(...)
 |      a.partition(kth, axis=-1, kind='introselect', order=None)
 |      
 |      Rearranges the elements in the array in such a way that the value of the
 |      element in kth position is in the position it would be in a sorted array.
 |      All elements smaller than the kth element are moved before this element and
 |      all equal or greater are moved behind it. The ordering of the elements in
 |      the two partitions is undefined.
 |      
 |      .. versionadded:: 1.8.0
 |      
 |      Parameters
 |      ----------
 |      kth : int or sequence of ints
 |          Element index to partition by. The kth element value will be in its
 |          final sorted position and all smaller elements will be moved before it
 |          and all equal or greater elements behind it.
 |          The order of all elements in the partitions is undefined.
 |          If provided with a sequence of kth it will partition all elements
 |          indexed by kth of them into their sorted position at once.
 |      axis : int, optional
 |          Axis along which to sort. Default is -1, which means sort along the
 |          last axis.
 |      kind : {'introselect'}, optional
 |          Selection algorithm. Default is 'introselect'.
 |      order : str or list of str, optional
 |          When `a` is an array with fields defined, this argument specifies
 |          which fields to compare first, second, etc. A single field can
 |          be specified as a string, and not all fields need to be specified,
 |          but unspecified fields will still be used, in the order in which
 |          they come up in the dtype, to break ties.
 |      
 |      See Also
 |      --------
 |      numpy.partition : Return a parititioned copy of an array.
 |      argpartition : Indirect partition.
 |      sort : Full sort.
 |      
 |      Notes
 |      -----
 |      See ``np.partition`` for notes on the different algorithms.
 |      
 |      Examples
 |      --------
 |      >>> a = np.array([3, 4, 2, 1])
 |      >>> a.partition(3)
 |      >>> a
 |      array([2, 1, 3, 4])
 |      
 |      >>> a.partition((1, 3))
 |      >>> a
 |      array([1, 2, 3, 4])
 |  
 |  prod(...)
 |      a.prod(axis=None, dtype=None, out=None, keepdims=False, initial=1, where=True)
 |      
 |      Return the product of the array elements over the given axis
 |      
 |      Refer to `numpy.prod` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.prod : equivalent function
 |  
 |  ptp(...)
 |      a.ptp(axis=None, out=None, keepdims=False)
 |      
 |      Peak to peak (maximum - minimum) value along a given axis.
 |      
 |      Refer to `numpy.ptp` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.ptp : equivalent function
 |  
 |  put(...)
 |      a.put(indices, values, mode='raise')
 |      
 |      Set ``a.flat[n] = values[n]`` for all `n` in indices.
 |      
 |      Refer to `numpy.put` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.put : equivalent function
 |  
 |  ravel(...)
 |      a.ravel([order])
 |      
 |      Return a flattened array.
 |      
 |      Refer to `numpy.ravel` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.ravel : equivalent function
 |      
 |      ndarray.flat : a flat iterator on the array.
 |  
 |  repeat(...)
 |      a.repeat(repeats, axis=None)
 |      
 |      Repeat elements of an array.
 |      
 |      Refer to `numpy.repeat` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.repeat : equivalent function
 |  
 |  reshape(...)
 |      a.reshape(shape, order='C')
 |      
 |      Returns an array containing the same data with a new shape.
 |      
 |      Refer to `numpy.reshape` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.reshape : equivalent function
 |      
 |      Notes
 |      -----
 |      Unlike the free function `numpy.reshape`, this method on `ndarray` allows
 |      the elements of the shape parameter to be passed in as separate arguments.
 |      For example, ``a.reshape(10, 11)`` is equivalent to
 |      ``a.reshape((10, 11))``.
 |  
 |  resize(...)
 |      a.resize(new_shape, refcheck=True)
 |      
 |      Change shape and size of array in-place.
 |      
 |      Parameters
 |      ----------
 |      new_shape : tuple of ints, or `n` ints
 |          Shape of resized array.
 |      refcheck : bool, optional
 |          If False, reference count will not be checked. Default is True.
 |      
 |      Returns
 |      -------
 |      None
 |      
 |      Raises
 |      ------
 |      ValueError
 |          If `a` does not own its own data or references or views to it exist,
 |          and the data memory must be changed.
 |          PyPy only: will always raise if the data memory must be changed, since
 |          there is no reliable way to determine if references or views to it
 |          exist.
 |      
 |      SystemError
 |          If the `order` keyword argument is specified. This behaviour is a
 |          bug in NumPy.
 |      
 |      See Also
 |      --------
 |      resize : Return a new array with the specified shape.
 |      
 |      Notes
 |      -----
 |      This reallocates space for the data area if necessary.
 |      
 |      Only contiguous arrays (data elements consecutive in memory) can be
 |      resized.
 |      
 |      The purpose of the reference count check is to make sure you
 |      do not use this array as a buffer for another Python object and then
 |      reallocate the memory. However, reference counts can increase in
 |      other ways so if you are sure that you have not shared the memory
 |      for this array with another Python object, then you may safely set
 |      `refcheck` to False.
 |      
 |      Examples
 |      --------
 |      Shrinking an array: array is flattened (in the order that the data are
 |      stored in memory), resized, and reshaped:
 |      
 |      >>> a = np.array([[0, 1], [2, 3]], order='C')
 |      >>> a.resize((2, 1))
 |      >>> a
 |      array([[0],
 |             [1]])
 |      
 |      >>> a = np.array([[0, 1], [2, 3]], order='F')
 |      >>> a.resize((2, 1))
 |      >>> a
 |      array([[0],
 |             [2]])
 |      
 |      Enlarging an array: as above, but missing entries are filled with zeros:
 |      
 |      >>> b = np.array([[0, 1], [2, 3]])
 |      >>> b.resize(2, 3) # new_shape parameter doesn't have to be a tuple
 |      >>> b
 |      array([[0, 1, 2],
 |             [3, 0, 0]])
 |      
 |      Referencing an array prevents resizing...
 |      
 |      >>> c = a
 |      >>> a.resize((1, 1))
 |      Traceback (most recent call last):
 |      ...
 |      ValueError: cannot resize an array that references or is referenced ...
 |      
 |      Unless `refcheck` is False:
 |      
 |      >>> a.resize((1, 1), refcheck=False)
 |      >>> a
 |      array([[0]])
 |      >>> c
 |      array([[0]])
 |  
 |  round(...)
 |      a.round(decimals=0, out=None)
 |      
 |      Return `a` with each element rounded to the given number of decimals.
 |      
 |      Refer to `numpy.around` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.around : equivalent function
 |  
 |  searchsorted(...)
 |      a.searchsorted(v, side='left', sorter=None)
 |      
 |      Find indices where elements of v should be inserted in a to maintain order.
 |      
 |      For full documentation, see `numpy.searchsorted`
 |      
 |      See Also
 |      --------
 |      numpy.searchsorted : equivalent function
 |  
 |  setfield(...)
 |      a.setfield(val, dtype, offset=0)
 |      
 |      Put a value into a specified place in a field defined by a data-type.
 |      
 |      Place `val` into `a`'s field defined by `dtype` and beginning `offset`
 |      bytes into the field.
 |      
 |      Parameters
 |      ----------
 |      val : object
 |          Value to be placed in field.
 |      dtype : dtype object
 |          Data-type of the field in which to place `val`.
 |      offset : int, optional
 |          The number of bytes into the field at which to place `val`.
 |      
 |      Returns
 |      -------
 |      None
 |      
 |      See Also
 |      --------
 |      getfield
 |      
 |      Examples
 |      --------
 |      >>> x = np.eye(3)
 |      >>> x.getfield(np.float64)
 |      array([[1.,  0.,  0.],
 |             [0.,  1.,  0.],
 |             [0.,  0.,  1.]])
 |      >>> x.setfield(3, np.int32)
 |      >>> x.getfield(np.int32)
 |      array([[3, 3, 3],
 |             [3, 3, 3],
 |             [3, 3, 3]], dtype=int32)
 |      >>> x
 |      array([[1.0e+000, 1.5e-323, 1.5e-323],
 |             [1.5e-323, 1.0e+000, 1.5e-323],
 |             [1.5e-323, 1.5e-323, 1.0e+000]])
 |      >>> x.setfield(np.eye(3), np.int32)
 |      >>> x
 |      array([[1.,  0.,  0.],
 |             [0.,  1.,  0.],
 |             [0.,  0.,  1.]])
 |  
 |  setflags(...)
 |      a.setflags(write=None, align=None, uic=None)
 |      
 |      Set array flags WRITEABLE, ALIGNED, (WRITEBACKIFCOPY and UPDATEIFCOPY),
 |      respectively.
 |      
 |      These Boolean-valued flags affect how numpy interprets the memory
 |      area used by `a` (see Notes below). The ALIGNED flag can only
 |      be set to True if the data is actually aligned according to the type.
 |      The WRITEBACKIFCOPY and (deprecated) UPDATEIFCOPY flags can never be set
 |      to True. The flag WRITEABLE can only be set to True if the array owns its
 |      own memory, or the ultimate owner of the memory exposes a writeable buffer
 |      interface, or is a string. (The exception for string is made so that
 |      unpickling can be done without copying memory.)
 |      
 |      Parameters
 |      ----------
 |      write : bool, optional
 |          Describes whether or not `a` can be written to.
 |      align : bool, optional
 |          Describes whether or not `a` is aligned properly for its type.
 |      uic : bool, optional
 |          Describes whether or not `a` is a copy of another "base" array.
 |      
 |      Notes
 |      -----
 |      Array flags provide information about how the memory area used
 |      for the array is to be interpreted. There are 7 Boolean flags
 |      in use, only four of which can be changed by the user:
 |      WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED.
 |      
 |      WRITEABLE (W) the data area can be written to;
 |      
 |      ALIGNED (A) the data and strides are aligned appropriately for the hardware
 |      (as determined by the compiler);
 |      
 |      UPDATEIFCOPY (U) (deprecated), replaced by WRITEBACKIFCOPY;
 |      
 |      WRITEBACKIFCOPY (X) this array is a copy of some other array (referenced
 |      by .base). When the C-API function PyArray_ResolveWritebackIfCopy is
 |      called, the base array will be updated with the contents of this array.
 |      
 |      All flags can be accessed using the single (upper case) letter as well
 |      as the full name.
 |      
 |      Examples
 |      --------
 |      >>> y = np.array([[3, 1, 7],
 |      ...               [2, 0, 0],
 |      ...               [8, 5, 9]])
 |      >>> y
 |      array([[3, 1, 7],
 |             [2, 0, 0],
 |             [8, 5, 9]])
 |      >>> y.flags
 |        C_CONTIGUOUS : True
 |        F_CONTIGUOUS : False
 |        OWNDATA : True
 |        WRITEABLE : True
 |        ALIGNED : True
 |        WRITEBACKIFCOPY : False
 |        UPDATEIFCOPY : False
 |      >>> y.setflags(write=0, align=0)
 |      >>> y.flags
 |        C_CONTIGUOUS : True
 |        F_CONTIGUOUS : False
 |        OWNDATA : True
 |        WRITEABLE : False
 |        ALIGNED : False
 |        WRITEBACKIFCOPY : False
 |        UPDATEIFCOPY : False
 |      >>> y.setflags(uic=1)
 |      Traceback (most recent call last):
 |        File "<stdin>", line 1, in <module>
 |      ValueError: cannot set WRITEBACKIFCOPY flag to True
 |  
 |  sort(...)
 |      a.sort(axis=-1, kind=None, order=None)
 |      
 |      Sort an array in-place. Refer to `numpy.sort` for full documentation.
 |      
 |      Parameters
 |      ----------
 |      axis : int, optional
 |          Axis along which to sort. Default is -1, which means sort along the
 |          last axis.
 |      kind : {'quicksort', 'mergesort', 'heapsort', 'stable'}, optional
 |          Sorting algorithm. The default is 'quicksort'. Note that both 'stable'
 |          and 'mergesort' use timsort under the covers and, in general, the
 |          actual implementation will vary with datatype. The 'mergesort' option
 |          is retained for backwards compatibility.
 |      
 |          .. versionchanged:: 1.15.0.
 |             The 'stable' option was added.
 |      
 |      order : str or list of str, optional
 |          When `a` is an array with fields defined, this argument specifies
 |          which fields to compare first, second, etc.  A single field can
 |          be specified as a string, and not all fields need be specified,
 |          but unspecified fields will still be used, in the order in which
 |          they come up in the dtype, to break ties.
 |      
 |      See Also
 |      --------
 |      numpy.sort : Return a sorted copy of an array.
 |      numpy.argsort : Indirect sort.
 |      numpy.lexsort : Indirect stable sort on multiple keys.
 |      numpy.searchsorted : Find elements in sorted array.
 |      numpy.partition: Partial sort.
 |      
 |      Notes
 |      -----
 |      See `numpy.sort` for notes on the different sorting algorithms.
 |      
 |      Examples
 |      --------
 |      >>> a = np.array([[1,4], [3,1]])
 |      >>> a.sort(axis=1)
 |      >>> a
 |      array([[1, 4],
 |             [1, 3]])
 |      >>> a.sort(axis=0)
 |      >>> a
 |      array([[1, 3],
 |             [1, 4]])
 |      
 |      Use the `order` keyword to specify a field to use when sorting a
 |      structured array:
 |      
 |      >>> a = np.array([('a', 2), ('c', 1)], dtype=[('x', 'S1'), ('y', int)])
 |      >>> a.sort(order='y')
 |      >>> a
 |      array([(b'c', 1), (b'a', 2)],
 |            dtype=[('x', 'S1'), ('y', '<i8')])
 |  
 |  squeeze(...)
 |      a.squeeze(axis=None)
 |      
 |      Remove single-dimensional entries from the shape of `a`.
 |      
 |      Refer to `numpy.squeeze` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.squeeze : equivalent function
 |  
 |  std(...)
 |      a.std(axis=None, dtype=None, out=None, ddof=0, keepdims=False)
 |      
 |      Returns the standard deviation of the array elements along given axis.
 |      
 |      Refer to `numpy.std` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.std : equivalent function
 |  
 |  sum(...)
 |      a.sum(axis=None, dtype=None, out=None, keepdims=False, initial=0, where=True)
 |      
 |      Return the sum of the array elements over the given axis.
 |      
 |      Refer to `numpy.sum` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.sum : equivalent function
 |  
 |  swapaxes(...)
 |      a.swapaxes(axis1, axis2)
 |      
 |      Return a view of the array with `axis1` and `axis2` interchanged.
 |      
 |      Refer to `numpy.swapaxes` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.swapaxes : equivalent function
 |  
 |  take(...)
 |      a.take(indices, axis=None, out=None, mode='raise')
 |      
 |      Return an array formed from the elements of `a` at the given indices.
 |      
 |      Refer to `numpy.take` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.take : equivalent function
 |  
 |  tobytes(...)
 |      a.tobytes(order='C')
 |      
 |      Construct Python bytes containing the raw data bytes in the array.
 |      
 |      Constructs Python bytes showing a copy of the raw contents of
 |      data memory. The bytes object can be produced in either 'C' or 'Fortran',
 |      or 'Any' order (the default is 'C'-order). 'Any' order means C-order
 |      unless the F_CONTIGUOUS flag in the array is set, in which case it
 |      means 'Fortran' order.
 |      
 |      .. versionadded:: 1.9.0
 |      
 |      Parameters
 |      ----------
 |      order : {'C', 'F', None}, optional
 |          Order of the data for multidimensional arrays:
 |          C, Fortran, or the same as for the original array.
 |      
 |      Returns
 |      -------
 |      s : bytes
 |          Python bytes exhibiting a copy of `a`'s raw data.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([[0, 1], [2, 3]], dtype='<u2')
 |      >>> x.tobytes()
 |      b'\x00\x00\x01\x00\x02\x00\x03\x00'
 |      >>> x.tobytes('C') == x.tobytes()
 |      True
 |      >>> x.tobytes('F')
 |      b'\x00\x00\x02\x00\x01\x00\x03\x00'
 |  
 |  tofile(...)
 |      a.tofile(fid, sep="", format="%s")
 |      
 |      Write array to a file as text or binary (default).
 |      
 |      Data is always written in 'C' order, independent of the order of `a`.
 |      The data produced by this method can be recovered using the function
 |      fromfile().
 |      
 |      Parameters
 |      ----------
 |      fid : file or str or Path
 |          An open file object, or a string containing a filename.
 |      
 |          .. versionchanged:: 1.17.0
 |              `pathlib.Path` objects are now accepted.
 |      
 |      sep : str
 |          Separator between array items for text output.
 |          If "" (empty), a binary file is written, equivalent to
 |          ``file.write(a.tobytes())``.
 |      format : str
 |          Format string for text file output.
 |          Each entry in the array is formatted to text by first converting
 |          it to the closest Python type, and then using "format" % item.
 |      
 |      Notes
 |      -----
 |      This is a convenience function for quick storage of array data.
 |      Information on endianness and precision is lost, so this method is not a
 |      good choice for files intended to archive data or transport data between
 |      machines with different endianness. Some of these problems can be overcome
 |      by outputting the data as text files, at the expense of speed and file
 |      size.
 |      
 |      When fid is a file object, array contents are directly written to the
 |      file, bypassing the file object's ``write`` method. As a result, tofile
 |      cannot be used with files objects supporting compression (e.g., GzipFile)
 |      or file-like objects that do not support ``fileno()`` (e.g., BytesIO).
 |  
 |  tolist(...)
 |      a.tolist()
 |      
 |      Return the array as an ``a.ndim``-levels deep nested list of Python scalars.
 |      
 |      Return a copy of the array data as a (nested) Python list.
 |      Data items are converted to the nearest compatible builtin Python type, via
 |      the `~numpy.ndarray.item` function.
 |      
 |      If ``a.ndim`` is 0, then since the depth of the nested list is 0, it will
 |      not be a list at all, but a simple Python scalar.
 |      
 |      Parameters
 |      ----------
 |      none
 |      
 |      Returns
 |      -------
 |      y : object, or list of object, or list of list of object, or ...
 |          The possibly nested list of array elements.
 |      
 |      Notes
 |      -----
 |      The array may be recreated via ``a = np.array(a.tolist())``, although this
 |      may sometimes lose precision.
 |      
 |      Examples
 |      --------
 |      For a 1D array, ``a.tolist()`` is almost the same as ``list(a)``, 
 |      except that ``tolist`` changes numpy scalars to Python scalars:
 |      
 |      >>> a = np.uint32([1, 2])
 |      >>> a_list = list(a)
 |      >>> a_list
 |      [1, 2]
 |      >>> type(a_list[0])
 |      <class 'numpy.uint32'>
 |      >>> a_tolist = a.tolist()
 |      >>> a_tolist
 |      [1, 2]
 |      >>> type(a_tolist[0])
 |      <class 'int'>
 |      
 |      Additionally, for a 2D array, ``tolist`` applies recursively:
 |      
 |      >>> a = np.array([[1, 2], [3, 4]])
 |      >>> list(a)
 |      [array([1, 2]), array([3, 4])]
 |      >>> a.tolist()
 |      [[1, 2], [3, 4]]
 |      
 |      The base case for this recursion is a 0D array:
 |      
 |      >>> a = np.array(1)
 |      >>> list(a)
 |      Traceback (most recent call last):
 |        ...
 |      TypeError: iteration over a 0-d array
 |      >>> a.tolist()
 |      1
 |  
 |  tostring(...)
 |      a.tostring(order='C')
 |      
 |      Construct Python bytes containing the raw data bytes in the array.
 |      
 |      Constructs Python bytes showing a copy of the raw contents of
 |      data memory. The bytes object can be produced in either 'C' or 'Fortran',
 |      or 'Any' order (the default is 'C'-order). 'Any' order means C-order
 |      unless the F_CONTIGUOUS flag in the array is set, in which case it
 |      means 'Fortran' order.
 |      
 |      This function is a compatibility alias for tobytes. Despite its name it returns bytes not strings.
 |      
 |      Parameters
 |      ----------
 |      order : {'C', 'F', None}, optional
 |          Order of the data for multidimensional arrays:
 |          C, Fortran, or the same as for the original array.
 |      
 |      Returns
 |      -------
 |      s : bytes
 |          Python bytes exhibiting a copy of `a`'s raw data.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([[0, 1], [2, 3]], dtype='<u2')
 |      >>> x.tobytes()
 |      b'\x00\x00\x01\x00\x02\x00\x03\x00'
 |      >>> x.tobytes('C') == x.tobytes()
 |      True
 |      >>> x.tobytes('F')
 |      b'\x00\x00\x02\x00\x01\x00\x03\x00'
 |  
 |  trace(...)
 |      a.trace(offset=0, axis1=0, axis2=1, dtype=None, out=None)
 |      
 |      Return the sum along diagonals of the array.
 |      
 |      Refer to `numpy.trace` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.trace : equivalent function
 |  
 |  transpose(...)
 |      a.transpose(*axes)
 |      
 |      Returns a view of the array with axes transposed.
 |      
 |      For a 1-D array this has no effect, as a transposed vector is simply the
 |      same vector. To convert a 1-D array into a 2D column vector, an additional
 |      dimension must be added. `np.atleast2d(a).T` achieves this, as does
 |      `a[:, np.newaxis]`.
 |      For a 2-D array, this is a standard matrix transpose.
 |      For an n-D array, if axes are given, their order indicates how the
 |      axes are permuted (see Examples). If axes are not provided and
 |      ``a.shape = (i[0], i[1], ... i[n-2], i[n-1])``, then
 |      ``a.transpose().shape = (i[n-1], i[n-2], ... i[1], i[0])``.
 |      
 |      Parameters
 |      ----------
 |      axes : None, tuple of ints, or `n` ints
 |      
 |       * None or no argument: reverses the order of the axes.
 |      
 |       * tuple of ints: `i` in the `j`-th place in the tuple means `a`'s
 |         `i`-th axis becomes `a.transpose()`'s `j`-th axis.
 |      
 |       * `n` ints: same as an n-tuple of the same ints (this form is
 |         intended simply as a "convenience" alternative to the tuple form)
 |      
 |      Returns
 |      -------
 |      out : ndarray
 |          View of `a`, with axes suitably permuted.
 |      
 |      See Also
 |      --------
 |      ndarray.T : Array property returning the array transposed.
 |      ndarray.reshape : Give a new shape to an array without changing its data.
 |      
 |      Examples
 |      --------
 |      >>> a = np.array([[1, 2], [3, 4]])
 |      >>> a
 |      array([[1, 2],
 |             [3, 4]])
 |      >>> a.transpose()
 |      array([[1, 3],
 |             [2, 4]])
 |      >>> a.transpose((1, 0))
 |      array([[1, 3],
 |             [2, 4]])
 |      >>> a.transpose(1, 0)
 |      array([[1, 3],
 |             [2, 4]])
 |  
 |  var(...)
 |      a.var(axis=None, dtype=None, out=None, ddof=0, keepdims=False)
 |      
 |      Returns the variance of the array elements, along given axis.
 |      
 |      Refer to `numpy.var` for full documentation.
 |      
 |      See Also
 |      --------
 |      numpy.var : equivalent function
 |  
 |  view(...)
 |      a.view(dtype=None, type=None)
 |      
 |      New view of array with the same data.
 |      
 |      Parameters
 |      ----------
 |      dtype : data-type or ndarray sub-class, optional
 |          Data-type descriptor of the returned view, e.g., float32 or int16. The
 |          default, None, results in the view having the same data-type as `a`.
 |          This argument can also be specified as an ndarray sub-class, which
 |          then specifies the type of the returned object (this is equivalent to
 |          setting the ``type`` parameter).
 |      type : Python type, optional
 |          Type of the returned view, e.g., ndarray or matrix.  Again, the
 |          default None results in type preservation.
 |      
 |      Notes
 |      -----
 |      ``a.view()`` is used two different ways:
 |      
 |      ``a.view(some_dtype)`` or ``a.view(dtype=some_dtype)`` constructs a view
 |      of the array's memory with a different data-type.  This can cause a
 |      reinterpretation of the bytes of memory.
 |      
 |      ``a.view(ndarray_subclass)`` or ``a.view(type=ndarray_subclass)`` just
 |      returns an instance of `ndarray_subclass` that looks at the same array
 |      (same shape, dtype, etc.)  This does not cause a reinterpretation of the
 |      memory.
 |      
 |      For ``a.view(some_dtype)``, if ``some_dtype`` has a different number of
 |      bytes per entry than the previous dtype (for example, converting a
 |      regular array to a structured array), then the behavior of the view
 |      cannot be predicted just from the superficial appearance of ``a`` (shown
 |      by ``print(a)``). It also depends on exactly how ``a`` is stored in
 |      memory. Therefore if ``a`` is C-ordered versus fortran-ordered, versus
 |      defined as a slice or transpose, etc., the view may give different
 |      results.
 |      
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([(1, 2)], dtype=[('a', np.int8), ('b', np.int8)])
 |      
 |      Viewing array data using a different type and dtype:
 |      
 |      >>> y = x.view(dtype=np.int16, type=np.matrix)
 |      >>> y
 |      matrix([[513]], dtype=int16)
 |      >>> print(type(y))
 |      <class 'numpy.matrix'>
 |      
 |      Creating a view on a structured array so it can be used in calculations
 |      
 |      >>> x = np.array([(1, 2),(3,4)], dtype=[('a', np.int8), ('b', np.int8)])
 |      >>> xv = x.view(dtype=np.int8).reshape(-1,2)
 |      >>> xv
 |      array([[1, 2],
 |             [3, 4]], dtype=int8)
 |      >>> xv.mean(0)
 |      array([2.,  3.])
 |      
 |      Making changes to the view changes the underlying array
 |      
 |      >>> xv[0,1] = 20
 |      >>> x
 |      array([(1, 20), (3,  4)], dtype=[('a', 'i1'), ('b', 'i1')])
 |      
 |      Using a view to convert an array to a recarray:
 |      
 |      >>> z = x.view(np.recarray)
 |      >>> z.a
 |      array([1, 3], dtype=int8)
 |      
 |      Views share data:
 |      
 |      >>> x[0] = (9, 10)
 |      >>> z[0]
 |      (9, 10)
 |      
 |      Views that change the dtype size (bytes per entry) should normally be
 |      avoided on arrays defined by slices, transposes, fortran-ordering, etc.:
 |      
 |      >>> x = np.array([[1,2,3],[4,5,6]], dtype=np.int16)
 |      >>> y = x[:, 0:2]
 |      >>> y
 |      array([[1, 2],
 |             [4, 5]], dtype=int16)
 |      >>> y.view(dtype=[('width', np.int16), ('length', np.int16)])
 |      Traceback (most recent call last):
 |          ...
 |      ValueError: To change to a dtype of a different size, the array must be C-contiguous
 |      >>> z = y.copy()
 |      >>> z.view(dtype=[('width', np.int16), ('length', np.int16)])
 |      array([[(1, 2)],
 |             [(4, 5)]], dtype=[('width', '<i2'), ('length', '<i2')])
 |  
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |  
 |  T
 |      The transposed array.
 |      
 |      Same as ``self.transpose()``.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([[1.,2.],[3.,4.]])
 |      >>> x
 |      array([[ 1.,  2.],
 |             [ 3.,  4.]])
 |      >>> x.T
 |      array([[ 1.,  3.],
 |             [ 2.,  4.]])
 |      >>> x = np.array([1.,2.,3.,4.])
 |      >>> x
 |      array([ 1.,  2.,  3.,  4.])
 |      >>> x.T
 |      array([ 1.,  2.,  3.,  4.])
 |      
 |      See Also
 |      --------
 |      transpose
 |  
 |  __array_finalize__
 |      None.
 |  
 |  __array_interface__
 |      Array protocol: Python side.
 |  
 |  __array_priority__
 |      Array priority.
 |  
 |  __array_struct__
 |      Array protocol: C-struct side.
 |  
 |  base
 |      Base object if memory is from some other object.
 |      
 |      Examples
 |      --------
 |      The base of an array that owns its memory is None:
 |      
 |      >>> x = np.array([1,2,3,4])
 |      >>> x.base is None
 |      True
 |      
 |      Slicing creates a view, whose memory is shared with x:
 |      
 |      >>> y = x[2:]
 |      >>> y.base is x
 |      True
 |  
 |  ctypes
 |      An object to simplify the interaction of the array with the ctypes
 |      module.
 |      
 |      This attribute creates an object that makes it easier to use arrays
 |      when calling shared libraries with the ctypes module. The returned
 |      object has, among others, data, shape, and strides attributes (see
 |      Notes below) which themselves return ctypes objects that can be used
 |      as arguments to a shared library.
 |      
 |      Parameters
 |      ----------
 |      None
 |      
 |      Returns
 |      -------
 |      c : Python object
 |          Possessing attributes data, shape, strides, etc.
 |      
 |      See Also
 |      --------
 |      numpy.ctypeslib
 |      
 |      Notes
 |      -----
 |      Below are the public attributes of this object which were documented
 |      in "Guide to NumPy" (we have omitted undocumented public attributes,
 |      as well as documented private attributes):
 |      
 |      .. autoattribute:: numpy.core._internal._ctypes.data
 |          :noindex:
 |      
 |      .. autoattribute:: numpy.core._internal._ctypes.shape
 |          :noindex:
 |      
 |      .. autoattribute:: numpy.core._internal._ctypes.strides
 |          :noindex:
 |      
 |      .. automethod:: numpy.core._internal._ctypes.data_as
 |          :noindex:
 |      
 |      .. automethod:: numpy.core._internal._ctypes.shape_as
 |          :noindex:
 |      
 |      .. automethod:: numpy.core._internal._ctypes.strides_as
 |          :noindex:
 |      
 |      If the ctypes module is not available, then the ctypes attribute
 |      of array objects still returns something useful, but ctypes objects
 |      are not returned and errors may be raised instead. In particular,
 |      the object will still have the ``as_parameter`` attribute which will
 |      return an integer equal to the data attribute.
 |      
 |      Examples
 |      --------
 |      >>> import ctypes
 |      >>> x
 |      array([[0, 1],
 |             [2, 3]])
 |      >>> x.ctypes.data
 |      30439712
 |      >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_long))
 |      <ctypes.LP_c_long object at 0x01F01300>
 |      >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_long)).contents
 |      c_long(0)
 |      >>> x.ctypes.data_as(ctypes.POINTER(ctypes.c_longlong)).contents
 |      c_longlong(4294967296L)
 |      >>> x.ctypes.shape
 |      <numpy.core._internal.c_long_Array_2 object at 0x01FFD580>
 |      >>> x.ctypes.shape_as(ctypes.c_long)
 |      <numpy.core._internal.c_long_Array_2 object at 0x01FCE620>
 |      >>> x.ctypes.strides
 |      <numpy.core._internal.c_long_Array_2 object at 0x01FCE620>
 |      >>> x.ctypes.strides_as(ctypes.c_longlong)
 |      <numpy.core._internal.c_longlong_Array_2 object at 0x01F01300>
 |  
 |  data
 |      Python buffer object pointing to the start of the array's data.
 |  
 |  dtype
 |      Data-type of the array's elements.
 |      
 |      Parameters
 |      ----------
 |      None
 |      
 |      Returns
 |      -------
 |      d : numpy dtype object
 |      
 |      See Also
 |      --------
 |      numpy.dtype
 |      
 |      Examples
 |      --------
 |      >>> x
 |      array([[0, 1],
 |             [2, 3]])
 |      >>> x.dtype
 |      dtype('int32')
 |      >>> type(x.dtype)
 |      <type 'numpy.dtype'>
 |  
 |  flags
 |      Information about the memory layout of the array.
 |      
 |      Attributes
 |      ----------
 |      C_CONTIGUOUS (C)
 |          The data is in a single, C-style contiguous segment.
 |      F_CONTIGUOUS (F)
 |          The data is in a single, Fortran-style contiguous segment.
 |      OWNDATA (O)
 |          The array owns the memory it uses or borrows it from another object.
 |      WRITEABLE (W)
 |          The data area can be written to.  Setting this to False locks
 |          the data, making it read-only.  A view (slice, etc.) inherits WRITEABLE
 |          from its base array at creation time, but a view of a writeable
 |          array may be subsequently locked while the base array remains writeable.
 |          (The opposite is not true, in that a view of a locked array may not
 |          be made writeable.  However, currently, locking a base object does not
 |          lock any views that already reference it, so under that circumstance it
 |          is possible to alter the contents of a locked array via a previously
 |          created writeable view onto it.)  Attempting to change a non-writeable
 |          array raises a RuntimeError exception.
 |      ALIGNED (A)
 |          The data and all elements are aligned appropriately for the hardware.
 |      WRITEBACKIFCOPY (X)
 |          This array is a copy of some other array. The C-API function
 |          PyArray_ResolveWritebackIfCopy must be called before deallocating
 |          to the base array will be updated with the contents of this array.
 |      UPDATEIFCOPY (U)
 |          (Deprecated, use WRITEBACKIFCOPY) This array is a copy of some other array.
 |          When this array is
 |          deallocated, the base array will be updated with the contents of
 |          this array.
 |      FNC
 |          F_CONTIGUOUS and not C_CONTIGUOUS.
 |      FORC
 |          F_CONTIGUOUS or C_CONTIGUOUS (one-segment test).
 |      BEHAVED (B)
 |          ALIGNED and WRITEABLE.
 |      CARRAY (CA)
 |          BEHAVED and C_CONTIGUOUS.
 |      FARRAY (FA)
 |          BEHAVED and F_CONTIGUOUS and not C_CONTIGUOUS.
 |      
 |      Notes
 |      -----
 |      The `flags` object can be accessed dictionary-like (as in ``a.flags['WRITEABLE']``),
 |      or by using lowercased attribute names (as in ``a.flags.writeable``). Short flag
 |      names are only supported in dictionary access.
 |      
 |      Only the WRITEBACKIFCOPY, UPDATEIFCOPY, WRITEABLE, and ALIGNED flags can be
 |      changed by the user, via direct assignment to the attribute or dictionary
 |      entry, or by calling `ndarray.setflags`.
 |      
 |      The array flags cannot be set arbitrarily:
 |      
 |      - UPDATEIFCOPY can only be set ``False``.
 |      - WRITEBACKIFCOPY can only be set ``False``.
 |      - ALIGNED can only be set ``True`` if the data is truly aligned.
 |      - WRITEABLE can only be set ``True`` if the array owns its own memory
 |        or the ultimate owner of the memory exposes a writeable buffer
 |        interface or is a string.
 |      
 |      Arrays can be both C-style and Fortran-style contiguous simultaneously.
 |      This is clear for 1-dimensional arrays, but can also be true for higher
 |      dimensional arrays.
 |      
 |      Even for contiguous arrays a stride for a given dimension
 |      ``arr.strides[dim]`` may be *arbitrary* if ``arr.shape[dim] == 1``
 |      or the array has no elements.
 |      It does *not* generally hold that ``self.strides[-1] == self.itemsize``
 |      for C-style contiguous arrays or ``self.strides[0] == self.itemsize`` for
 |      Fortran-style contiguous arrays is true.
 |  
 |  flat
 |      A 1-D iterator over the array.
 |      
 |      This is a `numpy.flatiter` instance, which acts similarly to, but is not
 |      a subclass of, Python's built-in iterator object.
 |      
 |      See Also
 |      --------
 |      flatten : Return a copy of the array collapsed into one dimension.
 |      
 |      flatiter
 |      
 |      Examples
 |      --------
 |      >>> x = np.arange(1, 7).reshape(2, 3)
 |      >>> x
 |      array([[1, 2, 3],
 |             [4, 5, 6]])
 |      >>> x.flat[3]
 |      4
 |      >>> x.T
 |      array([[1, 4],
 |             [2, 5],
 |             [3, 6]])
 |      >>> x.T.flat[3]
 |      5
 |      >>> type(x.flat)
 |      <class 'numpy.flatiter'>
 |      
 |      An assignment example:
 |      
 |      >>> x.flat = 3; x
 |      array([[3, 3, 3],
 |             [3, 3, 3]])
 |      >>> x.flat[[1,4]] = 1; x
 |      array([[3, 1, 3],
 |             [3, 1, 3]])
 |  
 |  imag
 |      The imaginary part of the array.
 |      
 |      Examples
 |      --------
 |      >>> x = np.sqrt([1+0j, 0+1j])
 |      >>> x.imag
 |      array([ 0.        ,  0.70710678])
 |      >>> x.imag.dtype
 |      dtype('float64')
 |  
 |  itemsize
 |      Length of one array element in bytes.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([1,2,3], dtype=np.float64)
 |      >>> x.itemsize
 |      8
 |      >>> x = np.array([1,2,3], dtype=np.complex128)
 |      >>> x.itemsize
 |      16
 |  
 |  nbytes
 |      Total bytes consumed by the elements of the array.
 |      
 |      Notes
 |      -----
 |      Does not include memory consumed by non-element attributes of the
 |      array object.
 |      
 |      Examples
 |      --------
 |      >>> x = np.zeros((3,5,2), dtype=np.complex128)
 |      >>> x.nbytes
 |      480
 |      >>> np.prod(x.shape) * x.itemsize
 |      480
 |  
 |  ndim
 |      Number of array dimensions.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([1, 2, 3])
 |      >>> x.ndim
 |      1
 |      >>> y = np.zeros((2, 3, 4))
 |      >>> y.ndim
 |      3
 |  
 |  real
 |      The real part of the array.
 |      
 |      Examples
 |      --------
 |      >>> x = np.sqrt([1+0j, 0+1j])
 |      >>> x.real
 |      array([ 1.        ,  0.70710678])
 |      >>> x.real.dtype
 |      dtype('float64')
 |      
 |      See Also
 |      --------
 |      numpy.real : equivalent function
 |  
 |  shape
 |      Tuple of array dimensions.
 |      
 |      The shape property is usually used to get the current shape of an array,
 |      but may also be used to reshape the array in-place by assigning a tuple of
 |      array dimensions to it.  As with `numpy.reshape`, one of the new shape
 |      dimensions can be -1, in which case its value is inferred from the size of
 |      the array and the remaining dimensions. Reshaping an array in-place will
 |      fail if a copy is required.
 |      
 |      Examples
 |      --------
 |      >>> x = np.array([1, 2, 3, 4])
 |      >>> x.shape
 |      (4,)
 |      >>> y = np.zeros((2, 3, 4))
 |      >>> y.shape
 |      (2, 3, 4)
 |      >>> y.shape = (3, 8)
 |      >>> y
 |      array([[ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 |             [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
 |             [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.]])
 |      >>> y.shape = (3, 6)
 |      Traceback (most recent call last):
 |        File "<stdin>", line 1, in <module>
 |      ValueError: total size of new array must be unchanged
 |      >>> np.zeros((4,2))[::2].shape = (-1,)
 |      Traceback (most recent call last):
 |        File "<stdin>", line 1, in <module>
 |      AttributeError: incompatible shape for a non-contiguous array
 |      
 |      See Also
 |      --------
 |      numpy.reshape : similar function
 |      ndarray.reshape : similar method
 |  
 |  size
 |      Number of elements in the array.
 |      
 |      Equal to ``np.prod(a.shape)``, i.e., the product of the array's
 |      dimensions.
 |      
 |      Notes
 |      -----
 |      `a.size` returns a standard arbitrary precision Python integer. This
 |      may not be the case with other methods of obtaining the same value
 |      (like the suggested ``np.prod(a.shape)``, which returns an instance
 |      of ``np.int_``), and may be relevant if the value is used further in
 |      calculations that may overflow a fixed size integer type.
 |      
 |      Examples
 |      --------
 |      >>> x = np.zeros((3, 5, 2), dtype=np.complex128)
 |      >>> x.size
 |      30
 |      >>> np.prod(x.shape)
 |      30
 |  
 |  strides
 |      Tuple of bytes to step in each dimension when traversing an array.
 |      
 |      The byte offset of element ``(i[0], i[1], ..., i[n])`` in an array `a`
 |      is::
 |      
 |          offset = sum(np.array(i) * a.strides)
 |      
 |      A more detailed explanation of strides can be found in the
 |      "ndarray.rst" file in the NumPy reference guide.
 |      
 |      Notes
 |      -----
 |      Imagine an array of 32-bit integers (each 4 bytes)::
 |      
 |        x = np.array([[0, 1, 2, 3, 4],
 |                      [5, 6, 7, 8, 9]], dtype=np.int32)
 |      
 |      This array is stored in memory as 40 bytes, one after the other
 |      (known as a contiguous block of memory).  The strides of an array tell
 |      us how many bytes we have to skip in memory to move to the next position
 |      along a certain axis.  For example, we have to skip 4 bytes (1 value) to
 |      move to the next column, but 20 bytes (5 values) to get to the same
 |      position in the next row.  As such, the strides for the array `x` will be
 |      ``(20, 4)``.
 |      
 |      See Also
 |      --------
 |      numpy.lib.stride_tricks.as_strided
 |      
 |      Examples
 |      --------
 |      >>> y = np.reshape(np.arange(2*3*4), (2,3,4))
 |      >>> y
 |      array([[[ 0,  1,  2,  3],
 |              [ 4,  5,  6,  7],
 |              [ 8,  9, 10, 11]],
 |             [[12, 13, 14, 15],
 |              [16, 17, 18, 19],
 |              [20, 21, 22, 23]]])
 |      >>> y.strides
 |      (48, 16, 4)
 |      >>> y[1,1,1]
 |      17
 |      >>> offset=sum(y.strides * np.array((1,1,1)))
 |      >>> offset/y.itemsize
 |      17
 |      
 |      >>> x = np.reshape(np.arange(5*6*7*8), (5,6,7,8)).transpose(2,3,1,0)
 |      >>> x.strides
 |      (32, 4, 224, 1344)
 |      >>> i = np.array([3,5,2,2])
 |      >>> offset = sum(i * x.strides)
 |      >>> x[3,5,2,2]
 |      813
 |      >>> offset / x.itemsize
 |      813
 |  
 |  ----------------------------------------------------------------------
 |  Data and other attributes defined here:
 |  
 |  __hash__ = None

In order to create the object arr (or instanciate the class np.ndarray), the constructor of the class is called with a set of parameters. This constructor returns the desired instance.

In [3]:
arr = np.ndarray(shape=(2, 2), dtype=int)
arr
Out[3]:
array([[       52116560, 139685221367808],
       [              0,               0]])

As seen in the documentation, the object arr has attributes:

In [4]:
print(arr.shape, arr.size)
(2, 2) 4

which charactarize the object, and methods (these are functions), often aimed at modifying the object or at returning an information given a particular knowledge:

In [5]:
arr.sort()
arr
Out[5]:
array([[       52116560, 139685221367808],
       [              0,               0]])
In [6]:
arr.mean(axis=0)
Out[6]:
array([2.60582800e+07, 6.98426107e+13])

All attributes and methods can be discovered in the documentation or with dir:

In [7]:
dir(arr)[-10:]
Out[7]:
['swapaxes',
 'take',
 'tobytes',
 'tofile',
 'tolist',
 'tostring',
 'trace',
 'transpose',
 'var',
 'view']

At the first look, attributes are features (floats, lists, strings…) stored in the object and methods are functions aimed at acting on the object. However, there may be an overlap between the roles of attributes and methods, as exemplified by the transposition of arrays:

In [8]:
arr.T
Out[8]:
array([[       52116560,               0],
       [139685221367808,               0]])
In [9]:
arr.transpose()
Out[9]:
array([[       52116560,               0],
       [139685221367808,               0]])
In [10]:
arr
Out[10]:
array([[       52116560, 139685221367808],
       [              0,               0]])

Both commands return exactly the same result, without modifying the object nor requiring an information, since transpose() is used without arguments (.T is actually defined by transpose()). This is so because the transposition of an array may be seen either as a feature of the object (user's point of view) or as an information requiring to browse the array (developper's point of view).

Creating a class

Let us start with a naive example.

In [11]:
class Student():
    """
    name: student name
    num: student number
    """
    def __init__(self, name, num=0):
        self.name = name
        self.num = num
        self.mark_ = None
        
    def __str__(self):
        return 'Student {} (number {})'.format(self.name, self.num)
    
    def set_mark(self, mark):
        self.mark_ = mark if type(mark) is str else None
        
    def get_mark(self):
        return 'not filled' if self.mark_ is None else self.mark_
In [12]:
help(Student)
Help on class Student in module __main__:

class Student(builtins.object)
 |  name: student name
 |  num: student number
 |  
 |  Methods defined here:
 |  
 |  __init__(self, name, num=0)
 |      Initialize self.  See help(type(self)) for accurate signature.
 |  
 |  __str__(self)
 |      Return str(self).
 |  
 |  get_mark(self)
 |  
 |  set_mark(self, mark)
 |  
 |  ----------------------------------------------------------------------
 |  Data descriptors defined here:
 |  
 |  __dict__
 |      dictionary for instance variables (if defined)
 |  
 |  __weakref__
 |      list of weak references to the object (if defined)

In [13]:
roger = Student('Roger', 42)
print(roger)
print('Mark:', roger.get_mark())
Student Roger (number 42)
Mark: not filled
In [14]:
roger.set_mark('A')
print('Mark:', roger.get_mark())
Mark: A

Basics of multiprocessing

This section is under construction. Please, refer to the official documentation.

The main idea of multiprocessing is to split a large amount of computations among differents computers or threads inside a computer. For now, we will focus on the latter case, called multithreading. It has the advantage to share memory on its own and not to be worried with broadcasting the data among various machines. In addition, it can quickly be set up on a powerful personnal server or on a virtual machine in the cloud.

Parallel computing particularly shines when:

  • there are few expensive tasks rather many cheap ones;
  • tasks are single threaded rather than multithreaded.

Below is a naive example of sequential versus parallel computation in a favorable situation.

In [15]:
n = 1000  # Matrix size

make_psd = lambda x: x.dot(x.T)
matrices = [make_psd(np.random.randn(n, n)) for _ in range(4)]

def naive_work(M):
    R = np.ndarray((n, n))
    for i in range(M.size):
        R.flat[i] = np.exp(-M.flat[i])
    return R
In [16]:
from time import time

t_in = time()

res = []
for M in matrices:
    res.append(naive_work(M))
    
t_out = time()

print(f'{t_out - t_in:.0f} seconds')
5 seconds
In [17]:
from multiprocessing import Pool

t_in = time()

with Pool() as p:
    res = p.map(naive_work, matrices)
    
t_out = time()

print(f'{t_out - t_in:.0f} seconds')
2 seconds

EDIT (thanks to M. Anakök):

On Windows, the execution of the previous cell may not end or may produce an error. If so, it may solve the issue to define workers in a seperate python file and to check the scope of the code:

In [19]:
%%writefile workers.py

def naive_work(M):
    R = np.ndarray((n, n))
    for i in range(M.size):
        R.flat[i] = np.exp(-M.flat[i])
    return R
Writing workers.py
In [20]:
import workers

if __name__ == '__main__':
    t_in = time()

    with Pool() as p:
        res = p.map(naive_work, matrices)

    t_out = time()

    print(f'{t_out - t_in:.0f} seconds')
2 seconds

When tasks are multithreaded, parallel computing is useless.

In [18]:
def np_work(M):
    return np.exp(-M)
        
t_in = time()

res = []
for M in matrices:
    res.append(np_work(M))
    
t_out = time()

print(f'{t_out - t_in:.1f} seconds')
0.2 seconds
In [19]:
t_in = time()

with Pool() as p:
    res = p.map(np_work, matrices)
    
t_out = time()

print(f'{t_out - t_in:.1f} seconds')
0.1 seconds

Machine learning in Python

Introduction

Machine learning is a vibrant scientific field at the connection between computer science, statistics, optimization and functional analysis:

  • computer science: machine learning is mainly interested in algorithms for solving tangible problems (hand-written text recognition, image classification, etc.);
  • statistics: it is all about estimating a property of the distribution from which the observed data is drawn;
  • optimization: estimation often relies on minimizing an empirical risk;
  • functional analysis: convergence guarantees for the algorithms as well as models for the estimator rely on this mathematical field.

The tasks tackled in machine learning are:

  • (semi-)supervised learning: discovering a link between an input data (observed) and an outcome (supposed unknown);
  • unsupervised learning: discovering hidden patterns in data;
  • reinforcement learning: learning to achieve a goal based on interacting with the environment.

The rest of this document will focus on supervised learning.

Relation to statistics

Machine learning has deep roots into statistics since its main goal is to estimate a property of a distribution based on an observed sample. However, machine learning defines a new paradigm of inference, to propose an answer to the difficulty of modeling the distribution of the data.

Unlike inferential statistics, in which practitioners define a statistical model (assumption on the distribution from which is drawn the observed sample), machine learners define a model for the estimator (no assumption on the underlying distribution).

To illustrate our words, let $(X, Y)$ be a couple of random variables with unknown joint distribution. The main tasks of supervised learning are:

  • regression: estimating $x \mapsto \mathbb E[Y ~|~ X=x]$, given that $Y$ has values in $\mathbb R$;
  • classification: estimating $x \mapsto \operatorname{sign}\left(\frac{\mathbb P (Y=1 ~|~ X=x)}{\mathbb P (Y=-1 ~|~ X=x)} -1\right)$, given that $Y$ has values in $\{-1, 1\}$.

These tasks of estimating the regression or classification function are done within a hypothesis class of functions (model on the estimator), regardless of the distribution of $(X, Y)$.

Machine learning in Python

Several packages are available for doing machine learning in Python, depending on the kind of methods the practitioner opts for:

Here, we focus on Scikit-learn, which offers a broad range of machine learning methods and has a simple and well documented API.

Scikit-learn API

The general procedure for using a model is:

  • creation (for instance of a decision tree):
clf = tree.DecisionTreeClassifier()
  • learning (estimation with data corresponding to $X$ and labels to $Y$):
clf.fit(data, labels)
  • prediction (of unknown labels):
clf.predict(data)

Example

We provide here a (counter)example of a machine learning method for classification. This one assumes that each class is normally distributed and estimates the Gaussian parameters using maximum likelihood.

In [20]:
import numpy as np
from sklearn.naive_bayes import GaussianNB
import matplotlib.pyplot as plt
%matplotlib inline

# Generate data
n = 200  # Sample size
x = np.r_[np.random.randn(n//2, 2) + [0, 1],
         np.random.randn(n//2, 2) + [3, -0.5]]  # Data
y = np.r_[np.ones(n//2), -np.ones(n//2)]  # Labels

# Fit and evaluate the model (classifier)
clf = GaussianNB()  # Model creation
clf.fit(x, y)  # Fit the model
y_pred = clf.predict(x)  # Predict with the model
s = clf.score(x, y)  # Compute the score
print("Classification score on training data is {0:0.0f}/100.".format(s*100))

# Plot data and prediction
fig, ax = plt.subplots()
mks = 10  # Marker size
alpha = 0.1
#ax.plot(x[y_pred>0, 0], x[y_pred>0, 1], 'bo', markersize=mks+5, alpha=alpha)  # Pred
#ax.plot(x[y_pred<0, 0], x[y_pred<0, 1], 'ro', markersize=mks+5, alpha=alpha)  # Pred
ax.plot(x[y>0, 0], x[y>0, 1], 'b.', markersize=mks)  # Ground truth
ax.plot(x[y<0, 0], x[y<0, 1], 'r.', markersize=mks);  # Ground truth

# Plot the frontier
X, Y = np.meshgrid(np.linspace(-5, 5, num=500), np.linspace(-5, 5, num=500))
X, Y = X.ravel(), Y.ravel()

Z = clf.predict_proba(np.c_[X, Y])[:, 0]
ind = np.where(np.fabs(Z-0.5) < 1e-2)
ax.plot(X[ind], Y[ind], 'g.', label="Frontier")

ax.legend(loc="best");
Classification score on training data is 95/100.

Exercises

Exercise 1

Inspired by scipy.stats, define a class Normal, that represents a random variable following a Gaussian distribution. The object should have attributes:

  • loc: location;
  • scale: scale;
  • mean: theoretical mean;
  • std: theoretical standard deviation;
  • var: theoretical variance;

and methods:

  • pdf(x): returns the pdf value for x;
  • rvs(size=1): returns an iid random sample of the specified sise.
In [ ]:
# Answer

How to redefine the previous class using inheritance?

In [ ]:
# Answer

Exercise 2

Compare time to solve 4 linear systems sequentially and using multithreading.

In [ ]:
# Answer

Exercise 3

We focus here on a classification problem. Propose a procedure to evaluate the accuracy of a classifer. Apply this procedure to compare a k-nearest neighbors algorithm with a decision tree on the Iris dataset.

In [ ]:
# Answer