.. _tutorial-interpolate_fitpack:

.. currentmodule:: scipy.interpolate

=================
Smoothing splines
=================

Spline smoothing in 1D
======================

For the interpolation problem, the task is to construct a curve which passes
through a given set of data points. This may be not appropriate if the data is
noisy: we then want to construct a smooth curve, :math:`g(x)`, which *approximates*
input data without passing through each point exactly.

To this end, `scipy.interpolate` allows constructing *smoothing splines* which
balance how close the resulting curve, :math:`g(x)`, is to the data, and 
the smoothness of :math:`g(x)`. Mathematically, the task is to solve
a penalized least-squares problem, where the penalty controls the smoothness of
:math:`g(x)`.

We provide two approaches to constructing smoothing splines, which differ in
(1) the form of the penalty term, and (2) the basis in which the smoothing
curve is constructed. Below we consider these two approaches.

The former variant is performed by the `make_smoothing_spline` function,
which is a clean-room reimplementation of the classic ``gcvspl`` Fortran package
by H. Woltring.
The latter variant is implemented by the `make_splrep` function, which is a
reimplementation of the Fortran FITPACK library by P. Dierckx. :ref:`A legacy interface 
to the FITPACK library <tutorial-fitpack-legacy>` is also available.


"Classic" smoothing splines and generalized cross-validation (GCV) criterion
----------------------------------------------------------------------------

Given the data arrays ``x`` and ``y`` and the array of
non-negative *weights*, ``w``, we look for a cubic spline function ``g(x)``
which minimizes

.. math::

        \sum\limits_{j=1}^n w_j \left\lvert y_j - g(x_j) \right\rvert^2 +
        \lambda\int\limits_{x_1}^{x_n} \left( g^{(2)}(u) \right)^2 d u

where :math:`\lambda \geqslant 0` is a non-negative penalty parameter, and :math:`g^{(2)}(x)`
is the second derivative of :math:`g(x)`. The summation in the first term runs
over the data points, :math:`(x_j, y_j)`, and the integral in the second term is
over the whole interval :math:`x \in [x_1, x_n]`.

Here the first term penalizes the deviation of the spline function from the data,
and the second term penalizes large values of the second derivative---which is
taken as the criterion for the smoothness of a curve.

The target function, :math:`g(x)`, is taken to be a natural cubic spline *with
knots at the data points*, :math:`x_j`, and the minimization is carried over
the spline coefficients at a given value of :math:`\lambda`.

Clearly, :math:`\lambda = 0` corresponds to the interpolation problem---the result
is a *natural interpolating spline*; in the opposite limit, :math:`\lambda \gg 1`,
the result :math:`g(x)` approaches a straight line (since the minimization
effectively zeros out the second derivative of :math:`g(x)`).

The smoothing function strongly depends on :math:`\lambda`,
and multiple strategies are possible for selecting an "optimal" value of the
penalty. One popular strategy is the so-called *generalized cross-validation* (GCV):
conceptually, this is equivalent to comparing the spline functions constructed on
reduced sets of data where we leave out one data point. Direct application of
this leave-one-out cross-validation procedure is very costly, and we use a
more efficient GCV algorithm.

To construct the smoothing spline given data and the penalty parameter,
we use the function `make_smoothing_spline`. Its interface is similar to the
constructor of interpolating splines, `make_interp_spline`: it accepts data arrays
and returns a callable `BSpline` instance. 

Additionally, it accepts an optional ``lam`` keyword argument to specify the penalty
parameter :math:`\lambda`. If omitted or set to ``None``, :math:`\lambda` is computed
via the GCV procedure.

To illustrate the effect of the penalty parameter, consider a toy example of a
sine curve with some noise:

.. plot::

   >>> import numpy as np
   >>> from scipy.interpolate import make_smoothing_spline

   Generate some noisy data:

   >>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/16)
   >>> rng = np.random.default_rng()
   >>> y =  np.sin(x) + 0.4*rng.standard_normal(size=len(x))

   Construct and plot smoothing splines for a series of values of the penalty
   parameter:

   >>> import matplotlib.pyplot as plt
   >>> xnew = np.arange(0, 9/4, 1/50) * np.pi
   >>> for lam in [0, 0.02, 10, None]:
   ...     spl = make_smoothing_spline(x, y, lam=lam)
   ...     plt.plot(xnew, spl(xnew), '-.', label=fr'$\lambda=${lam}')
   >>> plt.plot(x, y, 'o')
   >>> plt.legend()
   >>> plt.show()

We clearly see that ``lam=0`` constructs the interpolating spline; large values
of ``lam`` flatten out the resulting curve towards a straight line; and the GCV
result, ``lam=None``, is close to the underlying sine curve.


.. _tutorial-interpolate_GCV_batching:

Batching of ``y`` arrays
````````````````````````

`make_smoothing_spline` constructor accepts multidimensional ``y`` arrays and an
optional ``axis`` parameter and interprets them exactly the same way the
interpolating spline constructor, `make_interp_spline` does. See the
:ref:`interpolation section <tutorial-interpolate_batching>` for a discussion and
examples.


.. _tutorial_make_splrep:

Smoothing splines with automatic knot selection
-----------------------------------------------

As an addition to `make_smoothing_spline`, SciPy provides an alternative, in the
form of `make_splrep` and `make_splprep` routines. The former constructs spline
functions and the latter is for parametric spline curves in :math:`d > 1` dimensions.

While having a similar API (receive the data arrays, return a `BSpline` instance),
these differ from `make_smoothing_spline` in several ways:

- the functional form of the penalty term is different: these routines use
  *jumps* of the :math:`k`-th derivative instead of the integral of the
  :math:`(k-1)`-th derivative;
- instead of the penalty parameter :math:`\lambda`, a smoothness parameter :math:`s`
  is used;
- these routines automatically construct the knot vector; depending on inputs,
  resulting splines may have much fewer knots than data points.
- by default boundary conditions differ: while `make_smoothing_spline` constructs
  natural cubic splines, these routines use the not-a-knot boundary conditions by default. 

Let us consider the algorithm in more detail. First, **the smoothing criterion**.
Given a cubic spline function, :math:`g(x)`, defined by the knots, :math:`t_j`, and coefficients,
:math:`c_j`, consider the jumps of the third derivative at internal knots,

.. math::

    D_j = g^{(3)}(t_j + 0) - g^{(3)}(t_j - 0) .

(For degree-:math:`k` splines, we would have used jumps of the :math:`k`-th derivative.)

If all :math:`D_j = 0`, then :math:`g(x)` is a single polynomial on the whole
domain spanned by the knots. We thus consider :math:`g(x)` to be a piecewise
:math:`C^2`-differentiable spline function and use as the smoothing criterion
the sum of jumps,

.. math::

    \sum_j \left| D_j \right|^2 \to \mathrm{min} ,

where the minimization performed is over the spline coefficients, and, potentially,
the spline knots.

To make sure :math:`g(x)` approximates the input data, :math:`x_j` and :math:`y_j`,
we introduce the *smoothness parameter* :math:`s \geqslant 0` and add a constraint
that

.. math::

       \sum_{j=1}^m \left[w_j \times \left(g(x_j) - y_j\right) \right]^2 \leqslant s .

In this formulation, the smoothness parameter :math:`s` is a user input, much like
the penalty parameter :math:`\lambda` is for the classic smoothing splines.

Note that the limit ``s = 0`` corresponds to the interpolation problem where
:math:`g(x_j) = y_j`. Increasing ``s`` leads to smoother fits, and in the limit
of a very large ``s``, :math:`g(x)` degenerates into a single best-fit polynomial.

For a fixed knot vector and a given value of :math:`s`, the minimization problem is linear.
If we also minimize with respect to the knots, the problem becomes non-linear.
We thus need to specify an iterative minimization process to **construct the knot vector**
along with the spline coefficients.

We therefore use the following procedure:

- we start with a spline with no internal knots, and check the smoothness condition
  for the user-provided value of :math:`s`. If it is satisfied, we are done. Otherwise,

- iterate, where on each iteration we

  * add new knots by splitting the interval with the maximum deviation
    between the spline function :math:`g(x_j)` and the data :math:`y_j`.

  * construct the next approximation for :math:`g(x)` and check the smoothness
    criterion.

The iterations stop if either the smoothness condition is satisfied, or the
maximum allowed number of knots is reached. The latter can be either specified
by a user, or is taken as the default value ``len(x) + k + 1`` which corresponds
to the interpolation of the data array ``x`` with splines of degree ``k``.

Rephrasing and glossing over details, the procedure is to iterate over the
knot vectors generated by `generate_knots`, applying `make_lsq_spline` on each
step. In pseudocode::

    for t in generate_knots(x, y, s=s):
        g = make_lsq_spline(x, y, t=t)     # construct
        if ((y - g(x))**2).sum() < s:      # check smoothness
            break

.. note::

    For ``s=0``, we take a short-cut and construct the interpolating spline
    with the not-a-knot boundary condition instead of iterating.

The iterative procedure of constructing a knot vector is available through the
generator function `generate_knots`. To illustrate:

>>> import numpy as np
>>> from scipy.interpolate import generate_knots
>>> x = np.arange(7)
>>> y = x**4
>>> list(generate_knots(x, y, s=1))   # default is cubic splines, k=3
[array([0., 0., 0., 0., 6., 6., 6., 6.]),
 array([0., 0., 0., 0., 3., 6., 6., 6., 6.]),
 array([0., 0., 0., 0., 3., 5., 6., 6., 6., 6.]),
 array([0., 0., 0., 0., 2., 3., 4., 6., 6., 6., 6.])]

For ``s=0``, the generator cuts short:

>>> list(generate_knots(x, y, s=0))
[array([0, 0, 0, 0, 2, 3, 4, 6, 6, 6, 6])]

.. note::

    In general, knots are placed at data sites. The exception is even-order
    splines, where the knots can be placed away from data. This happens when
    ``s=0`` (interpolation), or when ``s`` is small enough so that the maximum number
    of knots is reached, and the routine switches to the ``s=0`` knot vector
    (sometimes known as Greville abscissae).

>>> list(generate_knots(x, y, s=1, k=2))   # k=2, quadratic spline
[array([0., 0., 0., 6., 6., 6.]),
 array([0., 0., 0., 3., 6., 6., 6.]),
 array([0., 0., 0., 3., 5., 6., 6., 6.]),
 array([0., 0., 0., 2., 3., 5., 6., 6., 6.]),
 array([0. , 0. , 0. , 1.5, 2.5, 3.5, 4.5, 6. , 6. , 6. ])]   # Greville sites

.. note::

    The heuristics for constructing the knot vector follows the algorithm used
    by the FITPACK Fortran library. The algorithm is the same, and small differences
    are possible due to floating-point rounding.

We now illustrate `make_splrep` results, using the same toy dataset as in the previous section

.. plot::

   >>> import numpy as np
   >>> from scipy.interpolate import make_splrep

   Generate some noisy data

   >>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/16)
   >>> rng = np.random.default_rng()
   >>> y =  np.sin(x) + 0.4*rng.standard_normal(size=len(x))

   Construct and plot smoothing splines for a series of values of the ``s``
   parameter:

   >>> import matplotlib.pyplot as plt
   >>> xnew = np.arange(0, 9/4, 1/50) * np.pi

   >>> plt.plot(xnew, np.sin(xnew), '-.', label='sin(x)')
   >>> plt.plot(xnew, make_splrep(x, y, s=0)(xnew), '-', label='s=0')
   >>> plt.plot(xnew, make_splrep(x, y, s=len(x))(xnew), '-', label=f's={len(x)}')
   >>> plt.plot(x, y, 'o')
   >>> plt.legend()
   >>> plt.show()

We see that the :math:`s=0` curve follows the (random) fluctuations of the data points,
while the :math:`s > 0` curve is close to the underlying sine function. Also note
that the extrapolated values vary wildly depending on the value of :math:`s`.

Finding a good value of :math:`s` is a trial-and-error process. If
the weights correspond to the inverse of standard deviations of the input data,
a "good" value of :math:`s` is expected to be somewhere between :math:`m - \sqrt{2m}`
and :math:`m + \sqrt{2m}`, where :math:`m` is the number of data points.
If all weights equal unity, a reasonable choice might be around :math:`s \sim m\,\sigma^2`,
where :math:`\sigma` is an estimate for the standard deviation of the data. 

.. note::

    The number of knots a very strongly dependent on ``s``. It is possible that
    small variations of ``s`` lead to drastic changes in the knot number.

.. note::

    Both `make_smoothing_spline` and `make_splrep` allow for weighted fits,
    where the user provides an array of weights, ``w``. Note that the definition
    differs somewhat: `make_smoothing_spline` squares the weights to be consistent
    with ``gcvspl``, while `make_splrep` does not---to be consistent with ``FITPACK``.



Smoothing spline curves in :math:`d>1`
--------------------------------------

So far we considered constructing smoothing spline functions, :math:`g(x)` given
data arrays ``x`` and ``y``. We now consider a related problem of constructing
a smoothing spline *curve*, where we consider the data as points on a plane,
:math:`\mathbf{p}_j = (x_j, y_j)`, and we want to construct a parametric function
:math:`\mathbf{g}(\mathbf{p}) = (g_x(u), g_y(u))`, where the values of the
parameter :math:`u_j` correspond to :math:`x_j` and :math:`y_j`.

Note that this problem readily generalizes to higher dimensions, :math:`d > 2`:
we simply have :math:`d` data arrays and construct a parametric function with
:math:`d` components.

Also note that the choice of parametrization cannot be automated, and different
parameterizations can lead to very different curves for the same data,
even for :ref:`interpolating curves<tutorial-interpolate_parametric>`.

Once a specific form of parametrization is chosen, the problem of constructing a
smoothing curve is conceptually very similar to constructing a smoothing spline
function. In a nutshell, 

- spline knots are added from the values of the parameter :math:`u`, and

- both the cost function to minimize and the constraint
  we considered for :ref:`spline functions <tutorial_make_splrep>` simply
  get an extra summation over the :math:`d` components.

The "parametric" generalization of the `make_splrep` function is `make_splprep`,
and its docstring spells out the precise mathematical formulation of the minimization
problem it solves.

The main user-visible difference of the parametric case is the user interface:

- instead of two data arrays, ``x`` and ``y``, `make_splprep` receives a single
  two-dimensional array, where the second dimension has size :math:`d` and each
  data array is stored along the first dimension (alternatively, you can supply
  a list of 1D arrays).

- the return value is pair: a `BSpline` instance and the array of parameter
  values, ``u``, which corresponds to the input data arrays.

By default, `make_splprep` constructs and returns the cord length parametrization
of input data (see the :ref:`Parametric spline curves <tutorial-interpolate_parametric>`
section for details). Alternatively, you can provide your own array of
parameter values, ``u``.

To illustrate the API, consider a toy problem: we have some data sampled from
a folium of Descartes *plus* some noise.

.. plot::

   >>> import numpy as np
   >>> from scipy.interpolate import make_splprep
   >>> th = np.linspace(-0.2, np.pi/2 + 0.2, 21)
   >>> r = 3 * np.sin(th) * np.cos(th) / (np.sin(th)**3 + np.cos(th)**3)
   >>> x, y = r * np.cos(th), r * np.sin(th)

   Add some noise and construct the interpolators

   >>> rng = np.random.default_rng()
   >>> xn = x + 0.1*rng.uniform(-1, 1, size=len(x))
   >>> yn = y + 0.1*rng.uniform(-1, 1, size=len(x))
   >>> spl, u = make_splprep([xn, yn], s=0)   # note the [xn, yn] argument
   >>> spl_n, u_n = make_splprep([xn, yn], s=0.1)

   And plot the results (the result of ``spl(u)`` is a 2D array, so we unpack it
   into a pair of ``x`` and ``y`` arrays for plotting).

   >>> import matplotlib.pyplot as plt
   >>> plt.plot(xn, yn, 'o')
   >>> plt.plot(*spl(u), '--')
   >>> plt.plot(*spl_n(u_n), '-')
   >>> plt.show()


Batching of ``y`` arrays
------------------------

Unlike :ref:`interpolating splines <tutorial-interpolate_batching>` and
:ref:`GCV smoothers <tutorial-interpolate_GCV_batching>`, `make_splrep` and `make_splprep`
do **not** allow multidimensional ``y`` arrays and require that ``x.ndim == y.ndim == 1``.

The technical reason for this limitation is that the length of the knot vector ``t``
depends on the ``y`` values, thus for a batched ``y``, the batched ``t`` could be a
ragged array, which `BSpline` is not equipped to handle.

Therefore if you need to handle batched inputs, you will need to loop over the
batch manually and construct a `BSpline` object per slice of the batch.
Having done that, you can mimic `BSpline` behavior for evaluations with a workaround
along the lines of

.. code-block:: python

    class BatchSpline:
        """BSpline-like class with batch behavior."""
        def __init__(self, x, y, axis, *, spline, **kwargs):
            y = np.moveaxis(y, axis, -1)
            self._batch_shape = y.shape[:-1]
            self._splines = [
                spline(x, yi, **kwargs) for yi in y.reshape(-1, y.shape[-1])
            ]
            self._axis = axis

        def __call__(self, x):
            y = [spline(x) for spline in self._splines]
            y = np.reshape(y, self._batch_shape + x.shape)
            return np.moveaxis(y, -1, self._axis) if x.shape else y


.. _tutorial-fitpack-legacy:

Legacy routines for spline smoothing in 1-D
-------------------------------------------

In addition to smoothing splines constructors we discussed in the previous sections,
`scipy.interpolate` provides direct interfaces for routines from the venerable FITPACK
Fortran library authored by P. Dierckx.

.. note::

    These interfaces should be considered *legacy*---while we do not
    plan to deprecate or remove them, we recommend that new code uses more modern
    alternatives, `make_smoothing_spline`, `make_splrep` or `make_splprep`, instead.

For historical reasons, `scipy.interpolate` provides two equivalent interfaces
for FITPACK, a interface and an object-oriented interface. While equivalent, these interfaces
have different defaults. Below we discuss them in turn, starting from the
functional interface.


.. _tutorial-interpolate_splXXX:

Procedural (`splrep`)
`````````````````````

Spline interpolation requires two essential steps: (1) a spline
representation of the curve is computed, and (2) the spline is
evaluated at the desired points. In order to find the spline
representation, there are two different ways to represent a curve and
obtain (smoothing) spline coefficients: directly and parametrically.
The direct method finds the spline representation of a curve in a 2-D
plane using the function :obj:`splrep`. The
first two arguments are the only ones required, and these provide the
:math:`x` and :math:`y` components of the curve. The normal output is
a 3-tuple, :math:`\left(t,c,k\right)` , containing the knot-points,
:math:`t` , the coefficients :math:`c` and the order :math:`k` of the
spline. The default spline order is cubic, but this can be changed
with the input keyword, *k.*

The knot array defines the interpolation interval to be ``t[k:-k]``, so that 
the first :math:`k+1` and last :math:`k+1` entries of the ``t`` array define
*boundary knots*. The coefficients are a 1D array of length at least
``len(t) - k - 1``. Some routines pad this array to have ``len(c) == len(t)``---
these additional coefficients are ignored for the spline evaluation. 

The ``tck``-tuple format is compatible with
:ref:`interpolating b-splines <tutorial-interpolate_bspl_basis>`: the output of
`splrep` can be wrapped into a `BSpline` object, e.g. ``BSpline(*tck)``, and
the evaluation/integration/root-finding routines described below
can use ``tck``-tuples and `BSpline` objects interchangeably.

For curves in N-D space the function
:obj:`splprep` allows defining the curve
parametrically. For this function only 1 input argument is
required. This input is a list of :math:`N` arrays representing the
curve in N-D space. The length of each array is the
number of curve points, and each array provides one component of the
N-D data point. The parameter variable is given
with the keyword argument, ``u``, which defaults to an equally-spaced
monotonic sequence between :math:`0` and :math:`1` (i.e., the :ref:`uniform
parametrization <tutorial-interpolate_parametric>`).

The output consists of two objects: a 3-tuple, :math:`\left(t,c,k\right)`
, containing the spline representation and the parameter variable
:math:`u.`

The coefficients are a list of :math:`N` arrays, where each array corresponds to
a dimension of the input data. Note that the knots, ``t`` correspond to the
parametrization of the curve ``u``. 

The keyword argument, ``s`` , is used to specify the amount of smoothing
to perform during the spline fit. The default value of :math:`s` is
:math:`s=m-\sqrt{2m}` where :math:`m` is the number of data points
being fit. Therefore, **if no smoothing is desired a value of**
:math:`\mathbf{s}=0` **should be passed to the routines.**

Once the spline representation of the data has been determined, it can be
evaluated either using the `splev` function or by wrapping
the `tck` tuple into a `BSpline` object, as demonstrated below.

We start by illustrating the effect of the ``s`` parameter on smoothing some
synthetic noisy data

.. plot::

   >>> import numpy as np
   >>> from scipy.interpolate import splrep, BSpline

   Generate some noisy data

   >>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/16)
   >>> rng = np.random.default_rng()
   >>> y =  np.sin(x) + 0.4*rng.standard_normal(size=len(x))

   Construct two splines with different values of ``s``.

   >>> tck = splrep(x, y, s=0)
   >>> tck_s = splrep(x, y, s=len(x))

   And plot them

   >>> import matplotlib.pyplot as plt
   >>> xnew = np.arange(0, 9/4, 1/50) * np.pi
   >>> plt.plot(xnew, np.sin(xnew), '-.', label='sin(x)')
   >>> plt.plot(xnew, BSpline(*tck)(xnew), '-', label='s=0')
   >>> plt.plot(xnew, BSpline(*tck_s)(xnew), '-', label=f's={len(x)}')
   >>> plt.plot(x, y, 'o')
   >>> plt.legend()
   >>> plt.show()

We see that the ``s=0`` curve follows the (random) fluctuations of the data points,
while the ``s > 0`` curve is close to the underlying sine function.
Also note that the extrapolated values vary wildly depending on the value of ``s``.

The default value of ``s`` depends on whether the weights are supplied or not,
and also differs for `splrep` and `splprep`. Therefore, we recommend always
providing the value of ``s`` explicitly.


Manipulating spline objects: procedural (``splXXX``)
````````````````````````````````````````````````````

Once the spline representation of the data has been determined,
functions are available for evaluating the spline
(:func:`splev`) and its derivatives
(:func:`splev`, :func:`spalde`) at any point
and the integral of the spline between any two points (
:func:`splint`). In addition, for cubic splines ( :math:`k=3`
) with 8 or more knots, the roots of the spline can be estimated (
:func:`sproot`). These functions are demonstrated in the
example that follows.

.. plot::
   :alt: " "

   >>> import numpy as np
   >>> import matplotlib.pyplot as plt
   >>> from scipy import interpolate

   Cubic spline

   >>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/8)
   >>> y = np.sin(x)
   >>> tck = interpolate.splrep(x, y, s=0)
   >>> xnew = np.arange(0, 2*np.pi, np.pi/50)
   >>> ynew = interpolate.splev(xnew, tck, der=0)

   Note that the last line is equivalent to ``BSpline(*tck)(xnew)``.

   >>> plt.figure()
   >>> plt.plot(x, y, 'x', xnew, ynew, xnew, np.sin(xnew), x, y, 'b')
   >>> plt.legend(['Linear', 'Cubic Spline', 'True'])
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('Cubic-spline interpolation')
   >>> plt.show()

   Derivative of spline

   >>> yder = interpolate.splev(xnew, tck, der=1)   # or BSpline(*tck)(xnew, 1)
   >>> plt.figure()
   >>> plt.plot(xnew, yder, xnew, np.cos(xnew),'--')
   >>> plt.legend(['Cubic Spline', 'True'])
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('Derivative estimation from spline')
   >>> plt.show()

   All derivatives of spline

   >>> yders = interpolate.spalde(xnew, tck)
   >>> plt.figure()
   >>> for i in range(len(yders[0])):
   ...    plt.plot(xnew, [d[i] for d in yders], '--', label=f"{i} derivative")
   >>> plt.legend()
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('All derivatives of a B-spline')
   >>> plt.show()

   Integral of spline

   >>> def integ(x, tck, constant=-1):
   ...     x = np.atleast_1d(x)
   ...     out = np.zeros(x.shape, dtype=x.dtype)
   ...     for n in range(len(out)):
   ...         out[n] = interpolate.splint(0, x[n], tck)
   ...     out += constant
   ...     return out

   >>> yint = integ(xnew, tck)
   >>> plt.figure()
   >>> plt.plot(xnew, yint, xnew, -np.cos(xnew), '--')
   >>> plt.legend(['Cubic Spline', 'True'])
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('Integral estimation from spline')
   >>> plt.show()

   Roots of spline

   >>> interpolate.sproot(tck)
   array([3.1416])  # may vary

   Notice that `sproot` may fail to find an obvious solution at the edge of the
   approximation interval, :math:`x = 0`. If we define the spline on a slightly
   larger interval, we recover both roots :math:`x = 0` and :math:`x = \pi`:

   >>> x = np.linspace(-np.pi/4, np.pi + np.pi/4, 51)
   >>> y = np.sin(x)
   >>> tck = interpolate.splrep(x, y, s=0)
   >>> interpolate.sproot(tck)
   array([0., 3.1416])

   Parametric spline

   >>> t = np.arange(0, 1.1, .1)
   >>> x = np.sin(2*np.pi*t)
   >>> y = np.cos(2*np.pi*t)
   >>> tck, u = interpolate.splprep([x, y], s=0)
   >>> unew = np.arange(0, 1.01, 0.01)
   >>> out = interpolate.splev(unew, tck)
   >>> plt.figure()
   >>> plt.plot(x, y, 'x', out[0], out[1], np.sin(2*np.pi*unew), np.cos(2*np.pi*unew), x, y, 'b')
   >>> plt.legend(['Linear', 'Cubic Spline', 'True'])
   >>> plt.axis([-1.05, 1.05, -1.05, 1.05])
   >>> plt.title('Spline of parametrically-defined curve')
   >>> plt.show()

Note that in the last example, `splprep` returns the spline coefficients as a
list of arrays, where each array corresponds to a dimension of the input data.
Thus to wrap its output to a `BSpline`, we need to transpose the coefficients
(or use ``BSpline(..., axis=1)``):

  >>> tt, cc, k = tck
  >>> cc = np.array(cc)
  >>> bspl = BSpline(tt, cc.T, k)    # note the transpose
  >>> xy = bspl(u)
  >>> xx, yy = xy.T   # transpose to unpack into a pair of arrays
  >>> np.allclose(x, xx)
  True
  >>> np.allclose(y, yy)
  True



Object-oriented (:class:`UnivariateSpline`)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The spline-fitting capabilities described above are also available via
an objected-oriented interface.  The 1-D splines are
objects of the `UnivariateSpline` class, and are created with the
:math:`x` and :math:`y` components of the curve provided as arguments
to the constructor.  The class defines :meth:`__call__ <UnivariateSpline.__call__>`, allowing the object
to be called with the x-axis values, at which the spline should be
evaluated, returning the interpolated y-values.  This is shown in
the example below for the subclass `InterpolatedUnivariateSpline`.
The :meth:`integral <UnivariateSpline.integral>`,
:meth:`derivatives <UnivariateSpline.derivatives>`, and
:meth:`roots <UnivariateSpline.roots>` methods are also available
on `UnivariateSpline` objects, allowing definite integrals,
derivatives, and roots to be computed for the spline.

The UnivariateSpline class can also be used to smooth data by
providing a non-zero value of the smoothing parameter `s`, with the
same meaning as the `s` keyword of the :obj:`splrep` function
described above.  This results in a spline that has fewer knots
than the number of data points, and hence is no longer strictly
an interpolating spline, but rather a smoothing spline.  If this
is not desired, the `InterpolatedUnivariateSpline` class is available.
It is a subclass of `UnivariateSpline` that always passes through all
points (equivalent to forcing the smoothing parameter to 0). This
class is demonstrated in the example below.

The `LSQUnivariateSpline` class is the other subclass of `UnivariateSpline`.
It allows the user to specify the number and location of internal
knots explicitly with the parameter `t`.  This allows for the creation
of customized splines with non-linear spacing, to interpolate in
some domains and smooth in others, or change the character of the
spline.


.. plot::
   :alt: " "

   >>> import numpy as np
   >>> import matplotlib.pyplot as plt
   >>> from scipy import interpolate

   InterpolatedUnivariateSpline

   >>> x = np.arange(0, 2*np.pi+np.pi/4, 2*np.pi/8)
   >>> y = np.sin(x)
   >>> s = interpolate.InterpolatedUnivariateSpline(x, y)
   >>> xnew = np.arange(0, 2*np.pi, np.pi/50)
   >>> ynew = s(xnew)

   >>> plt.figure()
   >>> plt.plot(x, y, 'x', xnew, ynew, xnew, np.sin(xnew), x, y, 'b')
   >>> plt.legend(['Linear', 'InterpolatedUnivariateSpline', 'True'])
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('InterpolatedUnivariateSpline')
   >>> plt.show()

   LSQUnivarateSpline with non-uniform knots

   >>> t = [np.pi/2-.1, np.pi/2+.1, 3*np.pi/2-.1, 3*np.pi/2+.1]
   >>> s = interpolate.LSQUnivariateSpline(x, y, t, k=2)
   >>> ynew = s(xnew)

   >>> plt.figure()
   >>> plt.plot(x, y, 'x', xnew, ynew, xnew, np.sin(xnew), x, y, 'b')
   >>> plt.legend(['Linear', 'LSQUnivariateSpline', 'True'])
   >>> plt.axis([-0.05, 6.33, -1.05, 1.05])
   >>> plt.title('Spline with Specified Interior Knots')
   >>> plt.show()


.. _tutorial-interpolate_2d_spline:

2-D smoothing splines
=====================

In addition to smoothing 1-D splines, the FITPACK library provides the means of
fitting 2-D *surfaces* to two-dimensional data. The surfaces can be thought of
as functions of two arguments, :math:`z = g(x, y)`, constructed as tensor products
of 1-D splines.

Assuming that the data is held in three arrays, ``x``, ``y`` and ``z``, 
there are two ways these data arrays can be interpreted. First---the *scattered*
interpolation problem---the data is assumed to be paired, i.e. the pairs of
values ``x[i]`` and ``y[i]`` represent the coordinates of the point ``i``, which
corresponds to ``z[i]``.

The surface :math:`g(x, y)` is constructed to satisfy

.. math::

    \sum_i \left[ w_i (g(x_i, y_i) - z_i)\right]^2 \leqslant s

where :math:`w_i` are non-negative weights, and ``s`` is the input parameter,
known as the *smoothing factor*, which controls the interplay between smoothness
of the resulting function ``g(x, y)`` and the quality of the approximation of
the data (i.e., the differences between :math:`g(x_i, y_i)` and :math:`z_i`). The
limit of :math:`s = 0` formally corresponds to interpolation, where the surface
passes through the input data, :math:`g(x_i, y_i) = z_i`. See the note below however.

The second case---the *rectangular grid* interpolation problem---is where the data
points are assumed to be on a rectangular grid defined by all pairs of the
elements of the ``x`` and ``y`` arrays. For this problem, the ``z`` array is
assumed to be two-dimensional, and ``z[i, j]`` corresponds to ``(x[i], y[j])``.
The bivariate spline function :math:`g(x, y)` is constructed to satisfy

.. math::

    \sum_i \sum_j \left[ (g(x_i, y_j) - z_{i,j})\right]^2 \leqslant s

where ``s`` is the smoothing factor. Here the limit of :math:`s=0` also
formally corresponds to interpolation, :math:`g(x_i, y_j) = z_{i, j}`.

.. note::
    Internally, the smoothing surface :math:`g(x, y)` is constructed by placing
    spline knots into the bounding box defined by the data arrays. The knots are
    placed automatically via the FITPACK algorithm until the desired smoothness
    is reached. 

    *The knots may be placed away from the data points.*

    While :math:`s=0` formally corresponds to a bivariate spline interpolation,
    the FITPACK algorithm is not meant for interpolation, and may lead to
    unexpected results.

    For scattered data interpolation, prefer `griddata`; for data on a regular
    grid, prefer `RegularGridInterpolator`. 


.. note::

    If the input data, ``x`` and ``y``, is such that input dimensions
    have incommensurate units and differ by many orders of magnitude, the
    interpolant :math:`g(x, y)` may have numerical artifacts. Consider
    rescaling the data before interpolation.

We now consider the two spline fitting problems in turn.


Bivariate spline fitting of scattered data
------------------------------------------

There are two interfaces for the underlying FITPACK library, a procedural
one and an object-oriented interface.


**Procedural interface (`bisplrep`)**


For (smooth) spline fitting to a 2-D surface, the function
:func:`bisplrep` is available. This function takes as required inputs
the **1-D** arrays ``x``, ``y``, and ``z``, which represent points on the
surface :math:`z=f(x, y).` The spline orders in ``x`` and ``y`` directions can
be specified via the optional parameters ``kx`` and ``ky``. The default is
a bicubic spline, ``kx=ky=3``.

The output of `bisplrep` is a list ``[tx ,ty, c, kx, ky]`` whose entries represent
respectively, the components of the knot positions, the coefficients
of the spline, and the order of the spline in each coordinate. It is
convenient to hold this list in a single object, ``tck``, so that it can
be passed easily to the function :obj:`bisplev`. The
keyword, ``s`` , can be used to change the amount of smoothing performed
on the data while determining the appropriate spline. The recommended values
for :math:`s` depend on the weights :math:`w_i`. If these are taken as :math:`1/d_i`,
with :math:`d_i` an estimate of the standard deviation of :math:`z_i`, a
good value of :math:`s` should be found in the range :math:`m- \sqrt{2m}, m + 
\sqrt{2m}`, where where :math:`m` is the number of data points in the ``x``,
``y``, and ``z`` vectors.

The default value is :math:`s=m-\sqrt{2m}`.  As a result, **if no smoothing is
desired, then ``s=0`` should be passed to `bisplrep`**. (See however the note above).

To evaluate the 2-D spline and its partial derivatives
(up to the order of the spline), the function
:obj:`bisplev` is required. This function takes as the
first two arguments **two 1-D arrays** whose cross-product specifies
the domain over which to evaluate the spline. The third argument is
the ``tck`` list returned from :obj:`bisplrep`. If desired,
the fourth and fifth arguments provide the orders of the partial
derivative in the :math:`x` and :math:`y` direction, respectively.

.. note::
    It is important to note that 2-D interpolation should not
    be used to find the spline representation of images. The algorithm
    used is not amenable to large numbers of input points. `scipy.signal`
    and `scipy.ndimage` contain more appropriate algorithms for finding
    the spline representation of an image. 

The 2-D interpolation commands are intended for use when interpolating a 2-D
function as shown in the example that follows. This
example uses the :obj:`mgrid <numpy.mgrid>` command in NumPy which is
useful for defining a "mesh-grid" in many dimensions. (See also the
:obj:`ogrid <numpy.ogrid>` command if the full-mesh is not
needed). The number of output arguments and the number of dimensions
of each argument is determined by the number of indexing objects
passed in :obj:`mgrid <numpy.mgrid>`.

.. plot::
   :alt: " "

   >>> import numpy as np
   >>> from scipy import interpolate
   >>> import matplotlib.pyplot as plt

   Define function over a sparse 20x20 grid

   >>> x_edges, y_edges = np.mgrid[-1:1:21j, -1:1:21j]
   >>> x = x_edges[:-1, :-1] + np.diff(x_edges[:2, 0])[0] / 2.
   >>> y = y_edges[:-1, :-1] + np.diff(y_edges[0, :2])[0] / 2.
   >>> z = (x+y) * np.exp(-6.0*(x*x+y*y))

   >>> plt.figure()
   >>> lims = dict(cmap='RdBu_r', vmin=-0.25, vmax=0.25)
   >>> plt.pcolormesh(x_edges, y_edges, z, shading='flat', **lims)
   >>> plt.colorbar()
   >>> plt.title("Sparsely sampled function.")
   >>> plt.show()

   Interpolate function over a new 70x70 grid

   >>> xnew_edges, ynew_edges = np.mgrid[-1:1:71j, -1:1:71j]
   >>> xnew = xnew_edges[:-1, :-1] + np.diff(xnew_edges[:2, 0])[0] / 2.
   >>> ynew = ynew_edges[:-1, :-1] + np.diff(ynew_edges[0, :2])[0] / 2.
   >>> tck = interpolate.bisplrep(x, y, z, s=0)
   >>> znew = interpolate.bisplev(xnew[:,0], ynew[0,:], tck)

   >>> plt.figure()
   >>> plt.pcolormesh(xnew_edges, ynew_edges, znew, shading='flat', **lims)
   >>> plt.colorbar()
   >>> plt.title("Interpolated function.")
   >>> plt.show()

..   :caption: Example of a 2-D spline interpolation.


**Object-oriented interface (`SmoothBivariateSpline`)**


The object-oriented interface for bivariate spline smoothing of scattered data,
`SmoothBivariateSpline` class, implements a subset of the functionality of the
`bisplrep` / `bisplev` pair, and has different defaults. 

It takes the elements of the weights array equal unity, :math:`w_i = 1`
and constructs the knot vectors automatically given the input value of the
smoothing factor `s`--- the default value is :math:`m`, the number of data points.

The spline orders in the ``x`` and ``y`` directions are controlled by the optional
parameters ``kx`` and ``ky``, with the default of ``kx=ky=3``.

We illustrate the effect of the smoothing factor using the following example:

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    from scipy.interpolate import SmoothBivariateSpline

    import warnings
    warnings.simplefilter('ignore')

    train_x, train_y = np.meshgrid(np.arange(-5, 5, 0.5), np.arange(-5, 5, 0.5))
    train_x = train_x.flatten()
    train_y = train_y.flatten()

    def z_func(x, y): 
        return np.cos(x) + np.sin(y) ** 2 + 0.05 * x + 0.1 * y

    train_z = z_func(train_x, train_y)
    interp_func = SmoothBivariateSpline(train_x, train_y, train_z, s=0.0)
    smth_func = SmoothBivariateSpline(train_x, train_y, train_z)

    test_x = np.arange(-9, 9, 0.01)
    test_y = np.arange(-9, 9, 0.01)
    grid_x, grid_y = np.meshgrid(test_x, test_y)

    interp_result = interp_func(test_x, test_y).T
    smth_result = smth_func(test_x, test_y).T
    perfect_result = z_func(grid_x, grid_y)

    fig, axes = plt.subplots(1, 3, figsize=(16, 8))
    extent = [test_x[0], test_x[-1], test_y[0], test_y[-1]]
    opts = dict(aspect='equal', cmap='nipy_spectral', extent=extent, vmin=-1.5, vmax=2.5)

    im = axes[0].imshow(perfect_result, **opts)
    fig.colorbar(im, ax=axes[0], orientation='horizontal')
    axes[0].plot(train_x, train_y, 'w.')
    axes[0].set_title('Perfect result, sampled function', fontsize=21)

    im = axes[1].imshow(smth_result, **opts)
    axes[1].plot(train_x, train_y, 'w.')
    fig.colorbar(im, ax=axes[1], orientation='horizontal')
    axes[1].set_title('s=default', fontsize=21)

    im = axes[2].imshow(interp_result, **opts)
    fig.colorbar(im, ax=axes[2], orientation='horizontal')
    axes[2].plot(train_x, train_y, 'w.')
    axes[2].set_title('s=0', fontsize=21)

    plt.tight_layout()
    plt.show()

Here we take a known function (displayed at the leftmost panel), sample it on
a mesh of points (shown by white dots), and construct the spline fit using the
default smoothing (center panel) and forcing the interpolation (rightmost panel).

Several features are clearly visible. First, the default value of ``s`` provides
too much smoothing for this data; forcing the interpolation condition, ``s = 0``,
allows to restore the underlying function to a reasonable accuracy. Second,
outside of the interpolation range (i.e., the area covered by
white dots) the result is extrapolated using a nearest-neighbor constant.
Finally, we had to silence the warnings (which is a bad form, yes!).

The warning here is emitted in the ``s=0`` case, and signals an internal difficulty
FITPACK encountered when we forced the interpolation condition. If you see
this warning in your code, consider switching to `bisplrep` and increase its
``nxest``, ``nyest`` parameters (see the `bisplrep` docstring for more details).


Bivariate spline fitting of data on a grid
------------------------------------------

For gridded 2D data, fitting a smoothing tensor product spline can be done
using the `RectBivariateSpline` class. It has the interface similar to that of
`SmoothBivariateSpline`, the main difference is that the 1D input arrays ``x``
and ``y`` are understood as defining a 2D grid (as their outer product),
and the ``z`` array is 2D with the shape of ``len(x)`` by  ``len(y)``.

The spline orders in the ``x`` and ``y`` directions are controlled by the optional
parameters ``kx`` and ``ky``, with the default of ``kx=ky=3``, i.e. a bicubic
spline.

The default value of the smoothing factor is ``s=0``. We nevertheless recommend
to always specify ``s`` explicitly.

.. plot::

    import numpy as np
    import matplotlib.pyplot as plt
    from scipy.interpolate import RectBivariateSpline

    x = np.arange(-5.01, 5.01, 0.25)        # the grid is an outer product
    y = np.arange(-5.01, 7.51, 0.25)        # of x and y arrays

    xx, yy = np.meshgrid(x, y, indexing='ij')
    z = np.sin(xx**2 + 2.*yy**2)            # z array needs to be 2-D

    func = RectBivariateSpline(x, y, z, s=0)

    xnew = np.arange(-5.01, 5.01, 1e-2)
    ynew = np.arange(-5.01, 7.51, 1e-2)
    znew = func(xnew, ynew)

    plt.imshow(znew)
    plt.colorbar()
    plt.show()


Bivariate spline fitting of data in spherical coordinates
---------------------------------------------------------

If your data is given in spherical coordinates, :math:`r = r(\theta, \phi)`,
`SmoothSphereBivariateSpline` and `RectSphereBivariateSpline` provide convenient
analogs of `SmoothBivariateSpline` and `RectBivariateSpline`, respectively.

These classes ensure the periodicity of the spline fits for
:math:`\theta \in [0, \pi]` and :math:`\phi \in [0, 2\pi]`, and offer some
control over the continuity at the poles. Refer to the docstrings of these
classes for details.

