Spline Interpolation Python

These coefficients are passed into splev() to actuallyevaluate the spline at the desired point x (in this example 1.25).x can also be an array. Calling f([1.0, 1.25, 1.5]) returns theinterpolated points at 1, 1.25, and 1,5, respectively.

This approach is admittedly inconvenient for single evaluations, but since the most common use case is to start with a handful of function evaluation points, then to repeatedly use the spline to find interpolated values, it is usually quite useful in practice.

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In a conclusion, this updated algorithm shall perform interpolation with better stability and faster than the previous code (O(n)). Associated with numba or cython, it shall be even very fast. Finally, it is totally independent of Scipy.Important, note that as most of algorithms, it is sometimes useful to normalize the data (e.g. against large or small number values) to get the best results. As well, in this code, I do not check nan values or ordered data.

There are several general facilities available in SciPy for interpolation andsmoothing for data in 1, 2, and higher dimensions. The choice of a specificinterpolation routine depends on the data: whether it is one-dimensional,is given on a structured grid, or is unstructured. One other factor is thedesired smoothness of the interpolator. In short, routines recommended forinterpolation can be summarized as follows:

In this example the cubic spline is used to interpolate a sampled sinusoid.You can see that the spline continuity property holds for the first andsecond derivatives and violates only for the third derivative.

In the second example, the unit circle is interpolated with a spline. Aperiodic boundary condition is used. You can see that the first derivativevalues, ds/dx=0, ds/dy=1 at the periodic point (1, 0) are correctlycomputed. Note that a circle cannot be exactly represented by a cubicspline. To increase precision, more breakpoints would be required.

The N-dimensional interpolation routines I have found are not quite good enough: I would prefer splines over LinearNDInterpolator for smoothness, and I have far too many data points (often over one million) for, e.g., a radial basis function to work.

As far as scaling and memory usage goes, map_coordinates will create a filtered copy of the array if you're using an order > 1 (i.e. not linear interpolation). If you just want to interpolate at a very small number of points, this is a rather large overhead. It doesn't increase with the number points you want to interpolate at, however. As long as have enough RAM for a single temporary copy of your input data array, you'll be fine.

If you can't store a copy of your data in memory, you can either a) specify prefilter=False and order=1 and use linear interpolation, or b) replace your original data with a filtered version using ndimage.spline_filter, and then call map_coordinates with prefilter=False.

You can try inverse distance weighted interpolation, see: Inverse Distance Weighted (IDW) Interpolation with Python .This should produce reasonably smooth results, and scale better than RBF to larger data sets.

In cubic spline interpolation (as shown in the following figure), the interpolating function is a set of piecewise cubic functions. Specifically, we assume that the points \((x_i, y_i)\) and \((x_{i+1}, y_{i+1})\) are joined by a cubic polynomial \(S_i(x) = a_i x^3 + b_i x^2 + c_i x + d_i\) that is valid for \(x_i \le x \le x_{i+1}\) for \(i = 1,\ldots, n-1\). To find the interpolating function, we must first determine the coefficients \(a_i, b_i, c_i, d_i\) for each of the cubic functions. For \(n\) points, there are \(n-1\) cubic functions to find, and each cubic function requires four coefficients. Therefore we have a total of \(4(n-1)\) unknowns, and so we need \(4(n-1)\) independent equations to find all the coefficients.

which gives us \(2(n-1)\) equations. Next, we want each cubic function to join as smoothly with its neighbors as possible, so we constrain the splines to have continuous first and second derivatives at the data points \(i = 2,\ldots,n-1\).

To determine the coefficients of each cubic function, we write out the constraints explicitly as a system of linear equations with \(4(n-1)\) unknowns. For \(n\) data points, the unknowns are the coefficients \(a_i, b_i, c_i, d_i\) of the cubic spline, \(S_i\) joining the points \(x_i\) and \(x_{i+1}\).

These equations are linear in the unknown coefficients \(a_i, b_i, c_i\), and \(d_i\). We can put them in matrix form and solve for the coefficients of each spline by left division. Remember that whenever we solve the matrix equation \(Ax = b\) for \(x\), we must make be sure that \(A\) is square and invertible. In the case of finding cubic spline equations, the \(A\) matrix is always square and invertible as long as the \(x_i\) values in the data set are unique.

It is a 1-D smoothing spline that fits a given group of data points. The scipy.interpolate.UnivariateSpline is used to fit a spline y = spl(x) of degree k to the provided x, y data. s specifies the number of knots by specifying a smoothing condition. The scipy.interpolate.UnivariateSpline. set_smoothing_factor: Spline computation with the given smoothing factor s and with the knots found at the last call.

SplineTransformer generates B-spline basisfunctions. A basis function of a B-spline is a piece-wise polynomial functionof degree degree that is non-zero only between degree+1 consecutiveknots. Given n_knots number of knots, this results in matrix ofn_samples rows and n_knots + degree - 1 columns:

This shows nicely that higher degree polynomials can fit the data better. Butat the same time, too high powers can show unwanted oscillatory behaviourand are particularly dangerous for extrapolation beyond the range of fitteddata. This is an advantage of B-splines. They usually fit the data as well aspolynomials and show very nice and smooth behaviour. They have also goodoptions to control the extrapolation, which defaults to continue with aconstant. Note that most often, you would rather increase the number of knotsbut keep degree=3.

In the left plot, we recognize the lines corresponding to simple monomialsfrom x**0 to x**3. In the right figure, we see the six B-splinebasis functions of degree=3 and also the four knot positions that werechosen during fit. Note that there are degree number of additionalknots each to the left and to the right of the fitted interval. These arethere for technical reasons, so we refrain from showing them. Every basisfunction has local support and is continued as a constant beyond the fittedrange. This extrapolating behaviour could be changed by the argumentextrapolation.

In the previous example we saw the limitations of polynomials and splines forextrapolation beyond the range of the training observations. In somesettings, e.g. with seasonal effects, we expect a periodic continuation ofthe underlying signal. Such effects can be modelled using periodic splines,which have equal function value and equal derivatives at the first and lastknot. In the following case we show how periodic splines provide a better fitboth within and outside of the range of training data given the additionalinformation of periodicity. The splines period is the distance betweenthe first and last knot, which we specify manually.

Periodic splines can also be useful for naturally periodic features (such asday of the year), as the smoothness at the boundary knots prevents a jump inthe transformed values (e.g. from Dec 31st to Jan 1st). For such naturallyperiodic features or more generally features where the period is known, it isadvised to explicitly pass this information to the SplineTransformer bysetting the knots manually.

My understanding is that in QuantLib the choice of the interpolation methods is given by the objects called, for example, PiecewiseLogCubicDiscount. In this case the cubic interpolation is performed on Log Discount Factors.

I would like to interpolate using Monotonic Cubic Spline on Log Discount Factor. I saw that in this file some interpolation are exported. I saw in the interpolation.i file that other interpolation methods are available. I just do not know how to access those. Thank you.

But if you get discount factors from dates that are not given by the instruments used to build the curve, you will get diferent values because you are using diferent interpolation methods on different variables:

First, step is given an X array and a Y array of equal length n (greater than 2), we want to build a tridiagonal matrix which we will then solve to produce the coefficients for the piece-wise spline. The goal of the spline is that it hits every point (x, y) and that the first and second derivatives match at these points too.

Now to put it all together and create a function to build the spline coefficients. The final part needed is to create a closure and wrap it up as a function which will find j and then evaluate the spline. There is an excellent library in Python, bisect which will do a binary search to find j easily and quickly.

Can some help point to what is the exact methodology/calculation used to generate these splines when we standardize the data. I have seen some post that point this is cubic splines, but i have not explanation how data is treated and spline generate when we use the standardized option.

The Scipy.org on-line documentation indicates that the optional bc_type argument in the scipy.interpolate.CubicSpline(x, y, axis=0, bc_type='not-a-knot', extrapolate=None) call is used to control the construction of the spline model. There is no argument corresponding to the JMP lambda argument that you can use to control the fit.

There are many kinds of splines and each kind usually has more than one method or implementation. I suggest that you search the vast literature about spline models if you are going to write your own code. For example, this chapter from a high level computer science course at Clemson University provides an overview of splines used in computer graphics. You might also examine the open source code such as available in Python or R for spline models that you like to learn about how to write such code. e24fc04721