Guide to numerical transformations

Following numerical transformations are currently defined in the HARDy

hardy.arbitrage.transformations.cumsum(raw_array)

The function return the cumulative sum of input array

Parameters:

raw_array: Input numpy array

Returns:

cumsum _array: np.ndarray

cumulative sum of values in the input array

Notes

\[Z = [Z_1, Z_1+Z_2, Z_1+Z_2+Z_3, Z_1+...+Z_n]\]

hardy.arbitrage.transformations.cwt_1d(raw_df, xy=0)

Transform to execute a “Continuous Wavelet Transform” on a 1d data array pass it a raw XY data and tell it which column to use for the transform. See Documentaion on CWT transform: https://docs.scipy.org/doc/scipy/reference/generated/ scipy.signal.cwt.html#scipy.signal.cwt Note: I need to do testing to understand the in/outputs here… Plan is to simply hard-code a certain type of Wavelet to use… and Output Data may not be able to be square… In that case, we will discuss how to integrate this result with the compression of the data.

Parameters:

raw_df: pandas.DataFrame or 1D array (Mx2 or Mx1)

the raw data which is to be transformed.

xy: boolean, or string ‘x’, or ‘y’

information on which dataframe column to transform. ignored if an 1D array is passed instead.

w_method: string or boolean?

input instructions guiding how to choose wavelet sizes. default should be linear, with options for log- or exponential? (Will have to experiment with data to discover best option)

Returns:

cwt_matrix: np.ndarray (MxM)

Square M-by-M matrix of the wavelet transform data (Not yet compressed to plottable 0-1 data)

hardy.arbitrage.transformations.derivative_1d(raw_array, spacing=0)

Function that outputs the gradient of 1-D array using numpy.gradient function

Parameters:

raw_array: numpy array

spacing: int representing the spacing between each datapoint

Returns:

derivative_array: np.ndarray

array representing gradient at each datapoint

Notes

\[f_{(0)}^{(1)} = \frac{f(x+1)-f(x-1)}{2h}\]

where \(h = [0, 1, 2, 3, ..., size-1]\)

hardy.arbitrage.transformations.derivative_2d(x, y, meta_data=None)

Function that outputs the slope between x and y data

Parameters:

x: numpy.array

array representing values on x-axis

y: numpy.array

array representing values on y-axis

Returns:

slope_array: numpy.array

array representing the slope between x and y

Notes

\[Z = [ \frac{Y_{(i+1)}-Y_{(i)}}{X_{(i+1)}-X_{(i)}}, ... ,0]\]

hardy.arbitrage.transformations.exp(raw_array)

Function that returns the exponent of individual elements in the array

Parameters:

raw_array: numpy.array

array representing data values

Returns:

exp_array: numpy.array

array representing the exponentials of data values

Notes

\[Z = \exp{Z}\]

hardy.arbitrage.transformations.log10(raw_array)

The function that outputs the natural log of input array

Parameters:

raw_array: Input numpy array

Returns:

log_array: np.ndarray

natural log values of each element in the input array

Notes

\[Z = \log{Z}\]

hardy.arbitrage.transformations.nlog(raw_array)

The function that outputs the natural log of input array

Parameters:

raw_array: Input numpy array

Returns:

log_array: np.ndarray

natural log values of each element in the input array

Notes

\[Z = \ln{Z}\]

hardy.arbitrage.transformations.power(x, y='None', meta_data=None)

Function that multiplies two arrays x^m & y^n, element by element. If y is None, it return x*x

Parameters:

x: numpy.array

numpy array representing the one array to be multiplied

y: numpy.array

numpy array representing the second array to be multiplied if None it the module will square the x array

meta_data: numpy.array

[m, n] where m is the power of x, and n is the power

of y. If none, m=1 and n=1

Returns:

multi_array: numpy.array

numpy array representing the one to one multiplication of two arrays

Notes

\[Z = X^m*Y^n\]

if \(Y=0\)

\[Z = X^m\]

hardy.arbitrage.transformations.raw(raw_array)

Function that provides returns data as it is

Parameters:

raw_array: numpy.array

array representing data values

Returns:

raw_array: numpy.array

array representing data values

Notes

\[Z = Z\]

hardy.arbitrage.transformations.reciprocal(raw_array)

The function the outputs the reciprocal of input array

Parameters:

raw_array: Input numpy array

Returns:

reciprocal_array: np.ndarray

reciprocal values of each element in the input array

Notes

\[Z = \frac{1}{Z}\]

Defining numerical transformations

User defined numerical transformations can be integrated in the HARDy by integrating the transformation definition inside the hardy.arbitrage.transformations module. The example transformation definition is shown below:

def transformation_function(args):
    y = f(x)

Most of the transformations defined in HARDy are one dimensional transformations i-e they require only one arguments. If the user want to include more than one argument and metadata (exponents), refer to function definition of power or derivative_2d defined in hardy.arbitrage.transformations module. If the metadata is different than the length of metadata in power, function apply_tform in hardy.arbitrage.arbitrage needs to be modified accordingly as well.