calculate synthetic ramps given a series of time points and coefficients of unit steps at each time. \n convolution ramps

synthetic_ramp_method.ipynb

synthetic_ramp_method.py

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# %% [markdown]
# synthetic ramp method
#
# calculate synthetic ramps using convolution for a given series of time points:
# * define matrix $A$ with the coefficients $a_{i,j}$ of the unit steps in $n$ different series $i$ for each time $t_j$.
# * define matrix $\sigma(t-t_j)*\sigma(t)=(t-t_j)\cdot \sigma(t-t_j)$ which is lower triangular with permutations of $(t-t_j)\cdot \sigma(t-t_j)$
#     * example with $t_0<t_1$: $(t_0-t_1)\cdot \sigma(t_0-t_1)=0$, and $(t_1-t_0)\cdot \sigma(t_1-t_0)=(t_1-t_0)$ due to step definition
# * the matrix $\sigma(t-t_j)\cdot A$ has then the slopes $\frac{dy_i}{dt}$ at times $t_j$ and $(t-t_j)\cdot \sigma(t-t_j)\cdot A$ has the synthetic ramps by convolution.
# $$
# \underbrace{\left[\begin{array}{cccc}
# 1 & 0 & 0 & ...\\
# 1 & 1 & 0 & ...\\
# 1 & 1 & 1 & ...\\
# ... & ... & ... & ...
# \end{array}\right]}_{\sigma(t-t_j)}\cdot 
# \underbrace{\left[\begin{array}{ccc}
# a_{1,0} & a_{2,0} & ...\\
# a_{1,1} & a_{2,1} & ...\\
# a_{1,2} & a_{2,2} & ...\\
# ... & ... & ...\\
# \end{array}\right]}_{A} = 
# \underbrace{\left[\begin{array}{ccc}
# a_{1,0} & a_{2,0} & ...\\
# a_{1,0}+a_{1,1} & a_{2,0}+a_{2,1} & ...\\
# a_{1,0}+a_{1,1}+a_{1,2} & a_{2,0}+a_{2,1}a_{2,2} & ...\\
# ... & ... & ...\\
# \end{array}\right]}_{\mathrm{slopes}}
# $$
#
# $$
# \underbrace{\left[\begin{array}{cccc}
# t_0-t_0 & 0 & 0 & ...\\
# t_1-t_0 & t_1-t_1 & 0 & ...\\
# t_2-t_0 & t_2-t_1 & t_2-t_2 & ...\\
# ... & ... & ... & ...
# \end{array}\right]}_{(t-t_j)\cdot \sigma(t-t_j)}\cdot 
# \underbrace{\left[\begin{array}{ccc}
# a_{1,0} & a_{2,0} & ...\\
# a_{1,1} & a_{2,1} & ...\\
# a_{1,2} & a_{2,2} & ...\\
# ... & ... & ...\\
# \end{array}\right]}_{A} = 
# \underbrace{\left[\begin{array}{ccc}
# a_{1,0}\cdot(t_0-t_0) & a_{2,0}\cdot(t_0-t_0) & ...\\
# a_{1,0}\cdot(t_1-t_0)+a_{1,1}\cdot(t_1-t_1) & a_{2,0}\cdot(t_1-t_0)+a_{2,1}\cdot(t_1-t_1) & ...\\
# a_{1,0}\cdot(t_2-t_0)+a_{1,1}\cdot(t_2-t_1)+a_{1,2}\cdot(t_2-t_2) & a_{2,0}\cdot(t_2-t_0)+a_{2,1}\cdot(t_2-t_1)+a_{2,2}\cdot(t_2-t_2) & ...\\
# ... & ... & ...\\
# \end{array}\right]}_{\mathrm{synth\ curves}}
# $$
#
# example below with two series $i=\{1,2\}$

# %%
from numpy import array,linspace
from matplotlib import pyplot as plt

steps_times=array([
    [0,0,0],
    [3,0,-1],
    [4,0,1],
    [6,3/5,0],
    [8,0,2],
    [8.5,0,-2],
    [11,-3/5,0],
    [14,-1.5/2,0],
    [16,-1.5/2,0.5],
    [17,1.5,-0.5],
    [20,0,0]
])
times=steps_times[:,0]
steps=steps_times[:,[1,2]]

lt=array([[times[i]-times[j] if times[i]-times[j]>=0 else 0 for j in range(times.shape[0])] for i in range(times.shape[0])])
synth_ramps=array([2,1])+lt.dot(steps)

times_extended=linspace(times.min(),times.max(),1000)
sigma=array([[1 if times_extended[i]-times[j]>=0 else 0 for j in range(times.shape[0])] for i in range(times_extended.shape[0])])
slopes=sigma.dot(steps)

print('lower triangular (t-tj)sigma(t-tj)) \n',lt)
fig,ax_list=plt.subplots(nrows=3,ncols=1,height_ratios=[1,1/2,1/2],constrained_layout=True)
ax_list[0].plot(times,synth_ramps,'-o')
for j in range(steps.shape[1]):
    ax_list[1].stem(times,steps[:,j],markerfmt=['blue','orange'][j],label=f'series i={j+1}')
    ax_list[2].plot(times_extended,slopes[:,j],label=f'series i={j+1}')
for j in range(len(times)):
    ax_list[2].axvline(times[j],linestyle='--',color=[0.7,0.7,0.7])
ax_list[-1].set_xlabel('time $t_j$')
ax_list[0].set_ylabel('$y_i$')
ax_list[1].set_ylabel('$a_{i,j}$')
ax_list[2].set_ylabel(r'$\frac{dy_i}{dt}$')
ax_list[0].legend([f'series i={j+1}' for j in range(steps.shape[1])],loc='center left',bbox_to_anchor=[1.01,0.5]);
ax_list[1].legend(loc='center left',bbox_to_anchor=[1.01,0.5])
ax_list[2].legend(loc='center left',bbox_to_anchor=[1.01,0.5]);