code: def test_piecewise_linear(self, backend):
      model = Prophet(stan_backend=backend)
  
      t = np.arange(11.0)
      m = 0
      k = 1.0
      deltas = np.array([0.5])
      changepoint_ts = np.array([5])
  
      y = model.piecewise_linear(t, deltas, k, m, changepoint_ts)
      y_true = np.array(
          [0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.5, 8.0, 9.5, 11.0, 12.5]
      )
      assert (y - y_true).sum() == 0.0
  
      t = t[8:]
      y_true = y_true[8:]
      y = model.piecewise_linear(t, deltas, k, m, changepoint_ts)
      assert (y - y_true).sum() == 0.0
  
imports: import numpy
  import pandas
  import pytest
  from prophet import Prophet
  from prophet.utilities import warm_start_params
label_desc: Use image node_id 5 for calling the TestProphetTrendComponent obj's underlying member method code with example usage: obj.test_piecewise_linear(backend) without return types