code: def test_piecewise_logistic(self, backend):
      model = Prophet(stan_backend=backend)
  
      t = np.arange(11.0)
      cap = np.ones(11) * 10
      m = 0
      k = 1.0
      deltas = np.array([0.5])
      changepoint_ts = np.array([5])
  
      y = model.piecewise_logistic(t, cap, deltas, k, m, changepoint_ts)
      y_true = np.array(
          [
              5.000000,
              7.310586,
              8.807971,
              9.525741,
              9.820138,
              9.933071,
              9.984988,
              9.996646,
              9.999252,
              9.999833,
              9.999963,
          ]
      )
  
      t = t[8:]
      y_true = y_true[8:]
      cap = cap[8:]
      y = model.piecewise_logistic(t, cap, deltas, k, m, changepoint_ts)
      assert (y - y_true).sum() == pytest.approx(0.0, abs=1e-5)
  
imports: import numpy
  import pandas
  import pytest
  from prophet import Prophet
  from prophet.utilities import warm_start_params
label_desc: Use image node_id 6 for calling the TestProphetTrendComponent obj's underlying member method code with example usage: obj.test_piecewise_logistic(backend) without return types