# Theta Lines transformer#

Computes Theta lines to be used in ThetaForecaster.

The Theta lines are obtained by modifying the distances between the points of the original time series. We apply coefficient θ (theta) to the second differences of the original observations, thus changing the local curvatures . Theta coefficient is a transformation parameter which creates series of the same mean and slope with that of the original data but different variances.

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```
import matplotlib.pyplot as plt
from sktime.datasets import load_airline
from sktime.transformations.series.theta import ThetaLinesTransformer
y = load_airline()
transformer = ThetaLinesTransformer()
transformer.fit(y)
y_thetas = transformer.transform(y)
fig, ax = plt.subplots()
y_thetas.plot(ax=ax, figsize=(12, 7))
plt.legend(["theta=0", "theta=2"])
```

```
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```

```
<matplotlib.legend.Legend at 0x1b808c49370>
```

## Special cases: theta = 0 and theta = 1#

Theta values **1 > theta > 0** *reduce* original time series curvatures. The smaller the value of Theta coefficient results in the larger deflation of time series.

Theta-line with **theta=0** gives a linear regression line.

```
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```

```
t = ThetaLinesTransformer([0, 0.25, 0.75, 1])
t.fit(y)
y_t = t.transform(y)
fig, ax = plt.subplots()
y_t.plot(ax=ax, figsize=(12, 7))
plt.legend(
["theta=0, linear regression", "theta=0.25", "theta=0.75", "theta=1, original ts"]
)
plt.ylim(0, 900)
```

```
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```

```
(0.0, 900.0)
```

Theta-line with **theta=1** returns original time series.

Theta values > 1 will *increase* local curvatures, and the larger theta coefficients will result in larger dilation.

```
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```

```
t_1 = ThetaLinesTransformer([0, 1, 2, 2.5])
t_1.fit(y)
y_t1 = t_1.transform(y)
fig, ax = plt.subplots()
y_t1.plot(ax=ax, figsize=(12, 7))
plt.legend(
["theta=0, linear regression", "theta=1, original ts", "theta=2", "theta=2.5"]
)
```

```
[17]:
```

```
<matplotlib.legend.Legend at 0x1b80996baf0>
```

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