Exponential Smoothing

import plotly
plotly.offline.init_notebook_mode()

import pandas as pd

df=pd.read_csv('weatherAUS.csv', encoding='utf-8', usecols=['Date', 'Location', 'MinTemp'])
df = df[df['Location'] == 'Albury']

import plotly.express as px
fig = px.line(df, x='Date', y='MinTemp')
fig.show()

Average Method

$$F_{t+1} = \frac{1}{t}\sum_{i=1}^{t}Y_{i}$$

Average Method - Moving Average

$$F_{t+1} = \frac{1}{k}\sum_{i=t-k+1}^{t}Y_{i}$$$$F_{t+2} = \frac{1}{k}\sum_{i=t-k+2}^{t+1}Y_{i}$$$$F_{t+2} = F_{t+1} + \frac{1}{k}(Y_{t+1} - Y_{t-k+1})$$

Single Exponential Smoothing (SES)

• a time series forecasting method for univariate data without a trend or seasonality
• F, forecast, Y, level value (observed value) $$F_{t+1} = F_{t} + \alpha(Y_{t} - F_{t})$$ $$F_{t+2} = F_{t+1} + \alpha(Y_{t+1} - F_{t+1})$$
• initial value $$F_{1} = Y_{1}$$ or $$F_{1} = \sum_{i=1}^{k}Y_{i}$$
• From known $Y_{1}$ to $Y_{t}$, only be able to predict $F_{t+1}$
• $\alpha$, smoothing_level, 0~1
  • close to 1 indicates fast learning, more weights on recent values
  • close to 0, more weights on distant values
  • 0.05 ≤ α ≤ 0.3 commonly used, values close to 1 are used rarely
• does not support trend and seasonality

from statsmodels.tsa.holtwinters import SimpleExpSmoothing

training = df['MinTemp'][:10]
model = SimpleExpSmoothing(training)
model_fit = model.fit(smoothing_level=0.4,optimized=True)

# conduct in sampe prediction
# conduct out of sample prediction on t+1
df['predict'] = model_fit.predict(start=0, end = 10) # 0-9 in sample prediction; 10 out of sample prediction

fig = px.line(df.iloc[:11], x='Date', y=['MinTemp', 'predict'])
fig.show()

Holt’s trend-corrected double exponential smoothing (DES)

• comes with the support for trend components, SES + Trend = DES
• F, forecast, Y, level value, T, trend (slop) $$F_{t+1} = Y_{t} + h*T_{t}$$ $$Y_{t} = \alpha F_{t} + (1-\alpha)(Y_{t-1}+T_{t-1})$$ $$T_{t} = \beta (Y_{t} - Y_{t-1}) + (1-\beta)T_{t-1}$$
• $\alpha$, smoothing level; $\beta$, trend coefficient
  • 0.02< $\alpha$, $\beta$<0.2
• h, step
• inital values $$ Y_{2}, observed$$ $$ T_{2} = Y_{2} - Y_{1}$$ $$ F_{3} = L_{2} + h* T_{2}$$

training = df['MinTemp'][:10]

from statsmodels.tsa.api import Holt
model = Holt(training)
model_fit = model.fit(smoothing_level=0.8, smoothing_trend=0.2, optimized=False)
df['predict'] = model_fit.predict(start=0, end = 12)

fig = px.line(df.iloc[:13], x='Date', y=['MinTemp', 'predict'])
fig.show()

import numpy as np
pd.DataFrame(np.c_[model_fit.level,model_fit.trend]).rename(columns={0:'level',1:'slope'}).plot(subplots=True)

array([<Axes: >, <Axes: >], dtype=object)

Triple Exponential Smoothing (TES)

• support seasonality, is also referred to as Holt-Winters Exponential Smoothing
• F, forecast, Y, level value, T, trend (slop), I, seaonality $$F_{t} = (Y_{t} + h*T_{t})I_{t+h+s}$$ $$Y_{t} = \alpha \frac{F_{t}}{I_{t-s}} + (1-\alpha)(Y_{t-1}+T_{t-1})$$ $$T_{t} = \beta (Y_{t} - Y_{t-1}) + (1-\beta)T_{t-1}$$ $$I_{t} = \gamma \frac{F_{t}}{Y_{t}} + (1-\gamma)I_{t-s}$$
• $\alpha$, smoothing level; $\beta$, smoothing trend; $\gamma$, smoothing seasonal
• h, step; s, # of periods in the seasonal cycles

from statsmodels.tsa.holtwinters import ExponentialSmoothing

training = df['MinTemp'][:10]

model = ExponentialSmoothing(training, seasonal_periods=2, trend='add', seasonal='add')
model_fit = model.fit(smoothing_level=0.8,smoothing_trend=0.2, smoothing_seasonal=0.8)
df['predict'] = model_fit.predict(start=0, end = 12)

fig = px.line(df.iloc[:13], x='Date', y=['MinTemp', 'predict'])
fig.show()

import numpy as np
pd.DataFrame(np.c_[model_fit.level, model_fit.trend, model_fit.season]).rename(columns={0:'level',1:'slope', 2:'season'}).plot(subplots=True)

array([<Axes: >, <Axes: >, <Axes: >], dtype=object)

Reference

• STAT 497 LECTURE NOTES 7
• A Gentle Introduction to Exponential Smoothing for Time Series Forecasting in Python
• A Tutorial on Exponential Smoothing and its Types
• Holt's trend method forecast
• Tutorial
• statsmodels user guide