Time-Series Forecasting
Introduction to time-series forecasting
What is time series?
A series of observations collected during some time intervals in know as time series. The observations which are collected are primarily dependent on the time at which it is collected. The sale of the items in a retail store like Walmart will be a time series. The periodicity of the the sale could be at daily level or weekly level. The number of people flying from Bangalore to Delhi on daily basis is a time series. Time-series forecasting uses historical time based data to forecast a variable for example
- At Supermarket the demand for a grocery product each day
- Count of the downloads of an App in a country for the coming month
- Product sales in units sold each day in a retail store
- Yearly unemployment rate
- The closing price of a stock
- Expected yearly yield of a crop
- Population growth in a country
- Daily demand for a bus/airline/train
Time series and a normal series?
Time component is important in time series. The time series is primarily dependent on the time. But a normal series say 1, 2, 3…100 has no time component to it. When a value that a series will take depends on the time it was recorded, it is called as time series.
How to define a time series object using R language
The function ts() helps to create time-series objects. As.ts() and is.ts() coerce an object to a time-series and test whether an object is a time series or not for example:
Frequency
The count of number of observations per unit of time is called as frequency. Frequency can further be distinguished as Frequency for yearly data is 1, Frequency for weekly data is 7, Frequency for quarterly data is 4 and so on…
Some useful softwares
forecast: For forecasting functions
tseries: For unit root tests and GARC models
Mcomp: Time series data from forecasting competitions
fma: For data
expsmooth: For data
fpp: For data
ARIMA
prophet
SPSS
prophet
Prophet represents a time-series as a combination of trend, seasonality and holidays. This decomposed time series model can be represented by the following equation:
Capture10
where:
g(t) defines the trend function that models the non-periodic changes in the time series
s(t) defines the periodic changes (e.g., weekly and yearly seasonality)
h(t) means the effect of holidays
εt: is the error notation representing information that was missed by the model
Trend
Prophet provides two variants for the trend function, g(t) which are logistic growth modeling non-linear growth and constant rate of growth or Piece wise Linear Model.
Seasonality
s(t) is called as the seasonal component which allows to flexibly model the periodic changes due to weekly and yearly seasonability. For example, a 5-day work week can produce effects on a time series that repeat each week, while vacation schedules and school breaks can produce effects that repeat each year.
Holidays and Events
There are events which can provide predictable outcome to business. These events are holidays or any other important ocasion such as black friday.Prophet provides list of such past and future events for modeling into time-series component.
library(Quandl)
library(quantmod)
library(prophet)
Quandl.api_key("YBM9uPgpnsaDYPkAn539")
#daily_gold = Quandl("WGC/GOLD_DAILY_INR",collapse="daily",start_date="2000-01-01",end_date = "2020-08-11",type="raw")
daily_gold = Quandl("LBMA/SILVER",collapse="daily",start_date="2020-01-01",end_date = "2020-08-12",type="raw")
keeps <- c("Date", "USD")
daily_gold = daily_gold[keeps]
names(daily_gold) <- c('ds','y')
daily_gold[2] <- daily_gold[2]
model = prophet(daily_gold)
future <- make_future_dataframe(model, periods = 10)
forecast <- predict(model, future)
print(forecast)## ds trend additive_terms additive_terms_lower
## 1 2020-01-02 18.63425 0.16652525 0.16652525
## 2 2020-01-03 18.59643 0.18912739 0.18912739
## 3 2020-01-06 18.48298 0.11055330 0.11055330
## 4 2020-01-07 18.44516 0.09853439 0.09853439
## 5 2020-01-08 18.40734 0.21000328 0.21000328
## 6 2020-01-09 18.36952 0.16652525 0.16652525
## 7 2020-01-10 18.33170 0.18912739 0.18912739
## 8 2020-01-13 18.21824 0.11055330 0.11055330
## 9 2020-01-14 18.18042 0.09853439 0.09853439
## 10 2020-01-15 18.14260 0.21000328 0.21000328
## 11 2020-01-16 18.10478 0.16652525 0.16652525
## 12 2020-01-17 18.06696 0.18912739 0.18912739
## 13 2020-01-20 17.95350 0.11055330 0.11055330
## 14 2020-01-21 17.91568 0.09853439 0.09853439
## 15 2020-01-22 17.87786 0.21000328 0.21000328
## 16 2020-01-23 17.84004 0.16652525 0.16652525
## 17 2020-01-24 17.80222 0.18912739 0.18912739
## 18 2020-01-27 17.68877 0.11055330 0.11055330
## 19 2020-01-28 17.65095 0.09853439 0.09853439
## 20 2020-01-29 17.61313 0.21000328 0.21000328
## 21 2020-01-30 17.57531 0.16652525 0.16652525
## 22 2020-01-31 17.53749 0.18912739 0.18912739
## 23 2020-02-03 17.42403 0.11055330 0.11055330
## 24 2020-02-04 17.38621 0.09853439 0.09853439
## 25 2020-02-05 17.34839 0.21000328 0.21000328
## 26 2020-02-06 17.31057 0.16652525 0.16652525
## 27 2020-02-07 17.27275 0.18912739 0.18912739
## 28 2020-02-10 17.15928 0.11055330 0.11055330
## 29 2020-02-11 17.12145 0.09853439 0.09853439
## 30 2020-02-12 17.08363 0.21000328 0.21000328
## 31 2020-02-13 17.04580 0.16652525 0.16652525
## 32 2020-02-14 16.99846 0.18912739 0.18912739
## 33 2020-02-17 16.85644 0.11055330 0.11055330
## 34 2020-02-18 16.80910 0.09853439 0.09853439
## 35 2020-02-19 16.76176 0.21000328 0.21000328
## 36 2020-02-20 16.70505 0.16652525 0.16652525
## 37 2020-02-21 16.64833 0.18912739 0.18912739
## 38 2020-02-24 16.47819 0.11055330 0.11055330
## 39 2020-02-25 16.42147 0.09853439 0.09853439
## 40 2020-02-26 16.36475 0.21000328 0.21000328
## 41 2020-02-27 16.30802 0.16652525 0.16652525
## 42 2020-02-28 16.25129 0.18912739 0.18912739
## 43 2020-03-02 16.08110 0.11055330 0.11055330
## 44 2020-03-03 16.02437 0.09853439 0.09853439
## 45 2020-03-04 15.96764 0.21000328 0.21000328
## 46 2020-03-05 15.91091 0.16652525 0.16652525
## 47 2020-03-06 15.85418 0.18912739 0.18912739
## 48 2020-03-09 15.68398 0.11055330 0.11055330
## 49 2020-03-10 15.62725 0.09853439 0.09853439
## 50 2020-03-11 15.57052 0.21000328 0.21000328
## 51 2020-03-12 15.51379 0.16652525 0.16652525
## 52 2020-03-13 15.45706 0.18912739 0.18912739
## 53 2020-03-16 15.28687 0.11055330 0.11055330
## 54 2020-03-17 15.23013 0.09853439 0.09853439
## 55 2020-03-18 15.17340 0.21000328 0.21000328
## 56 2020-03-19 15.11667 0.16652525 0.16652525
## 57 2020-03-20 15.05994 0.18912739 0.18912739
## 58 2020-03-23 14.88975 0.11055330 0.11055330
## 59 2020-03-24 14.83302 0.09853439 0.09853439
## 60 2020-03-25 14.77629 0.21000328 0.21000328
## 61 2020-03-26 14.75111 0.16652525 0.16652525
## 62 2020-03-27 14.72594 0.18912739 0.18912739
## 63 2020-03-30 14.65042 0.11055330 0.11055330
## 64 2020-03-31 14.62525 0.09853439 0.09853439
## 65 2020-04-01 14.60007 0.21000328 0.21000328
## 66 2020-04-02 14.61159 0.16652525 0.16652525
## 67 2020-04-03 14.62310 0.18912739 0.18912739
## 68 2020-04-06 14.65764 0.11055330 0.11055330
## 69 2020-04-07 14.66916 0.09853439 0.09853439
## 70 2020-04-08 14.68067 0.21000328 0.21000328
## 71 2020-04-09 14.71275 0.16652525 0.16652525
## 72 2020-04-14 14.87316 0.09853439 0.09853439
## 73 2020-04-15 14.90525 0.21000328 0.21000328
## 74 2020-04-16 14.93733 0.16652525 0.16652525
## 75 2020-04-17 14.96941 0.18912739 0.18912739
## 76 2020-04-20 15.07333 0.11055330 0.11055330
## 77 2020-04-21 15.10797 0.09853439 0.09853439
## 78 2020-04-22 15.14261 0.21000328 0.21000328
## 79 2020-04-23 15.17725 0.16652525 0.16652525
## 80 2020-04-24 15.21189 0.18912739 0.18912739
## 81 2020-04-27 15.32133 0.11055330 0.11055330
## 82 2020-04-28 15.35781 0.09853439 0.09853439
## 83 2020-04-29 15.39429 0.21000328 0.21000328
## 84 2020-04-30 15.43077 0.16652525 0.16652525
## 85 2020-05-01 15.46725 0.18912739 0.18912739
## 86 2020-05-04 15.57725 0.11055330 0.11055330
## 87 2020-05-05 15.61392 0.09853439 0.09853439
## 88 2020-05-06 15.65059 0.21000328 0.21000328
## 89 2020-05-07 15.68725 0.16652525 0.16652525
## 90 2020-05-11 15.83392 0.11055330 0.11055330
## 91 2020-05-12 15.87059 0.09853439 0.09853439
## 92 2020-05-13 15.90726 0.21000328 0.21000328
## 93 2020-05-14 15.94392 0.16652525 0.16652525
## 94 2020-05-15 15.98059 0.18912739 0.18912739
## 95 2020-05-18 16.09059 0.11055330 0.11055330
## 96 2020-05-19 16.12726 0.09853439 0.09853439
## 97 2020-05-20 16.16393 0.21000328 0.21000328
## 98 2020-05-21 16.20059 0.16652525 0.16652525
## 99 2020-05-22 16.23726 0.18912739 0.18912739
## 100 2020-05-26 16.38393 0.09853439 0.09853439
## 101 2020-05-27 16.42060 0.21000328 0.21000328
## 102 2020-05-28 16.45726 0.16652525 0.16652525
## 103 2020-05-29 16.49393 0.18912739 0.18912739
## 104 2020-06-01 16.60393 0.11055330 0.11055330
## 105 2020-06-02 16.64060 0.09853439 0.09853439
## 106 2020-06-03 16.67727 0.21000328 0.21000328
## 107 2020-06-04 16.71393 0.16652525 0.16652525
## 108 2020-06-05 16.75060 0.18912739 0.18912739
## 109 2020-06-08 16.86060 0.11055330 0.11055330
## 110 2020-06-09 16.89727 0.09853439 0.09853439
## 111 2020-06-10 16.93393 0.21000328 0.21000328
## 112 2020-06-11 16.97060 0.16652525 0.16652525
## 113 2020-06-12 17.00727 0.18912739 0.18912739
## 114 2020-06-15 17.11727 0.11055330 0.11055330
## 115 2020-06-16 17.15394 0.09853439 0.09853439
## 116 2020-06-17 17.19060 0.21000328 0.21000328
## 117 2020-06-18 17.22727 0.16652525 0.16652525
## 118 2020-06-19 17.26394 0.18912739 0.18912739
## 119 2020-06-22 17.37394 0.11055330 0.11055330
## 120 2020-06-23 17.45752 0.09853439 0.09853439
## 121 2020-06-24 17.54110 0.21000328 0.21000328
## 122 2020-06-25 17.62467 0.16652525 0.16652525
## 123 2020-06-26 17.70825 0.18912739 0.18912739
## 124 2020-06-29 17.95899 0.11055330 0.11055330
## 125 2020-06-30 18.14085 0.09853439 0.09853439
## 126 2020-07-01 18.32271 0.21000328 0.21000328
## 127 2020-07-02 18.50458 0.16652525 0.16652525
## 128 2020-07-03 18.68644 0.18912739 0.18912739
## 129 2020-07-06 19.23203 0.11055330 0.11055330
## 130 2020-07-07 19.41389 0.09853439 0.09853439
## 131 2020-07-08 19.59575 0.21000328 0.21000328
## 132 2020-07-09 19.77761 0.16652525 0.16652525
## 133 2020-07-10 19.95948 0.18912739 0.18912739
## 134 2020-07-13 20.50507 0.11055330 0.11055330
## 135 2020-07-14 20.68693 0.09853439 0.09853439
## 136 2020-07-15 20.86879 0.21000328 0.21000328
## 137 2020-07-16 21.05065 0.16652525 0.16652525
## 138 2020-07-17 21.23252 0.18912739 0.18912739
## 139 2020-07-20 21.77810 0.11055330 0.11055330
## 140 2020-07-21 21.95997 0.09853439 0.09853439
## 141 2020-07-22 22.14183 0.21000328 0.21000328
## 142 2020-07-23 22.32369 0.16652525 0.16652525
## 143 2020-07-24 22.50555 0.18912739 0.18912739
## 144 2020-07-27 23.05114 0.11055330 0.11055330
## 145 2020-07-28 23.23300 0.09853439 0.09853439
## 146 2020-07-29 23.41487 0.21000328 0.21000328
## 147 2020-07-30 23.59673 0.16652525 0.16652525
## 148 2020-07-31 23.77859 0.18912739 0.18912739
## 149 2020-08-03 24.32418 0.11055330 0.11055330
## 150 2020-08-04 24.50604 0.09853439 0.09853439
## 151 2020-08-05 24.68791 0.21000328 0.21000328
## 152 2020-08-06 24.86977 0.16652525 0.16652525
## 153 2020-08-07 25.05163 0.18912739 0.18912739
## 154 2020-08-10 25.59722 0.11055330 0.11055330
## 155 2020-08-11 25.77908 0.09853439 0.09853439
## 156 2020-08-12 25.96094 0.21000328 0.21000328
## 157 2020-08-13 26.14281 0.16652525 0.16652525
## 158 2020-08-14 26.32467 0.18912739 0.18912739
## 159 2020-08-15 26.50653 -0.38737181 -0.38737181
## 160 2020-08-16 26.68839 -0.38737181 -0.38737181
## 161 2020-08-17 26.87026 0.11055330 0.11055330
## 162 2020-08-18 27.05212 0.09853439 0.09853439
## 163 2020-08-19 27.23398 0.21000328 0.21000328
## 164 2020-08-20 27.41585 0.16652525 0.16652525
## 165 2020-08-21 27.59771 0.18912739 0.18912739
## 166 2020-08-22 27.77957 -0.38737181 -0.38737181
## additive_terms_upper weekly weekly_lower weekly_upper
## 1 0.16652525 0.16652525 0.16652525 0.16652525
## 2 0.18912739 0.18912739 0.18912739 0.18912739
## 3 0.11055330 0.11055330 0.11055330 0.11055330
## 4 0.09853439 0.09853439 0.09853439 0.09853439
## 5 0.21000328 0.21000328 0.21000328 0.21000328
## 6 0.16652525 0.16652525 0.16652525 0.16652525
## 7 0.18912739 0.18912739 0.18912739 0.18912739
## 8 0.11055330 0.11055330 0.11055330 0.11055330
## 9 0.09853439 0.09853439 0.09853439 0.09853439
## 10 0.21000328 0.21000328 0.21000328 0.21000328
## 11 0.16652525 0.16652525 0.16652525 0.16652525
## 12 0.18912739 0.18912739 0.18912739 0.18912739
## 13 0.11055330 0.11055330 0.11055330 0.11055330
## 14 0.09853439 0.09853439 0.09853439 0.09853439
## 15 0.21000328 0.21000328 0.21000328 0.21000328
## 16 0.16652525 0.16652525 0.16652525 0.16652525
## 17 0.18912739 0.18912739 0.18912739 0.18912739
## 18 0.11055330 0.11055330 0.11055330 0.11055330
## 19 0.09853439 0.09853439 0.09853439 0.09853439
## 20 0.21000328 0.21000328 0.21000328 0.21000328
## 21 0.16652525 0.16652525 0.16652525 0.16652525
## 22 0.18912739 0.18912739 0.18912739 0.18912739
## 23 0.11055330 0.11055330 0.11055330 0.11055330
## 24 0.09853439 0.09853439 0.09853439 0.09853439
## 25 0.21000328 0.21000328 0.21000328 0.21000328
## 26 0.16652525 0.16652525 0.16652525 0.16652525
## 27 0.18912739 0.18912739 0.18912739 0.18912739
## 28 0.11055330 0.11055330 0.11055330 0.11055330
## 29 0.09853439 0.09853439 0.09853439 0.09853439
## 30 0.21000328 0.21000328 0.21000328 0.21000328
## 31 0.16652525 0.16652525 0.16652525 0.16652525
## 32 0.18912739 0.18912739 0.18912739 0.18912739
## 33 0.11055330 0.11055330 0.11055330 0.11055330
## 34 0.09853439 0.09853439 0.09853439 0.09853439
## 35 0.21000328 0.21000328 0.21000328 0.21000328
## 36 0.16652525 0.16652525 0.16652525 0.16652525
## 37 0.18912739 0.18912739 0.18912739 0.18912739
## 38 0.11055330 0.11055330 0.11055330 0.11055330
## 39 0.09853439 0.09853439 0.09853439 0.09853439
## 40 0.21000328 0.21000328 0.21000328 0.21000328
## 41 0.16652525 0.16652525 0.16652525 0.16652525
## 42 0.18912739 0.18912739 0.18912739 0.18912739
## 43 0.11055330 0.11055330 0.11055330 0.11055330
## 44 0.09853439 0.09853439 0.09853439 0.09853439
## 45 0.21000328 0.21000328 0.21000328 0.21000328
## 46 0.16652525 0.16652525 0.16652525 0.16652525
## 47 0.18912739 0.18912739 0.18912739 0.18912739
## 48 0.11055330 0.11055330 0.11055330 0.11055330
## 49 0.09853439 0.09853439 0.09853439 0.09853439
## 50 0.21000328 0.21000328 0.21000328 0.21000328
## 51 0.16652525 0.16652525 0.16652525 0.16652525
## 52 0.18912739 0.18912739 0.18912739 0.18912739
## 53 0.11055330 0.11055330 0.11055330 0.11055330
## 54 0.09853439 0.09853439 0.09853439 0.09853439
## 55 0.21000328 0.21000328 0.21000328 0.21000328
## 56 0.16652525 0.16652525 0.16652525 0.16652525
## 57 0.18912739 0.18912739 0.18912739 0.18912739
## 58 0.11055330 0.11055330 0.11055330 0.11055330
## 59 0.09853439 0.09853439 0.09853439 0.09853439
## 60 0.21000328 0.21000328 0.21000328 0.21000328
## 61 0.16652525 0.16652525 0.16652525 0.16652525
## 62 0.18912739 0.18912739 0.18912739 0.18912739
## 63 0.11055330 0.11055330 0.11055330 0.11055330
## 64 0.09853439 0.09853439 0.09853439 0.09853439
## 65 0.21000328 0.21000328 0.21000328 0.21000328
## 66 0.16652525 0.16652525 0.16652525 0.16652525
## 67 0.18912739 0.18912739 0.18912739 0.18912739
## 68 0.11055330 0.11055330 0.11055330 0.11055330
## 69 0.09853439 0.09853439 0.09853439 0.09853439
## 70 0.21000328 0.21000328 0.21000328 0.21000328
## 71 0.16652525 0.16652525 0.16652525 0.16652525
## 72 0.09853439 0.09853439 0.09853439 0.09853439
## 73 0.21000328 0.21000328 0.21000328 0.21000328
## 74 0.16652525 0.16652525 0.16652525 0.16652525
## 75 0.18912739 0.18912739 0.18912739 0.18912739
## 76 0.11055330 0.11055330 0.11055330 0.11055330
## 77 0.09853439 0.09853439 0.09853439 0.09853439
## 78 0.21000328 0.21000328 0.21000328 0.21000328
## 79 0.16652525 0.16652525 0.16652525 0.16652525
## 80 0.18912739 0.18912739 0.18912739 0.18912739
## 81 0.11055330 0.11055330 0.11055330 0.11055330
## 82 0.09853439 0.09853439 0.09853439 0.09853439
## 83 0.21000328 0.21000328 0.21000328 0.21000328
## 84 0.16652525 0.16652525 0.16652525 0.16652525
## 85 0.18912739 0.18912739 0.18912739 0.18912739
## 86 0.11055330 0.11055330 0.11055330 0.11055330
## 87 0.09853439 0.09853439 0.09853439 0.09853439
## 88 0.21000328 0.21000328 0.21000328 0.21000328
## 89 0.16652525 0.16652525 0.16652525 0.16652525
## 90 0.11055330 0.11055330 0.11055330 0.11055330
## 91 0.09853439 0.09853439 0.09853439 0.09853439
## 92 0.21000328 0.21000328 0.21000328 0.21000328
## 93 0.16652525 0.16652525 0.16652525 0.16652525
## 94 0.18912739 0.18912739 0.18912739 0.18912739
## 95 0.11055330 0.11055330 0.11055330 0.11055330
## 96 0.09853439 0.09853439 0.09853439 0.09853439
## 97 0.21000328 0.21000328 0.21000328 0.21000328
## 98 0.16652525 0.16652525 0.16652525 0.16652525
## 99 0.18912739 0.18912739 0.18912739 0.18912739
## 100 0.09853439 0.09853439 0.09853439 0.09853439
## 101 0.21000328 0.21000328 0.21000328 0.21000328
## 102 0.16652525 0.16652525 0.16652525 0.16652525
## 103 0.18912739 0.18912739 0.18912739 0.18912739
## 104 0.11055330 0.11055330 0.11055330 0.11055330
## 105 0.09853439 0.09853439 0.09853439 0.09853439
## 106 0.21000328 0.21000328 0.21000328 0.21000328
## 107 0.16652525 0.16652525 0.16652525 0.16652525
## 108 0.18912739 0.18912739 0.18912739 0.18912739
## 109 0.11055330 0.11055330 0.11055330 0.11055330
## 110 0.09853439 0.09853439 0.09853439 0.09853439
## 111 0.21000328 0.21000328 0.21000328 0.21000328
## 112 0.16652525 0.16652525 0.16652525 0.16652525
## 113 0.18912739 0.18912739 0.18912739 0.18912739
## 114 0.11055330 0.11055330 0.11055330 0.11055330
## 115 0.09853439 0.09853439 0.09853439 0.09853439
## 116 0.21000328 0.21000328 0.21000328 0.21000328
## 117 0.16652525 0.16652525 0.16652525 0.16652525
## 118 0.18912739 0.18912739 0.18912739 0.18912739
## 119 0.11055330 0.11055330 0.11055330 0.11055330
## 120 0.09853439 0.09853439 0.09853439 0.09853439
## 121 0.21000328 0.21000328 0.21000328 0.21000328
## 122 0.16652525 0.16652525 0.16652525 0.16652525
## 123 0.18912739 0.18912739 0.18912739 0.18912739
## 124 0.11055330 0.11055330 0.11055330 0.11055330
## 125 0.09853439 0.09853439 0.09853439 0.09853439
## 126 0.21000328 0.21000328 0.21000328 0.21000328
## 127 0.16652525 0.16652525 0.16652525 0.16652525
## 128 0.18912739 0.18912739 0.18912739 0.18912739
## 129 0.11055330 0.11055330 0.11055330 0.11055330
## 130 0.09853439 0.09853439 0.09853439 0.09853439
## 131 0.21000328 0.21000328 0.21000328 0.21000328
## 132 0.16652525 0.16652525 0.16652525 0.16652525
## 133 0.18912739 0.18912739 0.18912739 0.18912739
## 134 0.11055330 0.11055330 0.11055330 0.11055330
## 135 0.09853439 0.09853439 0.09853439 0.09853439
## 136 0.21000328 0.21000328 0.21000328 0.21000328
## 137 0.16652525 0.16652525 0.16652525 0.16652525
## 138 0.18912739 0.18912739 0.18912739 0.18912739
## 139 0.11055330 0.11055330 0.11055330 0.11055330
## 140 0.09853439 0.09853439 0.09853439 0.09853439
## 141 0.21000328 0.21000328 0.21000328 0.21000328
## 142 0.16652525 0.16652525 0.16652525 0.16652525
## 143 0.18912739 0.18912739 0.18912739 0.18912739
## 144 0.11055330 0.11055330 0.11055330 0.11055330
## 145 0.09853439 0.09853439 0.09853439 0.09853439
## 146 0.21000328 0.21000328 0.21000328 0.21000328
## 147 0.16652525 0.16652525 0.16652525 0.16652525
## 148 0.18912739 0.18912739 0.18912739 0.18912739
## 149 0.11055330 0.11055330 0.11055330 0.11055330
## 150 0.09853439 0.09853439 0.09853439 0.09853439
## 151 0.21000328 0.21000328 0.21000328 0.21000328
## 152 0.16652525 0.16652525 0.16652525 0.16652525
## 153 0.18912739 0.18912739 0.18912739 0.18912739
## 154 0.11055330 0.11055330 0.11055330 0.11055330
## 155 0.09853439 0.09853439 0.09853439 0.09853439
## 156 0.21000328 0.21000328 0.21000328 0.21000328
## 157 0.16652525 0.16652525 0.16652525 0.16652525
## 158 0.18912739 0.18912739 0.18912739 0.18912739
## 159 -0.38737181 -0.38737181 -0.38737181 -0.38737181
## 160 -0.38737181 -0.38737181 -0.38737181 -0.38737181
## 161 0.11055330 0.11055330 0.11055330 0.11055330
## 162 0.09853439 0.09853439 0.09853439 0.09853439
## 163 0.21000328 0.21000328 0.21000328 0.21000328
## 164 0.16652525 0.16652525 0.16652525 0.16652525
## 165 0.18912739 0.18912739 0.18912739 0.18912739
## 166 -0.38737181 -0.38737181 -0.38737181 -0.38737181
## multiplicative_terms multiplicative_terms_lower multiplicative_terms_upper
## 1 0 0 0
## 2 0 0 0
## 3 0 0 0
## 4 0 0 0
## 5 0 0 0
## 6 0 0 0
## 7 0 0 0
## 8 0 0 0
## 9 0 0 0
## 10 0 0 0
## 11 0 0 0
## 12 0 0 0
## 13 0 0 0
## 14 0 0 0
## 15 0 0 0
## 16 0 0 0
## 17 0 0 0
## 18 0 0 0
## 19 0 0 0
## 20 0 0 0
## 21 0 0 0
## 22 0 0 0
## 23 0 0 0
## 24 0 0 0
## 25 0 0 0
## 26 0 0 0
## 27 0 0 0
## 28 0 0 0
## 29 0 0 0
## 30 0 0 0
## 31 0 0 0
## 32 0 0 0
## 33 0 0 0
## 34 0 0 0
## 35 0 0 0
## 36 0 0 0
## 37 0 0 0
## 38 0 0 0
## 39 0 0 0
## 40 0 0 0
## 41 0 0 0
## 42 0 0 0
## 43 0 0 0
## 44 0 0 0
## 45 0 0 0
## 46 0 0 0
## 47 0 0 0
## 48 0 0 0
## 49 0 0 0
## 50 0 0 0
## 51 0 0 0
## 52 0 0 0
## 53 0 0 0
## 54 0 0 0
## 55 0 0 0
## 56 0 0 0
## 57 0 0 0
## 58 0 0 0
## 59 0 0 0
## 60 0 0 0
## 61 0 0 0
## 62 0 0 0
## 63 0 0 0
## 64 0 0 0
## 65 0 0 0
## 66 0 0 0
## 67 0 0 0
## 68 0 0 0
## 69 0 0 0
## 70 0 0 0
## 71 0 0 0
## 72 0 0 0
## 73 0 0 0
## 74 0 0 0
## 75 0 0 0
## 76 0 0 0
## 77 0 0 0
## 78 0 0 0
## 79 0 0 0
## 80 0 0 0
## 81 0 0 0
## 82 0 0 0
## 83 0 0 0
## 84 0 0 0
## 85 0 0 0
## 86 0 0 0
## 87 0 0 0
## 88 0 0 0
## 89 0 0 0
## 90 0 0 0
## 91 0 0 0
## 92 0 0 0
## 93 0 0 0
## 94 0 0 0
## 95 0 0 0
## 96 0 0 0
## 97 0 0 0
## 98 0 0 0
## 99 0 0 0
## 100 0 0 0
## 101 0 0 0
## 102 0 0 0
## 103 0 0 0
## 104 0 0 0
## 105 0 0 0
## 106 0 0 0
## 107 0 0 0
## 108 0 0 0
## 109 0 0 0
## 110 0 0 0
## 111 0 0 0
## 112 0 0 0
## 113 0 0 0
## 114 0 0 0
## 115 0 0 0
## 116 0 0 0
## 117 0 0 0
## 118 0 0 0
## 119 0 0 0
## 120 0 0 0
## 121 0 0 0
## 122 0 0 0
## 123 0 0 0
## 124 0 0 0
## 125 0 0 0
## 126 0 0 0
## 127 0 0 0
## 128 0 0 0
## 129 0 0 0
## 130 0 0 0
## 131 0 0 0
## 132 0 0 0
## 133 0 0 0
## 134 0 0 0
## 135 0 0 0
## 136 0 0 0
## 137 0 0 0
## 138 0 0 0
## 139 0 0 0
## 140 0 0 0
## 141 0 0 0
## 142 0 0 0
## 143 0 0 0
## 144 0 0 0
## 145 0 0 0
## 146 0 0 0
## 147 0 0 0
## 148 0 0 0
## 149 0 0 0
## 150 0 0 0
## 151 0 0 0
## 152 0 0 0
## 153 0 0 0
## 154 0 0 0
## 155 0 0 0
## 156 0 0 0
## 157 0 0 0
## 158 0 0 0
## 159 0 0 0
## 160 0 0 0
## 161 0 0 0
## 162 0 0 0
## 163 0 0 0
## 164 0 0 0
## 165 0 0 0
## 166 0 0 0
## yhat_lower yhat_upper trend_lower trend_upper yhat
## 1 17.39110 20.25923 18.63425 18.63425 18.80078
## 2 17.45216 20.19620 18.59643 18.59643 18.78556
## 3 17.20842 19.90242 18.48298 18.48298 18.59353
## 4 17.04965 19.88040 18.44516 18.44516 18.54369
## 5 17.22148 19.95720 18.40734 18.40734 18.61734
## 6 17.25028 19.89888 18.36952 18.36952 18.53604
## 7 17.07936 19.91734 18.33170 18.33170 18.52082
## 8 16.85058 19.69068 18.21824 18.21824 18.32879
## 9 16.79554 19.62736 18.18042 18.18042 18.27895
## 10 16.91489 19.76443 18.14260 18.14260 18.35260
## 11 16.98485 19.65750 18.10478 18.10478 18.27131
## 12 17.07376 19.73210 18.06696 18.06696 18.25609
## 13 16.66025 19.54494 17.95350 17.95350 18.06406
## 14 16.61562 19.40317 17.91568 17.91568 18.01422
## 15 16.74405 19.48219 17.87786 17.87786 18.08787
## 16 16.63800 19.28093 17.84004 17.84004 18.00657
## 17 16.45650 19.35138 17.80222 17.80222 17.99135
## 18 16.41490 19.11742 17.68877 17.68877 17.79932
## 19 16.30239 19.19076 17.65095 17.65095 17.74948
## 20 16.49555 19.21424 17.61313 17.61313 17.82313
## 21 16.44063 19.12669 17.57531 17.57531 17.74183
## 22 16.33698 19.13289 17.53749 17.53749 17.72662
## 23 16.20231 18.92205 17.42403 17.42403 17.53458
## 24 16.08889 19.03030 17.38621 17.38621 17.48474
## 25 16.13654 18.98537 17.34839 17.34839 17.55839
## 26 16.10070 18.82362 17.31057 17.31057 17.47710
## 27 16.05406 18.88891 17.27275 17.27275 17.46187
## 28 15.74754 18.68187 17.15928 17.15928 17.26983
## 29 15.80537 18.63200 17.12145 17.12145 17.21999
## 30 15.87946 18.62283 17.08363 17.08363 17.29363
## 31 15.80637 18.73725 17.04580 17.04580 17.21233
## 32 15.82662 18.53327 16.99846 16.99846 17.18759
## 33 15.56886 18.39301 16.85644 16.85644 16.96700
## 34 15.59925 18.29529 16.80910 16.80910 16.90764
## 35 15.57295 18.33392 16.76176 16.76176 16.97177
## 36 15.51531 18.41282 16.70505 16.70505 16.87157
## 37 15.40651 18.27713 16.64833 16.64833 16.83746
## 38 15.21437 18.01811 16.47819 16.47819 16.58874
## 39 15.10931 17.80640 16.42147 16.42147 16.52000
## 40 15.05764 17.98101 16.36475 16.36475 16.57476
## 41 15.12667 17.86857 16.30802 16.30802 16.47455
## 42 15.10679 17.76927 16.25129 16.25129 16.44042
## 43 14.82536 17.63134 16.08110 16.08110 16.19165
## 44 14.88003 17.61923 16.02437 16.02437 16.12290
## 45 14.84379 17.65668 15.96764 15.96764 16.17764
## 46 14.70339 17.50947 15.91091 15.91091 16.07743
## 47 14.74935 17.44290 15.85418 15.85418 16.04330
## 48 14.37057 17.10590 15.68398 15.68398 15.79454
## 49 14.32009 17.17107 15.62725 15.62725 15.72579
## 50 14.43055 17.20218 15.57052 15.57052 15.78052
## 51 14.26057 17.18279 15.51379 15.51379 15.68031
## 52 14.36209 17.08260 15.45706 15.45706 15.64619
## 53 13.96497 16.69332 15.28687 15.28687 15.39742
## 54 13.91805 16.75307 15.23013 15.23013 15.32867
## 55 13.98803 16.69534 15.17340 15.17340 15.38341
## 56 13.93026 16.76769 15.11667 15.11667 15.28320
## 57 13.82984 16.64272 15.05994 15.05994 15.24907
## 58 13.61427 16.40292 14.88975 14.88975 15.00030
## 59 13.60436 16.43455 14.83302 14.83302 14.93155
## 60 13.54577 16.36273 14.77629 14.77629 14.98629
## 61 13.58181 16.31344 14.75111 14.75111 14.91764
## 62 13.50474 16.32659 14.72594 14.72594 14.91507
## 63 13.40274 16.09977 14.65042 14.65042 14.76097
## 64 13.36023 16.13587 14.62525 14.62525 14.72378
## 65 13.43379 16.13524 14.60007 14.60007 14.81008
## 66 13.40142 16.12089 14.61159 14.61159 14.77811
## 67 13.46894 16.24460 14.62310 14.62310 14.81223
## 68 13.46062 16.22295 14.65764 14.65764 14.76820
## 69 13.39023 15.98002 14.66916 14.66916 14.76769
## 70 13.45042 16.31242 14.68067 14.68067 14.89067
## 71 13.57711 16.31386 14.71275 14.71275 14.87928
## 72 13.54036 16.41653 14.87316 14.87316 14.97170
## 73 13.74125 16.47642 14.90525 14.90525 15.11525
## 74 13.73690 16.55975 14.93733 14.93733 15.10385
## 75 13.87325 16.55140 14.96941 14.96941 15.15854
## 76 13.78775 16.56646 15.07333 15.07333 15.18388
## 77 13.68995 16.58523 15.10797 15.10797 15.20650
## 78 13.91127 16.67812 15.14261 15.14261 15.35261
## 79 13.84532 16.79474 15.17725 15.17725 15.34377
## 80 13.97558 16.79610 15.21189 15.21189 15.40101
## 81 14.15406 16.87239 15.32133 15.32133 15.43188
## 82 14.10910 16.90622 15.35781 15.35781 15.45635
## 83 14.20282 16.95020 15.39429 15.39429 15.60430
## 84 14.24405 16.96866 15.43077 15.43077 15.59730
## 85 14.22044 17.05904 15.46725 15.46725 15.65638
## 86 14.22732 17.05510 15.57725 15.57725 15.68781
## 87 14.28519 17.08625 15.61392 15.61392 15.71246
## 88 14.47917 17.27375 15.65059 15.65059 15.86059
## 89 14.46733 17.27797 15.68725 15.68725 15.85378
## 90 14.60046 17.40840 15.83392 15.83392 15.94448
## 91 14.57386 17.29694 15.87059 15.87059 15.96912
## 92 14.72281 17.47311 15.90726 15.90726 16.11726
## 93 14.72607 17.54327 15.94392 15.94392 16.11045
## 94 14.84667 17.60594 15.98059 15.98059 16.16972
## 95 14.89165 17.60839 16.09059 16.09059 16.20115
## 96 14.81039 17.69530 16.12726 16.12726 16.22579
## 97 15.03676 17.88411 16.16393 16.16393 16.37393
## 98 15.00473 17.77557 16.20059 16.20059 16.36712
## 99 15.01168 17.86050 16.23726 16.23726 16.42639
## 100 15.04244 17.82622 16.38393 16.38393 16.48246
## 101 15.28753 18.04393 16.42060 16.42060 16.63060
## 102 15.11104 17.93390 16.45726 16.45726 16.62379
## 103 15.17068 18.05695 16.49393 16.49393 16.68306
## 104 15.39417 18.15347 16.60393 16.60393 16.71448
## 105 15.36179 18.13057 16.64060 16.64060 16.73913
## 106 15.56594 18.47054 16.67727 16.67727 16.88727
## 107 15.45947 18.33828 16.71393 16.71393 16.88046
## 108 15.60897 18.27906 16.75060 16.75060 16.93973
## 109 15.60659 18.31581 16.86060 16.86060 16.97115
## 110 15.68619 18.41948 16.89727 16.89727 16.99580
## 111 15.73514 18.52113 16.93393 16.93393 17.14394
## 112 15.70223 18.62811 16.97060 16.97060 17.13713
## 113 15.75834 18.62525 17.00727 17.00727 17.19640
## 114 15.87998 18.55898 17.11727 17.11727 17.22782
## 115 15.80991 18.68615 17.15394 17.15394 17.25247
## 116 16.07775 18.84311 17.19060 17.19060 17.40061
## 117 16.01030 18.82745 17.22727 17.22727 17.39380
## 118 16.08948 18.77623 17.26394 17.26394 17.45307
## 119 16.00668 18.92952 17.37394 17.37394 17.48449
## 120 16.09522 18.88058 17.45752 17.45752 17.55605
## 121 16.30634 19.11177 17.54110 17.54110 17.75110
## 122 16.33007 19.26722 17.62467 17.62467 17.79120
## 123 16.50474 19.39125 17.70825 17.70825 17.89738
## 124 16.63573 19.33872 17.95899 17.95899 18.06954
## 125 16.89582 19.63421 18.14085 18.14085 18.23939
## 126 17.06531 19.96433 18.32271 18.32271 18.53272
## 127 17.28920 20.06014 18.50458 18.50458 18.67110
## 128 17.43874 20.33050 18.68644 18.68644 18.87557
## 129 17.95697 20.76118 19.23203 19.23203 19.34258
## 130 18.03567 20.87520 19.41389 19.41389 19.51242
## 131 18.37302 21.31322 19.59575 19.59575 19.80576
## 132 18.55109 21.43097 19.77761 19.77761 19.94414
## 133 18.67997 21.52348 19.95948 19.95948 20.14860
## 134 19.23490 22.02983 20.50507 20.50507 20.61562
## 135 19.44373 22.23094 20.68693 20.68693 20.78546
## 136 19.61330 22.52100 20.86879 20.86879 21.07879
## 137 19.83782 22.51895 21.05065 21.05065 21.21718
## 138 20.06176 22.82972 21.23252 21.23252 21.42164
## 139 20.44451 23.29320 21.77810 21.77810 21.88866
## 140 20.65310 23.36781 21.95997 21.95997 22.05850
## 141 21.00160 23.72612 22.14183 22.14183 22.35183
## 142 21.18394 24.02696 22.32369 22.32369 22.49022
## 143 21.19914 24.03448 22.50555 22.50555 22.69468
## 144 21.72625 24.53404 23.05114 23.05114 23.16170
## 145 21.95588 24.71382 23.23300 23.23300 23.33154
## 146 22.23401 25.06013 23.41487 23.41487 23.62487
## 147 22.31558 25.15388 23.59673 23.59673 23.76326
## 148 22.39638 25.35407 23.77859 23.77859 23.96772
## 149 22.99761 25.71607 24.32418 24.32418 24.43473
## 150 23.23262 26.04558 24.50604 24.50604 24.60458
## 151 23.52933 26.33102 24.68791 24.68791 24.89791
## 152 23.64610 26.36515 24.86977 24.86977 25.03629
## 153 23.91204 26.58423 25.05163 25.05163 25.24076
## 154 24.35016 27.07859 25.59722 25.59722 25.70777
## 155 24.56891 27.28522 25.77908 25.77908 25.87762
## 156 24.79741 27.46271 25.96094 25.96094 26.17095
## 157 24.91051 27.74427 26.14281 26.14281 26.30933
## 158 25.18245 27.93028 26.32467 26.32467 26.51380
## 159 24.66304 27.50257 26.50401 26.51164 26.11916
## 160 24.96422 27.72828 26.67998 26.70194 26.30102
## 161 25.55810 28.33668 26.85449 26.89341 26.98081
## 162 25.74247 28.54675 27.02815 27.08556 27.15065
## 163 26.06872 28.82748 27.19966 27.28065 27.44399
## 164 26.21675 29.11274 27.36953 27.47522 27.58237
## 165 26.42933 29.23691 27.53924 27.67091 27.78684
## 166 26.04561 28.77800 27.70939 27.86783 27.39220plot(model,forecast)
prophet_plot_components(model,forecast)
Implementing forcasting using TimeSeries object in R
library(tseries)
library(zoo)
library(forecast)
library(normwhn.test)
dat = structure(
c(1100, 1150, 1200, 1150, 1150, 1200, 1300, 1100, 1200,1250, 1200, 1250,
1300,1400,1550,1400,1450,1450,1450,1450,1500,1550,1600,1650,
1650,1650,1650,1650,1750,1850,1750,1800,1850,1950,1950,2000,
1950,2000,2050,2100,2150,2200,2250,2450,2650,2700,2850,2850,
2750,2750,2750,2750,2800,2800,2850,2850,3000,2950,3000,2950,
2900,2850,2750,2750,2450,2400,2450,2500,2750,2550,2600,2300,
2470,2570,2570,2550,2500,2470,2550,2550,2470,2470,2370,2400,
2870,2648,2656,2600,2656,2608,2460,2644,2564,2508,2524,2508,
3006,3027,3129,3142,3119,2920,2902,2875,2880,2726,2543,2836,
2887,2978,2901,2946,2890,2947,2811,2899,3017,2989,2967,2902,
2994,3046,3074,3100,3141,3002,2934,2901,2908,3003,3097,3083,
3205,3253,3221,3192,3170,3240,3345,3480,3750,3725,3800,4000,
3800,4000,4200,4500,4800,5000,5100,5500,5500,5800,6000,6200), .Tsp = c(2008, 2020.91666666667, 12), class = "ts")
arimaModel <- arima(dat, order=c(0, 1, 1), list(order=c(0, 1, 0), period = 12))
summary(arimaModel)##
## Call:
## arima(x = dat, order = c(0, 1, 1), seasonal = list(order = c(0, 1, 0), period = 12))
##
## Coefficients:
## ma1
## -0.1067
## s.e. 0.0720
##
## sigma^2 estimated as 22135: log likelihood = -918.26, aic = 1840.53
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 14.32495 142.4436 105.2636 0.3245391 3.827641 1.189896 -0.02882344preds <- predict(arimaModel, n.ahead = 12)
print(preds)## $pred
## Jan Feb Mar Apr May Jun Jul Aug
## 2021 5998.214 6198.214 6398.214 6698.214 6998.214 7198.214 7298.214 7698.214
## Sep Oct Nov Dec
## 2021 7698.214 7998.214 8198.214 8398.214
##
## $se
## Jan Feb Mar Apr May Jun Jul Aug
## 2021 148.7772 199.4956 239.7131 274.0916 304.6146 332.3461 357.9354 381.8136
## Sep Oct Nov Dec
## 2021 404.2839 425.5694 445.8398 465.2279plot(dat, col = 'green', main = 'Acutal vs ARIMA Model',
ylab = 'Gold Rates', lwd = 3)
lines(arimaModel$fitted, col = 'black', lwd = 3, lty = 2)
legend('topleft', legend = c('Actual', 'ARIMA Fit'),
col = c('green','black'), lwd = c(3,3), lty = c(1,2))
ts.plot(dat, preds$pred, main = 'Actual vs ARIMA Predictions',
col = c('green','black'), lty = c(1,2), lwd = c(3,3))
legend('topleft', legend = c('Actual Data', 'ARIMA Predictions'),
col = c('green','black'), lwd = c(3,3), lty = c(1,2))
Summary
The main aim of time-series modeling is to carefully collect and regorously study the past observations of the time-series to develop an appropriate model which describes the inherent structure of the series. The model is then used to generate the future values for the series. Time series forecasting is thus can be termed as act of predicting the future by understanding the past.
