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Sulochana Kamshetty

2 years ago

- Introduction
- Key Observation

- Issues with Regressing on Time

- What is AR(p) models:- Auto Regressive model

- Drawbacks of AR model

- Moving Average or MA(q) models

- ARMA(p,q) model

- ARIMA(p,d,q) model

- ARIMAX Approach

- Spurious Regression

- What is the real goal?

- Granger Causality

- Difference between Regression and Causality

- Key Points To Take Away

We were working for the project Godrej Nature’s Basket, trying to manage its supply chain and delivery partners and would like to accurately forecast the sales for the period starting from “1st January 2019 to 15th January 2019”.

To know what it’s going to future prediction, of sales in these particular days. we use the time series models. for the most accepted results, using some important methodologies, like AR, MA, ARMA, ARIMA,ARIMAX, SPURIOUS & GRANGER CAUSALITY MODELS, In our last content we discussed about Time Series now, let’s understand

- What are these different models used in Timeseries?
- What are its uses?
- Where its applied and its details.

Lets begin with understanding what are:-AR, MA AND ARIMA MODELS,

using the same dataset with coding part simultaneously.

Since that we are going to predict the sales for 15 days will extract only those data and understand how its going to work with TimeSeries models.

Since that we are going to explore the sales, lets verify with basic of Timeseries steps those are Residual, Seasonal, Trend of the selected dataset.

- Seasonality looks more like additive seasonality.
- This is a strong indication of trend over 4 years and seasonality across months.
- Clearly there is yearly trend, with monthly seasonality in the data.

After walking through basic steps, like working with regression model, since it is easy and more flexible to work with Timeseries models, and then finding out the error matrices of seasonality & trend, important observation one should understand that trend required to capture all the movements of the data, If there is no trend or if seasonality, and fluctuations are more important than trend, then the coefficients behave weirdly. To avoid these major issues one has to work with advance method of regression which deals with capturing all the movements present in trend.

The term AR(Auto Regressive) in simple terms refers to working auto/self taking help of regression is called auto regressive.

It will help us to predict/to forecast the variable, of interest using linear regression, which is the combination of the past values of the variable. Auto regressive is so flexible to use wide range of different time series patterns.

Time series has got different styles of understanding the concept, but lets use the simplest, and more powerful methods in our content.

Where Auto-regressive model of order p

Ŷ t = α + β1 yt-1 + β2 yt-2 +… βp yt-p

- This above equation describes about calculations for future prediction using Ŷ which is the predicted value of y.
- We find the best value of parameters (β1 , β2,…) that minimize the errors in forecast of Ŷ t
- The order of the model p, is determined based on the number, beyond which PACF terms are zero.
- We normally restrict autoregressive models to stationary data, in which case some constraints on the values of the parameters are required.

Where the Timeseries was created out of integration, which although doesn’t get stationarize even after we difference it, which leads to quadratic differencing with second differencing, which is called (Integrated of order II). Which leads to capturing data in all the lags.

This method works with two different measures which considers the past error metrics, captures the regression of order of two coefficients. Where the largest non-zero terms speaks about the terms required to consider. Model attempts to predict future values using past error in predictions,Ʃ1 = Ŷ1 — Y1

- So MA(2) model is Where,

Ŷ t=µ+ϕ1Ʃt-1+ ϕ2Ʃt-2

Where µ, is the average value of the time series, it is the average value of the time series

• Again, the parameters (ϕ1 ,ϕ2 ) are determined so that prediction error is minimized.

• The number of terms, q, is determined from the ACF plot. Its the maximum lag beyond which the ACF is 0

which is called “Autoregressive moving average model”, which is the combination of both the models which takes two hypermeter, So a ARMA(2,1) model is which takes the two previous values of AR values and one error term for the regression. also requires two parameters with one coefficient.

Ŷ t = α + β1 yt-1 + β2 yt-2+ ϕ1Ʃt-1

Which is called “An autoregressive integrated moving average ”which is mostly used as an statistical tool, for the timeseries for better understanding of the data.

Where following are the different parameters used in ARIMA.

• p is the number of autoregressive terms, (a linear regression of the current value of the series against one or more prior values of the series.) — Maximum lag beyond which PACF is 0.

• d is the number of non-seasonal differences, (order of the differencing) used to make the time series stationary,

- q is the number of past prediction, error terms used for the future forecasts

Note:-The forecast is plotted in dark blue. The dark grey and light grey regions represent the 80% and 95% confidence intervals.

Before Automated functions were available, one used to use ACF plots to determine the best value of (p,d,q) for a given dataset

- Box–Jenkins Methodology: This method is used for Model identification and model selection, it make sure variables are stationary. also finds the difference as necessary to get a constant mean and transformations to get constant variance. Required to Check for seasonality, which Decays and spikes at regular intervals in ACF plots.
- Parameter estimation :- It Compute coefficients that best fit the selected model.

- Model checking:- This helps to Check if residuals are independent of each other and constant in mean and variance over time (white noise).

- Non-seasonal: ARIMA models are denoted ARIMA(p,d,q)

- Seasonal ARIMA: (SARIMA) models are denoted ARIMA(p,d,q)(P,D,Q)m, where m refers to the number of periods in each season and (P,D,Q) refer to the autoregressive, differencing and moving average terms of the seasonal part of the ARIMA model.

Identification Phase Step 1: Plot the data (transform data to stabilize variance, if required)

Step 2: Plot ACF and PACF to get preliminary understanding of the processes involved.(The suspension bridge pattern in ACF (also, positive and negative spikes in PACF) suggests non-stationarity and strong seasonality.)

Step 3: Perform a non-seasonal difference. We are getting read to build an ARIMA(x,1,y) model

Step 4: Check ACF and PACF of differenced data to explore remaining dependencies.(The differenced series looks somewhat stationary but has strong seasonal lags.)

Step 5: Perform seasonal differencing (t0 -t12, t1 -t13, etc.) on the original time series to get seasonal stationarity. This is the same as an ARIMA(p,0,q) (x,1,y) 12 model.

Step 6: Check ACF and PACF of seasonally differenced data to explore remaining dependencies and identify model(s). Strong positive autocorrelation indicates need for either an AR term or a non-seasonal differencing

Step 7: Perform a non-seasonal differencing on seasonally differenced data. This is like an ARIMA (p,1,q) (x,1,y) 12 model.

Step 8: Check ACF and PACF to explore remaining dependencies.: This indicates an ARIMA(1,1,1)(0,1,1)12 model. As the significant lag at seasonal period is negative, include a Seasonal MA(1) term.

Parameter Estimation Phase Step 9: Calculate parameters using the identified model(s). Use AIC to pick the best model.

Evaluation Phase Step 10: Check ACF and PACF of the residuals to evaluate model. The residuals indicate white noise. Indicates a good model that can be used for forecasting.

Evaluation Phase Step 10: The residuals indicate white noise. Can be checked using Ljung-Box test.

Important note: For non-seasonal time series, use h = min(10, n/5) For seasonal time series, use h = min(2m, n/5), where m is the seasonal period

h is the maximum lag being considered

n is the # of observations (length of the time series)

rk is the autocorrelation

If residuals are white noise (purely random),then Q has a Chi-Square distribution with h-p degrees of freedom, where p is the number of parameters estimated in the model. The residuals indicate white noise. Can be checked using Ljung-Box test.

Null hypothesis:-Residuals are random

Large p-value indicates, none hypothesis can be accepted.

• The number of parameters (p,d,q) needed to fit, depends on the dataset.

• There are techniques that automate model selection.

• auto.Arima command in R picks the best p,d & q parameters for ARIMA(p,d,q)

“Prediction is very difficult, especially if it’s about the future.” — Niels Bohr,

An ARIMAX (ARIMA with exogenous variables) model is simply a multiple regression with AR and/or MA terms.

when and why arimax is used lets understand with below live examples

- It is used for where daily data is provided, & to check what should be the frequency of the time series?
- If we find any annual spikes in that situation we can start by declaring the data as a timeseries object with frequency 365.

- If the data is not stationary, find out the difference of yt. then apply the same differencing to all exogenous variables, xt.
- Build a (multiple) regression model on the stationarized data.
- Check for Granger-causality. If xt does not Granger-cause yt, then do not proceed with ARIMAX. It will not do any better than ARIMA.

For example, yt-yt-1= β1(xt-xt-1)+nt

where nt are the residuals (white noise; i.e., constant mean and constant variance). also Check for white noise of residuals, insignificant exogenous variables,& multicollinearity among exogenous variables, signs, etc.

A version of ARIMAX is implemented in forecast package and can be called from the “auto.arima function”..

It is possible to estimate a regression and find a statistically significant relationship even if none exists. In time series analysis this is actually a common occurrence when data are not stationary, which converts the Univariate to Multivariate data.

So far, we discussed Time Series problems, with involving a single variable. there are few drawbacks involves with, and that is where spurious regression helps to resolve the issue with.

These are few situations where spurious, work better than regression models.

- We may be able to build better models if we have other causal variables as well.

- Often, people ignore the time-series property of the data and start build linear regression models in such cases. This could sometimes lead to misleading results.

- The R2 values could be high, even though the model might not have any predictive power.

which is working on predicting different aspects of price of stocks, and price movement etc. which helps to understand the variables that impact stock price of a company finds the Possible predictors: like GDP, Oil Price, Inflation, Commodity Prices.

Look at initial model Date Range: 1950–2017 and this predictions are been calculated taking some important dimensions from s&p and GDP

when we try executing with R or Python we require basic predictions which can be performed using simple calculations..

Ok then…

where R-squared gave the value of 0.8653, which is the measurement used to compare the values of previous predicted value.

- S&P 500 data has a strong trend (non-stationary)

Any other variable with a trend will also show large R2

lets check with some of the Spurious Regressions Some Examples/used technology.

• So, if directly regressing S&P500 with GDP is wrong, what is the right thing to do?

Our intent is to understand how change in GDP affects the S&P movements.

- S&P change vs GDP change.
- This is equivalent to stationarizing the data before we do the regression.

Granger causality is a statistical concept of causality that is based on prediction. According to Granger causality, if a signal X1 “Granger-causes” (or “G-causes”) a signal X2, then past values of X1 should contain information that helps predict X2 above and beyond the information contained in past values of X2 alone.

- Linear regression detects the presence of correlation between change in x vs change in y.
- The examples discussed show that high-correlation, does not imply causation.
- Sometimes, we want to know if there is a causal relationship.
- For eg: — Increased endorphins are associated with decreased stress. Does increase in endorphins actually cause decrease in stress or are they just correlated?
- Is there a way to detect causal relationship between two variables? • Existence of causal relationship would imply better predictive power for the models.
- Auto-regressive model of order p (RESTRICTED MODEL, RM)

Ŷ t = α + β1 yt-1 + β2 yt-2 +…+Ɣp yt-p

where p parameters (degrees of freedom) to be estimated

- The predictor is said to Granger-cause if can be better predicted using past values of xt.
- Simple premise: If X causes Y, then X must precede Y.
- This implies: — Lagged values of X should be significantly related to Y. — Lagged Values of Y should NOT be significantly related to X
- Tests the following H0 : xt does not Granger-cause yt i.e α1=α2=…..αp=0
- HA : Granger-causes yt, i.e., at least one of the lags of x is significant.
- Granger Causality is not true causality.
- It only says that past values of xt can help predict yt better; i.e.,x precedes y. For example, Diwali fireworks sales precede (i.e., Granger-cause) Diwali but they do not cause Diwali.
- Cannot overrule the possibility of a hidden predictor that is causing both xt and yt.

- Be suspicious of high R 2 in real-life complex problems, especially when time is a confounding factor. Possible spurious regression.
- Granger-Causality can help understand which variables have predictive influence.
- Granger-causality doesn’t necessarily mean real causality.
- You must remove autocorrelation (stationarize the data) before testing for Granger-causality

Finally we learnt all the necessary points which are required to cover in time series as well as its models.

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