1. Income Forecasting

2. Introduction
For some, budgeting while planning for the future can be difficult. In general, people don't know exactly what to expect of their income in the future. Because of this we decided to apply time series prediction algorithms to forecast probable future income.

3. Abstract
This project was decided to be a income forecasting program to find the probable income of a budget given the income history of the budget.
We looked into several algorithms to forecast over a time series including:
Forward-Backward
Moving average model (SMA)
Autoregressive integrated moving average (ARIMA)
Our goal is to measure the accuracy of their predictions over the time series, as well as compare their time complexity

4. Problem Statement Natural Language: Formal:
Given income history, forecast the probable future income using different algorithm approaches, while measuring their accuracy and execution time.
Formal:
The forecasting algorithms are measured by comparing their level of error between the forecasted values and the actual values. This difference between the forecast value and the actual value for the corresponding period is expressed as.
where the error (E) at period (i) is the absolute value of forecast (F) at period (i) minus the actual value (Y) at period (i). The lower the value E the more accurate the algorithm is for the given period (i)

5. Problem Statement(cont.)
The mean of these absolute error values, known as the mean absolute error(MAE), is given by the equation:
Where the number (n) is the absolute errors calculated and (E) is the absolute error calculated at period (i).
The MAE shows the average accuracy of each algorithm for a given set of data for the sake of comparison.

6. Introducing AR, MA and ARMA
Auto-Regressive: Model in which Y(t) depends only on its own past values.
Y(t) = φ1Y(t−1) + φ2Y(t−2) φpY(t−p) + ౬t .
Moving Average: Model in which Y(t) depends only on the random error terms, which follows the white noise process. Earthquake Example.
Y(t) = ౬t + θ1౬(t−1) θq౬(t−q).
ARMA(p,q) model: Provides simple tool in time series modeling.
Y(t) = φ1Y(t−1) + φ2Y(t−2) φpY(t−p) + ౬t + θ1౬(t−1) θq౬(t−q).

7. Pure ARIMA Models
General statistical model which is widely used in the field of time series analysis. Given a time series data Y(t), mathematically the pure model is written:

8. Autoregressive Integrated Moving Average(ARIMA)
ARIMA is an acronym for AutoRegressive Integrated Moving-Average. The order of an ARIMA model is usually denoted by ARIMA(p,d,q)
Where:
p (AR) is the order of the autoregressive part
d (I) is the order of differencing
q (MA) is the order of the moving-average process
-Where L is the lag Operator, alpha(i) are the parameters of autoregressive and theta(i), the parameters of the moving average and (E)t are errors

9. Implementing a Time Series Analysis
1) Visualize the time series
2) Stationarize the series
3) Plot ACF/PACF charts and find optimal parameters
4) Build the Arima Model
5)Make predictions

10. Using ARIMA model to forecast values
Years
Income($)
Forecast
2005
200,969
2006
213,189
2007
221,205
2008
215,543
2009
204,402
2010
213,178
2011
228,438
226,425
2012
239,493
236,729
2013
246,313
245,369
2014
250,775
252,692
2015
259,003
Using ARIMA model to forecast values
RunTime = secs

11. Simple Moving average model (SMA)
In a time series, it takes an average of the most recent (Y) values, for some integer n.This is the so-called moving average model (SMA), and its equation for predicting the value of Y at time t+1 based on data up to time t is:
n=number of periods in the moving average
n can have 3, 5 Periods or more depending on size of the data set
y=demand in periods of time

12. Simple Moving Average Cont.
Years
income
Forecast
2003
178,694
2004
190,253
2005
200,969
2006
213,189
189,972
2007
221,543
201,470
2008
215,543
211,900
2009
204,402
216,758
2010
213,178
213,829
2011
228,438
211,041
2012
239,493
215,339
2013
246,313
227,036
2014
250,775
238,081
2015
Unknown
245,527
Sdfd
RunTime =2.706e-06
3 time periods were used for calculations
SMA Complexity= O(n)

13. Forward-Backward Algorithm
Also known as hidden Markov models(HMM), evaluates the probability of a sequence of observations occurring when following a given sequence of states. This can be stated as:
Where A and B are matrices.
x= sequence of states
y=number of observation
I=current state
k= is the number of time steps
Xj=i=probability of being in state i at time j

14. Forward - Backward (HMM) Cont.
HMM is one in which you observe a sequence of emissions, but do not know the sequence of states the model went through to generate the emissions. Analyses of HMM seek to recover the sequence of states from the observed data.

15. Forward-backward implementation
Years
Income
Forecast (p)
2012
239,493
225,245
2013
246,313
240,625
2014
250,775
246,621
2015
P(x,y)
states = ('Future', 'Past')
end_state = 'E'
observations = ('260', '246', '250')
start_probability = {'Future': 0.97, 'Past': 0.3}
transition_probability = {
'Future' : {'Future': 0.69, 'Past': 0.3, 'E': 0.01},
'Past' : {'Future': 0.4, 'Past': 0.59, 'E': 0.01}, }
emission_probability = {
'Future' : {'260': 0.5, '246': 0.4, '250': 0.1},
'Past' : {'260': 0.1, '246': 0.3, '250': 0.6},
(years)
these are the probabilities 0f 15 states: {'Past': , 'Future': } {'Past': , 'Future': } {'Past': 0.01, 'Future': 0.01}
these are the probabilities : {'Past': , 'Future': } {'Past': , 'Future': } {'Past': , 'Future': }
Run Time = sec

16. Forward - backward cont
Backward probability
Assume that we start in a particular state (Xt=xi), T=transition
Uses a column vector
Years
Income
Forecast (p)
2012
239,493
225,245
2013
246,313
240,625
2014
250,775
246,621
2015
P(x,y)

17. Forecast Comparison

18. Error Comparison MSE for SMA: 18,380.5 MSE for ARIMA: 1,909.5
MSE for FB: 8,029.67

19. Conclusion ARIMA was the more accurate out of the three algorithms.
The ARIMA algorithm can be used to predict fairly accurate yearly income predictions.
-HMM FB was second best Model, however the results can vary depending on the initial states. Therefore the result can differ depending on the probability of reaching the next state. HMM is suitable for other types of predictions such as weather, shuffling or stocks since the states change constantly.

20. Other Work in Income Forecasting
Digit has an algorithm that “learns a user's spending and earning patterns”[3]
A website called Buxfer offers personal finance forecasting as well
There has been much work in comparing the algorithms themselves such as “Time Series Prediction Algorithms” by Kumara M.P.T.R.

21. Question
Q: What is the difference between using the ARMA and the ARIMA models?
A: The ARIMA model converts non-stationary data to stationary before operating on it.

22. Question
Q: Why is the simple moving average(SMA) less accurate than ARIMA?
A: Arima takes more constraints into account: central moving averages(CMA), weighted moving average (WMA), seasonality and lag (failure to maintain a desired pace) ,

23. Question Q: What is the Big(O) of HMM? A:
n is the number of hidden or latent variables
m is the number of observed sequences of observed variables

24. Questions?

25. References https://en.wikipedia.org/wiki/Forecasting
algorithms-literature-review
algorithm-do-your-saving-for-you-
gation

26. References cont.