Water Resources Systems - short answer type questions from AMIE exams (Winter 2019)

Define present worth factor and its mathematical formula.

The present value (PV) factor is used to derive the present value of a receipt of cash on a future date. The concept of the present value factor is based on the time value of money - that is, money received now is worth more than money received in the future, since money received now can be reinvested in an alternative investment to earn additional cash. 
The PV factor is greater for cash receipts scheduled for the near future, and smaller for receipts that are not expected until a later date. The factor is always a number less than one. The formula for calculating the present value factor is:

P = [1 / (1 + r)n]


P = The present value factor
r = The interest rate
n = The number of periods over which payments are made

What is cumulative distribution function and probability distribution function?

The Cumulative Distribution Function (CDF) of a real-valued random variable, say “X”, which is evaluated at x, is the probability that X takes a value less than or equal to x. It is defined for both discrete and random variables.

The CDF defined for a continuous random variable is given as;

{F_X}(x) = \int_{ - \infty }^x {{f_X}(t)dt}

A probability density function (PDF) is used to define the random variable’s probability coming within a distinct range of values, as opposed to taking on any one value. In other words, the probability density function produces the likelihood of values of the continuous random variable. Sometimes it is also called a probability distribution function or just a probability function. 

In the case of a continuous random variable, the probability taken by X on some given value x is always 0. In this case, if we find P(X = x), it does not work. Instead of this, we must calculate the probability of X lying in an interval (a, b). Now, we have to figure it for P(a< X< b), and we can calculate this using the formula of PDF. The Probability density function formula is given as,

P(a < X < b) = \int_a^b {f(x)dx}

Discuss moving average models. Define ARIMA.

In time-series, we sometimes observe similarities between past errors and present values. That’s because certain unpredictable events happen, and they need to be accounted for.

In other words, by knowing how far off our estimation yesterday was, compared to the actual value, we can tweak our model, so that it responds accordingly.

A moving average model is used for forecasting future values, while moving average smoothing is used for estimating the trend-cycle of past values.

An ARIMA model is a class of statistical models for analysing and forecasting time series data.

It explicitly caters to a suite of standard structures in time series data, and as such provides a simple yet powerful method for making skilful time series forecasts.

ARIMA is an acronym that stands for AutoRegressive Integrated Moving Average. It is a generalization of the simpler AutoRegressive Moving Average and adds the notion of integration.

Discuss gamma distribution.

The gamma distribution term is mostly used as a distribution which is defined as two parameters – shape parameter and inverse scale parameter, having continuous probability distributions. It is related to the normal distribution,  exponential distribution, chi-squared distribution and Erlang distribution. ‘Γ’ denotes the gamma function.

Enlist types of rain gauges.

The three major types of rain gauges are the standard gauge, tipping bucket gauge and weighing gauge.
The recording of rainfall using the standard or funnel rain gauge is generally done manually. These gauges work by catching the falling rain in a funnel-shaped collector that is attached to a measuring tube. Symons Rain Gauge Non-recording type rain gauge is the most common type of rain gauge used by meteorological department. For uniformity, the rainfall is measured every day at 8:30Am IST and is recorded as rainfall of the day.

Weighing bucket type rain gauge is the most common self-recording rain gauge. It consists of a receiver bucket supported by a spring or lever balance or some other weighing mechanism. The movement of the bucket due to its increasing weight is transmitted to a pen, which traces a record or some marking on a clock driven chart. Weighing bucket type rain gauge instrument gives a plot of the accumulated (increased) rainfall values against the elapsed time, and the curve so formed is called the mass curve.

A tipping bucket rain gauge consists of a cylindrical receiver 30 cm diameter with a funnel inside (Fig. 2.4). Just below the funnel, a pair of tipping buckets is pivoted such that when one of the bucket receives a rainfall of 0.25 mm it tips and empties into a tank below, while the other bucket takes its position and the process is repeated. The tipping of the bucket actuates on electric circuit which causes a pen to move on a chart wrapped round a drum which revolves by a clock mechanism. This type cannot record snow.

What are Kuhn-Tucker conditions ?

Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalises the method of Lagrange multipliers, which allows only equality constraints. 
KKT conditions is the necessary conditions for optimality in general constrained problem.
For a given nonlinear programming problem:

x* must satisfy all the following KKT conditions:

Discuss the working of Theissen polygon and Isohyetal method.

Theissen polygon method
This method attempts to allow for non-uniform distribution of gauges by providing a weighting factor for each gauge. The stations are plotted on a base map and are connected by straight lines. Perpendicular bisectors are drawn to the straight lines, joining adjacent stations to form polygons, known as Theissen polygons (see figure). 

Each polygon area is assumed to be influenced by the rain gauge station inside it, i.e., if P₁, P₂, P₃, .... are the rainfalls at the individual stations, and A₁, A₂, A₃, .... are the areas of the polygons surrounding these stations, (influence areas) respectively, the average depth of rainfall for the entire basin is given by

{P_{avg}} = \frac{{\sum {A_1}{P_1}}}{{\sum {A_1}}}

Where ΣA₁ = A = total area of the basin.

The Isohyetal method
In this method, the point rainfalls are plotted on a suitable base map and the lines of equal rainfall (isohyets) are drawn, giving consideration to orographic effects and storm morphology (see figure). 

The average rainfall between the successive isohyets taken as the average of the two isohyetal values are weighted with the area between the isohyets, added up and divided by the total area which gives the average depth of rainfall over the entire basin, i.e.,

{P_{avg}} = \frac{{\sum {A_{1 - 2}}{P_{1 - 2}}}}{{\sum {A_{1 - 2}}}}

where A1-2 = area between the two successive isohyets P₁ and P₂

{P_{1 - 2}} = \frac{{{P_1} + {P_2}}}{2}

Where ∑A1-2 = A = total area of the basin.

What is time series analysis.

Time series analysis accounts for the fact that data points taken over time may have an internal structure (such as autocorrelation, trend or seasonal variation) that should be accounted for.

Time series data analysis is the way to predict time series based on past behaviour. Prediction is made by analysing underlying patterns in the time-series data. 

There are many ways to model a time series in order to make predictions, some are given below.
  • moving average
  • exponential smoothing
Example of time series plot: The raw data used and the final historical time series of residential water use in Victoria. DWR refers to the Victorian Department of Water Resources, ABS is the Australian Bureau of Statistics and BRS is the Bureau of Rural Sciences

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