Bitcoin and a new world

I have coffee in Starbucks or Vienna everyday which cost me around 3-5$. And so do most of my friends here in US. This world has 7.8 billion people now. Out of them, about 3 billion people live below the poverty line which is around 3$. Every time I buy coffee, I never think about this. The realization that despite of all the wealth and innovation happening in this world, almost half of the population are unbanked, have no access to internet, world market, late alone credit card. I don’t envision that Bitcoin will solve the problem of poverty. But Bitcoin showed how currency can exist outside the realm of political and economic power which systemize the wealth hierarchy, define the line between poor and rich. Bitcoin has proven that a currency can be created out of the interest of people’s desire to trade between each other, definitely it happened because of the advent of computer, internet, open source software, you name it. I know it’s been speculated now as asset, get rich quick scheme (frankly I am also holding and speculating). Most general people are scared of the volatility, but I am talking about the core ideas that Satoshi Nakamoto had when he combined the idea of cryptocurrency and supply-demand economics for creating a digital currency for the world. Bitcoin movement is more than just money or a new innovation, it has the power of shifting the world. And that’s why I am a believer in a decentralized currency and frankly I don’t know how it will unravel in future and frankly nobody knows. Bitcoin will probably never be a currency, but it has shown the possibility. But it’s revolutionary to think that, every region in the world in future will have their own local cryptocurrency, born outside of politics and manipulation and the unbanked of the world have access to the economy. Well, there will always probably be politics, greed, manipulation; after all, we are sometimes just bunch of animals trying to beat each other for our own gains!! lol.. but I am looking at the big picture. I don’t envision a utopia but definitely I can see a better world. Also do you know what’s happening in Venezuela, everybody in Venezuela is buying Bitcoin because of the corruption of the government and the inflation. The problem is that it’s not that government is bad, ultimately we the people put officials there. It’s just that power and institutionalization corrupts mind and stagnates. We need a balance of the both, we need education, we need democracy, we need distribution of wealth and power. And as everyone says, it always depends on “Where is the money?” And Bitcoin showed but may be will be proven indirectly by many crypto currency movements over the future years, that money can exist outside of what we used to call money. And it’s mind boggling but we are living through the history that the future will talk about.
The political implications of bitcoin

Some thoughts on the inequality of Bitcoin ownership

Even if I am a proponent of Bitcoin, this bothers me. When Satoshi Nakamoto envisioned bitcoin as a payment protocol for everyone, one computer one vote, he was definitely unable to see how capitalism and the already accumulated money will also accumulate Bitcoin. I personally don’t identify myself as a capitalist, I actually dislike capitalism on many grounds, but at the same time, I know why and how it works, it’s not that there are many alternatives in the world. I understand Marxist, communist and socialist ideas but I fail to see how it can propel creativity and drive to move forward and change. It’s like a tradeoff and that’s why I probably like social democracy where they envision a balance of both. But with imperfection. Money is power and powerful always concentrate power and money for their own gains. Equal opportunity ought to be established but equal outcome we can never guarantee. The tension is always there. The concentration of Bitcoin in few places or few hands or few organizations go against the spirit of Bitcoin, as by protocol it is decentralized. But the fact is it is centralized in the big hands and this is the only reason I have against Bitcoin. And maybe that’s why I don’t believe completely that Bitcoin will be a currency for the future. I view it as a digital asset like digital gold which will be used as a store of value. But Bitcoin has shown us how money or value system or payment system can be created without a central money-making machine. I do believe that Bitcoin is better than government currency. I am against the spirit of nationalism and I think the national borders won’t make sense after few millennia and we will appreciate a global currency like Bitcoin. So, ultimately I want to say, Bitcoin will be there. But probably there will be a better currency which can surpass Bitcoin and which will probably not centralized. But then again that is Marxist, libertarian or anarchyst part of my brain talking. As I can see how capitalism is reigning in our world by the power, for the power, with the powerful, who knows Bitcoin might be the winner. Anyway, just some thoughts. I am not an astrologer who predicts future. I speculate and I miss.
The ethereum founder Vitalik Buterin also doesn’t believe in proof of work anymore. So, there are smart people who are already thinking about better solution already. That’s always our best bet for survival. Isn’t it? We need an einstein, we need a Nicola Tesla, we need few scientists, we need probably a work and innovation machine like Elon musk as most of us are just fooling around. Well, it’s too much for one little brain sometimes.

Which investment app to use?

Some of my friends ask me about investments in crypto space. You should already realize that Bitcoin/Crypto are high risk investments. So, you should be ready to lose money or be patient. If you are new into investment and want a much safer place to invest. My favorite apps are stash, robinhood and acorns. Each has its own advantage. Stash also has a very vibrant community that you can follow.

There’s a nice comparison in this video:

The volatile Bitcoin

Screen Shot 2018-06-14 at 2.04.11 AM
Technical Analysis of last 300 days of Bitcoin price data.
The top rectangangular frame shows the bar chart(green and orange bars) of the bitcoin price. A bar chart represents the OHLC(open, high, low, close) price for a time period (in this case 1 day) for an asset/stock/cryptocoin.
Bollinger Band: You can see a band (the dotted red lines above and below and a grey line in between) around the bitcoin OHLC price. The grey line represents the average price of 20 days. When the bars are above the grey line, the price is being appreciated, when the bars are below the grey line, the price is being depreciated.
Relative Strength Index(RSI): The blue line in second rectangular frame below shows RSI. Relative strength index(RSI) represents when the bitcoin has been overbought (more bought than being sold –> leads to price hike) or when the bitcoin has been oversold (more sold than being bought –> leads to price drop). When the blue RSI line is abobe 80 (check in the left y axis), it’s normally overbought (price rise) and you can see direct correlation of price hike comparing with the price chart above, when the blue RSI line is below 20, it’s normally oversold (price drop). You can see, right now everyone is panicking and everyone is selling, nobody almost is buying, so price is dropping rapidly. Bitcoin has hit the lowest RSI value of the year.
Moving Average Convergence Divergence(MACD): The third rectangular frame below shows MACD value with grey bars, grey dotted line and a red dotted line. The red dotted line is the average price of 26 days and the grey dotted line is the average price of 12 days. When the grey line is above the red line, it means that the short term price is higher than the long term price, but when it is below, it means that current price is being depreciated from the long term accepted price. MACD is another momentum oscillator which is similar to RSI. You can directly see how it is similar by comparing them. It’s probably easier to read the MACD chart than the RSI. Because there’s the zero line (in RSI it probably represents the value 50) when the grey bars are above(upwards) zero line, it means price appreciation, versus when the grey bars are below zero line(downwards), it means price depreciation. As you can see the price is highly depreciated now.
Commodity Channel Index (CCI): The fourth rectangular frame below shows CCI value with red line and filled red polygons. If in between 100 and -100, the price is not taking any huge turn (bull ride or bear drop). The filled polygons above 100 show the extreme price hike, where as the polygons below -100 show the recent extreme price drop. Because of these polygons, it’s very easy to see the price movement with CCI index.
Hope you enjoyed learning some of the common techniques used to understand market price.
Chart plotted by Abdullah Khan Zehady using R (quantmod, TTR library) language. Please contact if you want to reuse for your own purpose.
If you think it was helpful and you learnt valuable information which can help you guide your investment, you can contribute my effort and encourage me by sending small gift via Bitcoin or Ethereum.
BTC Address: 1HXSMFpN1C6ruAutwRNQoR4uzqPLuMHzi1
ETH Address: 0x22bC6a74e5B29167D8AE40197193762A9AC60F81

Methods in Fourier Spectral Analysis

Fourier Transform

It was a significant discovery in mathematics that any function can be expanded as a sum of harmonic functions (sines and cosines) and the resulting expression is known as Fourier series. A harmonic of repeating signals such as sunusoidal wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the first harmonic, the following harmonics are known as higher harmonics. Any function can also be expanded in terms of polynomials and the resulting expression is known as Taylor series. If the underlying forces are harmonic and there possibly exists some periodicity, then the use of harmonic series is more useful than using polynomials as it produces simpler equations. It is possible to discover a few dominating terms from such series expansion which may help identify the known natural forces with the same period.
Let the symbol h(t) represent a continuous function of time. The Fourier transform is a function of
frequency f.

H_T(f) = \int_{-\infty}^{\infty} h(t) e^{2 \pi i f t} dt
e^{2 \pi i f t} = cos(2\pi f t) + i sin(2\pi f t)

The amplitude and the phases of the sine waves can be found from this result. Given data h(t), we can find the Fourier transform H(f) using Inverse Fourier transform.

h(t) = \int_{-\infty}^{\infty} H_T(f) e^{-2 \pi i f t} dt

The spectral power P is defined as the square of the Fourier amplitude:

P_T(f) = |H_T(f)|^2

However, real data does not span infinite time and most likely be sampled only at a few discrete points over time. Suppose that, we received values of h(t) at times t_j, then an estimate of the Fourier transform is made by using summation. The inverse transform is also shown using the summation.

h_j \equiv h(t_j)
H(f) \equiv \sum_{j=0}^{N-1} h_j e^{2 \pi i f t_j}

The data are desired to be sampled from equally spaced time as nice statistical properties are available in such regular case. If the interval between equally spaced data points is \Delta t, then the highest frequency that will appear in the fourier transform is given by the Nyquist-Shannon sampling theorem. The theorem states “If a function f(t) contains no frequencies higher than f Hz, then it is completely determined by giving its ordinates at a series of points spaced \frac{1}{2f} seconds apart”. Therefore, the Nyquist frequency (highest frequency) is given by the following equation.

f_N = \frac{1}{2\Delta t}

The lowest frequency is the one that gives one full cycle in the time interval T. The other frequencies to evaluate is the multiples (f_k) of the low frequency f_L. And, also we can derive the symmetric pair of equations. Moreover, if h(t) is band-limited (no frequencies below f_L or above f_N), then there is a relationship between the continuous function h(t) and the discrete values H_k.

f_L = \frac{1}{T}
f_k = kf_L
H_k \equiv \sum_{j=0}^{N-1} h_j e^{2 \pi i jkf_L t_j}
h_j = \sum_{k=0}^{N-1} H_k e^{2 \pi i jkf_L t_j}
h(t) = \sum_{k=0}^{N-1} H_k e^{2 \pi i kf_L t} (when band limited)


Fourier transform give us the complex numbers and the square of the absolute value of these numbers represent the periodogram. This is the first form of numerical spectral analysis and is used to estimate spectral power. Even though the data points collected are at evenly spaced specific discrete time, it is possible to evaluate periodogram at any frequencies.

Fast Fourier Transform (FFT)

We can calculate the Fourier transform very efficiently by using FFT. It requires data at equally-spaced time points, and is most efficient when the number of points is an exact power of two. Interpolation is often used to produce the evenly-spaced data which may introduce additional bias and systematic eror. For real data consisting of N data points y_j, each taken at time t_j, the power spectrum outputs a set of N+1 data points. The first and the last data points are the same, and they represent the power at frequency zero. The second through to the N/2 + 1 data points represent the power at evenly-spaced frequencies up to the Nyquist frequency. The spectral power for a given frequency is distributed over several frequency bins, therefore an optimum determination of the power requires combining these information and proper investigation of leakage. FFT, generally, calculates the amplitude for a set of frequencies. N/2 complex amplitudes are calculated at N/2 different frequencies. Because, these may not be the true frequencies present in the record, we subtract the mean from the data and then pad it with zeros to overcome this challenge.


The time series consists of measurements made at a discrete, equally spaced, set of times on some phenomenon that is actually evolving continuously, or at least on a much finer time scale. For example, samples of Greenland Ice represent the temperature every 100 years, but if the sampling is not precisely spaced by a year, we will sometimes measure winter ice, and other times measure summer ice. Even without the existence of long-term variation in the temperature, fluctuations (jumping up and down) in the data can be noticed. So, there can be frequencies higher than the Nyquist frequency associated with the sampling interval. Thus a peak in the true spectrum at a frequncy beyond the Nyquist frequency may be strong enough to be seen(aliased) in the spectrum which may give the impression that a frequency is significant when it is not. Or, a peak may partly obscure another frequency of interest. This phenomenon is known as aliasing.


Fourier transform is defined for a function on a finite interval and the function needs to be periodic. But with the real data set, this requirment is not met as the data end suddenly at t=0 and t=T and can have discontinuities. This discontinuity introduces distortions (known as Gibbs phenomenon) in fourier transform and generates false high frequency in the spectrum. Tapering (using data window) is used to reduce these artificial presence. The data y=f(t) is multiplied by a taper function g(t) which is a simple, slowly varying function, often going towards zero
near the edges. Some of the popular tapers are:

1. Sine taper g(t) = sin(\pi t/T)
2. Hanning (offset cosine) taper g(t) = \frac{1}{2}(1-cos(2\pi t/T))
3. Hamming taper g(t) = 0.54 - 0.46cos(2 \pi t/T)
4. Parzen or Bartlett (triangle) window g(t) = 1 - (t - T/2)/(T/2)
5. Welch (parabolic) window g(t) = 1 - (t - T/2)^2/(T/2)^2
6. Daniell (untapered or rectangular) window g(t) = 1

The frequency resolution in the spectrum of the tapered data is degraded. If the primary interest is the resolution of peaks, then the untapered periodogram is superior. However, tapering significantly reduces the sidelobes and also the bias applied to other nearby peaks by the sidelobes of a strong peak. Because, the taper functions are broad and slowly varying and their fourier transform FT(g) are narrow. The effect of tapering the data is to convolve the fourier transform of the data with the narrow fourier transform of the taper function which amounts to smoothing the spectral values.

FT(fg) = FT(f) * FT(g)


p style=”text-align:justify;”>
// Sine taper
t <- seq(0,1, by=0.01)
T <- 1
g <- sin(pi * t * T)
plot(t, g, t='l', col=1, ylab='g(t)')

// Hanning (offset cosine) taper
g2 <- 1/2 * (1-cos(2*pi*t/T))
lines(t, g2, t=’l’, col=2)

// Hamming
g3 <- 0.54 – 0.46 * cos(2*pi*t/T)
lines(t, g3, t=’l’, col=3)

// Parzen or Bartlett (triangle) window
g4 0.5, 1 – (t-T/2)/(T/2), 2*t)
lines(t, g4, t=’l’, col=4)

// Welch (parabolic) window
g5 <- 1 – (t-T/2)^2/(T/2)^2
lines(t, g5, t=’l’, col=5)

// Daniell window
g6 <- rep(0.5, length(t))
g6 <- ifelse(t <= 0.2, 0, g6)
g6 = 0.8, 0, g6)
lines(t, g6, t=’l’, col=6)

legnd = c(‘Sine’, ‘Hanning’, ‘Hamming’, ‘Bartlett’, ‘Welch’, ‘Daniell(20%)’)
legend(‘topleft’, legend=legnd ,col=1:6, lty=1, cex=0.75)

Multitaper Analysis

We apply taper or data window to reduce the side lobes of the spectral lines. Basically we want to minimize the leakage of power from the strong peaks to other frequencies. In multitaper method, several different tapers are applied to the data and the resulting powers then averaged. Each data taper is multiplied element-wise by the signal to provide a windowed trial from which one estimates the power at each component frequency. As each taper is pairwise orthogonal to all other tapers, the windowed signals provide statistically independent estimates of the underlying spectrum. The final spectrum is obtained by averaging over all the tapered spectra. D. Thomson chose the Slepian or discrete prolate spheroidal sequences as tapers since these vectors are mutually orthogonal and possess desirable spectral concentration properties. Multitaper method can suppress sidelobes but have higher resolution. If we use few tapers, the resolution won’t be degraded, but then sidelobe reduction won’t happen much. So, there is a trade-off which is often misunderstood.

Blackman-Tuckey Method


Blackman and Tuckey prescribed some techniques to analyze a continuous spectrum that was biased by the presence of sidelobes of strong peaks in the ordinary periodogram. Blackman-Tuckey(BT) method was developed before 1958, prior to the FFT(Fast Fourier Transform) method. A discrete fourier transform of N points would
require the calculation of N^2 sines and cosines. With the slower computer in the pre-FFT days, the calculation of fourier transform was thus expensive. BT method has reduced the time by reducing the size of the dataset by a factor of the lag in the autocorrelation calculation. BT method is based on a fundamental theorem of Fourier transform that the Fourier transform of a correlation is equal to the product of the Fourier transforms. The correlation of two functions g(t) and h(t) is given by the first equation below.

C(\tau)=g\otimes h= \int_{-\infty}^{\infty} g(t) h(t+\tau) d\tau
FT(g\otimes h) = FT(g) FT(h)

When g = h, it is called Wiener-Khintchine theorem. Here, P is the spectral power.

FT(g\otimes g) = |FT(g)|^2 = P

The algorithm in BT method calculates partial autocorrelation function, defined by

A_{BT}(\tau) = \int_{0}^{N/l} f(t+\tau)f(t) dt

Here, N is the length of the data set but we integrate only upto N/l. $l$ is associated with the lag. When l=3 (recommended by Blackman and Tuckey) is used, we say that “a lag of 1/3” is used. Now the fourier transform of partial autocorrelation function A_{BT} gives us the spectral power. Moreover, the symmetric property of the partial autocorrelation function (A(-\tau) = A(\tau)) saves half of the computation time.

FT(A_{BT}) = \int_{-\infty}^{\infty} e^{2\pi i ft } A_{BT}(\tau) d \tau = P_{BT}(f)
P_{BT}(f) = 2 \int_{0}^{\infty} cos(2\pi f) A_{BT}(\tau)

If l=1, then it is basically the full autocorrelation function A(\tau) and gives the same answer as the ordinary periodogram.

P(f) = 2 \int_{0}^{\infty} cos(2\pi f) A(\tau) = FT(A)

Because we are using partial correlation function instead of the full correlation, the spectral power function gets smoother. Therefore, we lose resolution in the BT method. However, it averages the sidelobes into the main peak, and thereby gives a better estimate of the true power. The smoothing in BT method is different from the smoothing when we use a taper. With a taper, the fourier transform is smoothed, where as with Blackman-Tukey, it is the spectral power which gets smoothed. A spectral amplitude that is rapidly varying will be averaged to zero with a taper. But in BT method, a rapidly varying amplitude does not necessarily average to zero, since the process of squaring can make the function positive over the region of smoothing. The tapering does not
average the sidelobes into the main peak. Because, shift in the time scale behaves like phase modulation. The sidelobes, when tapering is applied, will not have the same phase, and if averaged in amplitude, they can reduce the strength of the peaks. A major challenge in the BT method is that we will have to estimate the proper lag to use before doing all the calculations. Blackman and Tukey recommended starting with the value 1/3 for the lag.

Lomb-Scargle Periodogram


The classic periodogram requires evenly spaced data, but we frequently encounter with unevenly spaced data in paleoclimatic research. Lomb and Scargle showed that if the cosine and sine coefficients are normalized separately, then the classic periodogram can be used with unevenly spaced data. If we have a data set (t_k, y_k), we first calculate the mean and variance:

\bar{y} = \frac{1}{N} \sum_{k=1}^{N}y_k
\sigma^2 = \frac{1}{N-1} \sum_{k=1}^{N}[y_k - \bar{y}]^2

For every frequency f, a time constant \tau is defined by

\tau = \frac{ \sum_{k=1}^{N}sin(4\pi f t_k)}{\sum_{k=1}^{N}cos(4\pi f t_k)}

Then the Lomb-Scargle periodogram of the spectral power P(f) at frequency f is given by

$P(f) = \frac{1}{2\sigma^2}\frac{ \sum_{k=1}^{N}(y_k – \bar{y} ) [cos(2\pi f (t_k-\tau))]^2}{\sum_{k=1}^{N}cos^2(2\pi f (t_k-\tau))} +
\frac{ \sum_{k=1}^{N}(y_k – \bar{y} ) [sin(2\pi f (t_k-\tau))]^2}{\sum_{k=1}^{N}cos^2(2\pi f (t_k-\tau))}$

With evenly spaced data, two signals of different frequencies can have identical values which is known as Aliasing. That is why the classic periodogram is usually shown with the frequency range from 0 to 0.5, as the rest is a mirrored version. But with Lomb-Scargle periodogram, the aliasing effect can be significantly reduced.

Maximum Likelihood Analysis

In maximum likelihood method, we adjust the parameter of the model and ultimately find the parameters with which our model have the maximum probability/likelihood of generating the data. To estimate the spectral power, we first select a false alarm probability and calculate the normalized periodogram. We identify the maximum peak and test it against the false alarm probability. If the maximum peak meets the false alarm test, we determine the amplitude and phase of the sinusoid representing the peak. Then we subtract the sinusoidal curve from the data which also removes the annoying sidelobes associated with that peak. After peak removal, the variance in the total record is also reduced. Now, with the new subtracted data, we continue finding the other stronger peaks following the same procedure. We stop when a peak does not meet the false alarm test. We need to carefully choose the false alarm probability, as if it is too low, we can miss some significant peaks; it is too low, we can mislabel noise as peaks.

Maximum Entropy Method

It is assumed that the true power spectrum can be approximated by an equation which has a power series. This method finds the spectrum which is closest to white noise (has the maximum randomness or “entropy”) while still having an autocorrelation function that agrees with the measured values – in the range for which there are measured values. It yields narrower spectral lines. This method is suitable for relatively smooth spectra. With noisy input functions, if very high order is chosen, there may occur spurious peaks. This method should be used in conjuction with other conservative methods, like periodograms, to choose the correct model order and to avoid getting false peaks.

Cross Spectrum and Coherency

If a climate proxy a(t) is influenced or dominated by a driving force b(t), we can use cross spectrum to see if their amplitudes are similar. Cross spectrum is given by the product of the fourier transform.

C(f) = A(f) B^*(f)

where A is the Fourier transform of a and B is the complex conjugate of the fourier transform of b. If we want to know whether two signals are in phase with each other, regardless of amplitude, then we can take the cross spectrum, square it, and divide by the spectral powers of individual signals using the following equation for coherency. Coherency measures only the phase relationship and is not sensitive to amplitude which is a big drawback.

c(f) = \frac{|C(f)|^2}{P_a(f) P_b(f)}

Coherency is valuable if two signals that are varying in time, stay in phase over a band of frequencies instead of a single frequency. Therefore, a band of adjacent frequancies are used in the averaging process to compute coherency:

coherency(f) = \gamma^2(f) = \frac{|<C(f)>|^2}{<P_a(f)> <P_b(f)>}


In bispectra, coherency relationship between several frequencies are used. A bispectrum shows a peak whenever (1) three frequencies f_1, f_2 and f_3 are present in the data such that $f_1 + f_2 = f_3$ and (2) the phase relationship between the three frequencies is coherent for at least a short averaging time for a band near these frequencies. If the nonlinear processes in driving force (e.g. eccentricity or inclination of the orbit of earth) has coherent frequency triplets, then the response (i.e. climate) is likely to contain same frequency triplet. For example, \delta ^{18}O is driven by eccentricity, we should be able to find eccentricity triplet. Thus, by comparing the bispectrum plot of climate proxy with the bispectrum plot of the driving forces, we can verify the influences of driving forces.

## Monte Carlo Simulation of Background
Monte carlo simulation is extremely useful to answer the questions like whether the data is properly tuned or not, whether the timescale is incorrect, whether some spectral power is being leaked to adjacent frequencies, whether the peak has real structure and also to understand the structures near the base of the peak (a shoulder) in a spectral analysis. Generally monte carlo simulation is run multiple times. For each simulation, a real signal(sinusoidal wave) is generated, then random background signal is added, then the spectral power is calculated to look for shoulders. In this way, the frequency of the shoulder occurence can be measured and the randomness can be realized. It is important to create background that behaves similarly to the background in real data. Dissimilar background will cause false conclusion. We also need to estimate the statistical significance of the peaks very carefully.

(This article is a quick review of the fourier spectral analysis from the book “Ice Ages And Astronomical Causes- (Data, Spectral Analysis and Mechanics) by Richard A. Muller and Gordon J. MacDonald

Statistics and Data Exploration: Quantiles, probability distribution, Box plot and Q-Q (Quantile-Quantile) plot

Statistics and Data Exploration: Quantiles, probability distribution, Box plot and Q-Q (Quantile-Quantile) plot


What are quantiles in statistics?

If the data is sorted from small to big, Quantiles are the points which divide the data/samples into equal sized, adjacent subgroups. Every data sample has maximum value, minimum value, median value(the middle value after you sort the data). The middle value in the sorted data is the 50% quantile because half of the data are below that point and half above that point. A 25% quantile is the cut point in the data where 1/4 -th of the data is below the point. IQR is inter-quartile range which contains half of the data which contains the median and are higher than the 25% low-value data point but less than the 25% high-value data point.

Box Plot

A box-plot can be a good representation to show the quantiles. Box plot can take different shapes depending on the data. Here is an example:

Screen Shot 2018-04-16 at 10.05.34 AM

(image source:

Example of Discrete/Continous Probability Distribution

In the figure below, you can see different frequency distribution. The blue data samples have most of it’s data near (0,1) interval, it’s left skewed. Check how the blue box is shifted to the left. The green data samples are normally distributed, meaning most of the data points are centered around zero. It also looks balanced. We find normal distribution in nature and in biological and social phenomena very often. The orange one shows almost a uniform distribution, where the data is spreaded across the range. And lastly a right skewed data. These are all discrete data points with discrete probability distribution. There are also very well known continuous probability distribution with continuous probability density function(

Screen Shot 2018-04-16 at 10.06.55 AM

  (Image source:

Below we can see the quantiles for the normal distribution- the cut points which divide the continuous range of points in equal probability area. The area over an interval (in x axis) under a continuous probability density function (like the normal distribution function below) represents the probability of the data falling into that range. In this case, the IQR is the blue box; data point in that interval has 50% probability of occurrence.

Screen Shot 2018-04-16 at 9.57.22 AM

Q-Q plot

We can use Q-Q plot to graphically compare two probability distributions. Q-Q plot stands for Quantile vs. Quantile plot. In Q-Q plotting, we basically compute the probabilities assuming a certain distribution (e.g. normal, gamma or poisson distribution) from the data and then compare it with theoritical quantiles. The steps used in Q-Q plotting is:

  1. Sort the data points from small to large
  2. For n data points, find n equally spaced points which serve as the probability using \frac{k}{n+1} where k=1, 2, ..., n
  3. Look at the data points, possibly plot it and assume the underlying probability distributions. Using the probabilities from the step 2, now you can calculate quantiles. Like in R language, you can use the quantile functions like qnorm or qgamma or qunif from the stats package.
  4. Now plot by putting the calculated quantiles in step 3 in x axis and putting the sorted data points in the y-axis. If the data points stay close to the y=x line, that means your assumption of the probability distribution was correct.

Below you can see one example, where the normal distribution is assumed for the ozone data. T

Screen Shot 2018-04-16 at 10.35.57 AM

Now you can see the gamma distribution fits better to the ozone data than the normal distribution.

Screen Shot 2018-04-16 at 10.37.45 AM

This is how you can check different probability distribution for your data using simple Q-Q plot. There is a fantastic Q-Q plot tutorial from which I collected the above image. For further reading, please check and


How do Paleoclimatologists investigate about ancient Earth? What are different Climate Proxies and what are their significance?

To know and understand about ancient climate, different climate proxies are generally used. We can measure the concentration of greenhouse gases by using entrapped air in the Greenland and Antarctic glaciers which give us samples of the atmosphere back to about 420 Kyr. The glaciers in North America and on mountains in tropical Andes can be estimated from scour marks, moraines and erratic boulders.Forams are microscopic organisms whose life cycles depend on local temperature and whose fossils preserve samples of ancient material. Some planktic forams (short for foraminifera) represent a “proxy” for sea surface temperature as they indirectly inform us about the temperature. One of the most remarkable proxies is the ratio of oxygen isotopes in benthic(bottom dwelling) forams in ancient sediment, which reflect the total amount of ice that existed on the Earth at the time the sea beds were formed. A scientist needs to be careful in their analysis as most proxies are dependent on more than one aspect of climate. Now I will discuss the primary proxies which have been used to investigate paleoclimate. Many of the samples come from seafloor cores, cores from Greenland or Anatarctic ice. The cores are named V22-174, RC13-110, DSXP-806 etc. In the geologic community, various of these prefixes are used some of which are enlisted below:

  • V: Vema, a converted yacht operated by Lamont-Doherty Earth Observatory of Columbia university.
  • RC: Research vessel Robert Conrad.
  • DSDP: Deep Sea Drilling Project operated from 1968 to 1983 by the Scripps Institution of Oceanography at University of California, San Diego.
  • ODP: Ocean Drilling Program as an international collaboration.\newline
  • GRIP: European based GReenland Ice-core Project.
  • GISP2: US-based Greenland Ice Sheet Project #2.
  • Vostok: Russian station on the East Antarctic ice plateau.
  • MD: The research vessel Marion Dufresne, operated by the French.

1. Oxygen Isotopes

The pattern of oxygen isotopes is remarkably similar in sea floor records around the world and this universality feature is very attractive for a climate proxy. The ratio of oxygen istopes found in ice, trapped air, benthic/planktic forams is widely used as a climate proxy. Oxygen consists of three stable istopes: 99.759% is ^{16}O, 0.037% is ^{17}O, and 0.204% is ^{18}O. The variation in the fraction of ^{18}Olatex can be measured with high accuracy. The fractional change, shown by the following equation, basically means that how much difference of the ratio of \frac{^{18}O}{^{16}O}latex exists in perts per thousand in the sample compared to the reference.

\delta^{18}O = \left(\frac{\left(\frac{^{18}O}{^{16}O}\right)_{Sample}}{\left(\frac{^{18}O}{^{16}O}\right)_{Reference}} - 1\right) \times 1000


Oxygen isotope separation occurs because of the isotopic differences in vapour pressure and chemical reaction rates, which depends on temperature. Some of the most important geophysical processes that lead to changes in \delta^{18}O are:

  1. Evaporated water is ligher than the remaining liquid. Water containing ^{16}O has higher vapor pressure than water containing ^{18}O, so it evaporates quickly.
  2. Precipitated water molecules are heavier than those in the residual vapor. H_2^{18}O condenses more readily than H_2^{16}O, so as water vapour is carried across to Greenland or to central Anatarctica, the residual becomes lighter.
  3. Oceanic \delta^{18}O in non-uniformly distributed. It means that the changes in the pattern of winds that carry vapor and change the source will also change \delta^{18}O. At present, the difference in surface water is 1.5% from pole to equator.
  4.  Biological activity enriches the heavy isotope. The \delta^{18}O in the calcium carbonate of shells is 40% greater, on average, than in the water in which the organism lives.


The net result of these effects is that glacial ice is light, with \delta^{18}O typically lower than seawater. So, in glacial ice containing more ^{16}O, \delta^{18}O is negative, where as in surface water containing more ^{18}O, \delta^{18}O is positive. However, when large volumes of ice are stored in ace-gage glaciers, then there can be considearable depletion of the light isotopes in the oceans.
In 1964, Dansgaard and colleageus showed that measurements of isotopic enrichment in ocean water as a function of latitude yield the following approximate relationship between temperature T and \delta^{18}O:

\delta^{18}O \equiv 0.7 T - 13.6
However, there can be other factors in the change of \delta^{18}O. Therefore, if we go back to earlier when the temperature was lower, \delta^{18}O might not be lower which contradicts the above equation. When several measurements are made at the same latitude, the effect is argued to depend on the amount of precipitation and not on temperature.
Moreover, depending on the source, we will have to consider other issues. In planktic fossils, we might expect \delta^{18}O to reflect surface conditions, and therefore be sensitive to temperature and salinity conditions. In benthic forams, \delta^{18}O must be more sensitive to global ice, since there is little temperature variation on the sea floor. In other samples (e.g. ice, trapper air or calcite), \delta^{18}O may represent the temperature, not ice volutme.
Several attempts have been made to extract the underlying \delta^{18}O signal that is common in the records. SPECMAP stack (Imbrie et al., 1984) was a combination of five $\delta^{18}O$ records from five cores: V30-40, RC11-120, V280238 and DSDP502b.

2. Deuterium – Temperature Proxy

Hydrogen generally contains only one proton in its nucleus and is lighter with atomic weight 1. Deuterim (D or ^2H), on the other hand, is one of the heavy isotopes of hydrogen which contains one proton and one neutron in its nucleus and thus the atomic weight is 2. Bonds formed with deuterium tend to be much more stable than those with light hydrogen. The deuterated water is more sensitive to temperature than that of ^{18}O. We can clearly see it in the “fractionation factor” which describes the equilibrium between liquid and vapour. The fractionation factor is defined to be the ratio of D/H in a liquid to the ratio of D/H in a vapor that is in equilibrium with that liquid. The fractionation factor for HDO is approximately 1.08 and it varies more rapidly with temperature compared to ^{18}O. Therefore, the condensation of the deuterised form of heavy water (HDO) is significantly more sensitive to temperature variation than is the ^{18}O form (H_2^{18}O). Therefore, deuterim is considered as a temperature proxy. A temperature scale was devised fro the Vostok ice core by assuming the equation:

\Delta T = \frac{\Delta \delta D_{ICE} - \Delta \delta^{18}O_{SW}}{9}
where, the $\delta^{18}O_{SW}$ refers to the sea floor isotope record.

3. Carbon-13

Carbon on the earth has two stable istopes, ^{12}C with an bundance of 98.9% and ^{13}C with an abundance of 1.1%. The ratio of these two isotopes is described by the quantity \delta^{13}C and defined by the equation below. The reference value is often taken to be a sample known as the “Peedee belemnite” (PDB); its \delta^{13}C value is very close to that of mean sea water.

\delta^{13}C = \left(\frac{\left(\frac{^{13}C}{^{12}C}\right)_{Sample}}{\left(\frac{^{13}C}{^{12}C}\right)_{Reference}} - 1\right) \times 1000

The lighter isotope, ^{12}C, is easily absorbed into the organic tissue of plants, leading to negative values for ^{13}C = -20% to -25%. In regions in which photosynthesis is active, this removes typically 10-20% of the dissolved inorganic carbon in seawater, leading to ^{13}C enrichment in surrounding water. Because different regions of the world have different activity, there is geographic variation. Warm surface water has the highest \delta^{13}C, where as deep Pacific water has the lowest \delta^{13}C. Thus \delta^{13}C can be used as a tracer for oceanic currents.
In contrast, there is only small separation of carbon istotopes that takes place in the formation of caclcium carbonate shells. Thus the measurement of \delta^{13}C reflects the composition of the ocean water at the time and location in which the shell grew.
^{13}C is extremely important isotope for paleoclimate studies, because it responds to the presence of life. \delta^{13}C can record climate change. During glacial periods, biological activity was reduced by advancing glaciers and colder temperature, and light carbon was released into the atmosphere and eventually mixed into the oceans. \delta^{13}C from benthic (bottom dwelling) forams is typically 0.35% lower during glacials than during interglacials. In contrast, planktic forms don’t show such changes.

4. Vostok

The ice core from the Vostok site in Antarctica (Petit et al., 1999) located at 78^oS and 107^oE, covers the longest period of time of any ice record. It reached a depth of 3623 metres. A untuned but unbiased timescale was derived based on ice accumulation and glacial flow models. Many proxies of climate interest were measured in the Vostok core, including atmospheric methane, atmospheric oxygen, deuterium in the ice, dust content and sea salt. Atmospheric methane is produced by the biological activity of anaerobic bacteria and it’s existence in paleoclimate data is presumed to reflect the area of the earth covered by swamps and wetlands. The observed dust (strong 100 Kyr cycle) in the Vostok dust record is beleived to reflect reduction in vegetation during those periods and accompanying increase in wind-blown erosion. Then, the sodium concentration reflects the presence of sea spray aerosols blowing over the Vostok region.

5. Atmospheric \delta^{18}O and Dole Effect

The atmospheric oxygen has a \delta^{18}O of +23.5% compared to that of mean ocean sea water due to the removal of lighter isotope ^{16}O from the atmosphere by biological activity. The difference is called the “Dole Effect” and it is assumed to be time-independent.

6. \delta^{18}O / $CO_2$ Mystery

The difference between ocean and atmospheric $\delta^{18}O$ is due to the biological activity. However, carbon dioxide, even though driven by biological processes, doesn’t show similar spectra. The strong peaks in the oxygen signal forced by precession parameter is absent in the carbon dioxide record which is mysterious and still under investigation.

7. Other Sea Floor Records

7.1 Terrigenous component

The terrigenous component of sea floor sediment is the fraction which has possibly come from land, in the form of wind-blown dust. The most significant frequencies which have been found in the spectrum of detuned terrigenous component Site 721 are marked with the periods: 41, 24, 22 and 19 Kyr. These periods indicate that the signals were dominated by solar insolation.

7.2 Foram size: the coarse, or “sand”, component

In the sea floor core, the main component of the sand is frequently large forams. Therefore, the coarse component reflects an interesting change in the ecology of the oceans. A clear eccentricity signal was detected in a core that already showed a clear absence of eccentricity in the \delta^{18}O component.

7.3 Lysocline: carbonate isopleths

Pressure varies in different depths of the ocean and which consequently influences the solubility of the calcium carbonate. At a certain depth, the shells of fossil plankton begin to dissolve, and this boundary is called lysocline. It can be quantified by the percentage of calcium carbonate in the sediment, as a function of depth. One can plot the depth at which the 60% lysocline is found, as a function of age and this depends on the depth of the oceans at that age. The signal apeears to be dominated by a 100 Kyr cycle, as would be expected if the primary driving force were the depth of the ocean, determined by the amount of ice accumulated on land.

Featured Image Courtesy:
Main Reference: Ice Ages and Astronomical Causes (Data, Spectral Analysis and Mechanisms) by Richard A. Muller and Gordon J. MacDonald.

Amazon book link:

Sharks on Bitcoin

What the big three sharks are saying. Somewhat dumb, somewhat great! Investor’s hat is definitely crucial, but not always the best. And I always love the entrepreneur’s hat. Who wouldn’t like to have the both hats with all different colors and taste!! 😉

Shark tank is a great platform that filters entrepreneurs, startup, growing business. If you haven’t known or watched shark tanks, you should definitely start. We human being are idea generators, everybody has ideas. Everytime you walk on the street or malls, you come up with ideas, it’s just the few who take the leap to materialize. Shark tank will give you some glimpse of what I just said, when you will see how innocuous, small ideas can go big if it’s under the right arm. Shark tank is one of the things that I like in both the fat, obese, dumb, racist in one hand and multicultural, smart, free, open, entrepreneurial North America on the other. LoL.;)
If you get the joke. Haha. You become what you choose to..
Here’s a great playlist for popular shark tank videos and roastings. Enjoy. 🙂

And subscribe to the Best of Shark Tank Channel to get more of these videos.

Recent Crypto Coin Dynamics of Bitcoin, Ethereum, Ripple, Cardano, Litecoin, Stellar and Monero

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You will realize the significance level of the top 7 cryptocurrency if you look at this chart.
1. Bitcoin: BTC (Yellow dotted curve)
2. Ethereum: ETH (Blue dotted curve)
3. Litecoin: LTC (Aliceblue dotted curve)
4. Ripple: XMR (Orange dotted curve)
5. Cardano: ADA ( Brown dotted curve)
6. Stellar Lumen: XLM (Light green dotted curve)
7. Monero: XMR (Pink dotted curve)

This chart shows you the price of these 7 major crypto coins from September till now. If you look at the pink rectangle, you can realize how Ethereum(dark blue), Monero(pink), Litecoin(alice blue), Stellar(Light green) has higher price than bitcoin (yellow). But, they were not always higher, look during november, everything was less valued than bitcoin (yellow curve is at the top in Nov 2017). As Bitcoin’s price had an upward trend during the end of december, all the coin price also saw upward trend. But as Bitcoin’s price is going down, some coins are holding their value but some are losing their value. You can see, underneath the pink rectangle, falls the two coin Cardano(brown) and Ripple(Orange), these two coins are losing their value. However, Ripple had higher values in early January as you can see if you look at the last peak for all coins near Jan 2018. I never had too much confidence in Ripple as it somehow defeats the idea of crypto currency with its centralized feature (But who knows it may see light again if it can deliver its promises). Ultimately, money will flock to services which have higher utility and can gain trust.
So, these should give you some rough idea about the coin dynamics. A successful coin economy depends on how the market share or daily volume of trade is going on that coin landscape. It’s kind of similar to how Forex (Foreign Exchange market) market where currencies are traded moves, how US dollar influences Euro or Japanese Yen, or Chinese Renminbi, or Bangladeshi Taka. Hope you have an idea of evaluating crypto coins and their potential gains or losses over time and can make educated investment decisions. And I guess you can see how crypto exchange market is going to look like in future.