The Noisy Pendulum

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The utilizations of the Kalman Filter in conjunction with the PID Controller will allow the system to maintain that position. While many Inverted Pendulum applications exist, in this research we propose the RapiBaBot; a robot that employs low-cost, easily obtained electronics to formulate a fairly stable inverted pendulum system.

In this paper we show how we solve issues that usually arise in inverted pendulum applications, namely, vibration and lack of mobility.

The Noisy Pendulum by Moshe Gitterman (ebook)

These issues have to be solved effectively for the stability and mobility of the system. We also explain how to construct the Kalman Filter and PID Controller into an effective and understandable form for the electronics to easily integrate with. Article :. This e-book comprises the final description of the mathematical pendulum topic to consistent torque, periodic and random forces.

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The latter seem in additive and multiplicative shape with their attainable correlation. For the underdamped pendulum pushed by way of periodic forces, a brand new phenomenon -- deterministic chaos -- comes into play, and the typical motion of this chaos and the impact of noise are taken under consideration. The inverted place of the pendulum might be stabilized both by way of periodic or random oscillations of the suspension axis or through putting a spring right into a inflexible rod, or through their mixture. It also means that if prices were moving in one direction in the past, they are likely to move in the opposite direction in the future.

Moreover, the greater the swing in one direction, the stronger is the reverse movement. They then traced the performance of the winner and loser portfolios for the subsequent three years. Using CRSP prices for the period between January and December , De Bondt and Thaler employed this procedure to construct winner and loser portfolios for 16 non-overlapping measurement periods. On average, 36 months after portfolio formation, the loser portfolios outperformed the market by The difference in cumulative average residual was The shape of the curve on Figure 2 gives us a rich description of the data.

In the first month after the portfolio formation there is a very strong mean reversion. This is known as short term mean reversion. Then for a horizon up to a year the winner stocks actually tend to slightly outperform the loser stocks. Just as there is momentum in a pendulum, there is intermediate term momentum in prices. Beyond the horizon of one year the winner stocks tend to underperform, while the loser stocks tend to outperform.

This is known as long-term mean reversion. Even though the De Bondt and Thaler chart stops three years after portfolio formation, long term mean reversion can be detected for as many as 10 years from the starting point. One potential explanation for the superior performance of extreme loser stocks is that they are riskier.

However, De Bondt and Thaler report that the average market betas of the securities in the winner portfolios 1. De Bondt and Thaler suggested an alternative, non-risk based explanation. Market participants might have overreacted to several years of subpar performance by the loser stocks, and underpriced them. Market participants also overreacted to several years of extraordinary performance on the part of the winner stocks, and overpriced them.

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In addition, numerous articles demonstrate that not only individual stocks but the stock market in general is subject to mean reversion. Among others Fama and French and Poterba and Summer present evidence that stocks mean-revert on the horizon up to five years.

Ralph C. Smith

Furthermore, high valuation multiples such as aggregate book-to-market or earnings-to-price ratios, which signal low current prices, have been found to forecast high subsequent stock market returns. See Campbell and Shiller [], among others. Mean Reversion is Related to the Value and Small Size Effects The mean reversion effect stands in an interesting relationship with the value and small size premia.

All three reflect the fact that low-price stocks tend to outperform high-price stocks. Value stocks have relatively low price-to-fundamentals ratios. And small stocks have lower market capitalizations. At the end of July , the median market capitalization of the Russell was U. A study by Arnott, Hsu, Liu, and Markowitz shows that a mispricing component in prices which the market eventually corrects can fully account for the mean reversion, value, and size effects observed in the data.

The farther the distance, the stronger the gravitational pull.

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The degree to which stocks are mispriced can vary over time, and when there is more mispricing there are greater opportunities for generating profits. Asness, Friedman, Krail, and Liew demonstrated that an estimate of the degree of aggregate mispricing can be used to forecast the value premium.

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The measures they chose were the spread in valuation multiples between a value portfolio and a growth portfolio the value spread and the spread in expected earnings growth between a growth portfolio and a value portfolio the earnings growth spread. The authors found that both measures are significant determinants of the difference in expected returns between value and growth strategies.

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The pale blue line shows the forecast implied by the relative cheapness of stocks with low valuations. Interestingly, the paper appeared at the height of the tech bubble, and the last point on the chart was forecasting a very high value premium.

This prediction came true when the tech bubble burst in